From cee17f9745665decd46bb25cda8d5a9e472359ab Mon Sep 17 00:00:00 2001 From: Vasily Ilin Date: Wed, 7 Oct 2026 21:18:40 -0700 Subject: [PATCH] Split the current project registry into independent cards --- LeanPool.lean | 294 +- LeanPool/projects.yml | 14106 ---------------- LeanPool/projects/1mfld.yaml | 33 + LeanPool/projects/2-coloring-1-round.yaml | 32 + ...lization-of-borel-determinacy-in-lean.yaml | 31 + LeanPool/projects/abc-exceptions.yaml | 40 + LeanPool/projects/acmax-conjecture.yaml | 51 + LeanPool/projects/agree-to-disagree.yaml | 41 + LeanPool/projects/aharonikorman.yaml | 28 + ...envectgors-perturbation-formalization.yaml | 49 + LeanPool/projects/anderson-conjecture.yaml | 27 + ...lous-diffusion-fractal-homogenization.yaml | 62 + LeanPool/projects/apportionment.yaml | 26 + .../projects/archon-firstproof-results.yaml | 34 + LeanPool/projects/artin-wedderburn.yaml | 32 + LeanPool/projects/asymptotic-moore-bound.yaml | 73 + .../projects/asymptotic-triangle-packing.yaml | 48 + LeanPool/projects/bannai-bannai-stanton.yaml | 31 + LeanPool/projects/beck-fiala-matrix.yaml | 50 + LeanPool/projects/besicovitchs-1-2.yaml | 47 + LeanPool/projects/bicausalot-palomar.yaml | 35 + LeanPool/projects/biswal.yaml | 43 + LeanPool/projects/bkarforestformula.yaml | 30 + .../projects/block-spectral-sensitivity.yaml | 61 + LeanPool/projects/bollobas-nikiforov.yaml | 84 + LeanPool/projects/boolean-isoperimetry.yaml | 45 + LeanPool/projects/boolean-multiplication.yaml | 74 + LeanPool/projects/brauergroup_new.yaml | 57 + LeanPool/projects/brill-noether-graphs.yaml | 39 + LeanPool/projects/brooks-subcubic.yaml | 37 + LeanPool/projects/brouwer.yaml | 59 + LeanPool/projects/bruhat-tits.yaml | 36 + LeanPool/projects/burkholder.yaml | 40 + .../projects/caffarelli-kohn-nirenberg.yaml | 52 + LeanPool/projects/cencov-petz.yaml | 46 + .../projects/centered-maximal-constant.yaml | 49 + LeanPool/projects/champernowne-normality.yaml | 49 + LeanPool/projects/channel-capacity.yaml | 51 + LeanPool/projects/chip-firing-with-lean.yaml | 36 + LeanPool/projects/chudnovsky.yaml | 40 + LeanPool/projects/chvatal-conjecture.yaml | 67 + LeanPool/projects/circuit-complexity.yaml | 46 + LeanPool/projects/circuitlib.yaml | 49 + LeanPool/projects/civ-axisymmetric.yaml | 56 + .../projects/classification-of-surfaces.yaml | 35 + LeanPool/projects/clawristotle.yaml | 31 + .../projects/common-neighbour-conjecture.yaml | 46 + LeanPool/projects/compact-spectral.yaml | 55 + .../projects/compactness-and-degeneracy.yaml | 44 + LeanPool/projects/composition-algebras.yaml | 49 + LeanPool/projects/computability.yaml | 45 + LeanPool/projects/computablereal.yaml | 46 + .../projects/concentration-inequalities.yaml | 44 + LeanPool/projects/connes-kreimer.yaml | 48 + LeanPool/projects/connes-rigidity.yaml | 38 + LeanPool/projects/conway-refinement.yaml | 30 + LeanPool/projects/conway-soifer-three.yaml | 42 + LeanPool/projects/cramer-wold.yaml | 50 + LeanPool/projects/critical-portraits.yaml | 41 + LeanPool/projects/cut-and-project.yaml | 50 + LeanPool/projects/dead-ends.yaml | 28 + LeanPool/projects/demazure-product.yaml | 57 + LeanPool/projects/demazureoperatorslean.yaml | 54 + LeanPool/projects/densityhalesjewett.yaml | 38 + LeanPool/projects/desargues.yaml | 41 + LeanPool/projects/dilatations.yaml | 100 + .../projects/directed-topology-lean-4.yaml | 42 + LeanPool/projects/distance-geometry.yaml | 48 + LeanPool/projects/domain-theory.yaml | 47 + ...oss-fractional-triangle-decomposition.yaml | 37 + LeanPool/projects/duality.yaml | 37 + LeanPool/projects/ec-tate-lean.yaml | 60 + LeanPool/projects/egrs75.yaml | 47 + LeanPool/projects/egz-formal-proof.yaml | 36 + .../projects/ehrhart-volume-inequality.yaml | 45 + LeanPool/projects/eilenberg-zilber.yaml | 53 + LeanPool/projects/elliptic-pde.yaml | 61 + .../projects/erdos-97-convex-octagon.yaml | 39 + .../erdos-moser-distinct-subset-sums.yaml | 45 + LeanPool/projects/erdos-tuza-valtr.yaml | 35 + LeanPool/projects/erdos1196.yaml | 29 + LeanPool/projects/erdos132-convex-k3.yaml | 113 + LeanPool/projects/erdos132-n14.yaml | 50 + LeanPool/projects/erdos132-three-chain.yaml | 136 + LeanPool/projects/erdos137.yaml | 44 + LeanPool/projects/erdos346.yaml | 40 + LeanPool/projects/erdos367.yaml | 55 + LeanPool/projects/erdos403.yaml | 35 + LeanPool/projects/erdos548.yaml | 57 + .../projects/erdos81-paper-i-contrib.yaml | 43 + .../erdos81-paper-ii-chordal-heredity.yaml | 45 + .../erdos81-paper-iii-sum-zero-triangles.yaml | 39 + LeanPool/projects/erdos865.yaml | 42 + .../projects/escauriaza-seregin-sverak.yaml | 70 + LeanPool/projects/euclidean-jordan.yaml | 37 + .../even-graph-cycle-decomposition.yaml | 36 + LeanPool/projects/event-structures.yaml | 59 + LeanPool/projects/exponential-chaos.yaml | 66 + LeanPool/projects/factorizationsystems.yaml | 38 + LeanPool/projects/fd1d-lean.yaml | 31 + LeanPool/projects/feige.yaml | 53 + LeanPool/projects/fel-conjecture.yaml | 29 + LeanPool/projects/fineqs.yaml | 37 + .../finite-graph-fundamental-group.yaml | 56 + .../projects/finite-max-flow-min-cut.yaml | 40 + LeanPool/projects/five-distance-sharp.yaml | 47 + LeanPool/projects/five-eighths-theorem.yaml | 36 + LeanPool/projects/flean.yaml | 46 + LeanPool/projects/flt-regular.yaml | 51 + LeanPool/projects/fo-zfc.yaml | 34 + LeanPool/projects/formal-learning-theory.yaml | 44 + .../projects/formalization-beyond-bethe.yaml | 36 + .../formalization-of-bounded-arithmetic.yaml | 57 + LeanPool/projects/formalizing-gmt.yaml | 53 + .../projects/four-ap-free-permutation.yaml | 38 + LeanPool/projects/frieze-patterns.yaml | 49 + .../frontiermath-open-hypergraphs.yaml | 52 + LeanPool/projects/fundamental-inequality.yaml | 41 + LeanPool/projects/gap-cvp.yaml | 51 + .../gaussian-moments-counterexamples.yaml | 54 + LeanPool/projects/gkp-density-one.yaml | 63 + .../projects/goemans-cost-counterexample.yaml | 57 + LeanPool/projects/granville-moore.yaml | 57 + LeanPool/projects/grothendieck-vanishing.yaml | 28 + LeanPool/projects/hadwiger-nelson-bounds.yaml | 43 + LeanPool/projects/hanson-wright.yaml | 52 + LeanPool/projects/hasse-minkowski.yaml | 44 + LeanPool/projects/hopf-problem.yaml | 47 + LeanPool/projects/hs-virial-nbc-identity.yaml | 42 + LeanPool/projects/hsd-interior-point-lp.yaml | 50 + LeanPool/projects/incompleteness.yaml | 32 + LeanPool/projects/infinitary-logic.yaml | 126 + .../projects/infinite-connes-rigidity.yaml | 44 + LeanPool/projects/inflation-termination.yaml | 36 + .../isograph-canonical-labelling.yaml | 42 + LeanPool/projects/isoperimetric.yaml | 32 + LeanPool/projects/istranscendentalpi.yaml | 34 + LeanPool/projects/jacobian-diffgeo.yaml | 48 + .../projects/johnson-lindenstrauss-lean.yaml | 54 + LeanPool/projects/kahn-kalai.yaml | 46 + LeanPool/projects/kalton-roberts.yaml | 41 + LeanPool/projects/kaltonpeck.yaml | 32 + LeanPool/projects/kasami-cyclic-additive.yaml | 50 + .../projects/koethe-conjecture-disproof.yaml | 64 + LeanPool/projects/komlos.yaml | 64 + LeanPool/projects/krafftsieve.yaml | 35 + LeanPool/projects/krohn-rhodes.yaml | 61 + LeanPool/projects/kuramoto.yaml | 41 + LeanPool/projects/kurosh.yaml | 63 + .../language-generation-characterization.yaml | 73 + LeanPool/projects/lattice-triangle.yaml | 29 + LeanPool/projects/lean-booleanfun.yaml | 28 + LeanPool/projects/lean-codes.yaml | 38 + LeanPool/projects/lean-malliavin.yaml | 37 + LeanPool/projects/lean-model-checking.yaml | 55 + LeanPool/projects/lean-operator-theory.yaml | 35 + LeanPool/projects/lean-poly-abc.yaml | 36 + LeanPool/projects/lean-quantumalg.yaml | 52 + LeanPool/projects/lean-scottish-book-155.yaml | 29 + LeanPool/projects/lean-scv.yaml | 41 + LeanPool/projects/lean-snumbers.yaml | 35 + LeanPool/projects/lean-spectral-theory.yaml | 36 + .../projects/lean-superorthogonality.yaml | 31 + LeanPool/projects/lean4-gl-coalgebras.yaml | 27 + LeanPool/projects/lean4-itree.yaml | 57 + LeanPool/projects/leancomplexanalysis.yaml | 41 + LeanPool/projects/leanmodularforms.yaml | 35 + LeanPool/projects/lehmer-e10.yaml | 40 + LeanPool/projects/lentil.yaml | 47 + .../li-criterion-rh-equivalence-lean.yaml | 42 + .../line-graph-colouring-matching.yaml | 57 + .../projects/list-edge-coloring-complete.yaml | 43 + LeanPool/projects/local-complex-geometry.yaml | 63 + .../projects/long-gaps-between-primes.yaml | 44 + .../projects/low-weight-pauli-dynamics.yaml | 63 + .../projects/lowdimsolvclassification.yaml | 32 + .../projects/main-theorem-of-polytopes.yaml | 34 + LeanPool/projects/markoff-modp.yaml | 28 + LeanPool/projects/markovprocess.yaml | 32 + LeanPool/projects/marshallhalltheorem.yaml | 45 + .../projects/matching-logic-completeness.yaml | 57 + LeanPool/projects/metric-codes.yaml | 50 + LeanPool/projects/microwave-networks.yaml | 50 + .../min-modulus-unique-multiset-sum.yaml | 40 + .../projects/minimum-degree-matchings.yaml | 58 + LeanPool/projects/misere-games.yaml | 41 + LeanPool/projects/monlib4.yaml | 72 + LeanPool/projects/monsky.yaml | 32 + .../projects/moser-lattice-colorings.yaml | 43 + LeanPool/projects/moving-sofa.yaml | 55 + LeanPool/projects/mriscx.yaml | 50 + .../projects/multicolor-triangle-ramsey.yaml | 40 + LeanPool/projects/nagatafactoriality.yaml | 35 + LeanPool/projects/nash-williams.yaml | 60 + LeanPool/projects/nashembedding.yaml | 46 + LeanPool/projects/naslund-counterexample.yaml | 52 + .../projects/navier-stokes-and-euler.yaml | 56 + LeanPool/projects/nemytskii-lebesgue.yaml | 83 + LeanPool/projects/neukirch.yaml | 50 + LeanPool/projects/nf-weak-partition.yaml | 55 + LeanPool/projects/nikodym.yaml | 37 + LeanPool/projects/nivat-conjecture.yaml | 80 + LeanPool/projects/non-sofic-group.yaml | 35 + LeanPool/projects/nrr-aak.yaml | 31 + .../odd-prime-valuation-distribution.yaml | 48 + LeanPool/projects/odlyzko-bound.yaml | 37 + LeanPool/projects/order-closures.yaml | 69 + LeanPool/projects/order-p-q.yaml | 34 + LeanPool/projects/osforgff.yaml | 32 + LeanPool/projects/paper-iv-clique-trees.yaml | 89 + LeanPool/projects/parallel-postulate.yaml | 57 + .../projects/parameter_free_gradient_p.yaml | 51 + .../partial-combinatory-algebras.yaml | 39 + LeanPool/projects/partial-regularity.yaml | 29 + LeanPool/projects/pcf-theory.yaml | 29 + .../projects/pdl-craig-interpolation.yaml | 73 + LeanPool/projects/pebbling-hypercube.yaml | 37 + .../pentagonal-number-theorem-analytic.yaml | 80 + .../projects/pentagonal-number-theorem.yaml | 37 + .../permanent-formula-lower-bound.yaml | 41 + LeanPool/projects/phaseretrieval.yaml | 53 + .../pl-accelerated-nesterov-lean.yaml | 54 + .../poincare-three-body-nonintegrability.yaml | 65 + LeanPool/projects/pointwise-birkhoff.yaml | 35 + .../projects/polya-enumeration-theorem.yaml | 27 + LeanPool/projects/polylean.yaml | 34 + LeanPool/projects/polylog-integrals.yaml | 169 + .../projects/polynomial-freiman-ruzsa.yaml | 94 + .../polynomial-method-restricted-sums.yaml | 55 + LeanPool/projects/puiseux.yaml | 48 + LeanPool/projects/pumping-cfg.yaml | 33 + ...ythagorean-polynomial-parametrization.yaml | 50 + LeanPool/projects/qec-certificates.yaml | 90 + ...quadratic-carleson-palomar-submission.yaml | 36 + LeanPool/projects/quadratic-iterates.yaml | 49 + .../projects/quantum-parallel-repetition.yaml | 39 + .../projects/quantum-query-foundations.yaml | 56 + LeanPool/projects/quasi-borel-spaces.yaml | 42 + LeanPool/projects/rado-theorem.yaml | 46 + LeanPool/projects/ramanujan-nagell.yaml | 32 + .../projects/ramanujan-tau-misses-primes.yaml | 37 + LeanPool/projects/redhill.yaml | 47 + LeanPool/projects/regts-sevenster.yaml | 45 + LeanPool/projects/rellich-kondrachov.yaml | 40 + .../projects/riemann-mapping-theorem.yaml | 36 + .../riemann-roch-function-fields.yaml | 62 + LeanPool/projects/rl-theory-in-lean.yaml | 26 + LeanPool/projects/root-system.yaml | 49 + .../projects/rr-equals-nonmeager-paper.yaml | 29 + .../runge-kutta-order-conditions.yaml | 50 + LeanPool/projects/rupert.yaml | 33 + LeanPool/projects/sabidussi.yaml | 42 + LeanPool/projects/salem-theorem.yaml | 35 + LeanPool/projects/sardmoreira.yaml | 34 + LeanPool/projects/schoenflies.yaml | 35 + LeanPool/projects/selberg-sieve4.yaml | 41 + LeanPool/projects/semicircle-catalan.yaml | 42 + LeanPool/projects/semicirclelaw.yaml | 42 + LeanPool/projects/sendov-conjecture.yaml | 111 + LeanPool/projects/sensitivity.yaml | 30 + LeanPool/projects/serre-mass-formula.yaml | 61 + LeanPool/projects/settheory.yaml | 43 + .../projects/shannon-1948-formalization.yaml | 36 + LeanPool/projects/singular-moduli.yaml | 60 + .../projects/small-undecidable-groups.yaml | 57 + LeanPool/projects/spectral-positivity.yaml | 49 + LeanPool/projects/sphere-packing.yaml | 50 + LeanPool/projects/stafford38-formal.yaml | 32 + LeanPool/projects/stallings-folding.yaml | 72 + .../projects/stationary-harmonic-maps.yaml | 57 + .../projects/structural-graph-classes.yaml | 73 + .../projects/sum-difference-exponent.yaml | 58 + LeanPool/projects/sumsthreesquares.yaml | 30 + LeanPool/projects/sundogcert.yaml | 48 + LeanPool/projects/synthetic-euclid.yaml | 47 + LeanPool/projects/three-gap-theorem.yaml | 37 + LeanPool/projects/turan3.yaml | 40 + LeanPool/projects/tutte-path.yaml | 45 + LeanPool/projects/ulms-theorem.yaml | 47 + .../unconditional-schauder-basis.yaml | 43 + LeanPool/projects/virasoroproject.yaml | 36 + .../projects/viscosity-solution-theory.yaml | 62 + .../vizing-equitable-edge-colouring.yaml | 49 + LeanPool/projects/vlasov-mean-field.yaml | 52 + LeanPool/projects/wallace-problem.yaml | 47 + LeanPool/projects/whitehead-theorem.yaml | 31 + LeanPool/projects/zeta-five-irrational.yaml | 53 + LeanPool/projects/zeta-h123.yaml | 47 + LeanPool/projects/zeta-zeros.yaml | 52 + .../zeta2-zeta3-linear-independence.yaml | 48 + LeanPool/projects/zeta_3_irrational.yaml | 42 + LeanPool/projects/zflean.yaml | 35 + LeanPool/projects/zhang-yeung-inequality.yaml | 31 + 293 files changed, 13777 insertions(+), 14398 deletions(-) delete mode 100644 LeanPool/projects.yml create mode 100644 LeanPool/projects/1mfld.yaml create mode 100644 LeanPool/projects/2-coloring-1-round.yaml create mode 100644 LeanPool/projects/a-formalization-of-borel-determinacy-in-lean.yaml create mode 100644 LeanPool/projects/abc-exceptions.yaml create mode 100644 LeanPool/projects/acmax-conjecture.yaml create mode 100644 LeanPool/projects/agree-to-disagree.yaml create mode 100644 LeanPool/projects/aharonikorman.yaml create mode 100644 LeanPool/projects/aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization.yaml create mode 100644 LeanPool/projects/anderson-conjecture.yaml create mode 100644 LeanPool/projects/anomalous-diffusion-fractal-homogenization.yaml create mode 100644 LeanPool/projects/apportionment.yaml create mode 100644 LeanPool/projects/archon-firstproof-results.yaml create mode 100644 LeanPool/projects/artin-wedderburn.yaml create mode 100644 LeanPool/projects/asymptotic-moore-bound.yaml create mode 100644 LeanPool/projects/asymptotic-triangle-packing.yaml create mode 100644 LeanPool/projects/bannai-bannai-stanton.yaml create mode 100644 LeanPool/projects/beck-fiala-matrix.yaml create mode 100644 LeanPool/projects/besicovitchs-1-2.yaml create mode 100644 LeanPool/projects/bicausalot-palomar.yaml create mode 100644 LeanPool/projects/biswal.yaml create mode 100644 LeanPool/projects/bkarforestformula.yaml create mode 100644 LeanPool/projects/block-spectral-sensitivity.yaml create mode 100644 LeanPool/projects/bollobas-nikiforov.yaml create mode 100644 LeanPool/projects/boolean-isoperimetry.yaml create mode 100644 LeanPool/projects/boolean-multiplication.yaml create mode 100644 LeanPool/projects/brauergroup_new.yaml create mode 100644 LeanPool/projects/brill-noether-graphs.yaml create mode 100644 LeanPool/projects/brooks-subcubic.yaml create mode 100644 LeanPool/projects/brouwer.yaml create mode 100644 LeanPool/projects/bruhat-tits.yaml create mode 100644 LeanPool/projects/burkholder.yaml create mode 100644 LeanPool/projects/caffarelli-kohn-nirenberg.yaml create mode 100644 LeanPool/projects/cencov-petz.yaml create mode 100644 LeanPool/projects/centered-maximal-constant.yaml create mode 100644 LeanPool/projects/champernowne-normality.yaml create mode 100644 LeanPool/projects/channel-capacity.yaml create mode 100644 LeanPool/projects/chip-firing-with-lean.yaml create mode 100644 LeanPool/projects/chudnovsky.yaml create mode 100644 LeanPool/projects/chvatal-conjecture.yaml create mode 100644 LeanPool/projects/circuit-complexity.yaml create mode 100644 LeanPool/projects/circuitlib.yaml create mode 100644 LeanPool/projects/civ-axisymmetric.yaml create mode 100644 LeanPool/projects/classification-of-surfaces.yaml create mode 100644 LeanPool/projects/clawristotle.yaml create mode 100644 LeanPool/projects/common-neighbour-conjecture.yaml create mode 100644 LeanPool/projects/compact-spectral.yaml create mode 100644 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mode 100644 LeanPool/projects/dilatations.yaml create mode 100644 LeanPool/projects/directed-topology-lean-4.yaml create mode 100644 LeanPool/projects/distance-geometry.yaml create mode 100644 LeanPool/projects/domain-theory.yaml create mode 100644 LeanPool/projects/dross-fractional-triangle-decomposition.yaml create mode 100644 LeanPool/projects/duality.yaml create mode 100644 LeanPool/projects/ec-tate-lean.yaml create mode 100644 LeanPool/projects/egrs75.yaml create mode 100644 LeanPool/projects/egz-formal-proof.yaml create mode 100644 LeanPool/projects/ehrhart-volume-inequality.yaml create mode 100644 LeanPool/projects/eilenberg-zilber.yaml create mode 100644 LeanPool/projects/elliptic-pde.yaml create mode 100644 LeanPool/projects/erdos-97-convex-octagon.yaml create mode 100644 LeanPool/projects/erdos-moser-distinct-subset-sums.yaml create mode 100644 LeanPool/projects/erdos-tuza-valtr.yaml create mode 100644 LeanPool/projects/erdos1196.yaml create mode 100644 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create mode 100644 LeanPool/projects/four-ap-free-permutation.yaml create mode 100644 LeanPool/projects/frieze-patterns.yaml create mode 100644 LeanPool/projects/frontiermath-open-hypergraphs.yaml create mode 100644 LeanPool/projects/fundamental-inequality.yaml create mode 100644 LeanPool/projects/gap-cvp.yaml create mode 100644 LeanPool/projects/gaussian-moments-counterexamples.yaml create mode 100644 LeanPool/projects/gkp-density-one.yaml create mode 100644 LeanPool/projects/goemans-cost-counterexample.yaml create mode 100644 LeanPool/projects/granville-moore.yaml create mode 100644 LeanPool/projects/grothendieck-vanishing.yaml create mode 100644 LeanPool/projects/hadwiger-nelson-bounds.yaml create mode 100644 LeanPool/projects/hanson-wright.yaml create mode 100644 LeanPool/projects/hasse-minkowski.yaml create mode 100644 LeanPool/projects/hopf-problem.yaml create mode 100644 LeanPool/projects/hs-virial-nbc-identity.yaml create mode 100644 LeanPool/projects/hsd-interior-point-lp.yaml 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LeanPool/projects/krohn-rhodes.yaml create mode 100644 LeanPool/projects/kuramoto.yaml create mode 100644 LeanPool/projects/kurosh.yaml create mode 100644 LeanPool/projects/language-generation-characterization.yaml create mode 100644 LeanPool/projects/lattice-triangle.yaml create mode 100644 LeanPool/projects/lean-booleanfun.yaml create mode 100644 LeanPool/projects/lean-codes.yaml create mode 100644 LeanPool/projects/lean-malliavin.yaml create mode 100644 LeanPool/projects/lean-model-checking.yaml create mode 100644 LeanPool/projects/lean-operator-theory.yaml create mode 100644 LeanPool/projects/lean-poly-abc.yaml create mode 100644 LeanPool/projects/lean-quantumalg.yaml create mode 100644 LeanPool/projects/lean-scottish-book-155.yaml create mode 100644 LeanPool/projects/lean-scv.yaml create mode 100644 LeanPool/projects/lean-snumbers.yaml create mode 100644 LeanPool/projects/lean-spectral-theory.yaml create mode 100644 LeanPool/projects/lean-superorthogonality.yaml create mode 100644 LeanPool/projects/lean4-gl-coalgebras.yaml create mode 100644 LeanPool/projects/lean4-itree.yaml create mode 100644 LeanPool/projects/leancomplexanalysis.yaml create mode 100644 LeanPool/projects/leanmodularforms.yaml create mode 100644 LeanPool/projects/lehmer-e10.yaml create mode 100644 LeanPool/projects/lentil.yaml create mode 100644 LeanPool/projects/li-criterion-rh-equivalence-lean.yaml create mode 100644 LeanPool/projects/line-graph-colouring-matching.yaml create mode 100644 LeanPool/projects/list-edge-coloring-complete.yaml create mode 100644 LeanPool/projects/local-complex-geometry.yaml create mode 100644 LeanPool/projects/long-gaps-between-primes.yaml create mode 100644 LeanPool/projects/low-weight-pauli-dynamics.yaml create mode 100644 LeanPool/projects/lowdimsolvclassification.yaml create mode 100644 LeanPool/projects/main-theorem-of-polytopes.yaml create mode 100644 LeanPool/projects/markoff-modp.yaml create mode 100644 LeanPool/projects/markovprocess.yaml create mode 100644 LeanPool/projects/marshallhalltheorem.yaml create mode 100644 LeanPool/projects/matching-logic-completeness.yaml create mode 100644 LeanPool/projects/metric-codes.yaml create mode 100644 LeanPool/projects/microwave-networks.yaml create mode 100644 LeanPool/projects/min-modulus-unique-multiset-sum.yaml create mode 100644 LeanPool/projects/minimum-degree-matchings.yaml create mode 100644 LeanPool/projects/misere-games.yaml create mode 100644 LeanPool/projects/monlib4.yaml create mode 100644 LeanPool/projects/monsky.yaml create mode 100644 LeanPool/projects/moser-lattice-colorings.yaml create mode 100644 LeanPool/projects/moving-sofa.yaml create mode 100644 LeanPool/projects/mriscx.yaml create mode 100644 LeanPool/projects/multicolor-triangle-ramsey.yaml create mode 100644 LeanPool/projects/nagatafactoriality.yaml create mode 100644 LeanPool/projects/nash-williams.yaml create mode 100644 LeanPool/projects/nashembedding.yaml create mode 100644 LeanPool/projects/naslund-counterexample.yaml create mode 100644 LeanPool/projects/navier-stokes-and-euler.yaml create mode 100644 LeanPool/projects/nemytskii-lebesgue.yaml create mode 100644 LeanPool/projects/neukirch.yaml create mode 100644 LeanPool/projects/nf-weak-partition.yaml create mode 100644 LeanPool/projects/nikodym.yaml create mode 100644 LeanPool/projects/nivat-conjecture.yaml create mode 100644 LeanPool/projects/non-sofic-group.yaml create mode 100644 LeanPool/projects/nrr-aak.yaml create mode 100644 LeanPool/projects/odd-prime-valuation-distribution.yaml create mode 100644 LeanPool/projects/odlyzko-bound.yaml create mode 100644 LeanPool/projects/order-closures.yaml create mode 100644 LeanPool/projects/order-p-q.yaml create mode 100644 LeanPool/projects/osforgff.yaml create mode 100644 LeanPool/projects/paper-iv-clique-trees.yaml create mode 100644 LeanPool/projects/parallel-postulate.yaml create mode 100644 LeanPool/projects/parameter_free_gradient_p.yaml create mode 100644 LeanPool/projects/partial-combinatory-algebras.yaml create mode 100644 LeanPool/projects/partial-regularity.yaml create mode 100644 LeanPool/projects/pcf-theory.yaml create mode 100644 LeanPool/projects/pdl-craig-interpolation.yaml create mode 100644 LeanPool/projects/pebbling-hypercube.yaml create mode 100644 LeanPool/projects/pentagonal-number-theorem-analytic.yaml create mode 100644 LeanPool/projects/pentagonal-number-theorem.yaml create mode 100644 LeanPool/projects/permanent-formula-lower-bound.yaml create mode 100644 LeanPool/projects/phaseretrieval.yaml create mode 100644 LeanPool/projects/pl-accelerated-nesterov-lean.yaml create mode 100644 LeanPool/projects/poincare-three-body-nonintegrability.yaml create mode 100644 LeanPool/projects/pointwise-birkhoff.yaml create mode 100644 LeanPool/projects/polya-enumeration-theorem.yaml create mode 100644 LeanPool/projects/polylean.yaml create mode 100644 LeanPool/projects/polylog-integrals.yaml create mode 100644 LeanPool/projects/polynomial-freiman-ruzsa.yaml create mode 100644 LeanPool/projects/polynomial-method-restricted-sums.yaml create mode 100644 LeanPool/projects/puiseux.yaml create mode 100644 LeanPool/projects/pumping-cfg.yaml create mode 100644 LeanPool/projects/pythagorean-polynomial-parametrization.yaml create mode 100644 LeanPool/projects/qec-certificates.yaml create mode 100644 LeanPool/projects/quadratic-carleson-palomar-submission.yaml create mode 100644 LeanPool/projects/quadratic-iterates.yaml create mode 100644 LeanPool/projects/quantum-parallel-repetition.yaml create mode 100644 LeanPool/projects/quantum-query-foundations.yaml create mode 100644 LeanPool/projects/quasi-borel-spaces.yaml create mode 100644 LeanPool/projects/rado-theorem.yaml create mode 100644 LeanPool/projects/ramanujan-nagell.yaml create mode 100644 LeanPool/projects/ramanujan-tau-misses-primes.yaml create mode 100644 LeanPool/projects/redhill.yaml create mode 100644 LeanPool/projects/regts-sevenster.yaml create mode 100644 LeanPool/projects/rellich-kondrachov.yaml create mode 100644 LeanPool/projects/riemann-mapping-theorem.yaml create mode 100644 LeanPool/projects/riemann-roch-function-fields.yaml create mode 100644 LeanPool/projects/rl-theory-in-lean.yaml create mode 100644 LeanPool/projects/root-system.yaml create mode 100644 LeanPool/projects/rr-equals-nonmeager-paper.yaml create mode 100644 LeanPool/projects/runge-kutta-order-conditions.yaml create mode 100644 LeanPool/projects/rupert.yaml create mode 100644 LeanPool/projects/sabidussi.yaml create mode 100644 LeanPool/projects/salem-theorem.yaml create mode 100644 LeanPool/projects/sardmoreira.yaml create mode 100644 LeanPool/projects/schoenflies.yaml create mode 100644 LeanPool/projects/selberg-sieve4.yaml create mode 100644 LeanPool/projects/semicircle-catalan.yaml create mode 100644 LeanPool/projects/semicirclelaw.yaml create mode 100644 LeanPool/projects/sendov-conjecture.yaml create mode 100644 LeanPool/projects/sensitivity.yaml create mode 100644 LeanPool/projects/serre-mass-formula.yaml create mode 100644 LeanPool/projects/settheory.yaml create mode 100644 LeanPool/projects/shannon-1948-formalization.yaml create mode 100644 LeanPool/projects/singular-moduli.yaml create mode 100644 LeanPool/projects/small-undecidable-groups.yaml create mode 100644 LeanPool/projects/spectral-positivity.yaml create mode 100644 LeanPool/projects/sphere-packing.yaml create mode 100644 LeanPool/projects/stafford38-formal.yaml create mode 100644 LeanPool/projects/stallings-folding.yaml create mode 100644 LeanPool/projects/stationary-harmonic-maps.yaml create mode 100644 LeanPool/projects/structural-graph-classes.yaml create mode 100644 LeanPool/projects/sum-difference-exponent.yaml create mode 100644 LeanPool/projects/sumsthreesquares.yaml create mode 100644 LeanPool/projects/sundogcert.yaml create mode 100644 LeanPool/projects/synthetic-euclid.yaml create mode 100644 LeanPool/projects/three-gap-theorem.yaml create mode 100644 LeanPool/projects/turan3.yaml create mode 100644 LeanPool/projects/tutte-path.yaml create mode 100644 LeanPool/projects/ulms-theorem.yaml create mode 100644 LeanPool/projects/unconditional-schauder-basis.yaml create mode 100644 LeanPool/projects/virasoroproject.yaml create mode 100644 LeanPool/projects/viscosity-solution-theory.yaml create mode 100644 LeanPool/projects/vizing-equitable-edge-colouring.yaml create mode 100644 LeanPool/projects/vlasov-mean-field.yaml create mode 100644 LeanPool/projects/wallace-problem.yaml create mode 100644 LeanPool/projects/whitehead-theorem.yaml create mode 100644 LeanPool/projects/zeta-five-irrational.yaml create mode 100644 LeanPool/projects/zeta-h123.yaml create mode 100644 LeanPool/projects/zeta-zeros.yaml create mode 100644 LeanPool/projects/zeta2-zeta3-linear-independence.yaml create mode 100644 LeanPool/projects/zeta_3_irrational.yaml create mode 100644 LeanPool/projects/zflean.yaml create mode 100644 LeanPool/projects/zhang-yeung-inequality.yaml diff --git a/LeanPool.lean b/LeanPool.lean index 421d34c308..7d43a4930c 100644 --- a/LeanPool.lean +++ b/LeanPool.lean @@ -1,294 +1,4 @@ module -- shake: keep-all --deprecated_module: ignore -public import LeanPool.ABCExceptions.Imports -public import LeanPool.ACMax.Imports -public import LeanPool.AFormalizationOfBorelDeterminacyInLean.Imports -public import LeanPool.AgreeToDisagree.Imports -public import LeanPool.AharoniKorman.Imports -public import LeanPool.AndersonConjecture.Imports -public import LeanPool.AnomalousDiffusion.Imports -public import LeanPool.Apportionment.Imports -public import LeanPool.ArchonFirstProofResults.Imports -public import LeanPool.ArtinWedderburn.Imports -public import LeanPool.AsymptoticTrianglePacking.Imports -public import LeanPool.BKARForestFormula.Imports -public import LeanPool.BannaiBannaiStanton.Imports -public import LeanPool.Basic.Imports -public import LeanPool.BeckFialaMatrix.Imports -public import LeanPool.Besicovitch.Imports -public import LeanPool.BeyondBethe.Imports -public import LeanPool.BicausalOT.Imports -public import LeanPool.Biswal.Imports -public import LeanPool.BlockSpectralSensitivity.Imports -public import LeanPool.BollobasNikiforov.Imports -public import LeanPool.BooleanIsoperimetry.Imports -public import LeanPool.BooleanMultiplication.Imports -public import LeanPool.BrauerGroupNew.Imports -public import LeanPool.BrillNoetherGraphs.Imports -public import LeanPool.BrooksSubcubic.Imports -public import LeanPool.Brouwer.Imports -public import LeanPool.BruhatTits.Imports -public import LeanPool.Burkholder.Imports -public import LeanPool.CIVAxisymmetric.Imports -public import LeanPool.CaffarelliKohnNirenberg.Imports -public import LeanPool.CarlsonFunctions.Imports -public import LeanPool.CencovPetz.Imports -public import LeanPool.CenteredMaximal.Imports -public import LeanPool.Champernowne.Imports -public import LeanPool.ChannelCapacity.Imports -public import LeanPool.ChipFiring.Imports -public import LeanPool.Chudnovsky.Imports -public import LeanPool.Chvatal.Imports -public import LeanPool.CircuitComplexity.Imports -public import LeanPool.Circuitlib.Imports -public import LeanPool.ClassificationOfSurfaces.Imports -public import LeanPool.Clawristotle.Imports -public import LeanPool.CommonNeighbourConjecture.Imports -public import LeanPool.CompactSpectral.Imports -public import LeanPool.CompactnessAndDegeneracy.Imports -public import LeanPool.CompositionAlgebras.Imports -public import LeanPool.Computability.Imports -public import LeanPool.ComputableReal.Imports -public import LeanPool.ConcentrationInequalities.Imports -public import LeanPool.ConnesKreimer.Imports -public import LeanPool.ConnesRigidity.Imports -public import LeanPool.ConwayRefinement.Imports -public import LeanPool.ConwaySoifer.Imports -public import LeanPool.CramerWold.Imports -public import LeanPool.CriticalPortraits.Imports -public import LeanPool.CutAndProject.Imports -public import LeanPool.DavisKahan.Imports -public import LeanPool.DeadEnds.Imports -public import LeanPool.DemazureOperatorsLean.Imports -public import LeanPool.DemazureProduct.Imports -public import LeanPool.DensityHalesJewett.Imports -public import LeanPool.Desargues.Imports -public import LeanPool.Dilatations.Imports -public import LeanPool.DirectedTopologyLean4.Imports -public import LeanPool.DistanceGeometry.Imports -public import LeanPool.DomainTheory.Imports -public import LeanPool.DrossFractionalTriangleDecomposition.Imports -public import LeanPool.Duality.Imports -public import LeanPool.EcTateLean.Imports -public import LeanPool.Egrs75.Imports -public import LeanPool.EhrhartVolumeInequality.Imports -public import LeanPool.EilenbergZilber.Imports -public import LeanPool.EllipticPDE.Imports -public import LeanPool.Erdos1196.Imports -public import LeanPool.Erdos132ConvexK3.Imports -public import LeanPool.Erdos132N14.Imports -public import LeanPool.Erdos132ThreeChain.Imports -public import LeanPool.Erdos137.Imports -public import LeanPool.Erdos346.Imports -public import LeanPool.Erdos367.Imports -public import LeanPool.Erdos403.Imports -public import LeanPool.Erdos548.Imports -public import LeanPool.Erdos81PaperIContrib.Imports -public import LeanPool.Erdos81PaperIIContrib.Imports -public import LeanPool.Erdos81PaperIIIContrib.Imports -public import LeanPool.Erdos865.Imports -public import LeanPool.Erdos97ConvexOctagon.Imports -public import LeanPool.ErdosGinzburgZiv.Imports -public import LeanPool.ErdosMoser.Imports -public import LeanPool.ErdosTuzaValtr.Imports -public import LeanPool.EscauriazaSereginSverak.Imports -public import LeanPool.EuclideanJordan.Imports -public import LeanPool.EvenGraphCycles.Imports -public import LeanPool.EventStructures.Imports -public import LeanPool.ExpChaotic.Imports -public import LeanPool.FactorizationSystems.Imports -public import LeanPool.Feige.Imports -public import LeanPool.FelConjecture.Imports -public import LeanPool.Fineqs.Imports -public import LeanPool.FiniteGraphFundamentalGroup.Imports -public import LeanPool.FiveEighthsTheorem.Imports -public import LeanPool.Flean.Imports -public import LeanPool.FltRegular.Imports -public import LeanPool.FoZfc.Imports -public import LeanPool.FormalLearningTheory.Imports -public import LeanPool.FormalizationOfBoundedArithmetic.Imports -public import LeanPool.FormalizingGMT.Imports -public import LeanPool.FourAP.Imports -public import LeanPool.FriezePatterns.Imports -public import LeanPool.FrontierMathOpenHypergraphs.Imports -public import LeanPool.FullyDynamicMatching.Imports -public import LeanPool.FundamentalInequality.Imports -public import LeanPool.GKPCarry.Imports -public import LeanPool.GapCVP.Imports -public import LeanPool.GaussianMomentsCounterexamples.Imports -public import LeanPool.GoemansFlow.Imports -public import LeanPool.GranvilleMoore.Imports -public import LeanPool.GraphColouringMatching.Imports -public import LeanPool.GrothendieckVanishing.Imports -public import LeanPool.HSDInteriorPointLP.Imports -public import LeanPool.HadwigerNelsonBounds.Imports -public import LeanPool.HansonWright.Imports -public import LeanPool.HardSphereNBC.Imports -public import LeanPool.HasseMinkowski.Imports -public import LeanPool.HopfProblem.Imports -public import LeanPool.Incompleteness.Imports -public import LeanPool.InfinitaryLogic.Imports -public import LeanPool.InfiniteConnesRigidity.Imports -public import LeanPool.InflationTermination.Imports -public import LeanPool.IsTranscendentalPi.Imports -public import LeanPool.IsoGraph.Imports -public import LeanPool.Isoperimetric.Imports -public import LeanPool.JacobianDiffgeo.Imports -public import LeanPool.JohnsonLindenstraussLean.Imports -public import LeanPool.KahnKalai.Imports -public import LeanPool.KaltonPeck.Imports -public import LeanPool.KaltonRoberts.Imports -public import LeanPool.KasamiCyclicAdditive.Imports -public import LeanPool.Koethe.Imports -public import LeanPool.Komlos.Imports -public import LeanPool.KrafftSieve.Imports -public import LeanPool.KrohnRhodes.Imports -public import LeanPool.Kuramoto.Imports -public import LeanPool.Kurosh.Imports -public import LeanPool.LanguageGeneration.Imports -public import LeanPool.LatticeTriangle.Imports -public import LeanPool.Lean4GlCoalgebras.Imports -public import LeanPool.Lean4Itree.Imports -public import LeanPool.LeanBooleanfun.Imports -public import LeanPool.LeanComplexAnalysis.Imports -public import LeanPool.LeanModelChecking.Imports -public import LeanPool.LeanModularForms.Imports -public import LeanPool.LeanPolyABC.Imports -public import LeanPool.LeanQuantumAlg.Imports -public import LeanPool.LeanStationaryHarmonicMaps.Imports -public import LeanPool.LehmerE10.Imports -public import LeanPool.Lentil.Imports -public import LeanPool.LiCriterion.Imports -public import LeanPool.ListEdgeColoringComplete.Imports -public import LeanPool.LocalComplexGeometry.Imports -public import LeanPool.LongGapsBetweenPrimes.Imports -public import LeanPool.LowDimSolvClassification.Imports -public import LeanPool.LowWeightPauliDynamics.Imports -public import LeanPool.MRiscX.Imports -public import LeanPool.Malliavin.Imports -public import LeanPool.MarkoffModP.Imports -public import LeanPool.MarkovProcess.Imports -public import LeanPool.MarshallHall.Imports -public import LeanPool.MassFormula.Imports -public import LeanPool.MatchingLogic.Imports -public import LeanPool.MaxFlowMinCut.Imports -public import LeanPool.MetricCodes.Imports -public import LeanPool.MicrowaveNetworks.Imports -public import LeanPool.MinModulusUniqueMultisetSum.Imports -public import LeanPool.MinimumDegreeMatching.Imports -public import LeanPool.MisereGames.Imports -public import LeanPool.Monlib4.Imports -public import LeanPool.Monsky.Imports -public import LeanPool.MooreBound.Imports -public import LeanPool.MoserLatticeColorings.Imports -public import LeanPool.MovingSofa.Imports -public import LeanPool.MulticolorTriangleRamsey.Imports -public import LeanPool.NFWeakPartition.Imports -public import LeanPool.NagataFactoriality.Imports -public import LeanPool.NandakumarRamanaRao.Imports -public import LeanPool.NashEmbedding.Imports -public import LeanPool.NashWilliams.Imports -public import LeanPool.NaslundCounterexample.Imports -public import LeanPool.NavierStokesAndEuler.Imports -public import LeanPool.NemytskiiLebesgue.Imports -public import LeanPool.Neukirch.Imports -public import LeanPool.Nikodym.Imports -public import LeanPool.Nivat.Imports -public import LeanPool.NonSoficGroup.Imports -public import LeanPool.OSforGFF.Imports -public import LeanPool.OddPrimeValuationDistribution.Imports -public import LeanPool.Odlyzko.Imports -public import LeanPool.OneManifold.Imports -public import LeanPool.OperatorTheory.Imports -public import LeanPool.OrderClosures.Imports -public import LeanPool.OrderPQ.Imports -public import LeanPool.PCFTheory.Imports -public import LeanPool.PDL.Imports -public import LeanPool.PFR.Imports -public import LeanPool.PLAcceleratedNesterovLean.Imports -public import LeanPool.PaperIVCliqueTree.Imports -public import LeanPool.ParallelPostulate.Imports -public import LeanPool.ParameterFreeGradient.Imports -public import LeanPool.PartialCombinatoryAlgebras.Imports -public import LeanPool.PartialRegularity.Imports -public import LeanPool.PebblingLean.Imports -public import LeanPool.PentagonalNumberTheorem.Imports -public import LeanPool.PentagonalNumberTheoremAnalytic.Imports -public import LeanPool.PermanentFormulaLowerBound.Imports -public import LeanPool.PhaseRetrieval.Imports -public import LeanPool.PoincareThreeBody.Imports -public import LeanPool.PointwiseBirkhoff.Imports -public import LeanPool.PolyaEnumerationTheorem.Imports -public import LeanPool.Polylean.Imports -public import LeanPool.PolylogIntegrals.Imports -public import LeanPool.PolynomialMethodRestrictedSums.Imports -public import LeanPool.Polytopes.Imports -public import LeanPool.Puiseux.Imports -public import LeanPool.PumpingCfg.Imports -public import LeanPool.PythagoreanPolynomialParametrization.Imports -public import LeanPool.QECCertificates.Imports -public import LeanPool.QuadraticCarleson.Imports -public import LeanPool.QuadraticIterates.Imports -public import LeanPool.QuantumParallelRepetition.Imports -public import LeanPool.QuantumQuery.Imports -public import LeanPool.QuasiBorelSpaces.Imports -public import LeanPool.Rado.Imports -public import LeanPool.RamanujanNagell.Imports -public import LeanPool.RamanujanTauMissesPrimes.Imports -public import LeanPool.RearrangementNumber.Imports -public import LeanPool.Redhill.Imports -public import LeanPool.RegtsSevenster.Imports -public import LeanPool.RellichKondrachov.Imports -public import LeanPool.RiemannMappingTheorem.Imports -public import LeanPool.RiemannRochFunctionFields.Imports -public import LeanPool.RlTheoryInLean.Imports -public import LeanPool.RootSystem.Imports -public import LeanPool.RungeKuttaOrderConditions.Imports -public import LeanPool.Rupert.Imports -public import LeanPool.SNumbers.Imports -public import LeanPool.Sabidussi.Imports -public import LeanPool.SalemTheorem.Imports -public import LeanPool.SardMoreira.Imports -public import LeanPool.Schoenflies.Imports -public import LeanPool.ScottishBook155.Imports -public import LeanPool.SelbergSieve4.Imports -public import LeanPool.SemicircleCheck.Imports -public import LeanPool.SemicircleLaw.Imports -public import LeanPool.Sendov.Imports -public import LeanPool.Sensitivity.Imports -public import LeanPool.SetTheory.Imports -public import LeanPool.SeveralComplexVariables.Imports -public import LeanPool.Shannon1948Formalization.Imports -public import LeanPool.SingularModuli.Imports -public import LeanPool.SmallUndecidableGroups.Imports -public import LeanPool.SpectralPositivity.Imports -public import LeanPool.SpectralTheory.Imports -public import LeanPool.SpherePacking.Imports -public import LeanPool.Stafford38.Imports -public import LeanPool.StallingsFolding.Imports -public import LeanPool.SteinhausThreeGap.Imports -public import LeanPool.StructuralGraphClasses.Imports -public import LeanPool.SumDifferenceExponent.Imports -public import LeanPool.SumsThreeSquares.Imports -public import LeanPool.Sundogcert.Imports -public import LeanPool.Superorthogonality.Imports -public import LeanPool.SyntheticEuclid4.Imports -public import LeanPool.ThreeGap.Imports -public import LeanPool.Turan3.Imports -public import LeanPool.TuttePath.Imports -public import LeanPool.TwoColoringOneRound.Imports -public import LeanPool.UlmsTheorem.Imports -public import LeanPool.UnconditionalSchauderBasis.Imports -public import LeanPool.VirasoroProject.Imports -public import LeanPool.ViscositySolutionTheory.Imports -public import LeanPool.Vizing.Imports -public import LeanPool.Vlasov.Imports -public import LeanPool.Wallace.Imports -public import LeanPool.WhiteheadTheorem.Imports -public import LeanPool.ZFLean.Imports -public import LeanPool.Zeta32.Imports -public import LeanPool.Zeta3Irrational.Imports -public import LeanPool.Zeta5Irrational.Imports -public import LeanPool.ZetaH123.Imports -public import LeanPool.ZetaZeros.Imports -public import LeanPool.ZhangYeungInequality.Imports +-- Lake builds every LeanPool module directly. Import a project's +-- Imports module to use its complete public API. diff --git a/LeanPool/projects.yml b/LeanPool/projects.yml deleted file mode 100644 index 9697f4c46d..0000000000 --- a/LeanPool/projects.yml +++ /dev/null @@ -1,14106 +0,0 @@ -projects: - - slug: structural-graph-classes - title: Split, cograph and threshold graph structures - summary: >- - Connects split partitions, cotrees and threshold creation sequences on - Mathlib SimpleGraph. Proves the finite forbidden-induced-subgraph - characterizations, split iff chordal and co-chordal, and completeness of - cotrees and creation sequences. Supplies certificate-preserving operations, - twins and nonempty union/join decompositions for cographs, maximum-clique - split partitions, and induced/isomorphic transport of existing chordal - elimination orders and clique trees. Classical graph theory is formalized; - no new mathematical theorem, canonical certificate or recognition-time - bound is claimed. - branch: graph theory - entry_module: LeanPool.StructuralGraphClasses - authors: - - Juan Pablo Traverso Gianini - source: - title: Structural graph classes, original Lean development checkpoint - url: https://github.com/jtraverso/lean-pool/tree/c431fcecd3c963bcca643c379d807260585711b8/LeanPool/StructuralGraphClasses - github_repo: jtraverso/lean-pool - commit: c431fcecd3c963bcca643c379d807260585711b8 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - SimpleGraph.isThreshold_iff_nonempty_creationRepresentation - main_results: - - declaration: SimpleGraph.isSplit_iff_forbidden_induced_subgraphs - informal: >- - A finite graph is split iff it contains no induced 2K2, C4 or C5. - source_ref: Földes and Hammer, Split graphs (1977). - - declaration: SimpleGraph.isSplit_iff_isChordal_and_compl - informal: >- - A finite graph is split iff both it and its complement are chordal. - source_ref: Földes and Hammer, Split graphs (1977). - - declaration: SimpleGraph.isCograph_iff_nonempty_cotreeRepresentation - informal: >- - A finite graph is P4-free iff it is isomorphic to the graph of a cotree, - whose constructors are empty, singleton, disjoint union and join. - source_ref: >- - Seinsche (1974), doi:10.1016/0095-8956(74)90063-X; - Corneil, Lerchs and Stewart Burlingham (1981), doi:10.1016/0166-218X(81)90013-5. - - declaration: SimpleGraph.isThreshold_iff_isSplit_and_isCograph - informal: >- - A graph has a split partition with nested independent-side - neighbourhoods iff it is both split and P4-free. No finiteness is required. - source_ref: Classical threshold characterization; local Threshold.lean and Cotree.lean. - - declaration: SimpleGraph.isThreshold_iff_forbidden_induced_subgraphs - informal: >- - A finite graph is threshold iff it contains no induced 2K2, C4 or P4. - source_ref: Chvátal and Hammer (1977), doi:10.1016/S0167-5060(08)70731-3. - - declaration: SimpleGraph.isThreshold_iff_nonempty_creationRepresentation - informal: >- - A finite graph is threshold iff it is isomorphic to a graph constructed - from the empty graph by successively adding an isolated or universal vertex. - source_ref: Classical threshold creation characterization; local ThresholdCreation.lean. - - declaration: SimpleGraph.IsCograph.exists_twins - informal: >- - A finite cograph with at least two vertices has two distinct vertices - with identical adjacency outside their pair, whether adjacent or not. - source_ref: Cotree induction in CographTwins.lean; classical cograph structure. - - declaration: SimpleGraph.IsCograph.exists_union_root - informal: >- - A finite cograph that is not preconnected admits a cotree representation - with a union root and two nonempty children. This is an existence claim, - not a claim about every unreduced cotree. - source_ref: Seinsche decomposition and CotreeRepresentation constructors. - - declaration: SimpleGraph.IsCograph.exists_join_root - informal: >- - A preconnected finite cograph with at least two vertices admits a cotree - representation with a join root and two nonempty children. - source_ref: Seinsche decomposition and CotreeRepresentation constructors. - - declaration: SimpleGraph.IsSplit.exists_partition_with_maximum_clique - informal: >- - Every finite split graph admits a split partition whose clique part - has maximum cardinality. Such a partition need not be unique. - source_ref: Direct finite maximization argument in SplitDecomposition.lean. - tags: - - graph-theory - - split-graphs - - cographs - - threshold-graphs - - structural-certificates - msc: - - 05C75 - - 05C05 - - slug: list-edge-coloring-complete - title: Häggkvist–Janssen list edge colouring of complete graphs - summary: >- - Proves that arbitrary lists of at least n colours on the edges of a - complete graph with n vertices admit a proper list edge colouring. - The proof constructs and proves uniqueness of the local rank certificate, - identifies a nonzero extended-star polynomial coefficient, and applies - Mathlib's combinatorial Nullstellensatz. This is an alternative proof - presentation, not a literal translation of the article's argument. - The conclusion uses Mathlib's - actual edge type and line-graph colouring, with no certificate hypothesis - and no restriction on the colour universe. This formalizes a classical - theorem, not a new bound or a solution of the general list-edge conjecture. - branch: graph theory - entry_module: LeanPool.ListEdgeColoringComplete - authors: - - Juan Pablo Traverso Gianini - source: - title: New Bounds on the List-Chromatic Index of the Complete Graph and Other Simple Graphs - authors: - - Roland Häggkvist - - Jeannette Janssen - doi: 10.1017/S0963548397002927 - url: https://github.com/jtraverso/haggkvist-janssen/tree/26d7b2ee0e14813f0e42c3f8628d6b216327c8df - github_repo: jtraverso/haggkvist-janssen - commit: 26d7b2ee0e14813f0e42c3f8628d6b216327c8df - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.ListEdgeColoringComplete.exists_listEdgeColoring_complete - main_results: - - declaration: LeanPool.ListEdgeColoringComplete.exists_listEdgeColoring_complete - informal: >- - For every finite vertex type V, arbitrary colour type and finite list - on each edge of the complete graph, if every list has cardinality at - least the number of vertices, there is a proper colouring of the line - graph choosing a colour from each list. This includes orders zero and one. - source_ref: Häggkvist–Janssen (1997), Theorems 1.1 and 3.1, complete-graph case. - tags: - - graph-theory - - list-edge-colouring - - complete-graphs - - combinatorial-nullstellensatz - msc: - - 05C15 - - slug: zeta2-zeta3-linear-independence - title: Linear independence of 1, zeta(2), and zeta(3) - summary: >- - Proves that 1, the Riemann zeta function at 2, and its value at 3 are - linearly independent over the rationals, using integer polynomial - approximation, p-adic coefficient bounds, prime-edge nonvanishing, - and logarithmic potential energy estimates for Hankel determinants. - Includes reusable irrationality criteria, contour integration, - Andreief identities, Newton-basis valuation bounds, and energy - inequalities for finite measures. The cited source is the completed - Lean proof; independent novelty of the informal argument is not claimed. - branch: analytic number theory - entry_module: LeanPool.Zeta32 - authors: - - Qian Tang - source: - title: Linear independence of 1, zeta(2), and zeta(3), Lean formalization - url: https://github.com/dtq1997/zeta2-zeta3-linear-independence/tree/669c92bf7a3728e0de9ad80c373bc29fb5ca3d58 - github_repo: dtq1997/zeta2-zeta3-linear-independence - commit: 669c92bf7a3728e0de9ad80c373bc29fb5ca3d58 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Zeta32.one_zeta_two_zeta_three_linearIndependent - main_results: - - declaration: Zeta32.one_zeta_two_zeta_three_linearIndependent - informal: >- - The three complex numbers 1, riemannZeta 2, and riemannZeta 3 - are linearly independent over the rational numbers. - source_ref: >- - Upstream Zeta32/Final.lean and Solution.lean at the pinned commit. - First public proof release: 2026-09-30 03:47:38 UTC, - commit ec4fcb5617126b0a861f9e0c16f95425b32d4638. - The upstream README and NOTICE disclose substantial AI assistance. - The polynomial argument is credited upstream to Fauzan; - this card cites the Lean artifact, not a separately verified paper. - - declaration: Zeta32Acceptance.one_zeta_two_zeta_three_linearIndependent_series - informal: >- - For rational a, b, and c, if a plus b times the real sum of - 1/(k+1)^2 plus c times the real sum of 1/(k+1)^3 is zero, - then all three coefficients are zero; both sums range over - natural numbers k. - - declaration: Zeta32.zeta3_sub_rat_mul_zeta2_irrational - informal: >- - For every rational r, the real series for zeta(3) minus r - times the real series for zeta(2) is irrational. - tags: - - zeta-values - - irrationality - - linear-independence - - p-adic-valuations - - potential-theory - msc: - - 11J72 - - 11M06 - - slug: parallel-postulate - title: Independence of Playfair's axiom via Euclidean and Klein models - summary: >- - Constructs Euclidean and Klein models of a synthetic Hilbert plane, - proving that Playfair's axiom is independent of incidence, Pasch order, - and congruence axioms, even with Archimedes and Dedekind continuity. - Develops reusable coordinate geometry over ordered fields, projective - motions, Klein angles and metric, smooth-path length and distance, - congruence constructions, and a coordinate curvature calculation. - Dedekind continuity replaces Hilbert's maximal completeness axiom; - their equivalence is not formalized. Curvature uses Brioschi's formula, - and distance uses C1 paths; no identification with Mathlib's Riemannian - distance or connection curvature is asserted. - branch: foundations of geometry - entry_module: LeanPool.ParallelPostulate - authors: - - Richard Sutton - source: - title: ParallelPostulate formalization by Richard Sutton - url: https://github.com/Richiesutie/ParallelPostulate/tree/8ee502804263194ae40babeca9773127f1e0a085 - github_repo: Richiesutie/ParallelPostulate - commit: 8ee502804263194ae40babeca9773127f1e0a085 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - ParallelPostulate.Hilbert.playfair_independent_continuous - main_results: - - declaration: ParallelPostulate.Hilbert.playfair_independent_continuous - informal: >- - There exists a Hilbert plane satisfying Archimedes and Dedekind - continuity and Playfair's axiom, and there exists another satisfying - the same continuity axioms and the negation of Playfair's axiom. - The models use real coordinate planes with bounds zero and minus one. - source_ref: >- - Upstream Hilbert/Independence.lean. First publicly announced - 2026-09-30 01:08:53 UTC at - https://leanprover.zulipchat.com/#narrow/channel/583339-AI-authored-projects/topic/Independence.20Parallel.20Postulate/near/627942879. - The author's follow-up discloses mostly AI-written proofs. - Pasch order and Dedekind continuity are the stated variants; - equivalence to plane separation and maximal completeness is not proved. - - declaration: ParallelPostulate.HyperbolicPlane.hyperbolic_parallels - informal: >- - Over any linearly ordered field, for every negative bound and every - line in the bounded coordinate plane, infinitely many distinct lines - pass through any point of the plane outside that line without meeting it. - - declaration: ParallelPostulate.HyperbolicPlane.kleinDist_eq_arsinh - informal: >- - For every negative real bound and every pair of points in its plane, - the infimum of C1-path lengths in Klein's coordinate metric equals - arsinh of the square root of minus the bound times apart, divided by - the square root of minus the bound. - - declaration: ParallelPostulate.HyperbolicPlane.kleinMetric_curvature - informal: >- - For every real bound, Brioschi's coordinate curvature formula applied - to the three coefficients of Klein's metric equals that bound at - every point of its coordinate plane. - tags: - - synthetic-geometry - - hyperbolic-geometry - - parallel-postulate - - klein-model - msc: - - 51F15 - - 51M10 - - 53A35 - - slug: moving-sofa - title: Optimality of Gerver's sofa - summary: >- - Formalizes Baek's solution of the moving sofa problem: Gerver's sofa attains - the supremum of the areas of closed connected shapes that can move continuously - around a right-angled corner in a unit-width hallway. Includes certified Gerver - parameters, admissibility, convex cap geometry, Stieltjes curve area, Jordan - separation, polygon approximation, and the concave quadratic upper bound. - branch: convex geometry - entry_module: LeanPool.MovingSofa - authors: - - Dean Cureton - - Dawid Trela - - Rado Kirov - - Kirill Alkhimov - source: - title: Optimality of Gerver's Sofa - authors: - - Jineon Baek - arxiv: '2411.19826' - url: https://github.com/deancureton/MovingSofa/tree/4d5569131940815f47a9ccf3e90a4c5043c56127 - github_repo: deancureton/MovingSofa - commit: 4d5569131940815f47a9ccf3e90a4c5043c56127 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - MovingSofa.sofaConstant_eq_volume_gerversSofa - main_results: - - declaration: MovingSofa.sofaConstant_eq_volume_gerversSofa - informal: >- - The supremum, in extended nonnegative real numbers, of the areas of all - admissible moving sofas equals the area of the explicitly defined Gerver sofa. - source_ref: >- - Baek, Optimality of Gerver's Sofa, Theorem 1.1. Upstream Lean completion - a0753860f1d2b4424e059c35e64ad611be71a415 (2026-09-20 10:55:24 UTC), announced - 2026-09-20 18:42:40 UTC at https://x.com/deancureton/status/2101743865392926967. - The imported commit has identical Lean sources; its later change is documentation only. - - declaration: MovingSofa.GerversSofa.ABφθSpec.existsUnique - informal: >- - Gerver's four defining equations have exactly one quadruple of parameters - with A and B nonnegative and 0 ≤ φ ≤ θ ≤ π/4. - source_ref: >- - Romik, Differential equations and exact solutions in the moving sofa problem, - equations (1)-(4), arXiv:1606.08111. The GerverSofaLean v1.1.0 release archive - (MIT, Dawid Trela) has SHA-256 - 7c496150f709ac709ae01f4836e9a702a18f752450efa69f15cf44264c0c2f77. - - declaration: MovingSofa.isMovingSofa_gerversSofa - informal: >- - Gerver's sofa is closed and connected and admits a continuous rigid motion - from the horizontal arm to the vertical arm, starting at the identity and - remaining within the unit-width hallway throughout. - tags: - - convex-geometry - - moving-sofa-problem - - geometric-optimization - - interval-arithmetic - msc: - - 52A40 - - 52A10 - - 49Q10 - - slug: krohn-rhodes - title: Aperiodic and simple-group wreath-division towers - summary: >- - Constructs a finite recursive wreath-division tower for every finite - monoid in universe zero, with finite aperiodic or finite simple-group - factors and every simple-group factor a monoid divisor of the original - monoid. Implements a variant of the Diekert–Kufleitner–Steinberg - local-divisor argument, with reusable monoid and semigroup division, - local divisors, action coverings, wreath reassociation, and the - Krasner–Kaloujnine group embedding. The endpoint permits arbitrary - aperiodic factors and recursive intermediate bases; it does not assert - the three-element flip-flop reduction, a single transformation-monoid - strong division, or quantitative decomposition bounds. - branch: algebra - entry_module: LeanPool.KrohnRhodes - authors: - - Aditya Rao - source: - title: Krohn–Rhodes formalization by Aditya Rao - url: https://github.com/adii800/krohn-rhodes-lean/tree/8152fef3507406d2b4f4ed3760404ea88298e382 - github_repo: adii800/krohn-rhodes-lean - commit: 8152fef3507406d2b4f4ed3760404ea88298e382 - arxiv: '1111.1585' - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.KrohnRhodes.krohn_rhodes_prime_decomposition - main_results: - - declaration: LeanPool.KrohnRhodes.krohn_rhodes_prime_decomposition - informal: >- - Every finite monoid M in universe zero admits a finite list of - finite monoid factors, each witnessed to be aperiodic or a simple - group, with DivTowerWreath M factors and every simple-group factor - a monoid divisor of M. An empty tower means M divides the trivial - monoid. At a nonempty step, M divides F wreath B for a finite - action of a finite monoid B that itself has the tail tower. - source_ref: >- - Upstream PrimeDecomposition.lean, completed September 28, 2026 - at 07:36:52 UTC and publicly released at 08:48:24 UTC. Variant of - Diekert–Kufleitner–Steinberg, arXiv:1111.1585, Sections 3–4, with - left actions, the larger local divisor cM intersect Mc, arbitrary - aperiodic factors, and recursive semigroup divisions. The stronger - flip-flop and transformation-monoid claims of Theorem 4.1 are not - asserted. Upstream CITATION.cff attributes the proofs primarily - to Claude Code under Aditya Rao's direction. - - declaration: LeanPool.KrohnRhodes.krasnerKaloujnine_injective - informal: >- - For any group G and normal subgroup N, the constructed monoid - homomorphism from G to Mathlib's regular wreath product of N and - G/N is injective. No finiteness assumption is required. - source_ref: >- - Upstream Foundations/KrasnerKaloujnine.lean; the classical - Krasner–Kaloujnine embedding theorem. - - declaration: LeanPool.KrohnRhodes.LocalDivisor.localDivides - informal: >- - For any monoid M and c in M, the local divisor on cM intersect Mc, - with its twisted multiplication and identity c, is a quotient of - a submonoid of M. - source_ref: >- - Diekert–Kufleitner–Steinberg, arXiv:1111.1585, Section 2.5; uses - the full local divisor rather than the smaller cMc union {c}. - tags: - - semigroup-theory - - finite-monoids - - wreath-products - - automata-theory - - group-extensions - msc: - - 20M10 - - 20M35 - - 20E22 - - slug: zeta-five-irrational - title: Irrationality of the Riemann zeta function at five - summary: >- - Proves that Mathlib's Riemann zeta function at five is an irrational - real number, following Aabir Fauzan's September 2026 preprint. - Constructs integer polynomials of degree 37n with positive values - at zeta(5) that decay exponentially in n squared. Includes a general - integer-polynomial irrationality criterion, Hankel determinant and - moment-integral identities, logarithmic-energy estimates, prime-adic - integrality bounds, and kernel-checked rational certificates. The - formalization uses a different normalizer and a positive decay - constant in place of the paper's specific constant 139/5. Reuses the - prime number theorem infrastructure already in the MooreBound project. - branch: number theory - entry_module: LeanPool.Zeta5Irrational - authors: - - Moritz Firsching - source: - title: Zeta5 formalization of Aabir Fauzan's zeta(5) is irrational - url: https://github.com/mo271/Zeta5/tree/f19a1960609f7d38e7b63fd2acb05e6f60a7b741 - github_repo: mo271/Zeta5 - commit: f19a1960609f7d38e7b63fd2acb05e6f60a7b741 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - Zeta5Irrational.irrational_five - main_results: - - declaration: Zeta5Irrational.irrational_five - informal: >- - There exists an irrational real number x whose complex coercion - equals Mathlib's Riemann zeta function evaluated at five. - source_ref: >- - Fauzan, zeta(5) is irrational, Theorem 1.1, September 17, 2026, - https://doi.org/10.5281/zenodo.22826419; upstream Apery/Main.lean. - The completed formalization was released on September 23, 2026. - - declaration: Zeta5Irrational.main_estimate - informal: >- - There is a positive real c such that, for all sufficiently large - n, an integer polynomial Q of degree 37n is a positive rational - multiple of the constructed determinant polynomial and satisfies - 0 < Q(zeta(5)) < exp(-c*n^2). - source_ref: >- - Upstream Apery/MainEstimate.lean; variant of the paper's Theorem 2.1 - with c = -800*(Aeff + U) instead of 139/5. - - declaration: Zeta5Irrational.irrational_of_eventually_exists_int_poly - informal: >- - A real xi is irrational if, eventually, integer polynomials of - degree at most d(n) have positive values at xi bounded by epsilon(n), - and b^d(n)*epsilon(n) tends to zero for every positive natural b. - source_ref: Upstream Apery/Criterion.lean; the paper's Section 1.1. - tags: - - irrationality - - zeta-function - - hankel-determinants - - potential-theory - - certified-computation - msc: - - 11J72 - - 11M06 - - 31A15 - - slug: block-spectral-sensitivity - title: Block sensitivity can exceed spectral sensitivity squared - summary: >- - Constructs a total Boolean function on 14011 blocks of 144 bits with - block sensitivity at least 14011 and squared spectral sensitivity at - most U = 7149 + 2 sqrt(150108). Proves the strict separation - lambda(f)^2.12 < bs(f), multiplicativity of spectral sensitivity under - arbitrary block composition, and an exponential lower bound with - factor 14011/U > 1.768 for the self-composed family. Includes reusable - finite product probability, an asymmetric Lovasz local lemma, partial - Boolean assignments, certificate-family spectral bounds, and a Paley - doubly regular tournament. The formalized explicit exponent is 2.12; - the exact logarithmic exponent is also retained. - branch: computational complexity - entry_module: LeanPool.BlockSpectralSensitivity - authors: - - Alex Meiburg - source: - title: Block Sensitivity can exceed Spectral Sensitivity Squared - arxiv: '2608.00851' - url: https://github.com/Timeroot/BS_Lam/tree/7bd39a8d41ee7910d3296d0477ad18f8fff9d870 - github_repo: Timeroot/BS_Lam - commit: 7bd39a8d41ee7910d3296d0477ad18f8fff9d870 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - BSLambda.Final.exists_bs_gt_lam_rpow - main_results: - - declaration: BSLambda.Final.exists_bs_gt_lam_rpow - informal: >- - There exists a total Boolean function on ZMod 14011 times Fin 144 - whose block sensitivity strictly exceeds its spectral sensitivity - raised to the real exponent 2.12. - source_ref: Upstream Main.lean; bs_lambda.txt Sections 5-12 and 14. - - declaration: BSLambda.Final.exists_counterexample - informal: >- - There exists a gate labelling whose certificate-union indicator f - has bs(f) at least 14011, lambda(f)^2 at most - U = 7149 + 2 sqrt(150108), and lambda(f)^alpha at most bs(f), - where alpha = 2 log(14011)/log(U) > 2. - source_ref: Upstream Main.lean; bs_lambda.txt Sections 5-12 and 14. - - declaration: BSLambda.Final.exists_ratio_blowup - informal: >- - A single gate labelling yields, for every natural m, an m-fold - self-composition F_m satisfying - (14011/U)^m * lambda(F_m)^2 <= bs(F_m). - The auxiliary theorem ratio_gt proves 14011/U > 1.768. - source_ref: Upstream Main.lean; bs_lambda.txt Section 14, division-free bound. - - declaration: BSLambda.lam_comp - informal: >- - For any Boolean functions on finite coordinate types, spectral - sensitivity of their block composition equals the product of - their spectral sensitivities, including the degenerate cases. - source_ref: Upstream Spectral/Multiplicative.lean; composition theorem used in Section 14. - - declaration: BSLambda.LLL.pr_avoid_pos - informal: >- - In a finite uniform product of nonempty alphabets, events supported - on specified coordinate sets have positive probability of all being - avoided if each event probability is at most its weight times the - product of one minus the weights of its dependent neighbours, - with every weight strictly between zero and one. - source_ref: Upstream LLL/Asymmetric.lean; asymmetric local lemma in Section 10. - tags: - - boolean-functions - - complexity - - probabilistic-method - msc: - - 68Q17 - - 05D40 - - 05C50 - - slug: low-weight-pauli-dynamics - title: Low-weight Pauli dynamics and truncation error - summary: >- - Derives a normalized Pauli 2-norm truncation-error bound for concrete - circuits of disjoint-support Hermitian Pauli rotations, with explicit - locality, angle, step-count, and cutoff hypotheses. Develops a faithful - matrix representation, Pauli coefficient reconstruction, finite Schur - bounds, and reusable scalar multi-jump ladder estimates. Also proves a - combinatorial light-cone bound and exact counterexamples showing why - repeated product-formula layers must count toward the layer exponent. - This is the norm-level analysis, not a formalization of expectation-value - accuracy, Hamiltonian approximation, or the paper's full algorithm. - branch: quantum information - entry_module: LeanPool.LowWeightPauliDynamics - authors: - - Jue Xu - source: - title: Classical Simulation of Noiseless Quantum Dynamics without Randomness - arxiv: '2601.15770' - url: https://github.com/Jue-Xu/Lean4LPD/tree/ad0561661c46ca4c331d21104468e5dc65b2a028 - github_repo: Jue-Xu/Lean4LPD - commit: ad0561661c46ca4c331d21104468e5dc65b2a028 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Lean4LPD.PauliString.pauliNorm_layerStep_error_le_model_of_source_regime - main_results: - - declaration: Lean4LPD.PauliString.pauliNorm_layerStep_error_le_model_of_source_regime - informal: >- - For an observable supported on Pauli weights at most ko, a block of - Γ positive-many disjoint-support layers of Hermitian rotations of - weight at most kh, kh ≥ 2, and |sin θ| ≤ a ≤ αt/r with α > 0 and t ≥ 0, - truncation to weight ko + m(kh - 1) after each of r blocks has Pauli - 2-norm error at most (2Γ(kh - 1)αt)^(m+1) (e(m+1))^(ko/(kh-1)) times - the input norm. The conditions include m ≥ 1, r ≥ 5, m ≤ r, - rΓ ≥ 8(m+1)^2, and r ≥ 8e^2(ko+kh-1)αt. Γ counts every layer of the - actual block, including repeated product-formula layers. - source_ref: >- - Upstream STATEMENTS.md and Pauli/LayerError.lean at the pinned commit; - norm-level counterpart of the revised supplement's truncation bound. - - declaration: Lean4LPD.l2_opNorm_le_schur - informal: >- - A finite rectangular matrix over any RCLike field, with absolute row - sums bounded by nonnegative R and absolute column sums bounded by - nonnegative C, has Euclidean operator norm at most sqrt(RC). - source_ref: Upstream Schur.lean, finite-dimensional Schur test. - - declaration: Lean4LPD.PauliString.reachable_weight_le - informal: >- - Every Pauli string reachable through a list of disjoint-support - kh-local layers, kh ≥ 1, has weight at most its initial weight times - kh raised to the number of layers. Reachability is combinatorial; - this theorem does not assert that a reachable coefficient is nonzero. - source_ref: Upstream Pauli/LayerWitness.lean, effective-layer light cone. - - declaration: Lean4LPD.LayerCounterexample.actual_discard_exceeds_gamma_bound - informal: >- - For the explicit eight-qubit second-order word L1,L2,L2,L1 and angles - 0 < θ < π/2, truncating the evolved weight-one input at weight one - discards a nonzero coefficient of weight greater than four. Thus the - two distinct layer types give an invalid bound 1*2^2; repetitions - must be counted. - source_ref: Upstream Pauli/LayerCounterexample.lean, exact symbolic witness. - tags: - - quantum-information - - pauli-operators - - numerical-analysis - - operator-norms - - truncation-error - msc: - - 81P68 - - 65L70 - - 47A30 - - slug: granville-moore - title: Integer Moore determinants and iterated Fermat quotients - summary: >- - Proves Granville's integrality and congruence formulas for iterated - Fermat quotients at odd primes, including the exceptional order p - 1. - For 1 ≤ d ≤ p and integers x_j prime to p, it proves that the nonzero - integer Moore determinant det(x_j^(p^i)) is divisible by p^C(d+1,3), - and that the next power does not divide it exactly when the Fermat - quotients q_p(x_j) are pairwise distinct modulo p. The development - supplies polynomial coefficient identities, denominator-clearing and - valuation estimates, a determinant reduction ladder, and weighted - Vandermonde formulas over general commutative rings. - branch: number theory - entry_module: LeanPool.GranvilleMoore - authors: - - Ken Ono - source: - title: The p-divisibility of the integer Moore determinant and iterated Fermat quotients - url: https://github.com/AxiomMath/GranvilleMoore - github_repo: AxiomMath/GranvilleMoore - commit: 16181f1f6a46d3001c1011c8947df13bcede948c - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - GranvilleMoore.not_pow_succ_dvd_det_mooreMatrix_iff - main_results: - - declaration: GranvilleMoore.exists_intCast_iteratedFermatQuot - informal: >- - For an odd prime p, an integer x not divisible by p, any j ≤ p - 1, - and any k, the iterated Fermat quotient F^(j)_k(x) is an integer. - source_ref: Upstream README, first main result; attributed there to Granville. - - declaration: GranvilleMoore.dvd_factorial_mul_sub_mul_pow - informal: >- - Under the same prime and unit hypotheses, for j ≤ p - 2 and any k, - j! F^(j)_k(x) is congruent to x q_p(x)^j modulo p, where - q_p(x) = (x^(p-1) - 1)/p. - source_ref: Upstream README, second main result. - - declaration: GranvilleMoore.dvd_factorial_mul_sub_exceptional - informal: >- - For an odd prime p and an integer x not divisible by p, - (p-1)! F^(p-1)_0(x) is congruent to x q_p(x)^(p-1) - x q_p(x) modulo p. - source_ref: Upstream README, exceptional congruence. - - declaration: GranvilleMoore.pow_dvd_det_mooreMatrix - informal: >- - For an odd prime p, 1 ≤ d ≤ p, and integers x_j not divisible by p - with nonzero Moore determinant, p^C(d+1,3) divides det(x_j^(p^i)). - source_ref: Upstream README, Moore determinant lower bound, in divisibility form. - - declaration: GranvilleMoore.not_pow_succ_dvd_det_mooreMatrix_iff - informal: >- - Under those determinant hypotheses, its p-adic valuation is exactly - C(d+1,3) if and only if the integer Fermat quotients q_p(x_j) have - pairwise distinct residues modulo p. The Lean statement expresses - exactness as failure of divisibility by p^(C(d+1,3)+1). - source_ref: Upstream README, equality criterion, including d = p. - tags: - - number-theory - - fermat-quotients - - determinants - - p-adic-valuations - - polynomial-identities - msc: - - 11A07 - - 11C20 - - 15A15 - - slug: kurosh - title: Kurosh and Schreier subgroup theorems via covering graphs - summary: >- - Proves the Kurosh subgroup theorem for arbitrary subgroups of indexed - free products, with factors given by actual intersections with conjugates - of the original factors and with the decomposition maps induced by their - inclusions. The development constructs Bass–Serre trees, quotient graphs, - graph-of-groups covers, path lifting, and normal forms. It also gives an - explicit finite Schreier basis and classifies automorphisms of a finite - regular Schreier action by its quotient group. The finite-graph basis - infrastructure is reused from LeanPool.FiniteGraphFundamentalGroup. - branch: combinatorial group theory - entry_module: LeanPool.Kurosh - authors: - - Arthur Freitas Ramos - - David Barros Hulak - - Ruy J. G. B. de Queiroz - source: - title: Schreier and Kurosh subgroup theorems via covering graphs - url: https://github.com/Arthur742Ramos/KuroshSubgroupTheorem - doi: 10.1007/BF01449159 - github_repo: Arthur742Ramos/KuroshSubgroupTheorem - commit: 911707126c8b9bb0c764bf853008fe1053c0aad9 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - GraphCoveringTheory.KuroshStatement.kurosh_decomposition_with_inclusions - main_results: - - declaration: GraphCoveringTheory.KuroshStatement.kurosh_decomposition_with_inclusions - informal: >- - Every subgroup H of an arbitrary indexed free product admits a free - product decomposition into a free group and subgroups of H that are - intersections with conjugates of the original factors. The isomorphism - to H is induced by the actual subgroup inclusions and a homomorphism - from the free factor. - source_ref: Kurosh subgroup theorem; upstream inclusion-preserving statement. - - declaration: GraphCoveringTheory.Kurosh.kurosh_decomposition - informal: >- - H is isomorphic to the free product of its nontrivial vertex stabilizers - in the quotient Bass–Serre graph and the free group of that graph. - - declaration: GraphCoveringTheory.schreier_basis_fintype - informal: >- - A finite-index subgroup H of a free group on any finite generator type - of cardinality r has an explicit free basis of size d*r + 1 - d, - where d is its index and subtraction is natural-number subtraction. - This formulation also covers the empty generator type. - - declaration: GraphCoveringTheory.schreier_basis_fintype_proved - informal: >- - If the finite generator type is nonempty, the basis has the classical - Schreier size 1 + d*(r - 1). - source_ref: Schreier index formula; DOI 10.1007/BF02952517. - - declaration: GraphCoveringTheory.finiteIndexQuotientDeckGroupEquiv - informal: >- - For a finite-index normal subgroup H of a group G, the automorphism - group of the finite regular Schreier G-action is isomorphic to G/H - by right translations. This is a theorem about the action, not a - classification of general topological covers. - - declaration: GraphCoveringTheory.Kurosh.kurosh_subgroup_is_free_of_trivial_stabilizers - informal: >- - A subgroup with trivial stabilizers at every vertex of its quotient - Bass–Serre graph is a free group. - tags: - - group-theory - - free-products - - bass-serre-theory - - schreier-index-formula - - covering-graphs - msc: - - 20E07 - - 05C25 - - 18B40 - - slug: komlos - title: The Komlós and Beck–Fiala bounds with constant 36 - summary: >- - Proves that finitely many real vectors of Euclidean norm at most one admit - signs whose sum has every coordinate bounded by 36, and deduces the - Beck–Fiala discrepancy bound 36 times the square root of the maximum - vertex degree for finite hypergraphs. The Karingula–Lovett splitting - argument is supported by reusable finite-distribution, total-variation, - transport, convex-hull, and matrix-discrepancy infrastructure. The - near-invariant distribution uses discrete squared tent weights, and - real vectors are handled by coordinatewise truncation toward zero. - branch: discrepancy theory - entry_module: LeanPool.Komlos - authors: - - Gabriel Dahia - source: - title: An elementary proof of the Komlós conjecture - authors: - - Sankeerth Rao Karingula - - Shachar Lovett - url: https://eccc.weizmann.ac.il/report/2026/188/ - github_repo: gdahia/Komlos - commit: d802857449234318d557361b1dbb0a27e0258528 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Komlos.discrepancy_le_of_forall_sum_sq_le_one - main_results: - - declaration: Komlos.discrepancy_le_of_forall_sum_sq_le_one - informal: >- - For a real matrix with arbitrary finite row and column index types, - if each column has sum of squared entries at most one, its minimum - signed-column discrepancy is at most 36. - source_ref: Theorem 1.2, matrix form. - - declaration: Komlos.exists_isColouring_forall_abs_sum_apply_le - informal: >- - Any finite family of vectors in a finite-dimensional real Euclidean - space, each of norm at most one, admits signs in {-1, 1} whose signed - sum has absolute value at most 36 in every coordinate. - source_ref: Theorem 1.2. - - declaration: Hypergraph.exists_isColouring_forall_abs_finsum_le - informal: >- - If a hypergraph has finitely many vertices and each vertex belongs - to at most t edges, there is a {-1, 1} colouring of all vertices - whose sum on each edge has absolute value at most 36 times sqrt(t). - source_ref: Beck–Fiala consequence discussed in Section 1. - - declaration: Komlos.exists_isColouring_mean_add_sum_mem_convexHull - informal: >- - For a finitely supported probability distribution on any real - module and n vectors, if translation by six times each vector has - total variation distance at most one third, some signed sum added - to the mean belongs to the convex hull of the distribution's support. - source_ref: Lemma 1.4, generalized to arbitrary real modules. - - declaration: Komlos.exists_nearInvariant - informal: >- - Every positive-denominator integer grid in finite-dimensional real - coordinate space admits a finitely supported mean-zero probability - distribution in [-6, 6]^d whose total variation distance under any - grid translation of Euclidean norm at most one is at most one third. - source_ref: Lemma 1.5, using discrete squared tent weights. - tags: - - discrepancy - - vector-balancing - - hypergraphs - - probability - - total-variation - msc: - - 11K38 - - 05C65 - - 05D40 - - slug: sum-difference-exponent - title: The optimal exponent relating sumsets and difference sets - summary: >- - For a finite integer set A with at least two elements, proves that - C(A) = log(|A+A|/|A|) / log(|A-A|/|A|) is strictly below 2, while - explicit finite sets have exponents tending to 2. Thus 2 is the - supremum and is never attained. Also gives an explicit witness within - 10^(-999) of 2. This imports the upstream base-39 row-and-column - variant, not the paper's base-12 automaton construction or its - quantitative cardinality estimates. Supporting infrastructure includes - injective digit expansions and the strict Pluennecke-Petridis - equality argument for finite integer sets. - branch: additive combinatorics - entry_module: LeanPool.SumDifferenceExponent - authors: - - Haowei Lin - - Shanda Li - source: - title: Settling the Optimal Exponent Relating Sumsets and Difference Sets - authors: - - Haowei Lin - - Shanda Li - arxiv: '2607.27199' - url: https://github.com/linhaowei1/sum-diff-proof - github_repo: linhaowei1/sum-diff-proof - commit: f53f84223b48457ac922836b661227309d9340c3 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - SumDifferenceExponent.sSup_admissibleExponents - main_results: - - declaration: SumDifferenceExponent.growthExponent_lt_two - informal: >- - For every finite set A of integers with at least two elements, - log(|A+A|/|A|) / log(|A-A|/|A|) is strictly less than 2. - - declaration: SumDifferenceExponent.tendsto_growthExponent_approximatingSet - informal: >- - The growth exponents of the explicit finite integer sets - approximatingSet l converge to 2 as l tends to infinity; these - sets have at least two elements whenever l is positive. - - declaration: SumDifferenceExponent.isLUB_admissibleExponents - informal: >- - The number 2 is the least upper bound of the growth exponents of - all finite integer sets with at least two elements. - - declaration: SumDifferenceExponent.sSup_admissibleExponents - informal: >- - The supremum of the growth exponents of all finite integer sets - with at least two elements equals 2. - - declaration: SumDifferenceExponent.supremum_not_attained - informal: >- - No finite integer set with at least two elements has growth - exponent equal to 2. - - declaration: SumDifferenceExponent.quantitative_example - informal: >- - The explicitly defined finite integer set quantitativeSet has - growth exponent strictly between 2 - 10^(-999) and 2. - tags: - - additive-combinatorics - - sumsets - - difference-sets - - Pluennecke-Ruzsa - - optimal-exponents - msc: - - 11B13 - - 11B30 - - 05D10 - - slug: four-ap-free-permutation - title: A 4AP-free permutation of the positive integers - summary: >- - Formalizes Boon Suan Ho's construction of a permutation of the positive - integers with no four-term arithmetic progression as a subsequence, - proving the binary-order lemmas, extension lemma, and computable limit. - branch: additive combinatorics - entry_module: LeanPool.FourAP - authors: - - Boon Suan Ho - source: - title: A 4AP-free permutation of the positive integers - authors: - - Boon Suan Ho - arxiv: '2609.12780' - url: https://github.com/boonsuan/4ap - github_repo: boonsuan/4ap - commit: 4630aefbbcb7f6537aceaeb1afac8dd3af455871 - license: MIT - status: verified - provenance: human - main_declarations: - - FourAP.exists_fourAPFree_positive_permutation - main_results: - - declaration: FourAP.exists_fourAPFree_positive_permutation - informal: >- - There exists a bijection of the positive integers such that no four - increasing positions map to a nonconstant four-term arithmetic - progression, including negative common differences. - - declaration: FourAP.explicitPositivePermutation_apFree - informal: >- - The explicit computable positive-integer permutation constructed in - the development has the stated four-term progression avoidance property. - tags: - - additive-combinatorics - - arithmetic-progressions - - Erdos-problem-196 - - permutations - msc: - - 11B25 - - 05D10 - - slug: koethe-conjecture-disproof - title: Disproof of the Köthe conjecture (Krempa's matrix form) - summary: >- - Refutes Köthe's 1930 conjecture in Krempa's matrix form: there is a ring - R (the unitization of a nil algebra of three weighted backward shifts - over the algebraic closure of F_2) with a nil two-sided ideal I such - that the matrix ideal M_2(I) contains a non-nilpotent matrix. The - compared theorem is exactly the negation of the research-open statement - `Koethe.KotherConjecture.variants.general_matrix` from Google DeepMind's - Formal Conjectures, in every universe. The supporting library proves a - multiprojective common-zero theorem via the Segre cone, the - Nullstellensatz and Krull's height theorem, a mask-mortality theorem for - one-row matrix pencils, a diagonal construction of universal mortal - sequences, and a linearization theorem reducing nilness of a - three-generator algebra to nilness of its root-row pencils. - branch: noncommutative ring theory - entry_module: LeanPool.Koethe - authors: - - Tom Adamczewski - source: - title: A counterexample to Köthe's conjecture and a question of Rowen - authors: - - Tom Adamczewski - - Bernhard Böhmler - - Rene Marczinzik - arxiv: '2609.07996' - url: https://github.com/tadamcz/koethe - github_repo: tadamcz/koethe - commit: a94a72aec957cb28d76aa30fb97a01d440990d7f - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Koethe.KotherConjecture.variants.general_matrix.disproof - main_results: - - declaration: Koethe.KotherConjecture.variants.general_matrix.disproof - informal: >- - It is not the case that for every ring R, every nil two-sided ideal - I of R and every finite index type n, the matrix ideal M_n(I) of - M_n(R) is nil. This is the negation of Krempa's matrix form of - Köthe's conjecture, stated verbatim as in Formal Conjectures. - - declaration: KoetheCounterexample.counterexample - informal: >- - In every universe there is a ring R with a nil two-sided ideal I - such that some 2 x 2 matrix with entries in I is not nilpotent. - - declaration: KoetheMultiProjective.exists_common_zero - informal: >- - Over an algebraically closed field, at most 2N polynomials that are - multihomogeneous of a common positive degree in each of N blocks of - three variables have a common zero with no block equal to zero. - - declaration: KoetheCounterexample.maskMortality - informal: >- - Over an algebraically closed field, every one-row matrix pencil and - every periodic mask with more than half of its residues free admit a - mask-compatible word of nonzero letters whose forward product is - the zero matrix. - - declaration: KoetheCounterexample.exists_universalMortalSequence - informal: >- - Over a countable field satisfying mask mortality there is a - sequence of nonzero letter triples such that every pencil has all - its windows of some fixed positive length equal to zero. - - declaration: KoetheCounterexample.nil_of_all_pencils_nil - informal: >- - If every root-row pencil lifted along three elements of an algebra - is nilpotent, then the non-unital subalgebra they generate is nil. - tags: - - ring-theory - - nil-ideals - - koethe-conjecture - - counterexample - - matrix-rings - msc: - - 16N40 - - 16S50 - - slug: riemann-roch-function-fields - title: Riemann–Roch for algebraic function fields - summary: Formalizes the Riemann–Roch theorem for algebraic function fields of one variable - through the adelic route. Divisors are indexed by intrinsic places (discrete rank-one - valuation subrings containing the constants), with an equivalence to a two-chart - coordinate model via Ostrowski's theorem. The development builds Riemann–Roch spaces, - the genus, an intrinsic adele space, Weil differentials and the canonical class, and - proves Riemann's inequality, the duality theorem, Riemann–Roch, Clifford's inequality, - and the standard corollaries. As an application it shows that a Weierstrass elliptic - curve has genus one and identifies its degree-one places with the class group of the - coordinate ring compatibly with the group law. - branch: algebraic curves and function fields - entry_module: LeanPool.RiemannRochFunctionFields - authors: - - Guanghao Li - source: - title: Riemann–Roch for function fields, in Lean 4 - authors: - - Guanghao Li - url: https://github.com/vaca22/riemann-roch-function-fields - github_repo: vaca22/riemann-roch-function-fields - commit: dbca5beed1da77e2ecd1eec207d0451fa57e8aa6 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - FunctionField.riemann_roch - main_results: - - declaration: FunctionField.riemann_roch - informal: For a function field K over its full constant field k, a canonical divisor W, - and any divisor D indexed by intrinsic places, ℓ(D) = deg D + 1 − g + ℓ(W − D). - - declaration: FunctionField.duality - informal: For a canonical divisor W and any divisor D, ℓ(W − D) equals the index of - specialty i(D), the codimension of L(D) plus K inside the adele space. - - declaration: FunctionField.riemann_ineq - informal: Riemann's inequality ℓ(D) ≥ deg D + 1 − g holds for every divisor D. - - declaration: FunctionField.clifford - informal: Clifford's inequality 2(ℓ(D) − 1) ≤ deg D holds for every divisor D with - 0 ≤ deg D ≤ 2g − 2, ℓ(D) > 0, and ℓ(W − D) > 0 for a canonical divisor W. The degree - bounds are not needed for the inequality and are dropped in - FunctionField.clifford_of_ell_pos. - - declaration: FunctionField.exists_isCanonical - informal: Every function field admits a canonical divisor, namely the divisor of a - nonzero Weil differential. - - declaration: WeierstrassCurve.Affine.genus_eq_one - informal: The function field of a Weierstrass elliptic curve over a field k that is its - full constant field (for example any algebraically closed k, in any characteristic) - has genus one. - - declaration: WeierstrassCurve.Affine.picTorsor_compat_groupLaw - informal: The bijection between degree-one places of an elliptic function field and the - class group of the coordinate ring is compatible with the elliptic group law. - tags: - - riemann-roch - - function-fields - - algebraic-curves - - weil-differentials - - elliptic-curves - msc: - - 14H05 - - 11R58 - - 14H52 - - slug: long-gaps-between-primes - title: Improved long gaps between consecutive primes - summary: >- - Formalizes the unconditional lower bound showing that, for all - sufficiently large X, the largest gap between consecutive primes below - X is at least a positive constant times - log X (log log X)^2 log log log log X / (log log log X)^2. The - development includes the short-translates sieve proposition, divisor - coefficient estimates, truncated product weights, moment bounds, and - the Erdos--Rankin covering argument used to derive the headline result. - branch: analytic number theory - entry_module: LeanPool.LongGapsBetweenPrimes - authors: - - OpenAI - source: - title: Improved Long Gaps Between Primes - authors: - - OpenAI - url: https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/long_gaps.pdf - github_repo: openai/LongGapsBetweenPrimes - commit: 03a1190d0bc5502d9f54eeb60ad3e45e22b0df0b - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LongGapsBetweenPrimes.long_prime_gaps - main_results: - - declaration: LongGapsBetweenPrimes.long_prime_gaps - informal: >- - There are a positive constant c and a real threshold X0 such that - for every X at least X0, some pair of consecutive indexed primes - below X has gap greater than c times - log X (log log X)^2 log log log log X / (log log log X)^2. - - declaration: LongGapsBetweenPrimes.long_gap_theorem - informal: >- - For all sufficiently large X, there are consecutive primes p < q - with q at most X and q - p at least a fixed positive multiple of - the improved long-gap scale. - - declaration: LongGapsBetweenPrimes.short_translates - informal: >- - For all sufficiently large x, every sufficiently sparse subset S - of an interval of length between x and x(log x)^2 has a translate - b + Q(x)t + S consisting entirely of composite numbers, with - 1 <= t <= exp x. - tags: - - analytic-number-theory - - prime-gaps - - sieve-theory - - erdos-rankin - msc: - - "11N05" - - slug: wallace-problem - title: The Wallace problem in ZFC - summary: >- - Formalizes in ZFC a Hausdorff countably compact group topology without - nontrivial convergent sequences on every torsion-free Abelian group of - cardinality continuum. It specializes the construction to the rational - direct sum and the additive real group, and derives a commutative - Tychonoff countably compact cancellative semigroup that is not a group. - branch: set-theoretic topology and topological algebra - entry_module: LeanPool.Wallace - authors: - - Juliane Trianon Fraga - - Vinicius de Oliveira Rodrigues - source: - title: The Wallace problem and countably compact torsion-free Abelian groups in ZFC - authors: - - Juliane Trianon Fraga - - Vinicius de Oliveira Rodrigues - arxiv: "2608.17317" - doi: 10.48550/arXiv.2608.17317 - url: https://github.com/vo-rodrigues/wallace-problem-zfc-paper - github_repo: vo-rodrigues/wallace-problem-zfc-paper - commit: 23756864de1f14e272dafb070b97fcdc7cbc75c5 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Wallace.commutativeTychonoffWallaceCounterexampleExists - main_results: - - declaration: Wallace.torsionFreeAbelianGroup_mainTheorem_exact - informal: >- - Every torsion-free Abelian group of cardinality continuum admits a - Hausdorff countably compact group topology with no nontrivial - convergent sequences. - - declaration: Wallace.continuumRationalGroup_and_real_mainTheorem - informal: >- - Both the continuum-indexed rational direct sum and the additive - group of real numbers admit such Hausdorff countably compact group - topologies without nontrivial convergent sequences. - - declaration: Wallace.commutativeTychonoffWallaceCounterexampleExists - informal: >- - There exists a commutative Tychonoff countably compact topological - semigroup with two-sided cancellation which is not a group. - tags: - - wallace-problem - - topological-groups - - set-theoretic-topology - msc: - - "22A05" - - "54D30" - - "20K20" - - slug: zeta-zeros - title: Simple zeros of the Riemann zeta function - summary: >- - Formalizes Lamzouri's Hilbert-space proof of lower bounds for simple - and distinct zeros of the Riemann zeta function. From explicit - Riemann--von Mangoldt and unconditional pair-correlation hypotheses, - the development proves that eventually more than 67.25% of nontrivial - zeros are simple and on the critical line and more than 83.625% are - distinct. It includes the finite-multiset Hilbert-space inequality, - zeta-zero multiplicity symmetries, an admissible cutoff construction, - and certified trigonometric estimates. - branch: analytic number theory - entry_module: LeanPool.ZetaZeros - authors: - - Kenny Lau - source: - title: >- - A new proof that more than 2/3 of the zeros of the Riemann zeta - function are simple and on the critical line - authors: - - Youness Lamzouri - arxiv: 2609.02882 - doi: 10.48550/arXiv.2609.02882 - url: https://github.com/AxiomMath/ZetaZeros - github_repo: AxiomMath/ZetaZeros - commit: 4bcaf70e544506c311d83a5a5b143a134b9fc5f7 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - ZetaZeros.simple_proportion_d4 - main_results: - - declaration: ZetaZeros.simple_proportion_d4 - informal: >- - Assuming the stated Riemann--von Mangoldt and unconditional - pair-correlation formulas, beyond some height more than 67.25% of - nontrivial zeta zeros are simple and lie on the critical line. - - declaration: ZetaZeros.distinct_proportion_d5 - informal: >- - Under the same two explicit analytic hypotheses, beyond some height - more than 83.625% of nontrivial zeta zeros are distinct. - - declaration: ZetaZeros.card_simpleRealPart_lower - informal: >- - Bounds from below the number of multiplicity-one real points in a - finite nonempty conjugation-invariant multiset using the squared - test-kernel sum of any admissible function. - - declaration: ZetaZeros.card_lower - informal: >- - Bounds from below the number of distinct points in a finite nonempty - conjugation-invariant multiset using the same Hilbert-space kernel - method. - tags: - - analytic-number-theory - - riemann-zeta-function - - critical-line - - simple-zeros - - pair-correlation - msc: - - "11M06" - - "11M26" - - slug: common-neighbour-conjecture - title: Common-neighbour counterexamples for Saxl graphs - summary: >- - Proves that for every base size B at least two and every prescribed - degree bound, there is a finite faithful primitive permutation group - of at least that degree and exact base size B whose generalized Saxl - graph contains two nonadjacent vertices with no common neighbor. The - supporting library constructs an explicit infinite family from binary - deleted permutation modules over finite fields and proves the required - base-size, irreducibility, primitivity, and graph-theoretic properties. - branch: permutation groups and algebraic graph theory - entry_module: LeanPool.CommonNeighbourConjecture - authors: - - Aluna Rizzoli - - Adam R. Thomas - source: - title: Common neighbour conjectures for Saxl graphs fail at every base size - authors: - - Aluna Rizzoli - - Adam R. Thomas - doi: 10.5281/zenodo.22231393 - url: https://github.com/alunik/common-neighbour-conjecture - github_repo: alunik/common-neighbour-conjecture - commit: b2c0a8f95aa4dca76a034a0e769a8c8249f38ae2 - license: MIT - status: verified - provenance: human - main_declarations: - - SaxlCounterexamples.MainTheorems.mainTheorem - main_results: - - declaration: SaxlCounterexamples.MainTheorems.mainTheorem - informal: >- - For every natural number B at least two and every degree bound, - there exists a finite faithful primitive permutation group of - degree at least the bound and base size exactly B whose generalized - Saxl graph has two nonadjacent vertices with no common neighbor. - - declaration: SaxlCounterexamples.EveryBase.hq_coreFamily - informal: >- - For every odd d with 3 ≤ d and every tail, the explicit affine - wreath-product family has exact base size tail + 2, acts - primitively, and its displayed pair has no common neighbor in the - corresponding generalized Saxl graph. - tags: - - permutation-groups - - group-actions - - algebraic-graph-theory - - saxl-graphs - - counterexamples - msc: - - "20B15" - - "05C25" - - slug: kahn-kalai - title: Kahn–Kalai expectation-threshold theorem - summary: >- - Formalizes Tran and Vu's short proof of the Park–Pham theorem resolving - the Kahn–Kalai expectation-threshold conjecture. For an ℓ-bounded - family of subsets of a finite ground set, it proves the strengthened - covering theorem with the explicit constant L = 1000 and derives the - threshold bound p_c(F) ≤ K q(F) log₂ ℓ with the explicit witness - K = 100000. The supporting library develops finite-set cover cost, - level-density, double-counting, binomial-tail, and numeric estimates. - branch: probabilistic combinatorics - entry_module: LeanPool.KahnKalai - authors: - - Dan Clemens Posch - source: - title: A Short Proof of Kahn-Kalai Conjecture - authors: - - Phuc Tran - - Van Vu - arxiv: 2303.02144 - doi: 10.37236/12266 - url: https://github.com/dcposch/kahn-kalai-lean - github_repo: dcposch/kahn-kalai-lean - commit: 641aa75f8e873d31442f2f7c317b2e7582a26d94 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - KahnKalai.covering_theorem - - KahnKalai.park_pham - main_results: - - declaration: KahnKalai.covering_theorem - informal: >- - If ℓ ≤ N, p lies in [0, 1], and an ℓ-bounded family H has cover - cost at least 1/2 - 2^(-(ℓ+2)), then its upset has at least - (2/3 + 2^(-(ℓ+2))) choose(N, m) members of cardinality m, where - m = floor(1000 p N log₂(ℓ+1)). For m ≤ N this is the corresponding - level-density bound; for N < m both cardinalities are zero. - - declaration: KahnKalai.park_pham - informal: >- - There is an absolute positive constant K such that every - ℓ-bounded finite family with ℓ at least two satisfies - p_c(F) ≤ K q(F) log₂ ℓ; the proof supplies K = 100000. - tags: - - probabilistic-combinatorics - - random-structures - - threshold-phenomena - - set-systems - msc: - - "05C80" - - "60C05" - - slug: quadratic-iterates - title: Galois groups of quadratic polynomial iterates - summary: >- - Formalizes Stoll's criterion for when the Galois group of the n-th iterate - of X² + a over the rationals is the full n-fold iterated wreath product of - C₂. It proves the criterion in terms of two-independence of a signed - critical-orbit sequence and its pairwise-coprime Möbius factors, and derives the paper's - congruence families for which maximality holds at every level. The - supporting library develops reusable multiquadratic-extension, iterated - wreath-product, Möbius-factor, and valuation results. - branch: arithmetic dynamics and Galois theory - entry_module: LeanPool.QuadraticIterates - authors: - - Michael Stoll - source: - title: Galois groups over ℚ of some iterated polynomials - authors: - - Michael Stoll - doi: 10.1007/BF01197321 - url: https://github.com/MichaelStollBayreuth/QuadraticIterates - github_repo: MichaelStollBayreuth/QuadraticIterates - commit: b0413c173972ddf121cb4394cb06f9d2b83e501f - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - QuadraticIterates.section3_main - main_results: - - declaration: QuadraticIterates.section1_equiv - informal: >- - For each n, assuming -a is not a rational square, the Galois group of - the n-th iterate of X² + a is the full n-fold iterated wreath product - of C₂ exactly when either the first n terms of the signed sequence - c₁ = -a, cₖ = fₖ(0) for k ≥ 2, or the first n integer Möbius - factors of that sequence are independent modulo rational - squares. - - declaration: QuadraticIterates.section1_squarefree - informal: >- - If -a is not a rational square and none of the absolute Möbius factors - from levels 2 through n is a rational square, then the n-th iterate - has the full iterated-wreath-product Galois group. - - declaration: QuadraticIterates.section3_main - informal: >- - The Galois group is the full iterated wreath product at every positive - level when a is positive and congruent to 1 or 2 modulo 4, or when a - is negative and divisible by 4 with -a nonsquare. - tags: - - arithmetic-dynamics - - galois-theory - - iterated-polynomials - msc: - - "11R32" - - "12F10" - - "37P05" - - slug: multicolor-triangle-ramsey - title: Superexponential lower bounds for multicolor triangle Ramsey numbers - summary: >- - Resolves Erdős problem 183 by proving that the k-color triangle Ramsey - number has a k-th root tending to infinity. The development constructs - saturated palette families and recursive edge colorings, derives an - explicit superexponential lower bound, and proves matching order-of-growth - bounds for the logarithm of the Ramsey number. - branch: extremal combinatorics and Ramsey theory - entry_module: LeanPool.MulticolorTriangleRamsey - authors: - - OpenAI - source: - title: Ten Advances in Mathematics and Theoretical Computer Science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - ErdosProblems.MulticolourTriangleRamsey.erdos_problem_183_explicit - main_results: - - declaration: ErdosProblems.MulticolourTriangleRamsey.erdos_183 - informal: >- - The k-th root of the k-color triangle Ramsey number tends to infinity, - resolving Erdős problem 183. - - declaration: ErdosProblems.MulticolourTriangleRamsey.erdos_problem_183_explicit - informal: >- - For every k at least two, the k-color triangle Ramsey number is at least - ((k^(1/3))/(6 exp(38) log k))^k, and its k-th root tends to infinity. - - declaration: ErdosProblems.MulticolourTriangleRamsey.triangleRamseyNumber_log_isTheta - informal: >- - The logarithm of the k-color triangle Ramsey number has order - k log k. - tags: - - extremal-combinatorics - - ramsey-theory - - graph-coloring - - erdos-problems - msc: - - "05D10" - - "05C55" - - slug: compactness-and-degeneracy - title: Counterexamples to graph compactness and two-degenerate extremal bounds - summary: >- - Formalizes two counterexamples in extremal graph theory. The first gives - a finite family of connected bipartite cyclic graphs whose family - extremal number grows strictly slower than every member's extremal - number, disproving the Erdős--Simonovits compactness conjecture. The - second constructs a connected bipartite two-degenerate graph with - extremal number at least c n^(3/2+ε), disproving the proposed universal - n^(3/2) upper bound under the stated bipartition degree condition. - branch: extremal combinatorics and graph theory - entry_module: LeanPool.CompactnessAndDegeneracy - authors: - - OpenAI - - Dean Cureton - source: - title: Ten Advances in Mathematics and Theoretical Computer Science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - CompactnessConjecture.quantitativeCompactnessCounterexample - main_results: - - declaration: CompactnessConjecture.quantitativeCompactnessCounterexample - informal: >- - There is a nonempty finite family of connected bipartite cyclic - graphs such that every member has a uniform positive - n^(4/3)-scale extremal lower bound, while every family-free host on n - vertices has its sixteenth power of the edge count bounded by a - constant times n^21. Consequently the family is not compact and the - Erdős--Simonovits compactness conjecture is false. - - declaration: TwoDegenerateGraphs.twoDegenerateExtremalCounterexample - informal: >- - There is a connected bipartite two-degenerate finite graph H such - that each side of every two-coloring has a vertex of degree greater - than two, yet ex(n,H) is eventually at least c n^(3/2+ε) for some - positive constants c and ε. - tags: - - extremal-combinatorics - - extremal-graph-theory - - degenerate-graphs - - counterexamples - msc: - - "05C35" - - "05C75" - - slug: connes-rigidity - title: Conditional formalization of Zhou's Connes-rigidity counterexample - summary: >- - Constructs Zhou's two countable discrete groups and, assuming the project's - universe-polymorphic property-(T) predicate for EL₃(𝔽₂[t]), proves that both - satisfy the same predicate, are ICC and nonisomorphic, while their group von - Neumann algebras are trace-preservingly normally star-isomorphic. - branch: operator algebras and geometric group theory - entry_module: LeanPool.ConnesRigidity - authors: - - Utensil Song - source: - title: ICC property(T) groups without W*-superrigidity - authors: - - Shuoxing Zhou - arxiv: "2608.02327" - url: https://github.com/utensil/connes-rigidity - github_repo: utensil/connes-rigidity - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Connes.theoremA - main_results: - - declaration: Connes.theoremA - informal: >- - At any Hilbert-space universe, assuming property-(T) for EL₃(𝔽₂[t]), - constructs two countable discrete ICC groups satisfying property-(T) - at that universe which are nonisomorphic although their group von Neumann - algebras are trace-preservingly normally star-isomorphic. - tags: - - operator-algebras - - von-neumann-algebras - - property-t - - group-theory - - connes-rigidity - msc: - - "46L10" - - "20F65" - - "22D10" - - slug: concentration-inequalities - title: Bennett--Bernstein, Freedman, and Hoeffding concentration inequalities - summary: >- - Develops variance-sensitive sub-gamma moment-generating-function estimates, Bennett and - Bernstein bounds, a Freedman--Bernstein upper tail for finite adapted sums, conditional and - independent-variable Hoeffding inequalities. - branch: probability theory and probabilistic combinatorics - entry_module: LeanPool.ConcentrationInequalities - authors: - - Juan Pablo Traverso Gianini - - Aristotle - source: - title: >- - Reusable concentration-inequality components accompanying Linear-Error Clique Partitions - of Split Graphs via Structured Triangle Packing - authors: - - Juan Pablo Traverso Gianini - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/b3423f3e8c8db7c2d1b279673293ef3079faf903/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Contrib/FreedmanBernstein.lean - github_repo: jtraverso/erdos-81-chordal-clique-partitions - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Contrib.Freedman.subgamma_bernstein_tail - main_results: - - declaration: Contrib.Bennett.hasSubgammaMGF_of_bounded_above - informal: >- - A centered real random variable bounded above by a positive constant, with bounded second - moment, satisfies a Bennett--Bernstein sub-gamma moment-generating-function estimate. - - declaration: Contrib.Freedman.subgamma_bernstein_tail - informal: >- - A finite adapted sum of conditionally sub-gamma increments satisfies the corresponding - variance-sensitive Freedman--Bernstein upper-tail bound. - - declaration: Contrib.ConditionalHoeffding.hasCondSubgaussianMGF_of_mem_Icc - informal: >- - A bounded random variable with zero conditional expectation satisfies a conditional - Hoeffding sub-Gaussian moment-generating-function estimate. - - declaration: Contrib.Hoeffding.hoeffding_upper - informal: >- - A sum of finitely many independent random variables valued in the unit interval exceeds - its mean by t with probability at most exp(-2 t squared divided by the number of terms). - tags: - - probability-theory - - concentration-inequalities - - martingales - msc: - - "60E15" - - "60G42" - - slug: erdos81-paper-i-contrib - title: Finitely generated cones and finite LP duality - summary: >- - Proves that finitely generated cones in real normed spaces are closed, including a conic - Carathéodory reduction and PointedCone API forms. Derives a finite-dimensional inequality - form of Farkas' lemma and proves packing attainment with equality to the covering infimum - for finite covering and packing linear programs. The Farkas and LP statements overlap the - pooled Duality development; the generic cone-closedness results provide - the project's distinct reusable API. - branch: convex geometry and optimization - entry_module: LeanPool.Erdos81PaperIContrib - authors: - - Juan Pablo Traverso Gianini - source: - title: Affine Profile Reduction for Fractional Triangle Packings in Split Graphs - authors: - - Juan Pablo Traverso Gianini - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/tree/main/preprints/PAPER_I - github_repo: jtraverso/erdos-81-chordal-clique-partitions - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - LeanPool.Erdos81PaperIContrib.fg_cone_isClosed - main_results: - - declaration: LeanPool.Erdos81PaperIContrib.fg_cone_isClosed - informal: >- - The cone generated by a finite family in a real normed vector space is a closed set. - - declaration: LeanPool.Erdos81PaperIContrib.farkas_ge - informal: >- - An infeasible finite system of nonnegative linear inequalities admits a nonnegative - separating Farkas certificate. - - declaration: LeanPool.Erdos81PaperIContrib.covering_packing_duality - informal: >- - Given a finite nonnegative matrix with a strictly positive entry in every column and - nonnegative capacities, there is a feasible nonnegative packing whose unit-objective - value is maximal and equals the infimum cost of nonnegative coverings with unit demand. - tags: - - convex-geometry - - linear-programming - - duality - - farkas-lemma - msc: - - "52A20" - - "90C05" - - slug: boolean-isoperimetry - title: Boolean Isoperimetry and Conway--Guy Coherent Gaps - summary: >- - Formalizes Harper's vertex-isoperimetric theorem for the Boolean cube: - among all families of a fixed cardinality, the simplicial initial segment - has the smallest closed radius-one Hamming neighborhood. Develops the - supporting binomial-cascade identities, Kruskal--Katona shadow bounds, - coordinate compressions, and the final dimension induction. Applies the - coherent-gap certificate method to prove normalized consecutive-gap - rigidity for the Conway--Guy sequence from its exact recurrence. - branch: extremal and additive combinatorics - entry_module: LeanPool.BooleanIsoperimetry - authors: - - Alexey Milovanov - - Egor Lyfar - source: - title: >- - Optimal numberings and isoperimetric problems on graphs; A sum packing - problem of Erdős and the Conway--Guy sequence - authors: - - L. H. Harper - - Tom Bohman - doi: "10.1016/S0021-9800(66)80059-5" - url: "https://doi.org/10.1090/S0002-9939-96-03653-2" - github_repo: AlexeyMilovanov/BooleanIsoperimetry - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - BooleanIsoperimetry.harper_theorem - - BooleanIsoperimetry.conwayGuyGapRigidity - main_results: - - declaration: BooleanIsoperimetry.harper_theorem - informal: >- - For every family of vertices in an n-dimensional Boolean cube, the - simplicial initial segment of the same cardinality has a closed - radius-one Hamming neighborhood no larger than that of the family. - - declaration: >- - BooleanIsoperimetry.conwayGuyGapRigidity - informal: >- - In every dimension, any real row whose consecutive subset-sum gaps - relative to the Conway--Guy reference row are at least one dominates - that row coordinatewise. - tags: - - additive-combinatorics - - isoperimetry - - boolean-cube - - subset-sums - - coherent-orders - msc: - - "05D05" - - "05C35" - - slug: biswal - title: Chebyshev Quotients and Demazure Multiplicities - summary: >- - Formalizes positivity results for the coefficients of Chebyshev-type - polynomial quotients attached to integer partitions, modelling Demazure - multiplicities. Proves that the relevant generating function is a - polynomial and that its coefficients are eventually positive (Theorem 1), - gives explicit Cauchy-product and matrix-inverse formulas for the - coefficients (Theorem 2), and establishes their nonnegativity through a - Dyck-path lattice counting model (Theorem 3). - branch: algebraic combinatorics - entry_module: LeanPool.Biswal - authors: - - Jujian Zhang - source: - title: "Chebyshev quotients, Demazure multiplicities, and Dyck-path models" - authors: - - Evan Chen - - Kenny Lau - - Seewoo Lee - - Ken Ono - - Jujian Zhang - arxiv: "2604.25246" - github_repo: AxiomMath/Biswal - license: MIT - status: verified - provenance: AI - main_declarations: - - Biswal.Theorem23.thm_manifest_coeff_nonneg - main_results: - - declaration: Biswal.Theorem1.genFun_is_polynomial - informal: >- - Proves that the generating function attached to a partition is a - polynomial whenever its parts are bounded by the parameter `m`. - - declaration: Biswal.Theorem1.genFun_coeff_eventually_pos - informal: >- - Proves that the coefficients of the generating function attached to a - partition are eventually strictly positive. - - declaration: Biswal.Theorem23.thm_manifest_coeff_nonneg - informal: >- - Proves that every coefficient of the generating function is - nonnegative, via the explicit Dyck-path lattice counting model. - tags: - - algebraic-combinatorics - - chebyshev - - demazure - msc: - - "05E10" - - slug: bannai-bannai-stanton - title: The Bannai-Bannai-Stanton Bound on Distance Sets - summary: >- - Formalizes the Bannai-Bannai-Stanton theorem: a distance set `S` in `ℝ^d` - realizing `s` distinct distances has size at most `Nat.choose (d + s) s`. - Follows the short polynomial-method proof of Petrov and Pohoata via a - generalized Croot-Lev-Pach rank lemma, with auxiliary results on the - diagonal linear-independence argument for polynomials, monomial counting - of bounded degree, and a rank bound for sums of matrices. - branch: combinatorics - entry_module: LeanPool.BannaiBannaiStanton - authors: - - Antoine du Fresne von Hohenesche - source: - github_repo: AntoineduFresne/Bannai-Bannai-Stanton_Theorem - url: https://github.com/AntoineduFresne/Bannai-Bannai-Stanton_Theorem - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - BannaiBannaiStanton.bannai_bannai_stanton_bound - main_results: - - declaration: BannaiBannaiStanton.bannai_bannai_stanton_bound - informal: >- - A distance set `S` in Euclidean space `ℝ^d` realizing `s` distinct - distances satisfies `S.encard ≤ Nat.choose (d + s) s`. - tags: - - combinatorics - - distance-sets - - polynomial-method - msc: - - "52C10" - - "05B30" - - slug: pythagorean-polynomial-parametrization - title: Polynomial parametrizations of Pythagorean triples - summary: Formalizes Frisch and Vaserstein's result that Pythagorean triples are not parametrizable by one triple of integer-coefficient polynomials in any number of variables, but are parametrizable by a single triple of integer-valued rational polynomials, including the positive and sixteen-parameter variants. - branch: number theory - entry_module: LeanPool.PythagoreanPolynomialParametrization - authors: - - Lazar Milikic - source: - title: Parametrization of Pythagorean triples by a single triple of polynomials - authors: - - Sophie Frisch - - Leonid Vaserstein - arxiv: "0706.0290" - doi: "10.1016/j.jpaa.2007.05.019" - github_repo: epfl-lara/AutoformalizedProjects - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.PythagoreanPolynomialParametrization.exists_int_valued_parametrization - main_results: - - declaration: LeanPool.PythagoreanPolynomialParametrization.no_int_poly_parametrization - informal: Proves that no single triple of integer-coefficient polynomials in any number of variables parametrizes all Pythagorean triples. - - declaration: LeanPool.PythagoreanPolynomialParametrization.exists_int_valued_parametrization - informal: Gives the explicit four-variable integer-valued polynomial parametrization of all Pythagorean triples. - - declaration: LeanPool.PythagoreanPolynomialParametrization.positive_triples_parametrization - informal: Formalizes the positive Pythagorean-triple parametrization with positive parameters and a nonnegative auxiliary parameter. - - declaration: LeanPool.PythagoreanPolynomialParametrization.exists_16_param_parametrization - informal: Converts the positive parametrization to an unrestricted sixteen-integer-parameter parametrization using the four-square theorem. - - declaration: LeanPool.PythagoreanPolynomialParametrization.single_intValued_parametrization_yields_finite_intPoly_parametrization - informal: Proves the cited reduction from one integer-valued polynomial tuple to finitely many integer-coefficient polynomial tuples. - - declaration: LeanPool.PythagoreanPolynomialParametrization.integerValued_polynomial_ring_not_uniqueFactorization - informal: Records the integer-valued polynomial ring non-UFD phenomenon behind the paper's motivating discussion. - tags: - - number-theory - - pythagorean-triples - - integer-valued-polynomials - msc: - - "11D09" - - "11D85" - - "13F20" - - slug: cramer-wold - title: The Cramer-Wold theorem - summary: Formalizes the finite-dimensional Cramer-Wold theorem for measures on Euclidean spaces, with Banach-space continuous-dual generalizations, Hilbert-space inner-product versions, and random-variable law corollaries for identical distributions and `HasLaw`. - branch: probability theory - entry_module: LeanPool.CramerWold - authors: - - Lazar Milikic - source: - title: Some theorems on distribution functions - authors: - - Harald Cramer - - Herman Wold - doi: "10.1112/jlms/s1-11.4.290" - github_repo: Lemmy00/lean-pool - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.CramerWold.euclideanSpace_measure_eq_of_forall_inner_map_eq - main_results: - - declaration: LeanPool.CramerWold.euclideanSpace_measure_eq_of_forall_inner_map_eq - informal: Proves the classical finite-dimensional Cramer-Wold theorem for finite Borel measures on `EuclideanSpace ℝ (Fin n)`, showing equality when all scalar projections have equal pushforward measures. - - declaration: LeanPool.CramerWold.measure_eq_of_forall_dual_map_eq - informal: Generalizes the extensionality criterion to finite Borel measures on complete second-countable real normed spaces, using all continuous-linear real projections. - - declaration: LeanPool.CramerWold.measure_eq_of_forall_inner_map_eq - informal: Specializes Cramer-Wold extensionality to real Hilbert spaces, using scalar projections of the form `x ↦ ⟪t, x⟫_ℝ`. - - declaration: LeanPool.CramerWold.identDistrib_of_forall_dual_comp_identDistrib - informal: Proves that two finite-measure random variables are identically distributed when every continuous-linear real projection of them is identically distributed. - - declaration: LeanPool.CramerWold.identDistrib_of_forall_inner_comp_identDistrib - informal: Gives the Hilbert-space random-variable version using all scalar inner-product projections. - - declaration: LeanPool.CramerWold.hasLaw_of_forall_dual_comp_hasLaw - informal: Proves that a random variable has a target law when every continuous-linear real projection has the corresponding projected target law. - tags: - - probability - - measure-theory - - characteristic-functions - - cramer-wold - msc: - - "60B11" - - "60E10" - - "28A33" - - slug: cencov-petz - title: Finite Čencov-Petz Uniqueness - summary: Formalizes the finite/discrete Čencov-Petz uniqueness theorem, showing that any continuous metric family on finite probability simplexes that is monotone under Markov morphisms is a scalar multiple of the Fisher information metric. - branch: information geometry - entry_module: LeanPool.CencovPetz - authors: - - Adam Benenson - source: - title: Statistical Decision Rules and Optimal Inference - authors: - - N. N. Chentsov - url: https://bookstore.ams.org/mmono-53 - github_repo: abenenson/cencov-petz - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_continuous - main_results: - - declaration: LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_continuous - informal: Proves that every continuous monotone metric family on a finite probability simplex is a scalar multiple of the Fisher information metric. - source_ref: "N. N. Chentsov, Statistical Decision Rules and Optimal Inference, AMS Translations of Mathematical Monographs 53, 1982." - - declaration: LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_isRational - informal: Establishes the scalar Fisher-form identity at rational simplex points using split representations. - - declaration: LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_isSplitRepresentable - informal: Proves the scalar Fisher-form identity for simplex points represented as pushforwards of uniform distributions under splitting maps. - - declaration: LeanPool.CencovPetz.MonotoneMetricFamily.uniformScalar_eq_uniformScalar_two - informal: Shows that the uniform-point scalar is independent of simplex dimension once the dimension is at least two. - - declaration: LeanPool.CencovPetz.TangentFin.Bilin.B_eq_smul_fisherBilin_uniform - informal: Proves that the permutation-invariant bilinear form at a uniform finite simplex point is a scalar multiple of the Fisher bilinear form. - tags: - - information-geometry - - fisher-information - - markov-morphisms - - finite-simplex - msc: - - "62B10" - - "53C21" - - slug: demazureoperatorslean - title: Demazure Operators and Lean - summary: Formalizes Demazure operators on multivariate polynomials and their nilpotence, braid, commutation, and symmetric-polynomial linearity relations, with auxiliary Coxeter-theory support including strong exchange and a conditional Matsumoto theorem. - branch: algebraic combinatorics - entry_module: LeanPool.DemazureOperatorsLean - authors: - - Óscar Álvarez Sánchez - source: - title: "Schubert cells, and the cohomology of the spaces G/P" - authors: - - I. N. Bernstein - - I. M. Gelfand - - S. I. Gelfand - doi: "10.1070/RM1973v028n03ABEH001557" - github_repo: bolito2/DemazureOperatorsLean - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - Demazure.Dem - - CoxeterSystem.strongExchangeProperty - main_results: - - declaration: Demazure.Dem - informal: Defines the Demazure operator as a linear map over the subalgebra of symmetric polynomials. - - declaration: Demazure.demazure_order_two - informal: Proves that applying the Demazure operator twice gives zero. - - declaration: Demazure.demazure_commutes_adjacent - informal: Proves the braid relation for adjacent Demazure operators. - - declaration: Demazure.demazure_commutes_non_adjacent - informal: Proves that non-adjacent Demazure operators commute. - - declaration: Demazure.demazure_mul_symm - informal: Proves that symmetric polynomials act as scalars with respect to Demazure operators. - - declaration: CoxeterSystem.strongExchangeProperty - informal: Proves the strong exchange property for Coxeter systems. - source_ref: "J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990." - - declaration: CoxeterSystem.matsumoto_reduced - informal: Auxiliary theorem proving, under the extra MatsumotoCondition hypothesis, that reduced words for the same Coxeter group element are connected by braid moves. - source_ref: "J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990." - tags: - - algebraic-combinatorics - - demazure-operators - - polynomials - - representation-theory - msc: - - "05E05" - - "13P10" - - "20F55" - - slug: demazure-product - title: Extended Demazure Product on ASP Permutations - summary: Formalizes the extended Demazure product on almost-sign-preserving integer permutations, including slipface models, residuals, greedy and stingy characterizations, reduction theorems, adjacent-transposition formulas, and 321-avoiding factorization results. - branch: algebraic combinatorics - entry_module: LeanPool.DemazureProduct - authors: - - Nathan Pflueger - source: - title: An extended Demazure product on integer permutations via min-plus matrix multiplication - authors: - - Nathan Pflueger - arxiv: "2206.14227" - github_repo: npflueger/demazure - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - LeanPool.DemazureProduct.AspPerm.star - main_results: - - declaration: LeanPool.DemazureProduct.AspPerm.star - informal: Defines the extended Demazure product on almost-sign-preserving permutations. - - declaration: LeanPool.DemazureProduct.AspPerm.star_assoc - informal: Proves associativity of the extended Demazure product on almost-sign-preserving permutations. - - declaration: LeanPool.DemazureProduct.Reduction.starGreedy - informal: Characterizes the Demazure product as the Bruhat-maximum of ordinary products of smaller factors. - - declaration: LeanPool.DemazureProduct.Reduction.reduce - informal: Reduces inequalities against a Demazure product to ordinary products of Bruhat-smaller factors with fixed shift. - - declaration: LeanPool.DemazureProduct.Reduction.lres_stingy - informal: Gives the stingy characterization of the left residual as a Bruhat-minimum of ordinary products. - - declaration: LeanPool.DemazureProduct.ReducedProducts.star_eq_mul_iff_reducedProduct - informal: Shows that the Demazure product equals ordinary multiplication exactly for reduced ASP products. - - declaration: LeanPool.DemazureProduct.Transpositions.starSigma - informal: Proves the adjacent-transposition formula for multiplying an ASP permutation by a no-consecutive-pair involution. - - declaration: LeanPool.DemazureProduct.ASP321a.dprod_eq_iff - informal: Characterizes Demazure factorizations of 321-avoiding ASP permutations by isolated inversion-set decompositions. - - declaration: LeanPool.DemazureProduct.Tableaux.setValuedTableauEquivHeckeFactorization - informal: Relates fixed-shift set-valued tableaux to Hecke factorizations of 321-avoiding ASP permutations. - tags: - - algebraic-combinatorics - - demazure-product - - bruhat-order - - permutations - msc: - - "05E05" - - "20F55" - - slug: zeta_3_irrational - title: Irrationality of ζ(3) - summary: Formalizes Beukers' proof of Apéry's theorem for ζ(3), including the integral identities, denominator estimates, positivity, upper bounds, and final irrationality contradiction. - branch: analytic number theory - entry_module: LeanPool.Zeta3Irrational - authors: - - Junqi Liu - - Jujian Zhang - source: - title: A note on the irrationality of ζ(2) and ζ(3) - authors: - - Frits Beukers - doi: "10.1112/blms/11.3.268" - arxiv: "2503.07625" - github_repo: ahhwuhu/zeta_3_irrational - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - LeanPool.Zeta3Irrational.zeta3_irrational - main_results: - - declaration: LeanPool.Zeta3Irrational.zeta3_irrational - informal: Proves that ζ(3) is irrational. - - declaration: LeanPool.Zeta3Irrational.linear_int - informal: Clears denominators in the Apéry integral to produce a linear form in 1 and ζ(3). - - declaration: LeanPool.Zeta3Irrational.JJ_pos - informal: Proves positivity of the transformed Beukers-Apéry integral. - - declaration: LeanPool.Zeta3Irrational.JJ_upper - informal: Bounds the Beukers-Apéry integral by a geometric factor times the ζ(3) series. - - declaration: LeanPool.Zeta3Irrational.fun1_tendsto_zero - informal: Shows that the denominator-cleared Apéry linear forms tend to zero. - tags: - - number-theory - - analysis - - zeta-functions - msc: - - "11M06" - - "11J72" - - slug: zeta-h123 - title: Thakur's hypotheses on power sums of F_q[t] - summary: Formalizes four components of Thakur's hypotheses on power sums over finite polynomial rings, including uniqueness and monotonicity results for the carry-free digit optimization problems and the Laurent-series expansion of finite-field power sums. - branch: number theory - entry_module: LeanPool.ZetaH123 - authors: - - Evan Chen - - Kenny Lau - - Ken Ono - - Jujian Zhang - source: - title: Thakur's hypotheses on power sums of F_q[t] - authors: - - Evan Chen - - Ken Ono - arxiv: "2606.16239" - github_repo: AxiomMath/zeta-h123 - license: MIT - status: verified - provenance: AI - main_declarations: - - ZetaH123.H1.main_theorem - - ZetaH123.Lem41.main - main_results: - - declaration: ZetaH123.H1.main_theorem - informal: Proves uniqueness of the admissible base-q decomposition maximizing the H1 power-sum weight functional. - - declaration: ZetaH123.H2.main - informal: Proves the H2 unique-minimizer statement for the recursive digit/carry-free optimization problem defining the values s_d(k). - - declaration: ZetaH123.H3.main_theorem - informal: Proves the H3 monotonicity statement that s_d(k) is strictly less than s_d(k+1) when p does not divide k. - - declaration: ZetaH123.Lem41.main - informal: Proves the Lemma 4.1 Laurent-series expansion for S_d(k), with nonzero coefficients indexed by the carry-free tuples. - tags: - - number-theory - - function-fields - - zeta-functions - msc: - - "11M38" - - "11T55" - - "11G09" - - slug: bruhat-tits - title: Formalisation of the Bruhat-Tits Tree - summary: Formalizes the Bruhat-Tits tree for GL₂ over the fraction field of a discrete valuation ring. - branch: algebraic geometry - entry_module: LeanPool.BruhatTits - authors: - - Judith Ludwig - - Christian Merten - source: - title: Trees - authors: - - Jean-Pierre Serre - url: https://doi.org/10.1007/978-3-642-61856-7 - github_repo: chrisflav/bruhat-tits - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - BruhatTits.BTtree - - BruhatTits.BTlaplace_surjective - main_results: - - declaration: BruhatTits.BTtree - informal: Proves that the Bruhat-Tits graph is a tree. - - declaration: BruhatTits.btgraph_regular - informal: Proves regularity of the Bruhat-Tits graph when the residue field is finite. - - declaration: BruhatTits.BTlaplace_surjective - informal: Proves surjectivity of the Bruhat-Tits Laplacian on harmonic cochains. - tags: - - algebraic-geometry - - graph-theory - - discrete-valuation-rings - msc: - - "20E42" - - "05C25" - - "13H05" - - slug: ramanujan-tau-misses-primes - title: ABC implies that Ramanujan's tau function misses almost all primes - summary: Formalizes the ABC-conditional proof that Ramanujan's tau function takes prime values only rarely. - branch: analytic number theory - entry_module: LeanPool.RamanujanTauMissesPrimes - authors: - - Evan Chen - - Kenny Lau - - Ken Ono - - Jujian Zhang - source: - arxiv: "2603.29970" - github_repo: AxiomMath/ramanujan-tau-misses-primes - license: MIT - status: verified - provenance: AI - main_declarations: - - main_theorem - - reduction_lemma - - abc_bound_E2 - - abc_bound_E4 - main_results: - - declaration: main_theorem - informal: ABC implies that Ramanujan's tau function misses almost all primes. - - declaration: reduction_lemma - informal: Reduces the main prime-value statement to the key analytic estimates. - - declaration: abc_bound_E2 - informal: ABC-bound input for the Eisenstein-series term E2. - - declaration: abc_bound_E4 - informal: ABC-bound input for the Eisenstein-series term E4. - tags: - - number-theory - - modular-forms - msc: - - "11F30" - - "11N05" - - slug: fel-conjecture - title: Fel's Conjecture on Syzygies of Numerical Semigroups - summary: Formalizes Fel's conjectural formula for syzygies of numerical semigroups. - branch: commutative algebra - entry_module: LeanPool.FelConjecture - authors: - - Evan Chen - - Kenny Lau - - Ken Ono - - Jujian Zhang - source: - arxiv: "2602.03716" - github_repo: AxiomMath/fel-polynomial - license: MIT - status: verified - provenance: AI - main_declarations: - - fels_conjecture - main_results: - - declaration: fels_conjecture - informal: Proves Fel's conjecture for the polynomial attached to numerical-semigroup syzygies. - tags: - - number-theory - - combinatorics - - polynomials - msc: - - "20M14" - - "13D02" - - slug: dead-ends - title: Dead Ends in Square-Free Digit Walks - summary: Formalizes the density formula for base-b dead ends in square-free digit walks. - branch: analytic number theory - entry_module: LeanPool.DeadEnds - authors: - - Evan Chen - - Kenny Lau - - Seewoo Lee - - Ken Ono - - Jujian Zhang - source: - arxiv: "2602.05095" - github_repo: AxiomMath/dead-ends - license: MIT - status: verified - provenance: AI - main_declarations: - - LeanPool.DeadEnds.baseBDeadEnd_density_formula - main_results: - - declaration: LeanPool.DeadEnds.baseBDeadEnd_density_formula - informal: Proves the inclusion-exclusion density formula for base-b dead ends. - tags: - - number-theory - - combinatorics - msc: - - "11N25" - - slug: formal-learning-theory - title: Formal Learning Theory Kernel - summary: Formalizes major equivalences and separations in PAC, VC, Gold-style, and online learning theory. - branch: computational learning theory - entry_module: LeanPool.FormalLearningTheory - authors: - - Dhruv Gupta - source: - url: https://github.com/Zetetic-Dhruv/formal-learning-theory-kernel - github_repo: Zetetic-Dhruv/formal-learning-theory-kernel - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - fundamental_theorem - - littlestone_characterization - - pac_not_implies_online - main_results: - - declaration: vc_characterization - informal: Characterizes PAC learnability by finite VC dimension. - - declaration: fundamental_theorem - informal: Bundles the central PAC, VC, compression, Rademacher, and sample-complexity equivalences. - - declaration: fundamental_vc_compression - informal: Relates finite VC dimension to compression with side information. - - declaration: gold_theorem - informal: States the Gold-style characterization for identification in the limit. - - declaration: mind_change_characterization - informal: Characterizes learnability by a bounded mind-change ordinal. - - declaration: littlestone_characterization - informal: Characterizes online learnability by finite Littlestone dimension. - - declaration: pac_not_implies_online - informal: Separates PAC learnability from online learnability. - - declaration: ex_not_implies_pac - informal: Separates explanatory learnability from PAC learnability. - tags: - - learning-theory - - probability - - combinatorics - - online-learning - msc: - - "68Q32" - - "68T05" - - slug: rl-theory-in-lean - title: RL Theory in Lean - summary: Develops probability, Markov-chain, and stochastic-matrix infrastructure for reinforcement-learning theory. - branch: probability theory - entry_module: LeanPool.RlTheoryInLean - authors: - - Shangtong Zhang - source: - arxiv: "2511.03618" - github_repo: ShangtongZhang/rl-theory-in-lean - license: MIT - status: verified - provenance: mix - main_declarations: - - StochasticMatrix.stationary_distribution_exists - main_results: - - declaration: StochasticMatrix.stationary_distribution_exists - informal: Proves existence of a stationary distribution for a stochastic matrix. - tags: - - probability - - reinforcement-learning - - stochastic-matrices - msc: - - "62L20" - - "60J10" - - slug: clawristotle - title: "Clawristotle: Vlasov-Maxwell-Landau steady-state classification" - summary: Formalizes a steady-state classification result for the Vlasov-Maxwell-Landau system with Coulomb kernel. - branch: kinetic theory - entry_module: LeanPool.Clawristotle - authors: - - Vasily Ilin - source: - arxiv: "2603.15929" - github_repo: Vilin97/Clawristotle - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - VML.Theorem42 - - VML.CoulombConcreteTheorem42 - main_results: - - declaration: VML.Theorem42 - informal: States the abstract steady-state classification theorem for the Vlasov-Maxwell-Landau framework. - - declaration: VML.CoulombConcreteTheorem42 - informal: Specializes the steady-state classification theorem to the Coulomb kernel. - tags: - - pde - - kinetic-theory - - mathematical-physics - msc: - - "35Q83" - - "82C40" - - "35Q61" - - slug: partial-regularity - title: Almost all primes are partially regular - summary: Formalizes an asymptotic theorem about partial regularity of primes via Bernoulli-number irregularity. - branch: analytic number theory - entry_module: LeanPool.PartialRegularity - authors: - - Evan Chen - - Kenny Lau - - Seewoo Lee - - Ken Ono - - Jujian Zhang - source: - arxiv: "2602.05090" - github_repo: AxiomMath/partial-regularity - license: MIT - status: verified - provenance: AI - main_declarations: - - LeanPool.PartialRegularity.Extension.irregularPrimes_isBigO - main_results: - - declaration: LeanPool.PartialRegularity.Extension.irregularPrimes_isBigO - informal: Proves the asymptotic big-O bound behind the partial-regularity theorem. - tags: - - number-theory - - asymptotics - msc: - - "11B68" - - "11N05" - - slug: riemann-mapping-theorem - title: Riemann Mapping Theorem - summary: Formalizes a proof of the Riemann mapping theorem for proper simply connected plane domains with primitives. - branch: complex analysis - entry_module: LeanPool.RiemannMappingTheorem - authors: - - Vincent Beffara - source: - url: https://github.com/vbeffara/RMT4 - github_repo: vbeffara/RMT4 - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - RMT - - main - - montel - - hurwitz - main_results: - - declaration: RMT - informal: Proves the Riemann mapping theorem for the project's domain hypotheses. - - declaration: main - informal: Constructs a conformal map from a good domain onto the unit disk. - - declaration: montel - informal: Proves a Montel compactness theorem for uniformly bounded holomorphic families. - - declaration: hurwitz - informal: Proves the Hurwitz theorem used to preserve injectivity under limits. - tags: - - complex-analysis - - conformal-maps - - schwarz-lemma - msc: - - "30C35" - - "30C20" - - slug: archon-firstproof-results - title: Archon-FirstProof-Results - summary: Collects formal solutions from the Archon FirstProof benchmark, including polynomial and graph-theoretic results. - branch: problem-solving mathematics - entry_module: LeanPool.ArchonFirstProofResults - authors: - - FrenzyMath - source: - arxiv: "2602.05192" - github_repo: frenzymath/Archon-FirstProof-Results - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Problem4.harmonic_mean_inequality_full - - Problem6.exists_eps_light_subset - main_results: - - declaration: Problem4.harmonic_mean_inequality_full - informal: Proves the benchmark harmonic-mean inequality for real-rooted polynomial transforms. - - declaration: Problem6.exists_eps_light_subset - informal: Proves the benchmark light-subset existence theorem in the graph Laplacian setting. - tags: - - polynomials - - analysis - - combinatorics - - linear-algebra - - graph-theory - msc: - - "26D15" - - "05C50" - - "15A42" - - slug: erdos1196 - title: Primitive Sets Above x (Erdos Problem 1196) - summary: Formalizes Math Inc.'s proof around primitive sets above x for Erdos Problem 1196. - branch: analytic number theory - entry_module: LeanPool.Erdos1196 - authors: - - Math Inc. - source: - arxiv: "2605.00301" - github_repo: math-inc/Erdos1196 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - PrimitiveSetsAboveX.mainTheorem - - Erdos1196.erdos_1196 - main_results: - - declaration: PrimitiveSetsAboveX.mainTheorem - informal: Proves the main primitive-sets-above-x theorem. - - declaration: Erdos1196.erdos_1196 - informal: States the formalized version of Erdos Problem 1196. - tags: - - number-theory - - combinatorics - - analysis - msc: - - "11N25" - - "05D05" - - slug: erdos403 - title: Sums of Distinct Factorials That Are Powers of Two (Erdos Problem 403) - summary: >- - Reconstructs the finiteness theorem for Erdos Problem 403: only finitely many sums of - distinct factorials are powers of two, and every such power is at most 2^7 = 128 (the - extremal solution 2! + 3! + 5! = 128). The sharp bound is proved in the factorial number - system via a single fixed modulus 12!, with all computations kernel-pure. - branch: number theory - entry_module: LeanPool.Erdos403 - authors: - - Trevor Morris - source: - title: "Erdos Problem 403: sums of distinct factorials that are powers of two" - authors: - - Trevor Morris - url: "https://www.erdosproblems.com/403" - github_repo: gotrevor/erdos-403 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Erdos403.erdos_403_sharp - - Erdos403.erdos_403_finite - main_results: - - declaration: Erdos403.erdos_403_sharp - informal: >- - Every power of two that is a sum of distinct factorials is at most 2^7, with 128 the - largest solution. - - declaration: Erdos403.erdos_403_finite - informal: >- - Only finitely many sums of distinct factorials are powers of two. - tags: - - number-theory - - factorials - - erdos-problems - msc: - - "11B83" - - slug: erdos865 - title: "A sharp 5/8 bound for Erdős Problem 865" - summary: >- - Formalizes the resolution of Erdős Problem 865 on pairwise sums. For A ⊆ {1,…,N}, a - pairwise-sum triple is a set of distinct a, b, c ∈ A with a+b, a+c, b+c all in A, and A - is triple-free when it has none. The development proves that every triple-free A ⊆ [1,N] - satisfies 8·|A| ≤ 5·N + 53, that any A exceeding this bound contains a triple, and — via - the construction A = [M,2M] ∪ [4M,8M] for N = 8M — that the constant 5/8 is optimal, so - f₃(N) = 5N/8 + O(1). The upper bound runs through a folded additive lemma over ℤ/mℤ (a - four-set union bound) and a folding lemma, closed by strong induction on N. - branch: additive combinatorics - entry_module: LeanPool.Erdos865 - authors: - - Ricky Cipollini - source: - title: "A sharp 5/8 bound for an Erdős–Sós pairwise-sums problem" - url: "https://github.com/mrricky22/erdos-865-lean" - github_repo: mrricky22/erdos-865-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Erdos865.erdos865_upper_bound - - Erdos865.sharpness - main_results: - - declaration: Erdos865.erdos865_upper_bound - informal: >- - Every triple-free set A ⊆ [1,N] satisfies 8·|A| ≤ 5·N + 53, i.e. |A| ≤ (5/8)·N + O(1). - - declaration: Erdos865.erdos865_contains_triple - informal: >- - Every A ⊆ [1,N] with 5·N + 53 < 8·|A| contains a pairwise-sum triple (the paper's - Theorem 1.1 form). - - declaration: Erdos865.sharpness - informal: >- - For N = 8M with M ≥ 1 there is a triple-free A ⊆ [1,N] with 8·|A| = 5·N + 16, so the - constant 5/8 is optimal. - tags: - - additive-combinatorics - - erdos-problems - - sum-free-sets - - combinatorics - msc: - - "11B75" - - "11B13" - - slug: erdos137 - title: "Erdős Problem #137: powerful products of consecutive integers" - summary: >- - Formalizes the abc-conditional finiteness results for Erdős Problem #137 (Erdős–Selfridge): - a number N is powerful if every prime dividing N has its square dividing N, and the question - is whether the product F(k,n) = n(n+1)⋯(n+k-1) of k ≥ 3 consecutive integers can be powerful - for infinitely many n. Each route isolates its abc input (a radical lower bound) as a single - explicit hypothesis and proves everything else outright, concluding that, conditionally, - F(k,n) is powerful for only finitely many n for each fixed k ≥ 3. - branch: number theory - entry_module: LeanPool.Erdos137 - authors: - - Scott D. Hughes - source: - title: "Erdős Problem #137: powerful products of consecutive integers" - authors: - - Scott D. Hughes - url: "https://www.erdosproblems.com/137" - github_repo: scottdhughes/erdos137 - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - Erdos137.erdos137_finite - - Erdos137.erdos137_eventually_not_powerful - main_results: - - declaration: Erdos137.erdos137_finite - informal: >- - Under the Granville–Langevin radical bound RadLB k (an abc-conditional hypothesis) and - k ≥ 3, the set of n ≥ 1 for which F(k,n) is powerful is finite. - - declaration: Erdos137.erdos137_eventually_not_powerful - informal: >- - Under RadLB k and k ≥ 3, there is a threshold beyond which F(k,n) is never powerful. - - declaration: Erdos137.lemma_star - informal: >- - Unconditionally, the radical squared times the powerful part of m is at most m squared - (rad m ^ 2 * B2 m ≤ m ^ 2). - - declaration: Erdos137.powerful_rad_sq_le - informal: >- - A nonzero powerful number N satisfies rad N ^ 2 ≤ N. - tags: - - number-theory - - powerful-numbers - - erdos-problems - msc: - - "11A51" - - "11N25" - - slug: lattice-triangle - title: On the paucity of lattice triangles - summary: Formalizes an analytic lower bound used in the paucity theorem for lattice triangles. - branch: additive combinatorics - entry_module: LeanPool.LatticeTriangle - authors: - - Evan Chen - - Kenny Lau - - Ken Ono - - Jujian Zhang - source: - arxiv: "2603.23928" - github_repo: AxiomMath/lattice-triangle - license: MIT - status: verified - provenance: AI - main_declarations: - - analyticEngine_lower_bound - main_results: - - declaration: analyticEngine_lower_bound - informal: Proves the analytic-engine lower bound underlying the lattice-triangle result. - tags: - - number-theory - - combinatorics - msc: - - "11H06" - - "11B30" - - slug: grothendieck-vanishing - title: Grothendieck's Vanishing Theorem - summary: Formalizes Grothendieck's vanishing theorem for sheaf cohomology on Noetherian topological spaces. - branch: algebraic geometry - entry_module: LeanPool.GrothendieckVanishing - authors: - - Vasily Ilin - - Brian Nugent - source: - doi: "10.1007/978-1-4757-3849-0" - url: https://github.com/Vilin97/Clawristotle/tree/grothendieck-vanishing - github_repo: Vilin97/Clawristotle - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - GrothendieckVanishing - main_results: - - declaration: GrothendieckVanishing - informal: Proves vanishing of sheaf cohomology above the topological Krull dimension. - tags: - - algebraic-geometry - - sheaf-theory - - topology - msc: - - "14F06" - - "18F20" - - slug: sensitivity - title: "Sensitivity Conjecture: sqrt(deg) <= sensitivity" - summary: Formalizes the quantitative Nisan-Szegedy/Huang sensitivity theorem for Boolean functions. - branch: extremal combinatorics - entry_module: LeanPool.Sensitivity - authors: - - Samuel Schlesinger - source: - arxiv: "1907.00847" - doi: "10.4007/annals.2019.190.3.6" - github_repo: SamuelSchlesinger/sensitivity-conjecture - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - LeanPoolSensitivity.sensitivity_ge_sqrt_degree - main_results: - - declaration: LeanPoolSensitivity.sensitivity_ge_sqrt_degree - informal: Proves that the square root of multilinear degree is bounded above by sensitivity. - - declaration: LeanPoolSensitivity.degree_le_sensitivity_sq - informal: Derives the degree bound by the square of sensitivity. - tags: - - combinatorics - - boolean-functions - - computational-complexity - msc: - - "06E30" - - "68Q17" - - slug: isoperimetric - title: Prekopa-Leindler, Brunn-Minkowski, and the isoperimetric inequality - summary: Formalizes the Prekopa-Leindler inequality, Brunn-Minkowski inequality, and an isoperimetric inequality. - branch: geometric analysis - entry_module: LeanPool.Isoperimetric - authors: - - Jonathan Ho - source: - url: https://terrytao.wordpress.com/2011/09/16/the-brunn-minkowski-inequality-for-nilpotent-groups/ - github_repo: hojonathanho/isoperimetric - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - prekopa_leindler - - brunn_minkowski - - isoperimetric_inequality - main_results: - - declaration: prekopa_leindler - informal: Proves the Prekopa-Leindler inequality. - - declaration: brunn_minkowski - informal: Proves the Brunn-Minkowski inequality. - - declaration: isoperimetric_inequality - informal: Proves the isoperimetric inequality derived from Brunn-Minkowski. - tags: - - measure-theory - - geometric-inequalities - msc: - - "28A75" - - "52A40" - - "49Q20" - - slug: leancomplexanalysis - title: Formalized Complex Analysis in Lean - summary: Formalizes Poisson integral formulas on the unit disc and arbitrary centered discs, Herglotz–Riesz representation, and Harnack's inequality on the unit disc, plus infrastructure for univalent function classes S and Σ. - branch: complex analysis - entry_module: LeanPool.LeanComplexAnalysis - authors: - - seb488 - - Aristotle - source: - url: https://github.com/seb488/LeanComplexAnalysis - github_repo: seb488/LeanComplexAnalysis - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - LeanPool.LeanComplexAnalysis.harnack_ineq - main_results: - - declaration: LeanPool.LeanComplexAnalysis.poisson_integral_of_harmonicOn_unitDisc_continuousOn_closedUnitDisc - informal: Proves the Poisson integral formula for harmonic functions on the unit disc. - - declaration: LeanPool.LeanComplexAnalysis.poisson_integral_of_harmonicOn_disc_continuousOn_closedDisc - informal: Proves the Poisson integral formula for harmonic functions on arbitrary centered discs. - - declaration: LeanPool.LeanComplexAnalysis.HerglotzRiesz_representation_harmonic - informal: Proves that every positive harmonic function on the unit disc with value 1 at 0 has a unique Herglotz–Riesz representation by a probability measure on the unit circle. - - declaration: LeanPool.LeanComplexAnalysis.harnack_ineq - informal: Proves Harnack's inequality for positive harmonic functions on the unit disc. - tags: - - complex-analysis - - harmonic-functions - - poisson-integral - - univalent-functions - msc: - - "30A99" - - "31A05" - - "30C55" - - slug: partial-combinatory-algebras - title: Partial Combinatory Algebras - summary: Formalises partial combinatory algebras with the K/S combinators, programming combinators, the free combinatory algebra, and the graph model. - branch: logic and computability - entry_module: LeanPool.PartialCombinatoryAlgebras - authors: - - Andrej Bauer - source: - title: "Realizability: An Introduction to its Categorical Side" - authors: - - Jaap van Oosten - url: https://doi.org/10.1016/S0049-237X(08)80003-1 - github_repo: andrejbauer/partial-combinatory-algebras - license: MIT - status: verified - provenance: human - main_declarations: - - LeanPool.PartialCombinatoryAlgebras.PCA - main_results: - - declaration: LeanPool.PartialCombinatoryAlgebras.PCA - informal: Defines the class of partial combinatory algebras (PCAs) with K and S combinators. - - declaration: LeanPool.PartialCombinatoryAlgebras.CA - informal: Defines total combinatory algebras and exhibits every total combinatory algebra as a PCA. - - declaration: LeanPool.PartialCombinatoryAlgebras.FreeCA - informal: Constructs the free total combinatory algebra as a quotient of formal expressions. - - declaration: LeanPool.PartialCombinatoryAlgebras.GraphModel.isCA - informal: Constructs the graph-model combinatory algebra on the powerset of any Listing-typed carrier. - tags: - - combinatory-algebra - - lambda-calculus - - computability - msc: - - "03B40" - - "03D75" - - slug: duality - title: Duality theory in linear optimization and its extensions - summary: Formalizes Farkas-like theorems and strong LP duality, including extensions to a linearly ordered field with bot and top. - branch: linear optimization - entry_module: LeanPool.Duality - authors: - - Martin Dvorak - source: - arxiv: "2409.08119" - doi: "10.46298/afm.14253" - github_repo: madvorak/duality - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - extendedFarkas - - StandardLP.strongDuality - main_results: - - declaration: equalityFarkas - informal: Farkas' lemma for linear equalities over nonnegative variables. - - declaration: inequalityFarkas - informal: Farkas' lemma for linear inequalities over nonnegative variables. - - declaration: extendedFarkas - informal: Extended Farkas theorem for matrices and vectors over a linearly ordered field with bot and top. - - declaration: ValidELP.strongDuality - informal: Strong duality theorem for extended linear programs. - - declaration: StandardLP.strongDuality - informal: Strong duality theorem for linear programs in standard form. - tags: - - linear-programming - - optimization - - farkas-lemma - msc: - - "90C05" - - "90C46" - - slug: sumsthreesquares - title: Sums of Three Squares - summary: Formalizes a blueprint step in the three-squares theorem using quadratic forms and geometry of numbers. - branch: number theory - entry_module: LeanPool.SumsThreeSquares - authors: - - Bhavik Mehta - - Pietro Monticone - - Abel Donate Munoz - source: - doi: "10.1090/S0002-9939-1957-0085275-8" - github_repo: pitmonticone/SumsThreeSquares - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - LeanPool.SumsThreeSquares.blueprint_case_mod8_eq3 - main_results: - - declaration: LeanPool.SumsThreeSquares.blueprint_case_mod8_eq3 - informal: Proves the mod-8-equals-3 blueprint case for the three-squares theorem. - source_ref: "H. Davenport, The geometry of numbers, The Mathematical Gazette 31 (1947), 206-210, doi:10.2307/3608159." - tags: - - number-theory - - quadratic-forms - - geometry-of-numbers - msc: - - "11E25" - - "11H06" - - slug: formalization-of-bounded-arithmetic - title: Strengthened V0 Bounded Arithmetic Interfaces - summary: Formalizes first-order and two-sorted bounded-arithmetic model infrastructure, including arithmetic model laws and string-addition laws for strengthened bundled V0-style interfaces with propositional induction, order, cancellation, decidability, and algebraic laws assumed as fields rather than derived from a separate raw V0 class. - branch: mathematical logic - entry_module: LeanPool.FormalizationOfBoundedArithmetic - authors: - - ruplet - source: - title: Logical Foundations of Proof Complexity - authors: - - Stephen Cook - - Phuong Nguyen - doi: "10.1017/CBO9780511676277" - github_repo: ruplet/formalization-of-bounded-arithmetic - license: MIT - status: verified - provenance: human - main_declarations: - - V0Model.ind_strengthened_v0 - - str_add_assoc_strengthened_v0 - main_results: - - declaration: IOPENModel.add_assoc - informal: Proves associativity of addition in IOPEN models. - - declaration: IOPENModel.add_comm - informal: Proves commutativity of addition in IOPEN models. - - declaration: IDelta0Model.pred_exists - informal: Shows that nonzero elements in IDelta0 models have predecessors. - - declaration: IDelta0Model.mul_cancel_right - informal: Proves right cancellation for multiplication by a nonzero element in IDelta0 models. - - declaration: V0Model.xind - informal: Exposes string induction inside the strengthened bundled V0Model structure, whose fields include the comprehension and induction/order principles used downstream. - - declaration: V0Model.ind_strengthened_v0 - informal: Exposes propositional induction inside the strengthened bundled V0Model structure, rather than deriving it from only the raw finite V0 axiom list. - - declaration: str_add_comm_strengthened_v0 - informal: Proves commutativity of string addition in V0 extension models extending the strengthened V0Model bundle. - - declaration: str_add_assoc_strengthened_v0 - informal: Proves associativity of string addition in V0 extension models extending the strengthened V0Model bundle, using the bundled propositional induction field. - - declaration: str_succ_assoc_strengthened_v0 - informal: Proves that string addition associates with string successor in V0 extension models extending the strengthened V0Model bundle. - tags: - - logic - - bounded-arithmetic - - model-theory - - computational-complexity - msc: - - "03F30" - - "03D15" - - slug: polylean - title: Polylean Unit Conjecture Counterexample - summary: Formalizes Gardam's counterexample to Kaplansky's Unit Conjecture, including the Promislow group, group rings, torsion-freeness of the group, and the non-trivial unit in 𝔽₂[P]. - branch: algebra - entry_module: LeanPool.Polylean - authors: - - Siddhartha Gadgil - - Anand Rao - source: - arxiv: "2102.11818" - doi: "10.4007/annals.2021.194.3.9" - github_repo: siddhartha-gadgil/Polylean - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - LeanPool.Polylean.P.torsionFree - - LeanPool.Polylean.Gardam.GardamTheorem - main_results: - - declaration: LeanPool.Polylean.P.torsionFree - informal: Proves that the Promislow group used in Gardam's construction is torsion-free. - - declaration: LeanPool.Polylean.Gardam.GardamTheorem - informal: Proves Gardam's counterexample to Kaplansky's Unit Conjecture by constructing a non-trivial unit in the group ring 𝔽₂[P]. - tags: - - algebra - - group-theory - - ring-theory - - unit-conjecture - msc: - - "16S34" - - "20F65" - - slug: directed-topology-lean-4 - title: Directed Topology in Lean 4 - summary: Foundational library for directed topological spaces, dipaths, dihomotopy, the fundamental category functor, and a directed Van Kampen pushout theorem. - branch: algebraic topology - entry_module: LeanPool.DirectedTopologyLean4 - authors: - - Dominique Lawson - - Henning Basold - - Peter Bruin - source: - arxiv: "2312.06506" - doi: "10.4230/LIPIcs.ITP.2024.8" - github_repo: Dominique-Lawson/Directed-Topology-Lean-4 - license: MIT - status: verified - provenance: human - main_declarations: - - DirectedSpace - - Dipath - - DirectedVanKampen.directed_van_kampen - main_results: - - declaration: DirectedSpace - informal: Definition of a directed topological space as a topological space with a class of directed paths closed under constant paths, concatenation, and monotone reparametrization. - - declaration: Dipath - informal: A directed path in a directed space. - - declaration: Dipath.Dihomotopic - informal: Dihomotopy equivalence relation on dipaths between fixed endpoints. - - declaration: DirectedVanKampen.directed_van_kampen - informal: Proves the directed Van Kampen theorem, showing that the directed fundamental category functor sends the standard open cover square to a pushout in the category of categories. - tags: - - directed-topology - - algebraic-topology - - category-theory - msc: - - "55U40" - - "55Q05" - - "18A30" - - slug: selberg-sieve4 - title: Selberg Sieve - summary: Formalizes the Selberg sieve with applications to prime-counting bounds and a Brun-Titchmarsh-style estimate. - branch: analytic number theory - entry_module: LeanPool.SelbergSieve4 - authors: - - Arend Mellendijk - source: - title: Sieve Methods - authors: - - Heini Halberstam - - Hans-Egon Richert - url: https://archive.org/details/sievemethods0000halb - github_repo: amellendijk/selberg-sieve4 - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - fundamental_theorem_simple - - primeCounting_isBigO_atTop - - primesBetween_le - main_results: - - declaration: fundamental_theorem_simple - informal: Proves the simple Selberg sieve upper-bound theorem. - - declaration: primeCounting_isBigO_atTop - informal: Derives the asymptotic prime-counting upper bound. - - declaration: primeCounting_le_mul - informal: Gives an eventual explicit constant upper bound for the prime-counting function. - - declaration: primesBetween_le - informal: Bounds the number of primes in an interval using the sieve estimate. - tags: - - number-theory - - analytic-number-theory - - sieve-theory - - prime-counting - msc: - - "11N35" - - "11N05" - - "11N13" - - slug: abc-exceptions - title: Exceptional Set in the abc Conjecture - summary: Formalizes reductions and optimization bounds for a power-saving estimate on exceptional triples in the abc conjecture. - branch: analytic number theory - entry_module: LeanPool.ABCExceptions - authors: - - Bhavik Mehta - - Arend Mellendijk - source: - title: Bounds on the exceptional set in the abc conjecture - authors: - - Tim Browning - - Jared Duker Lichtman - - Joni Teräväinen - arxiv: "2410.12234" - github_repo: b-mehta/ABC-Exceptions - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - abcConjecture_iff_countTriples - - thm_4_point_3 - main_results: - - declaration: abcConjecture_iff_countTriples - informal: Recasts the abc conjecture as a uniform big-O bound for bounded exceptional triples. - - declaration: refinedCountTriplesStar_isBigO_B - informal: Bounds the refined exceptional-triple count by the dyadic structured-box supremum. - - declaration: thm_4_point_3 - informal: Derives the Section 4 optimization bound from the determinant, geometry, and Fourier hypotheses. - tags: - - number-theory - - analytic-number-theory - - abc-conjecture - msc: - - "11D75" - - "11N37" - - slug: lean-booleanfun - title: Analysis of Boolean functions in Lean - summary: Walsh-Fourier analysis on Boolean functions, including Arrow's theorem and BLR linearity testing. - branch: combinatorics - entry_module: LeanPool.LeanBooleanfun - authors: - - Joris Roos - source: - arxiv: "2105.10386" - doi: "10.1017/CBO9781139814782" - github_repo: roos-j/lean-booleanfun - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - LeanPool.LeanBooleanfun.BooleanFun.BV.dictator_of_condorcet_and_unanimous - main_results: - - declaration: LeanPool.LeanBooleanfun.BooleanFun.BV.dictator_of_condorcet_and_unanimous - informal: Arrow's theorem for two-candidate Boolean voting rules. - tags: - - boolean-functions - - fourier-analysis - - social-choice - msc: - - "06E30" - - "42C10" - - "91B14" - - slug: polya-enumeration-theorem - title: "Pólya's enumeration theorem" - summary: Formalizes Pólya's enumeration theorem counting orbits of colorings under a group action. - branch: combinatorics - entry_module: LeanPool.PolyaEnumerationTheorem - authors: - - Luka Opravš - source: - url: "https://github.com/Luka-O/polya-enumeration-theorem" - github_repo: Luka-O/polya-enumeration-theorem - license: MIT - status: verified - provenance: human - main_declarations: - - LeanPool.PolyaEnumerationTheorem.polya_theorem - main_results: - - declaration: LeanPool.PolyaEnumerationTheorem.polya_theorem - informal: Counts distinct colorings of a finite set under a group action by averaging powers of the number of colors over cycle indices. - tags: - - combinatorics - - group-theory - - enumeration - msc: - - "05A15" - - "20B30" - - slug: fineqs - title: FinEqs - reducing equations defining a subset of n-space over a finite field - summary: Formalizes the finite-field equation-reduction theorem and sharpness examples. - branch: number theory - entry_module: LeanPool.Fineqs - authors: - - Stefan Barańczuk - - Aristotle - source: - arxiv: "1906.11174" - doi: "10.5802/afst.1766" - github_repo: nasqret/fineqs - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.Fineqs.theorem_1 - - LeanPool.Fineqs.prop_1 - main_results: - - declaration: LeanPool.Fineqs.theorem_1 - informal: Reduces equations on a small finite set to at most n equations in the same span. - - declaration: LeanPool.Fineqs.prop_1 - informal: Shows the cardinality bound in the reduction theorem is sharp. - - declaration: LeanPool.Fineqs.corollary_2 - informal: Applies the reduction theorem to affine space over a finite field. - - declaration: LeanPool.Fineqs.corollary_3 - informal: Applies the reduction theorem to nonempty projective zero loci. - - declaration: LeanPool.Fineqs.remark_example - informal: Shows n - 1 coordinate-function combinations have at least q common zeros. - tags: - - number-theory - - finite-fields - - algebraic-geometry - msc: - - "14G15" - - "11T06" - - slug: virasoroproject - title: Virasoro Project - summary: Formalizes the Virasoro algebra, its Witt-algebra central extension, Verma modules, and the basic bosonic Sugawara construction. - branch: representation theory - entry_module: LeanPool.VirasoroProject - authors: - - Kalle Kytölä - source: - title: Infinite-Dimensional Lie Algebras - authors: - - Victor Kac - url: https://doi.org/10.1017/CBO9780511626234 - github_repo: kkytola/VirasoroProject - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - VirasoroProject.VirasoroAlgebra - - VirasoroProject.sugawaraRepresentation - main_results: - - declaration: VirasoroProject.VirasoroAlgebra - informal: Defines the Virasoro algebra as a one-dimensional central extension of the Witt algebra. - - declaration: VirasoroProject.sugawaraRepresentation - informal: Constructs the basic bosonic Sugawara representation of the Virasoro algebra from locally truncated Heisenberg actions. - tags: - - representation-theory - - lie-algebras - - mathematical-physics - - virasoro-algebra - msc: - - "17B68" - - "17B56" - - "81R10" - - - slug: lowdimsolvclassification - title: Classification of low-dimensional solvable Lie algebras - summary: Formalises the classification of solvable Lie algebras up to dimension three. - branch: lie theory - entry_module: LeanPool.LowDimSolvClassification - authors: - - Viviana del Barco - - Gustavo Infanti - - Exequiel Rivas - - Paul Schwahn - source: - title: Classification and Identification of Lie Algebras - authors: - - Pavel Winternitz - - Libor Šnobl - doi: "10.1090/crmm/033" - url: https://bookstore.ams.org/crmm-33 - github_repo: LieLean/LowDimSolvClassification - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - LieAlgebra.Dim3.classification - main_results: - - declaration: LieAlgebra.Dim3.classification - informal: Classifies three-dimensional solvable Lie algebras over arbitrary fields. - tags: - - lie-algebras - - solvable - - classification - msc: - - "17B30" - - slug: artin-wedderburn - title: Artin-Wedderburn Theorem - summary: Formalizes the Artin-Wedderburn theorem for prime and simple Artinian rings. - branch: ring theory - entry_module: LeanPool.ArtinWedderburn - authors: - - Matevz Miščič - - Maša Žaucer - - Job Petrovčič - source: - title: A First Course in Noncommutative Rings - authors: - - T. Y. Lam - arxiv: "2405.04588" - doi: "10.1007/978-1-4419-8616-0" - github_repo: JobPetrovcic/ArtinWedderburn - license: MIT - status: verified - provenance: human - main_declarations: - - LeanPool.ArtinWedderburn.ArtinWedderburnForPrime - main_results: - - declaration: LeanPool.ArtinWedderburn.ArtinWedderburnForPrime - informal: A nontrivial prime Artinian ring is isomorphic to a matrix ring over a division ring. - tags: - - ring-theory - - noncommutative-algebra - - artinian-rings - msc: - - "16K20" - - "16P20" - - slug: order-p-q - title: Classification of groups of order p * q - summary: Classification of noncyclic finite groups of order p*q, including the prime-square direct-product case and the p- - Perron-Frobenius theory, Jentzsch's theorem, and matrix/operator positivity - in Lean 4. At the matrix level it proves that nonnegative matrices have - nonnegative powers, the Metzler decomposition and positivity of the matrix - exponential of a Metzler matrix, M-matrix inverse positivity, and the - Perron-Frobenius theorem for irreducible nonnegative matrices (a positive - dominant eigenvalue with a strictly positive eigenvector). At the operator - level it proves Jentzsch's theorem for compact self-adjoint - positivity-improving operators on L2 — a simple, strictly positive ground - state with a spectral gap — together with a kernel positivity criterion. - branch: linear algebra - entry_module: LeanPool.SpectralPositivity - authors: - - Michael R. Douglas - source: - title: "spectral-positivity: Perron-Frobenius, Jentzsch, and matrix positivity in Lean 4" - authors: - - Michael R. Douglas - url: "https://github.com/mrdouglasny/spectral-positivity" - github_repo: mrdouglasny/spectral-positivity - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - allpos_has_pos_eigenvec - - perron_frobenius - - ground_state_strictly_positive - main_results: - - declaration: allpos_has_pos_eigenvec - informal: >- - A matrix with all strictly positive entries has a positive eigenvalue - with a strictly positive eigenvector (Perron-Frobenius for positive - matrices, via the Collatz-Wielandt characterization). - - declaration: perron_frobenius - informal: >- - An irreducible (primitive) nonnegative matrix has a positive eigenvalue - with a strictly positive eigenvector. - - declaration: ground_state_strictly_positive - informal: >- - The ground eigenfunction of a compact self-adjoint positivity-improving - operator on L2 is strictly positive almost everywhere (Jentzsch). - - declaration: SpectralPositivity.Matrix.MMatrix.inverse_pos - informal: >- - The inverse of a nonsingular irreducible M-matrix is entrywise strictly - positive. - tags: - - linear-algebra - - perron-frobenius - - positivity - msc: - - "15B48" - - "15A18" - - slug: unconditional-schauder-basis - title: Unconditional Schauder Bases - summary: >- - Develops an API for Schauder bases and unconditional Schauder bases of - complete normed spaces, including ordered partial-sum convergence, - rearrangement criteria for unconditionality, and an arbitrary-index - variant built on unconditional summability, then proves a finite sign - criterion: a family with dense closed span, nonzero vectors, and a uniform - finite sign bound is the basis of an unconditional Schauder basis. - branch: functional analysis - entry_module: LeanPool.UnconditionalSchauderBasis - authors: - - Daniel Smania - source: - title: Classical Banach Spaces I and II - authors: - - Joram Lindenstrauss - - Lior Tzafriri - doi: "10.1007/978-3-662-53294-2" - github_repo: SmaniaD/UnconditionalSchauderBasis - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - UnconditionalCriterion.exists_unconditionalSchauderBasis_of_finiteSignBound - main_results: - - declaration: UnconditionalCriterion.exists_unconditionalSchauderBasis_of_finiteSignBound - informal: >- - Finite sign criterion for sequences: if `x : ℕ → E` has dense closed - linear span, every `x n` is nonzero, and finite sign changes are - uniformly bounded by `C`, then `x` is the basis sequence of an - unconditional Schauder basis. - source_ref: "J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II, Chapter 1, Section c, unconditional bases, doi:10.1007/978-3-662-53294-2." - - declaration: UnconditionalCriterion.exists_unconditionalSchauderBasisAbstractIndex_of_finiteSignBound - informal: >- - Abstract-index form of the finite sign criterion: under the same dense - span, nonzero, and uniform finite sign bound hypotheses for an - arbitrarily indexed family, there is an abstractly indexed - unconditional Schauder basis whose basis family is exactly `x`. - source_ref: "J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II, Chapter 1, Section c, unconditional bases, doi:10.1007/978-3-662-53294-2." - tags: - - functional-analysis - - banach-spaces - - schauder-bases - msc: - - "46B15" - - slug: compact-spectral - title: Spectral theorem for compact self-adjoint operators - summary: >- - Formalizes the spectral theorem for compact self-adjoint operators on a Hilbert space: such - an operator admits a Hilbert basis of eigenvectors. The development passes from weak - compactness of Hilbert closed balls and Rayleigh-quotient extrema to an eigenvector at the - operator norm, finiteness of the point spectrum outside any neighbourhood of zero, finite-rank - cutoff approximation, and finally the orthogonal decomposition into eigenspaces. - branch: functional analysis - entry_module: LeanPool.CompactSpectral - authors: - - Adam Benenson - source: - title: Advanced Real Analysis - authors: - - Anthony W. Knapp - doi: "10.3792/euclid/9781429799911-2" - github_repo: abenenson/compact-spectral - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - CompactSelfAdjoint.exists_hasEigenvector_iSup_or_iInf_of_isCompactOperator - main_results: - - declaration: CompactSelfAdjoint.exists_hilbertBasis_hasEigenvector_of_isCompactOperator_of_isSelfAdjoint - informal: >- - A compact self-adjoint operator on a Hilbert space admits a Hilbert basis of - eigenvectors. - source_ref: "A. W. Knapp, Advanced Real Analysis, Chapter II, Section 2, Theorem 2.3, doi:10.3792/euclid/9781429799911-2." - - declaration: CompactSelfAdjoint.iSup_eigenspace_orthogonal_eq_bot_of_isCompactOperator_of_isSelfAdjoint - informal: >- - The eigenspaces of a compact self-adjoint operator span a dense subspace: the orthogonal - complement of the supremum of all eigenspaces is trivial. - source_ref: "A. W. Knapp, Advanced Real Analysis, Chapter II, Section 2, Theorem 2.3, doi:10.3792/euclid/9781429799911-2." - - declaration: CompactSelfAdjoint.exists_hasEigenvector_iSup_or_iInf_of_isCompactOperator - informal: >- - A compact self-adjoint operator has an eigenvector whose eigenvalue realizes either the - supremum or the infimum of its Rayleigh quotient. - source_ref: "A. W. Knapp, Advanced Real Analysis, Chapter II, Section 2, Theorem 2.2, doi:10.3792/euclid/9781429799911-2." - - declaration: CompactSelfAdjoint.exists_unit_norm_eq_opNorm_of_isCompactOperator - informal: >- - A compact operator attains its operator norm on the unit sphere of the Hilbert space. - source_ref: "A. W. Knapp, Advanced Real Analysis, Chapter II, Section 1, compactness preliminaries for compact operators, doi:10.3792/euclid/9781429799911-2." - - declaration: CompactSelfAdjoint.countable_set_hasEigenvalue_ne_zero_of_isCompactOperator_of_isSelfAdjoint - informal: >- - The set of nonzero eigenvalues of a compact self-adjoint operator is countable. - source_ref: "A. W. Knapp, Advanced Real Analysis, Chapter II, Section 2, spectral theorem consequences, doi:10.3792/euclid/9781429799911-2." - tags: - - spectral-theory - - functional-analysis - - compact-operators - msc: - - "47A75" - - "47B07" - - slug: hsd-interior-point-lp - title: Homogeneous self-dual interior-point method for linear programming - summary: Formalizes a homogeneous self-dual interior-point method for linear programming, including primal-dual data, Newton block solvability from full row rank, fixed Ye-Todd-Mizuno predictor/corrector local estimates, contraction bounds, and logarithmic stopping certificates for generated iterates. - branch: linear optimization - entry_module: LeanPool.HSDInteriorPointLP - authors: - - Makoto Yamashita - source: - title: "An O(√nL)-iteration homogeneous and self-dual linear programming algorithm" - authors: - - Yinyu Ye - - Michael J. Todd - - Shinji Mizuno - doi: "10.1287/moor.19.1.53" - github_repo: makoto-yamashita/proof-on-a-homogeneous-self-dual-interior-point-method-for-linear-programming - license: MIT - status: verified - provenance: human - main_declarations: - - HSDInteriorPointLP.YTM_fixed_local_theory_from_paper - main_results: - - declaration: HSDInteriorPointLP.YTM_fixed_local_theory_from_paper - informal: Packages the fixed Ye-Todd-Mizuno local theory, combining predictor and corrector estimates for the tight and wide central neighborhoods. - source_ref: "Y. Ye, M. J. Todd, and S. Mizuno, An O(√nL)-iteration homogeneous and self-dual linear programming algorithm, Mathematics of Operations Research 19(1):53-67, 1994." - - declaration: HSDInteriorPointLP.HSDGeneratedAlgorithm.gap_stop_even_at_logPairBound - informal: Proves that the generated even iterate satisfies the gap stopping test at the logarithmic pair bound. - - declaration: HSDInteriorPointLP.HSDGeneratedAlgorithm.mu_stop_even_at_logPairBound - informal: Proves that the generated even iterate satisfies the duality-measure stopping test at the scaled logarithmic pair bound. - - declaration: HSDInteriorPointLP.HLP_newton_block_system_solvable_from_full_row_rank - informal: Shows solvability of the Newton block system from the full-row-rank assumption through finite-dimensional linear algebra. - - declaration: HSDInteriorPointLP.YTM_contraction_power_le_exp - informal: Bounds the fixed YTM contraction power by the corresponding exponential decay estimate. - tags: - - linear-programming - - interior-point-methods - - optimization - - homogeneous-self-dual - msc: - - "90C05" - - "90C51" - - slug: connes-kreimer - title: Connes-Kreimer Hopf algebra of rooted trees - summary: Formalizes the planar Connes-Kreimer/Foissy Hopf algebra of rooted trees, its cut coproduct, bialgebra and Hopf algebra structures, the commutative quotient, Adams operators, and the Eulerian idempotent on the quotient. - branch: algebraic combinatorics - entry_module: LeanPool.ConnesKreimer - authors: - - Carles Marín - source: - title: "How a Tree Remembers Its Cuts: the Connes-Kreimer-Foissy Hopf Algebra of Rooted Trees, Machine-Checked in Lean 4" - authors: - - Carles Marín - doi: "10.5281/zenodo.20762280" - github_repo: karlesmarin/connes-kreimer-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - CK.instCKHopf - - CK.instHabHopf - - CK.eulerian1_idem_ab - main_results: - - declaration: CK.instCKHopf - informal: Constructs the planar rooted-tree Connes-Kreimer/Foissy Hopf algebra as a Mathlib `HopfAlgebra`. - - declaration: CK.instHabHopf - informal: Constructs the Hopf algebra structure on the commutative quotient of the planar rooted-tree algebra. - - declaration: CK.eulerian1_idem_ab - informal: Proves that the first Eulerian idempotent is idempotent on the commutative quotient. - - declaration: CK.coassoc_map - informal: Lifts the list-level cut coassociativity proof to equality of linear maps for the coproduct. - - declaration: PowerSeries.logOf_mul - informal: Proves additivity of `PowerSeries.logOf` for power series with constant coefficient one. - tags: - - hopf-algebras - - rooted-trees - - renormalization - - combinatorics - msc: - - "16T05" - - "05C05" - - "81T15" - - slug: burkholder - title: Burkholder Martingale Transform Inequality - summary: >- - Formalizes Burkholder's sharp Lp inequality for real-valued martingale - transforms: if `f` is a discrete martingale and `w` a predictable - multiplier bounded by `1`, then the Lp norm of the transform `w ⋆ₘ f` is - at most `(pStar p - 1)` times the Lp norm of `f`, where `pStar p` is the - maximum of `p` and its conjugate exponent. The proof builds the Burkholder - special function as a majorant of the relevant payoff in each exponent - regime (`1 < p < 2`, `p = 2`, `p > 2`). - branch: probability - entry_module: LeanPool.Burkholder - authors: - - Daniel Smania - source: - title: "Boundary value problems and sharp inequalities for martingale transforms" - authors: - - Donald L. Burkholder - doi: "10.1214/aop/1176993220" - github_repo: SmaniaD/Burkholder - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - MeasureTheory.Lp_Burkholder_inequality_martingaleTransform - main_results: - - declaration: MeasureTheory.Lp_Burkholder_inequality_martingaleTransform - informal: >- - The sharp Lp Burkholder inequality: for a discrete martingale `f` and a - predictable multiplier `w` with `|w n| ≤ 1`, the Lp norm of the - martingale transform `(w ⋆ₘ f) n` is bounded by `(pStar p - 1)` times - the Lp norm of `f n`. - - declaration: MeasureTheory.martingaleTransform_martingale - informal: >- - The martingale transform of a martingale by a strongly predictable - bounded multiplier is itself a martingale. - - declaration: Majorants.exists_majorant_p_g_1 - informal: >- - For every exponent `p > 1` there exists a Burkholder majorant, the - special function dominating the payoff and driving the inequality. - tags: - - probability - - martingales - - burkholder-inequality - msc: - - "60G42" - - slug: erdos367 - title: "Erdős Problem #367" - summary: Formalizes upper and lower bound results for powerful parts of products of consecutive integers, including abc-conditional upper bounds, an unconditional odd r-full lower bound, a Pell-style limsup theorem, and polynomial arithmetic cores. - branch: number theory - entry_module: LeanPool.Erdos367 - authors: - - Scott D. Hughes - source: - title: Powerful parts of consecutive integers and Davenport-Zannier polynomials - authors: - - Scott D. Hughes - url: https://github.com/scottdhughes/erdos367 - github_repo: scottdhughes/erdos367 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Erdos367.erdos_367_k3 - - RFullOdd.erdos367_iv - - Erdos367.erdos367 - main_results: - - declaration: Erdos367.erdos_367_k3 - informal: Proves an abc-conditional k = 3 upper bound for the product of square-full parts of three consecutive integers. - - declaration: GeneralK.B2_upper_bound - informal: Proves a general-k upper bound for the square-full part of a product of consecutive integers from an explicit radical lower-bound hypothesis. - - declaration: RFullOdd.erdos367_iv - informal: Gives an unconditional odd r construction where the product of r-full parts of two consecutive integers is larger than the stated power of n. - - declaration: Erdos367.erdos367 - informal: Proves an unconditional limsup form showing that the product of square-full parts of three consecutive integers exceeds any fixed multiple of n squared times log n. - - declaration: Core139.T3 - informal: Proves the integer-form arithmetic core yielding the E3 at least 13/9 lower-bound construction under explicit prime-supply data. - - declaration: Core4027.IDENT - informal: Verifies the degree-27 polynomial identity used in the Davenport-Zannier construction. - - declaration: Core4027.T3 - informal: Proves the integer-form arithmetic core yielding the E3 at least 40/27 lower-bound construction under explicit prime-supply data. - tags: - - number-theory - - erdos-problems - - powerful-numbers - - abc-conjecture - - pell-equations - msc: - - "11D09" - - "11N25" - - "11D45" - - slug: critical-portraits - title: Counting Critical Portraits - summary: >- - Formalizes the census of degree-d critical portraits, modeled as - level-canonical (d-1)-subsets of the cyclic group ZMod (d*m), and proves - that their number equals C(d*m, d-1)/d for all d, via the - Dvoretzky-Motzkin cycle lemma together with an explicit - level-canonicalization bijection (injectivity and surjectivity of the - delete-minimum map). - branch: combinatorics - entry_module: LeanPool.CriticalPortraits - authors: - - Keston Aquino-Michaels - source: - title: "Counting critical portraits via the cycle lemma" - authors: - - Keston Aquino-Michaels - doi: "10.5281/zenodo.20737896" - github_repo: no-way-labs/lean-critical-portraits - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - CriticalPortraits.card_portraits - - CriticalPortraits.Cycle.cycle_lemma - main_results: - - declaration: CriticalPortraits.card_portraits - informal: >- - The number of degree-d critical portraits at level m equals - C(d*m, d-1)/d. - - declaration: CriticalPortraits.Cycle.cycle_lemma - informal: >- - The Dvoretzky-Motzkin cycle lemma: a period-n integer sequence with - period-sum 1 has exactly one good cyclic shift. - - declaration: CriticalPortraits.T_injOn - informal: >- - The delete-minimum map is injective on critical portraits. - - declaration: CriticalPortraits.T_surjOn - informal: >- - The delete-minimum map surjects onto the level-canonical subsets. - tags: - - combinatorics - - cycle-lemma - - critical-portraits - - enumeration - msc: - - "05A15" - - "37F20" - - slug: domain-theory - title: DomainTheory - summary: Formalizes Dana Scott's domain theory through Scott information systems and PRG-19 neighbourhood systems, including approximable maps, products, function spaces, fixed points, recursive presentations, and initial-algebra constructions for denotational semantics. - branch: domain theory - entry_module: LeanPool.DomainTheory - authors: - - Catskills Research Company - source: - title: "Domains for Denotational Semantics" - authors: - - Dana Scott - doi: "10.1007/BFb0012801" - url: "https://doi.org/10.1007/BFb0012801" - github_repo: catskillsresearch/domain_theory - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - InfoSys - - Domain.Neighborhood.NeighborhoodSystem - main_results: - - declaration: InfoSys - informal: Defines Scott information systems, with a distinguished bottom token, finite consistency predicate, entailment relation, and Scott's six axioms. - - declaration: Domain.Neighborhood.NeighborhoodSystem - informal: Defines PRG-19 neighbourhood systems, Scott's neighbourhood presentation of domains. - - declaration: Domain.Neighborhood.ApproximableMap - informal: Defines approximable maps between neighbourhood systems as relations preserving finite approximations. - - declaration: Domain.Neighborhood.trianglelefteq_of_isInitial - informal: Proves that an initial algebra for a functor embeds into every other solution through Scott's subdomain relation. - - declaration: ScottMap.fix_eq - informal: Proves the fixed-point equation for Scott-continuous self maps on complete lattices. - tags: - - domain-theory - - denotational-semantics - - information-systems - msc: - - "03B70" - - "06B35" - - "68Q55" - - slug: lean-quantumalg - title: Lean-QuantumAlg - summary: Provides a Mathlib-based library of formally verified quantum-algorithm definitions, primitives, and endpoints, including QFT, QSP, quantum kernels, amplitude amplification, Grover search, phase estimation, order finding, teleportation, and a Lie-algebraic barren-plateau witness. - branch: quantum computing - entry_module: LeanPool.LeanQuantumAlg - authors: - - QudeLeap Team - source: - title: Lean-QuantumAlg - authors: - - QudeLeap Team - url: https://github.com/QudeLeap/Lean-QuantumAlg - github_repo: QudeLeap/Lean-QuantumAlg - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - QuantumAlg.GroverSearch.main_success_probability - main_results: - - declaration: QuantumAlg.QuantumFourierTransform.main - informal: Proves the QFT action on computational-basis states and records exact circuit counts. - - declaration: QuantumAlg.ReflectionBasedQuantumSignalProcessing.main - informal: Characterizes reflection-based single-qubit QSP transforms in the Chebyshev basis. - - declaration: QuantumAlg.TrigonometricQuantumSignalProcessing.main - informal: Characterizes trigonometric single-qubit QSP transforms in the YZZYZ form. - - declaration: QuantumAlg.QuantumKernel.main - informal: Proves that fidelity quantum-kernel Gram matrices are positive semidefinite. - - declaration: QuantumAlg.QuantumKernelAdvantage.separation - informal: States the conditional quantum-kernel learning separation from discrete-log hardness. - - declaration: QuantumAlg.fullControllability_hasBarrenPlateau - informal: >- - Exhibits an exponentially vanishing loss-variance family for a fully controllable - dynamical Lie algebra, conditional on the assumed Ragone second-moment variance law - (the Haar/Weingarten averaging identity Var = P_g(rho)*P_g(O)/dim, taken as a - hypothesis rather than derived from a circuit twirl). - - declaration: QuantumAlg.GroverSearch.main_success_probability - informal: Derives Grover search success probability from the amplitude-amplification closed form. - tags: - - quantum-computing - - quantum-algorithms - - quantum-signal-processing - - quantum-kernels - msc: - - "81P68" - - "81P45" - - "68Q12" - - slug: five-distance-sharp - title: Sharp Five-Distance and Sup-Norm Gap Theorems - summary: >- - Formalizes sharp gap theorems for Kronecker sequences on tori: the Euclidean - five-distance theorem on the 2-torus and the sup-norm gap bound, together with explicit - arithmetic witnesses realizing the extremal cases - five distinct nearest-neighbour - distances for the Euclidean and sup-norm metrics in dimension two, and nine distinct - sup-norm distances in dimension three (Haynes-Marklof-Ramirez), via exact modular - arithmetic on best simultaneous Diophantine approximations. - branch: number theory - entry_module: LeanPool.ThreeGap - authors: - - Vico Bonfioli - source: - title: "Higher dimensional gap theorems for the maximum metric" - authors: - - Alan Haynes - - Jens Marklof - - Ana Cristina Moreira Ramirez - arxiv: "2010.08842" - github_repo: ElVec1o/five-distance-sharp - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - ThreeGap.LinftyRecords3.nine_attained - main_results: - - declaration: ThreeGap.EuclideanRecords.sharp_attained - informal: >- - Exhibits a base point on the 2-torus with an irrational coordinate and an N whose - Euclidean nearest-neighbour distances realize exactly five distinct values, showing the - Euclidean five-distance bound is sharp. - source_ref: "A. Haynes and J. Marklof, A five distance theorem for Kronecker sequences, International Mathematics Research Notices 2022(24):19747-19789, doi:10.1093/imrn/rnab205, arXiv:2009.08444." - - declaration: ThreeGap.LinftyRecords.sharp_attained - informal: >- - Exhibits a base point on the 2-torus with an irrational coordinate and an N whose sup-norm - nearest-neighbour distances realize exactly five distinct values, showing the sup-norm gap - bound is sharp in dimension two. - - declaration: ThreeGap.LinftyRecords3.nine_attained - informal: >- - Exhibits a base point on the 3-torus with an irrational coordinate and an N whose sup-norm - nearest-neighbour distances realize at least nine distinct values, showing the sup-norm gap - bound is sharp in dimension three. - tags: - - number-theory - - kronecker-sequences - - three-gap - msc: - - "11J71" - - slug: misere-games - title: Misere combinatorial games - summary: Formalizes a combinatorial game theory library for misere play, including comparison modulo universes, p-free forms, outcome-stability, Property X, and concrete rulesets such as Hackenbush. - branch: combinatorics - entry_module: LeanPool.MisereGames - authors: - - Tomasz Maciosowski - - Alfie Davies - source: - title: misere-games - authors: - - Tomasz Maciosowski - - Alfie Davies - - Violeta Hernandez Palacios - - Yuyang Zhao - url: https://github.com/t4ccer/misere-games - github_repo: t4ccer/misere-games - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - MisereGames.Form.Promain.of_universe - main_results: - - declaration: MisereGames.Form.Promain.of_universe - informal: Shows that comparison modulo a universe is characterized by the maintenance-proviso test. - - declaration: MisereGames.IntegerInvertible.isPFree_of_propertyX - informal: Proves that Property X makes the sum of two short p-free games p-free in an outcome-stable, hereditary, integer-invertible monoid. - - declaration: MisereGames.Form.Preservation.endLikeSeedToUniverse_bijective - informal: Proves that universal closure gives a bijection from bounded end-like seeds to universes. - tags: - - combinatorial-games - - misere-play - - game-theory - msc: - - "91A46" - - "05C57" - - slug: frontiermath-open-hypergraphs - title: FrontierMath Ramsey Hypergraphs - summary: Formalizes lower bounds for the Brian-Larson hypergraph extremal function H(n), including a uniform 26/25 improvement over the recursive benchmark k_n for n ≥ 15 and a Lubell-frame asymptotic lower bound. - branch: combinatorics - entry_module: LeanPool.FrontierMathOpenHypergraphs - authors: - - Dean Cureton - source: - title: Choosing between incompatible ideals - authors: - - Will Brian - - Paul B. Larson - doi: "10.1016/j.ejc.2021.103349" - arxiv: "1908.10914" - github_repo: math-inc/FrontierMathOpen-Hypergraphs - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - HypergraphLowerBound.thm_main_H_lower_bound - main_results: - - declaration: HypergraphLowerBound.thm_main_H_lower_bound - informal: Proves H(n) ≥ 26 * k(n) / 25 for every n ≥ 15. - source_ref: "Epoch AI FrontierMath, A Ramsey-style Problem on Hypergraphs." - - declaration: HypergraphLowerBound.uniform_26_25 - informal: Proves the uniform recursive construction bound A(n) ≥ 26 * k(n) / 25 for every n ≥ 15. - - declaration: HypergraphLowerBound.frame_recurrence - informal: The substitution-frame recurrence that combines block witnesses and a valid support frame into a larger witness. - - declaration: HypergraphLowerBound.lubell_is_frame - informal: Proves that the Lubell support family forms a valid frame for every arity t ≥ 2. - - declaration: HypergraphLowerBound.fixed_t_bound - informal: Derives a fixed-t lower bound for the exact witness supremum from Lubell frames. - - declaration: HypergraphLowerBound.asymptotic_lower_bound - informal: Proves the asymptotic lower bound liminf H(n) / k(n) ≥ 2 * log 2. - - declaration: HypergraphLowerBound.thm_asymptotic - informal: States the asymptotic consequence in terms of H(n) divided by n log n. - tags: - - combinatorics - - hypergraphs - - ramsey-theory - - extremal-combinatorics - - frontiermath - msc: - - "05C65" - - "05D10" - - "03E02" - - slug: fundamental-inequality - title: The Fundamental Inequality of Valued Fields - summary: Defines the ramification index and residue degree of an extension of valued fields and proves the fundamental inequality e * f ≤ finrank K L for finite extensions, via the valuation independence of products of coset representatives and residue-field basis lifts. - branch: number theory - entry_module: LeanPool.FundamentalInequality - authors: - - Alessandro Linzi - source: - title: "Valued Fields" - authors: - - Franz-Viktor Kuhlmann - url: "https://github.com/linzialessandro/FundamentalInequality" - github_repo: linzialessandro/FundamentalInequality - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - Valuation.fundamentalInequality - - Valuation.ramificationIndex - main_results: - - declaration: Valuation.ramificationIndex - informal: Defines the ramification index e(w/v) as the relative index of the value group of the restricted valuation inside the value group of the extension. - - declaration: Valuation.residueDegree - informal: Defines the residue degree f(w/v) as the rank of the residue field extension. - - declaration: Valuation.valuation_independence - informal: Elements with valuations in distinct cosets and residues that are linearly independent give products that are linearly independent over the base field. - - declaration: Valuation.fundamentalInequality - informal: The fundamental inequality e(w/v) * f(w/v) ≤ finrank K L for finite extensions of valued fields. - source_ref: "F.-V. Kuhlmann, Valued Fields, Chapter 4, §4.4." - tags: - - valued-fields - - number-theory - - valuation-theory - msc: - - "12J20" - - slug: pebbling-hypercube - title: Optimal Pebbling Number of the Hypercube - summary: >- - Formalizes that the optimal pebbling number of the n-dimensional hypercube - Q_n is Theta((4/3)^n). Builds a weight-function lower bound of (4/3)^n and - a probabilistic recursive upper bound C * (4/3)^n with an explicit constant, - combining them into a two-sided big-Theta witness. - branch: combinatorics - entry_module: LeanPool.PebblingLean - authors: - - Lior Pachter - source: - title: "Optimal pebbling number of the hypercube" - authors: - - Lior Pachter - url: "https://github.com/pachterlab/P_2026_2" - github_repo: pachterlab/P_2026_2 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - PebblingLean.Hypercube.Paper.optimalPebblingNumber_theta - main_results: - - declaration: PebblingLean.Hypercube.Paper.optimalPebblingNumber_theta - informal: >- - The optimal pebbling number of Q_n has order (4/3)^n: there is a positive - constant C with (4/3)^n <= o(Q_n) <= C * (4/3)^n for all n. - - declaration: PebblingLean.Hypercube.Paper.lower_bound - informal: >- - The optimal pebbling number of Q_n is at least (4/3)^n, proved by a - weight-function argument summing one-pebble weights over the cube. - - declaration: PebblingLean.Hypercube.Paper.upper_bound - informal: >- - The optimal pebbling number of Q_n is at most C * (4/3)^n for an explicit - constant C, via a probabilistic recursive splitting construction. - tags: - - combinatorics - - pebbling - - hypercube - msc: - - "05C57" - - slug: krafftsieve - title: "A Conditional Sieve Criterion for Twin Primes via Krafft Geometry" - summary: Introduces an original conditional sieve criterion for the Twin Prime Conjecture. Using a centered modular alignment (termed the Krafft alignment), proves that if the Rayleigh-Ritz quotient over the orthogonal unit sphere satisfies muMin < 1, then a twin prime pair must exist in the corresponding interval. - branch: analytic number theory - entry_module: LeanPool.KrafftSieve - authors: - - Fernando Portela - source: - doi: "10.5281/zenodo.19763833" - github_repo: ElNando888/KrafftSieve - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - KrafftSieve.mu_min_lt_one_implies_tpc - main_results: - - declaration: KrafftSieve.krafft_sieve_guarantee_with_mu_min - informal: Proves that muMin n < 1 is a sufficient condition for the existence of a twin prime pair in the corresponding interval. - - declaration: KrafftSieve.mu_min_lt_one_implies_tpc - informal: If there are infinitely many intervals where the optimal multidimensional weight achieves a ratio strictly less than 1, then there are infinitely many twin primes. This is a conditional reduction; the hypothesis muMin n < 1 is not established here for any n. - tags: - - analytic-number-theory - - sieve-theory - - twin-primes - - optimization - msc: - - "11N05" - - "11N35" - - slug: singular-moduli - title: Quadratic-order prime classification - summary: Formalizes foundational algebra for quadratic orders associated to discriminants, including their power basis, norm and discriminant identities, odd-prime splitting/ramification/inertness criteria via Legendre symbols, and root-counting results for quadratic congruences modulo odd prime powers. - branch: algebraic number theory - entry_module: LeanPool.SingularModuli - authors: - - Caleb L. Geiger - source: - title: Singular Moduli and the Ideal Class Group - authors: - - C. Geiger - github_repo: ElodinLaarz/lean-thesis - url: https://github.com/ElodinLaarz/lean-thesis - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - QuadraticOrder.prime_inert_iff - - QuadraticOrder.prime_split_iff - main_results: - - declaration: QuadraticOrder.prime_inert_iff - informal: Proves that for an odd rational prime p and a valid discriminant d, the ideal (p) is maximal in the quadratic order O_d exactly when the Legendre symbol (d/p) is -1. - source_ref: "C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.2, Proposition 3.2.1." - - declaration: QuadraticOrder.prime_ramified_iff - informal: Proves that for an odd rational prime p and a valid discriminant d, the ideal (p) is not radical in O_d exactly when p divides d. - source_ref: "C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.2, Proposition 3.2.1." - - declaration: QuadraticOrder.prime_split_iff - informal: Proves that for an odd rational prime p and a valid discriminant d, the ideal (p) is radical but not maximal exactly when the Legendre symbol (d/p) is 1. - source_ref: "C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.2, Proposition 3.2.1." - - declaration: QuadraticOrder.cardSqrts_prime_pow_coprime - informal: Counts roots of x^2 = c modulo p^n for odd prime p and p not dividing c, giving two roots when the Legendre symbol is 1 and none otherwise. - source_ref: "C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.1." - - declaration: QuadraticOrder.cardSqrts_prime_pow_even_val - informal: Counts roots of x^2 = p^(2r) u modulo p^n for odd prime p, p not dividing u, and 2r < n, reducing to the coprime case and multiplying by p^r. - source_ref: "C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.1." - - declaration: QuadraticOrder.tau_sub_tauConj_sq_of_valid_disc - informal: Proves the discriminant identity (tau - tauConj)^2 = d for valid quadratic-order discriminants d congruent to 0 or 1 modulo 4. - source_ref: "C. Geiger, Singular Moduli and the Ideal Class Group, Section 2." - tags: - - algebraic-number-theory - - quadratic-orders - - legendre-symbol - msc: - - "11R11" - - "11R29" - - "11A15" - - slug: agree-to-disagree - title: "Aumann's Agreement Theorem" - summary: >- - Formalizes Aumann's agreement theorem for information partitions and a - common-prior setting, together with a p-belief variant bounding posterior - disagreement by 2 * (1 - p). - branch: probability theory - entry_module: LeanPool.AgreeToDisagree - authors: - - Ruiz Chen - - Ben Eltschig - - Ken Ono - - Jujian Zhang - source: - title: "We Can't Agree to Disagree, Formally: Aumann's Theorem and Assumption Accounting in Lean" - url: "https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6837298" - github_repo: AxiomMath/AgreeToDisagree - license: MIT - status: verified - provenance: AI - main_declarations: - - AgreeToDisagree.agreeToDisagree - - AgreeToDisagree.agreeToDisagree_beliefs - main_results: - - declaration: AgreeToDisagree.agreeToDisagree - informal: >- - If it is common knowledge at a state that two agents assign fixed - posterior probabilities to an event, and the common-knowledge cell has - equal posterior under their priors, then the two posterior - probabilities are equal. - - declaration: AgreeToDisagree.agreeToDisagree' - informal: >- - Generalizes the agreement theorem to a family of agents indexed by an - arbitrary type. - - declaration: AgreeToDisagree.agreeToDisagree_beliefs - informal: >- - Proves that common p-belief of fixed posteriors bounds the difference - between any two agents' real-valued posteriors by 2 * (1 - p). - tags: - - probability - - game-theory - - epistemic-logic - msc: - - "60A10" - - "91A40" - - slug: kalton-roberts - title: Halving the Kalton-Roberts upper bound - summary: Formalizes the paper-style Boolean-algebra theorem that every approximately additive real-valued set function is uniformly close to a finitely additive signed measure, proving the certified Kalton-Roberts constant bound below 19.838. - branch: analysis - entry_module: LeanPool.KaltonRoberts - authors: - - Ho Boon Suan - source: - title: Halving the original Kalton-Roberts upper bound for nearly additive set functions - authors: - - Ho Boon Suan - - Tomasz Kania - arxiv: "2606.06807" - url: https://github.com/boonsuan/KaltonRoberts - github_repo: boonsuan/KaltonRoberts - license: MIT - status: verified - provenance: AI - main_declarations: - - KaltonRoberts.KR_constant_lt - main_results: - - declaration: KaltonRoberts.KR_constant_lt - informal: Proves the headline certified bound that the Kalton-Roberts constant is strictly less than 9919/500, i.e. 19.838. - - declaration: KaltonRoberts.KR_constant_le_C₂ - informal: Proves the exact rational upper bound C₂ = 694198146664396294486127753 / 34994834677886019996000000 for the Kalton-Roberts constant. - - declaration: KaltonRoberts.set_algebra_bound_C₂ - informal: States the paper-style set-algebra formulation, where every approximately additive function on a Boolean subalgebra of sets is within C₂ times the error of a finitely additive signed measure. - tags: - - functional-analysis - - finitely-additive-measures - - kalton-roberts - msc: - - "46B20" - - "28A12" - - "05C35" - - slug: chudnovsky - title: A Detailed Proof of the Chudnovsky Formula - summary: >- - Formalizes the Chudnovsky brothers' rapidly convergent series for 1/π, - proving Mathlib's `chudnovskySum = π⁻¹`. Follows Milla's elementary - complex-analytic proof: Weierstrass and Eisenstein theory, Liouville and - parallelogram-residue calculus, the Kummer hypergeometric bridge and the - main modular identity relating the j-invariant to the lattice periods, - Baker's division-value machinery, and the singular-moduli inputs at the CM - point τ = (1 + √-163)/2 — integrality of j(τ) via modular polynomials and - Kronecker's congruence, and rationality of the Weierstrass s₂ invariant - via Masser's Appendix A.1. - branch: number theory - entry_module: LeanPool.Chudnovsky - authors: - - Xuanji Li - source: - title: A Detailed Proof of the Chudnovsky Formula - authors: - - Lorenz Milla - arxiv: "1809.00533" - github_repo: ldct/lean-eval-chudnovsky - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Chudnovsky.chudnovskySum_eq_pi_inv - main_results: - - declaration: Chudnovsky.chudnovskySum_eq_pi_inv - informal: >- - Proves the Chudnovsky series identity, i.e. Mathlib's - `chudnovskySum = π⁻¹`: the series - (12 / 640320^(3/2)) · ∑ₙ (-1)ⁿ (6n)! (545140134 n + 13591409) - / ((3n)! (n!)³ 640320^(3n)) sums to 1/π. - tags: - - number-theory - - pi - - chudnovsky - - modular-forms - - complex-multiplication - msc: - - "11Y60" - - "11F03" - - "33C05" - - slug: rellich-kondrachov - title: Rellich–Kondrachov compact embedding theorem - summary: >- - Formalizes the Rellich–Kondrachov compact embedding theorem: the Sobolev embedding H¹ → L² - is a compact operator on compact Riemannian manifolds. The proof builds the Euclidean core - (compactness of the H¹ → L² inclusion on a fixed compact support via the Fréchet–Kolmogorov - criterion and the H¹ translation estimate ‖τₐu − u‖₂ ≤ ‖a‖ · ‖∇u‖₂), glues chart - contributions over a finite atlas, and specializes to the Levi-Civita Riemannian volume - measure. - branch: analysis - entry_module: LeanPool.RellichKondrachov - authors: - - Adam Benenson - source: - title: "Rellich–Kondrachov compact Sobolev embedding theorem" - authors: - - Adam Benenson - url: "https://github.com/abenenson/rellich-kondrachov" - github_repo: abenenson/rellich-kondrachov - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - RellichKondrachov.Geometry.Manifold.Sobolev.exists_riemannianFiniteChartData - main_results: - - declaration: RellichKondrachov.Geometry.Manifold.Sobolev.RiemannianFiniteChartData.isCompactOperator_h1ToL2_riemannianVolume - informal: >- - On a compact Riemannian manifold, the Sobolev inclusion H¹ → L² for the Levi-Civita - volume measure is a compact operator. - - declaration: RellichKondrachov.Analysis.FunctionalSpaces.Sobolev.Euclidean.isCompactOperator_h1OnToL2 - informal: >- - Euclidean Rellich–Kondrachov compactness: on a fixed compact support K, the H¹ → L² - inclusion restricted to functions supported in K is a compact operator. - - declaration: RellichKondrachov.Geometry.Manifold.Sobolev.exists_riemannianFiniteChartData - informal: >- - On a compact Riemannian manifold there exists finite chart data equipped with explicit - Lipschitz neighborhoods and a subordinate partition of unity, the input to the manifold - Rellich glue. - tags: - - analysis - - pde - - sobolev-embedding - msc: - - "46E35" - - slug: semicircle-catalan - title: Genus-Zero Pairings and Catalan Numbers - summary: >- - Formalizes the finite combinatorics behind the genus-zero term of the - Wigner semicircle law: a pairing of Fin (2n), viewed as a fixed-point-free - involution, has genus zero (total cycle count of the rotation product - maximal) exactly when it is noncrossing, and these genus-zero pairings are - counted by the Catalan number Cₙ. The Catalan recurrence is obtained from - an explicit equivalence decomposing a noncrossing pairing by the partner of - vertex 0 into independent inside and outside pairings. - branch: combinatorics - entry_module: LeanPool.SemicircleCheck - authors: - - Wondermonger-daydreaming - source: - title: "semicircle-catalan: genus-zero pairings and the Catalan moments of the Wigner semicircle law" - authors: - - Wondermonger-daydreaming - url: "https://github.com/Wondermonger-daydreaming/semicircle-catalan" - github_repo: Wondermonger-daydreaming/semicircle-catalan - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - card_noncrossingPairing_eq_catalan - - Pairing.genus_zero_count - main_results: - - declaration: card_noncrossingPairing_eq_catalan - informal: >- - The number of noncrossing pairings on 2n points equals the Catalan - number catalan n. - - declaration: Pairing.genus_zero_count - informal: >- - The number of genus-zero pairings of Fin (2n) equals the Catalan - number catalan n. - - declaration: Pairing.genus_zero_iff_noncrossing - informal: >- - A pairing has genus zero if and only if it is noncrossing. - tags: - - combinatorics - - catalan-numbers - - noncrossing-partitions - msc: - - "05A15" - - "05A18" - - slug: kuramoto - title: Finite-N Kuramoto Synchronization - summary: >- - Formalizes the finite-N Kuramoto model of coupled phase oscillators: the order - parameter and its unit bound, the potential/force gradient identities, smoothness - and local Lipschitz continuity of the vector field with ODE existence and - uniqueness, Lyapunov descent along trajectories, pairwise and semicircle - contraction estimates, Hebbian phase/weight dynamics, and full convergence to - synchrony under all-to-all coupling from initial phases in an open semicircle. - branch: dynamical systems - entry_module: LeanPool.Kuramoto - authors: - - Ben Cassie - source: - title: "Finite-N Kuramoto synchronization: order-parameter bounds and Lyapunov descent" - authors: - - Ben Cassie - url: "https://github.com/velvetmonkey/kuramoto-lean" - github_repo: velvetmonkey/kuramoto-lean - license: MIT - status: verified - provenance: AI - main_declarations: - - kuramotoR_norm_le_one - - weighted_lyapunov_descent - main_results: - - declaration: kuramotoR_norm_le_one - informal: >- - The Kuramoto order parameter has norm at most one for any phase configuration. - - declaration: weighted_lyapunov_descent - informal: >- - The weighted Kuramoto potential is non-increasing along the gradient flow, since - the sum of force times potential-derivative over all oscillators is at most zero. - - declaration: allToAll_convergence_to_synchrony - informal: >- - Under all-to-all coupling with positive strength and initial phases in an open - semicircle, the order parameter norm tends to one, i.e. the oscillators synchronize. - tags: - - dynamical-systems - - synchronization - - kuramoto - msc: - - "34D06" - - slug: min-modulus-unique-multiset-sum - title: Minimum modulus for the unique multiset-sum problem - summary: >- - Formalizes the exact minimum modulus - `N_min(n) = 2^n - 2^{floor(log_2 n)}` for the unique multiset-sum - problem on the super-increasing residue set `{2^i - 1 : 0 <= i < n}`, - and proves the sharp elementary-abelian-2-group lower bound `k >= n - 1` - for unique multiset sums in `(Z/2Z)^k`. - branch: additive combinatorics - entry_module: LeanPool.MinModulusUniqueMultisetSum - authors: - - José A. R. Fonollosa - source: - title: Minimum modulus for the unique multiset-sum problem, with an application to computing the permanent - authors: - - José A. R. Fonollosa - arxiv: "2607.08366" - github_repo: jarfo/min-modulus - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - MinModulus.nmin_eq - - MinModulus.elementaryAbelianTwoGroups_optimal - main_results: - - declaration: MinModulus.nmin_eq - informal: >- - Proves that among moduli `N >= 2` the least one making the - super-increasing residue set `{2^i - 1 : 0 <= i < n}` valid for the - unique multiset-sum problem is `2^n - 2^{floor(log_2 n)}`, for every - `n >= 2` (stated as `IsLeast`). - - declaration: MinModulus.elementaryAbelianTwoGroups_optimal - informal: >- - Proves that if `g_0, ..., g_{n-1}` in `(Z/2Z)^k` have unique - multiset sums, then `k >= n - 1`: an elementary abelian 2-group - admitting such a family has order at least `2^(n - 1)`. - tags: - - additive-combinatorics - - multiset-sums - - finite-groups - - permanent - msc: - - "11B75" - - "05D99" - - slug: lehmer-e10 - title: Lehmer's Polynomial and the E10 Coxeter Element - summary: >- - Proves that Lehmer's degree-10 polynomial — the integer polynomial of - smallest known Mahler measure above 1 — is irreducible over the - integers, and that it is the characteristic polynomial of a Coxeter - element of the E10 Weyl group (the rank-10 instance of McMullen). - Against Mathlib's Mahler-measure API it computes M(L) exactly as the - Salem root with the unconditional bounds 1 < M(L) < 7/5, states - Lehmer's conjecture as a defined Prop (never assumed) with the strong - form implying the gap form, and contrasts the E8 Coxeter element - (order exactly 30) with the E10 element (infinite order in GL(10,Z)): - the E-series crosses from torsion to a Salem matrix at rank 10, and - the crossing value is Lehmer's number. - branch: number theory - entry_module: LeanPool.LehmerE10 - authors: - - Dillon Ryan - source: - github_repo: dillon-11/lehmer-E10 - url: https://github.com/dillon-11/lehmer-E10 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - LehmerE10.main_theorem - main_results: - - declaration: LehmerE10.main_theorem - informal: >- - Lehmer's polynomial L(x) = x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - - x^3 + x + 1 is irreducible over the integers, and the - characteristic polynomial of a Coxeter element of the E10 Weyl - group equals L. - - declaration: LehmerE10.lehmer_mahlerMeasure - informal: >- - The Mahler measure of L equals its Salem root mu, with the - unconditional bounds 1 < M(L) < 7/5. - tags: - - number-theory - - mahler-measure - - salem-numbers - - coxeter-groups - msc: - - "11R06" - - "11C08" - - "20F55" - - slug: sabidussi - title: "Sabidussi's compatibility conjecture for Eulerian multigraphs" - summary: >- - Formalizes Sabidussi's compatibility conjecture for finite labelled endpoint multigraphs - with loops and parallel edges, where a loop contributes two half-edge incidences to the - degree of its vertex. Given an Euler tour and minimum degree at least four, the main theorem - constructs a partition of all edges into nonempty, edge-connected, 2-regular ordinary - circuits such that consecutive edges of the Euler tour never lie in the same circuit. The - proof runs a four-colour (F_2^2) algebra over the cyclic transition word, uses an odd - balancing parity lemma to choose compatible local frames, and transports the resulting - colouring back to a compatible ordinary circuit decomposition. - branch: graph theory - entry_module: LeanPool.Sabidussi - authors: - - Nikolay Ulyanov - source: - title: "Graph Puzzles III.1: A proof of Sabidussi's compatibility conjecture" - authors: - - Nikolay Ulyanov - url: "https://github.com/gexahedron/sabidussi-lean/blob/032e640df0cc41d4d3d217f7c68628ab19cb3767/sabidussi_proof.pdf" - github_repo: gexahedron/sabidussi-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Sabidussi.LoopMultigraph.loop_sabidussi_compatibility_ordinary - main_results: - - declaration: Sabidussi.LoopMultigraph.loop_sabidussi_compatibility_ordinary - informal: >- - For any finite endpoint multigraph with an Euler tour and minimum degree at least four, - the edges partition into ordinary, nonempty, connected, 2-regular circuits that are - compatible with the tour: no circuit contains two consecutive tour edges. - - declaration: Sabidussi.LoopMultigraph.loop_sabidussi_compatibility - informal: >- - The same compatibility statement phrased with the intrinsic even-edge-set circuit - decomposition, from which the ordinary-circuit form is derived. - tags: - - graph-theory - - eulerian-graphs - - circuit-decomposition - - sabidussi - msc: - - "05C45" - - slug: three-gap-theorem - title: The Three-Gap (Steinhaus) Theorem - summary: >- - Formalizes the three-gap (Steinhaus, three-distance) theorem: for every real `a` and - every `N`, the `N`-point orbit `{i·a mod 1 : i < N}` partitions `[0,1)` into gaps taking - at most three distinct lengths, and when exactly three occur the largest is the sum of - the other two. The proof works in `[0,1)` via `Int.fract` (not `AddCircle`) through the - two first-return times `η⁺` and `η⁻`, and is uniform in the rotation number, covering - rational and irrational `a` with no case split. - branch: number theory - entry_module: LeanPool.SteinhausThreeGap - authors: - - Dirk Kunert - source: - title: "The Three Gap Theorem (Steinhaus Conjecture)" - authors: - - Tony van Ravenstein - doi: "10.1017/S1446788700031062" - github_repo: dkunert/three-gap-theorem-lean - license: MIT - status: verified - provenance: AI - main_declarations: - - SteinhausThreeGap.three_gap_card_le_three - main_results: - - declaration: SteinhausThreeGap.three_gap_card_le_three - informal: >- - For every real `a` and natural `N`, the gaps of the orbit `{i·a mod 1 : i < N}` take - at most three distinct lengths. - - declaration: SteinhausThreeGap.three_gap_lengths_eq - informal: >- - When the gaps take exactly three distinct lengths, those lengths are precisely the two - first returns `η⁺`, `η⁻`, and their sum `η⁺ + η⁻`; in particular the largest gap is the - sum of the other two. - tags: - - number-theory - - three-gap-theorem - - equidistribution - msc: - - "11K06" - - slug: stationary-harmonic-maps - title: Monotonicity Formula for Stationary Harmonic Maps - summary: >- - Formalizes the monotonicity formula for stationary `W^{1,2}_{loc}` maps with Euclidean - target. The proof is target independent: it uses only the displayed weak gradient and a - vanishing domain first variation, so no target-manifold structure enters the argument. - From a bundled stationary Sobolev witness `StationaryW12LocMap u Omega` it proves, for - `0 < s <= r < R0`, that the normalized energy `Theta(Du, a, rho) = rho^(2-n) * integral - over B(a, rho) of |Du|^2` is monotone: `weakTheta Du a s <= weakTheta Du a r`, together - with the exact increment formula `weakTheta Du a r - weakTheta Du a s = weakMonotonicityRhs - Du a s r`. The Sobolev layer is project-defined from Mathlib's `MemLp`, weak-derivative - identities, Bochner integration, and Euclidean measure theory, following the De - Giorgi-Nash-Moser engineering choice; the internal proof routes through a radial - vector-field first variation, a coarea/Radon-Nikodym radius representation, smooth radial - cutoffs, and an absolute-continuity boundary step. - branch: analysis - entry_module: LeanPool.LeanStationaryHarmonicMaps - authors: - - Wei Wang - source: - title: "Theorems on Regularity and Singularity of Energy Minimizing Maps" - authors: - - Leon Simon - doi: "10.1007/978-3-0348-9193-6" - github_repo: BrookWW/LeanStationaryHarmonicMaps - license: MIT - status: verified - provenance: mix - main_declarations: - - LeanStationaryHarmonicMaps.StationaryHarmonicMap.StationaryW12LocMap - main_results: - - declaration: LeanStationaryHarmonicMaps.StationaryHarmonicMap.StationaryW12LocMap - informal: >- - The public input package: a domain-variation stationary `W^{1,2}_{loc}` Sobolev map - bundling a chosen weak gradient (`W12LocMapWitness`) with the vanishing domain - first-variation identity. This is the object users construct to invoke the - monotonicity theorem. - - declaration: LeanStationaryHarmonicMaps.StationaryHarmonicMap.stationaryW12LocMonotonicity_euclidean - informal: >- - Witness-style monotonicity: from a stationary `W^{1,2}_{loc}` map package, the - normalized weak energy `weakTheta` is monotone in the radius, `weakTheta Du a s <= - weakTheta Du a r` for `0 < s <= r < R0`. - - declaration: LeanStationaryHarmonicMaps.StationaryHarmonicMap.stationaryW12LocMonotonicityFormula_euclidean - informal: >- - Witness-style monotonicity formula: the increment of `weakTheta` equals the - nonnegative annular radial-energy term, `weakTheta Du a r - weakTheta Du a s = - weakMonotonicityRhs Du a s r` for `0 < s < r < R0`. - - declaration: LeanStationaryHarmonicMaps.StationaryHarmonicMap.stationarySobolevMonotonicity_euclidean - informal: >- - Componentwise convenience wrapper: the same monotonicity inequality stated directly - from the local `L^2`, measurability, weak-gradient, and first-variation hypotheses, - without first constructing the witness package. - tags: - - analysis - - pde - - harmonic-maps - - monotonicity-formula - - sobolev - msc: - - "58E20" - - "35J50" - - slug: gkp-density-one - title: Density-One GKP Divisibility and Its Carry-Language Characterization - summary: >- - Proves that the exponents k satisfying the Graham--Knuth--Patashnik divisibility - condition have natural density one. The proof gives an exact all-depth modular sieve, - quantitative bounds for possible failures in initial intervals, and a density-zero - limit for the failure proportion. It also develops the Kummer digit-sum bridge, the - binary-popcount reduction, and a verified ternary carry automaton. The universal GKP - conjecture remains open because the possible density-zero failure set may be nonempty. - branch: number theory - entry_module: LeanPool.GKPCarry - authors: - - Egor Lyfar - source: - title: "Concrete Mathematics: A Foundation for Computer Science, Second Edition" - authors: - - Ronald L. Graham - - Donald E. Knuth - - Oren Patashnik - url: https://cs.stanford.edu/~knuth/gkp.html - github_repo: lyfar/gkp-carry-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - GKPCarry.tendsto_gkpSuccessProportion_one - main_results: - - declaration: GKPCarry.tendsto_gkpSuccessProportion_one - informal: >- - The proportion of exponents k below N for which nine divides the central binomial - coefficient indexed by 2^k tends to one as N tends to infinity. - - declaration: GKPCarry.card_gkpFailureExponentsBelow_le - informal: >- - At each ternary depth n+3, the number of possible failures below N is at most - (N divided by 2*3^(n+2), plus one) times (n+9)*2^n. - - declaration: GKPCarry.gkp_exact_modular_sieve - informal: >- - At depth n+3, exactly (n+9)*2^n of the 2*3^(n+2) exponent classes remain - uncertified; every other class gives divisibility by nine and avoids the deficient - carry language for all exponents in that class. - - declaration: GKPCarry.gkpConjecture_iff_badCarryLanguageExclusion - informal: >- - The GKP conjecture is equivalent to the assertion that every nonexceptional power - of two avoids the explicitly defined deficient-carry regular language. This is an - exact reformulation and does not prove the remaining avoidance assertion. - - declaration: GKPCarry.nine_dvd_centralBinom_iff_two_le_ternaryDoubleCarryCount - informal: >- - For every natural number n, its central binomial coefficient is divisible by nine - exactly when doubling the canonical ternary expansion of n creates at least two - carries. - - declaration: GKPCarry.badCarryLanguage_true_iff_shape - informal: >- - The deficient-carry automaton accepts exactly three concrete little-endian ternary - word shapes: all digits zero or one; a unique top digit two; or a unique two followed - by a higher zero with all other digits zero or one. - - declaration: GKPCarry.card_badCarryWords - informal: >- - Among all ternary words of length n+2, exactly (n+7) times 2^n create fewer than two - carries when doubled. - - declaration: GKPCarry.nine_dvd_centralBinom_four_pow_in_finite_range - informal: >- - For every m from 27 through 6560, nine divides the central binomial coefficient - indexed by 4^m; this is a bounded computational corollary, not the headline result. - tags: - - number-theory - - finite-automata - - asymptotic-density - msc: - - "11A63" - - "11B65" - - "68Q45" - - slug: moser-lattice-colorings - title: Geometric Four-Colorings of the Moser Lattice and Ring - summary: >- - Formalizes the two explicit parity colorings from Dúcz's 2026 note on the - Moser lattice. The development derives the exact quadratic norm formula, - proves that the integer-coordinate embedding is faithful, proves both - colorings proper and geometric on the entire lattice, and descends them - through localization by powers of three to the Moser ring. As a structural - corollary, every unit-distance graph realized entirely in the Moser ring is - four-colorable. The uniqueness classification of geometric four-colorings - and the chromatic number of the full Euclidean plane are not claimed. - branch: combinatorics - entry_module: LeanPool.MoserLatticeColorings - authors: - - Egor Lyfar - source: - title: "A note on geometric colorings of the Moser lattice" - authors: - - Ákos Dúcz - arxiv: "2606.12325" - github_repo: lyfar/moser-lattice-colorings-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.MoserLatticeColorings.MoserRing.both_colorings_are_geometric - main_results: - - declaration: LeanPool.MoserLatticeColorings.MoserRing.both_colorings_are_geometric - informal: >- - The two parity maps defined by Dúcz descend to proper four-colorings of - the full Moser ring, and equality of colors for each map depends only - on Euclidean distance. - source_ref: "Theorem 3.1 and the Moser-ring extension following it." - - declaration: LeanPool.MoserLatticeColorings.MoserRing.graph_colorable_four_of_moserRing_realization - informal: >- - Every abstract graph with a realization in the Moser ring that sends - each edge to a unit-distance pair is four-colorable. - source_ref: "Corollary of the geometric four-coloring of the Moser ring." - tags: - - combinatorics - - graph-coloring - - discrete-geometry - - unit-distance-graphs - - moser-lattice - msc: - - "05C15" - - slug: hadwiger-nelson-bounds - title: Known Bounds for the Hadwiger-Nelson Problem - summary: >- - Formalizes the currently known bounds 5 <= chi(R^2) <= 7 for the - chromatic number of the Euclidean plane. The upper bound constructs an - explicit Isbell-style seven-coloring. The lower bound checks Parts' - 481-vertex coloring certificate, develops a finite doubled - triangular-lattice forcing gadget, proves its exact Euclidean embedding, - and closes the four-color contradiction with a spindle construction. - The exact chromatic number remains open. - branch: graph theory and discrete geometry - entry_module: LeanPool.HadwigerNelsonBounds - authors: - - Egor Lyfar - source: - title: A human-verifiable proof of the chromatic number of the plane being at least 5 - authors: - - Jaan Parts - arxiv: "2010.12661" - url: https://www.erdosproblems.com/508 - github_repo: lyfar/hadwiger-nelson-bounds-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - HadwigerNelsonBounds.hadwiger_nelson_known_bounds - main_results: - - declaration: HadwigerNelsonBounds.hadwiger_nelson_known_bounds - informal: >- - The chromatic number of the unit-distance graph on the Euclidean - plane lies between five and seven, inclusive. - - declaration: HadwigerNelsonBounds.unitDistanceGraph_not_colorable_four - informal: >- - The unit-distance graph on the Euclidean plane admits no proper - coloring with four colors. - - declaration: HadwigerNelsonBounds.chromaticNumber_le_seven - informal: >- - An explicit Isbell-style coloring proves that the unit-distance graph - on the Euclidean plane is colorable with seven colors. - tags: - - graph-theory - - geometric-graph-theory - - graph-coloring - - hadwiger-nelson - msc: - - "05C15" - - "52C10" - - slug: distance-geometry - title: Euclidean Distance Geometry - summary: >- - Develops the basepoint-centered Gram approach to finite Euclidean distance geometry. - Proves both directions of Schoenberg's characterization, including a - rank-controlled factorization of positive-semidefinite matrices; proves that - three affinely independent centers in Euclidean three-space determine at most - two points with prescribed distances; and verifies the Cayley--Menger - determinant for segments and triangles, including the triangle-area identity - underlying Heron's formula. - branch: geometry - entry_module: LeanPool.DistanceGeometry - authors: - - Egor Lyfar - source: - title: Remarks to Maurice Fréchet's article Sur la définition axiomatique d'une classe d'espace distanciés vectoriellement applicable sur l'espace de Hilbert - authors: - - Isaac Jacob Schoenberg - doi: "10.2307/1968654" - github_repo: lyfar/distance-geometry-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - DistanceGeometry.schoenberg - - DistanceGeometry.trilateration_le_two - main_results: - - declaration: DistanceGeometry.schoenberg - informal: >- - For n at least one, a symmetric hollow n-by-n matrix is a squared-distance - matrix of points in k-dimensional Euclidean space exactly when its - basepoint-centered Gram matrix is positive semidefinite and has rank at most k. - - declaration: DistanceGeometry.encard_setOf_forall_dist_eq_le_two - informal: >- - In a finite-dimensional real inner-product affine space, prescribed - distances to centers whose affine span has codimension one admit at most - two solutions. - - declaration: DistanceGeometry.trilateration_le_two - informal: >- - Three affinely independent centers in three-dimensional Euclidean space - admit at most two points with any prescribed triple of distances. - - declaration: DistanceGeometry.cayleyMenger_det_heron - informal: >- - The Cayley--Menger determinant of three planar points equals minus sixteen - times the squared area of their triangle. - tags: - - distance-geometry - - euclidean-geometry - - linear-algebra - - cayley-menger - msc: - - "51K05" - - "52C99" - - "15A18" - - slug: odd-prime-valuation-distribution - title: Exact odd-prime distributions of central-binomial valuations - summary: >- - Formalizes the exact distribution of the p-adic valuation of the central binomial - coefficient binom(2n,n) over every complete block 0 <= n < p^k for every odd prime p. - The development proves Kummer correctness for padded digit words, constructs the - two-state doubling carry enumerator, identifies it with the arithmetic valuation - polynomial, derives its second-order recurrence, proves the rational bivariate - generating function, and obtains closed formulas for valuations zero and one. - branch: number theory - entry_module: LeanPool.OddPrimeValuationDistribution - authors: - - Egor Lyfar - source: - title: "Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen" - authors: - - Ernst Eduard Kummer - doi: "10.1515/crll.1852.44.93" - github_repo: lyfar/gkp-carry-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - OddPrimeValuationDistribution.oddPrimeValuationGeneratingSeries_identity - main_results: - - declaration: OddPrimeValuationDistribution.oddPrimeValuationGeneratingSeries_identity - informal: >- - For every odd prime p = 2h+1, proves the denominator-cleared identity for the - bivariate series whose coefficient of T^k X^r counts n < p^k with - nu_p(binomial(2n,n)) = r; equivalently the series is - (1-XT)/(1-(h+1)(1+X)T+pXT^2). - source_ref: "A. Straub, V. H. Moll, T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (2009), 31-42." - - declaration: OddPrimeValuationDistribution.oddPrimeValuationPolynomial_recurrence - informal: >- - Proves that the exact valuation polynomials satisfy - F_{k+2}(X)+pX F_k(X)=(h+1)(1+X)F_{k+1}(X), with F_0=1 and - F_1=(h+1)+hX. - - declaration: OddPrimeValuationDistribution.padicValNat_oddPrime_centralBinom_ofDigits_eq_carryCount - informal: >- - Proves Kummer correctness for every valid padded base-p word: the p-adic valuation - of its central binomial coefficient equals the number of carries produced by - doubling the word. - - declaration: OddPrimeValuationDistribution.card_oddPrimeBlock_valuation_one - informal: >- - Gives a closed formula for the number of n below p^(k+2) whose central binomial - coefficient has p-adic valuation exactly one. - tags: - - central-binomial-coefficients - - p-adic-valuations - - digit-carries - - generating-functions - msc: - - "11A63" - - "11B65" - - slug: erdos-97-convex-octagon - title: The Convex-Octagon Case of Erdős Problem 97 - summary: >- - Proves that among eight labelled points in the Euclidean plane in convex position, at least - one point has no four other points at a common distance. The development reduces a geometric - counterexample to a finite incidence model, classifies normalized row patterns up to symmetry, - proves geometric and algebraic obstructions, and excludes seven canonical first-row masks with - a kernel-audited exhaustive search over the remaining legal row masks and checked geometric - obstruction witnesses. This is the modern four-equidistant formulation; Erdős's 1946 paper - posed the stronger three-equidistant version. This is the convex-octagon case only; the - general Erdős Problem 97 remains open. - branch: discrete geometry - entry_module: LeanPool.Erdos97ConvexOctagon - authors: - - Egor Lyfar - source: - title: On Sets of Distances of n Points - authors: - - Paul Erdős - doi: "10.1080/00029890.1946.11991674" - url: "https://www.erdosproblems.com/97" - github_repo: lyfar/erdos-97-octagon-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Erdos97Octagon.erdos97_convex_octagon - main_results: - - declaration: Erdos97Octagon.erdos97_convex_octagon - informal: >- - Among eight labelled points in the Euclidean plane in convex position, some point has no - four other points at a common distance. - tags: - - discrete-geometry - - distance-geometry - - erdos-problems - - convexity - msc: - - "51K05" - - "52A10" - - slug: erdos132-convex-k3 - title: Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures - summary: >- - Proves that every finite cyclically strictly convex planar configuration - has a vertex incident to at most six edges from its three largest - distance classes. The proof selects a maximal-gap vertex and coordinated - majorants, closes all exceptional inner branches, performs the five-row - enumeration, and transports all thirteen canonical cover words to four - shared full-two-rung, anti-saturation, terminal-cage, and four-edge-cage - predicates. It also proves exceptional-count rigidity and an explicit - heptagon refutation of a tempting stronger majorant claim, with rational - regression witnesses. This settles the all-n k=3 instance of the paper's - suggested degree-2k convex bound; Morić--Pritchard proved the corresponding - top-three edge bound for sufficiently large n. It does not settle the - general-k suggestion or the general Erdős problem 132. - branch: discrete geometry - entry_module: LeanPool.Erdos132ConvexK3 - authors: - - Egor Lyfar - source: - title: On the Graph of Large Distances - authors: - - Paul Erdős - - László Lovász - - Katalin Vesztergombi - doi: "10.1007/BF02187746" - url: https://www.erdosproblems.com/132 - github_repo: lyfar/erdos-132-convex-k3-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.Erdos132ConvexK3.convex_top_three_degree_six - main_results: - - declaration: LeanPool.Erdos132ConvexK3.convex_top_three_degree_six - informal: >- - For every nonempty finite cyclically strictly convex planar - configuration with three strictly ordered largest squared-distance - classes d₁, d₂, and d₃, some vertex is incident to at most six pairs - whose squared distance is one of those three classes. - source_ref: >- - Erdős--Lovász--Vesztergombi, printed p. 542, suggest the stronger - convex bound degree at most 2k. Morić--Pritchard, "Counting large - distances in convex polygons" (arXiv:1103.0412), Theorem 1.4, prove - m₁ + m₂ + m₃ ≤ 3n for sufficiently large n, which gives the same - degree-six consequence in that range. The novelty here is the all-n - theorem: it removes the sufficiently-large-n restriction. ErLV89 - Theorem 2.7, printed p. 548, proves only degree at most 3k-1, which - specializes to eight. - - declaration: LeanPool.Erdos132ConvexK3.convex_top_three_draft_reduction_complete - informal: >- - Cyclic strict convexity and the three-largest-distance-class - hypothesis construct the full maximal-gap, thirteen-word reduction - package together with route-specific closure evidence, without any - remaining coordinated-majorant or geometric-adapter assumption. - source_ref: >- - New proof-producing assembly theorem in this project. It combines - the ErLV89 majorant framework with the project's maximal-gap tail, - terminal-color, five-row, and thirteen-word transport lemmas. - - declaration: LeanPool.Erdos132ConvexK3.global_thirteen_word_closure_complete - informal: >- - Every one of the thirteen canonical maximal-gap cover words - transports to its routed full-two-rung, anti-saturation, - terminal-cage, or four-edge-cage realization, and hence supplies - the corresponding degree bound. - source_ref: >- - New geometric transport theorem in this project. The thirteen-word - classification and its four-predicate closure interface are not - statements of ErLV89. - - declaration: LeanPool.Erdos132ConvexK3.concrete_word_closures - informal: >- - A thirteen-tag routing interface over four shared geometric - realization predicates maps B:3→2 and D:3→2 to - Row1B32WordRealization, B:3→1, BA, D:3→1, and DC to - OnePenultimateWordGeometry, DD to Row4DDWordRealization, and the - remaining six tags to FullTwoRungGeometry. Each inhabited routed - predicate supplies a vertex of degree at most six, with the - one-penultimate route proving the stronger bound five. - source_ref: >- - New derived variant contributed and proved in this Lean Pool - project. The thirteen-word classification and its closures are not - statements of ErLV89. - - declaration: LeanPool.Erdos132ConvexK3.fullTwoRung_realization_degree_le_six - informal: >- - Every raw full-two-rung realization has its displayed vertex of - degree at most six. A rigid normalization derives the canonical - coordinates, the diameter branch is impossible, the long branch is - counted from four raw boundary arcs, and the short branch contradicts - the exact two-rung identity and metric dichotomy. - - declaration: LeanPool.Erdos132ConvexK3.convex_k3_degree_six_of_reduction - informal: >- - This implication from the abstract reduction interface is retained - as the final assembly map. Its hypothesis is now discharged by - convex_top_three_draft_reduction_complete, yielding the unconditional - theorem convex_top_three_degree_six; ErLV89 itself proves only the - degree-eight bound obtained from 3k−1 when k=3. - source_ref: >- - Erdős--Lovász--Vesztergombi, printed p. 542: “Perhaps in the convex - case there always exists an x_i such that the degree of x_i is at - most 2k.” Alongside it, Theorem 2.7, printed p. 548, proves only: - “If S is the set of vertices of a convex polygon then the graph - G(S, k) has a point of degree at most 3k−1.” Thus degree six for - k=3 is the stronger suggested variant; the proved source bound is - eight, while this project's global closure proves degree six. - - declaration: LeanPool.Erdos132ConvexK3.coordinated_exceptional_count_rigidity - informal: >- - Every exceptional jointly minimal pair lies in case (1,2), (2,1), - or (2,2). In (1,2), every first substitute has at least one right - move while every second substitute has exactly two left moves and - zero right moves. In (2,1), every first substitute has exactly two - right moves and zero left moves while every second substitute has at - least one left move. Only in (2,2) are those exact substitute counts - proved on both sides. - - declaration: LeanPool.Erdos132ConvexK3.coordinated_exceptional_rank_and_exchange_obstruction - informal: >- - Assuming the three top distance classes and TopThreeAdjacent for - both starting edges z–x and t–u, each forced two-move chain in an - exceptional inner-move case has the rank ladder from the - third-largest through the second-largest to the largest distance, - while its proposed intermediate exchange edge remains nonterminal. - - declaration: LeanPool.Erdos132ConvexK3.heptagon_nondiameter_edge_is_majorant - informal: >- - In an explicit rational convex heptagon, a second-largest-distance - edge is already terminal under both permitted cover moves, refuting - the stronger claim that every non-diameter top-three edge admits a - longer cover. - - declaration: LeanPool.Erdos132ConvexK3.Witnesses.ninePoint_top_three - informal: >- - An explicit rational nine-point regression configuration realizes - squared-distance classes 855625, 838525, and 811165 as its three - largest classes. - - declaration: LeanPool.Erdos132ConvexK3.Witnesses.rationalHexagon_top_three - informal: >- - An explicit rational six-point regression configuration realizes the - three stated rational squared-distance values as its largest classes. - tags: - - discrete-geometry - - distance-graphs - - erdos-problems - - convexity - msc: - - "52C10" - - "05C12" - - slug: erdos132-n14 - title: A Conditional Fourteen-Point Case of Erdős Problem 132 - summary: >- - Formalizes the planar Hopf--Pannwitz diameter bound and proves the - fourteen-point low-multiplicity conclusion conditional on exactly two - remaining published interfaces: the known maximum-cardinality bound for - planar sets with at most six distances and the Szöllősi--Östergård - classification of thirteen-point planar six-distance sets. Internally - derives the exact failure profile and endpoint deletion, constructs the - three classified templates, eliminates the nonregular cases by - multiplicity, and excludes the regular tridecagon extension by explicit - second and fourth moments. The general Erdős problem remains open. - branch: discrete geometry - entry_module: LeanPool.Erdos132N14 - authors: - - Egor Lyfar - source: - title: Constructions of maximum few-distance sets in Euclidean spaces - authors: - - Ferenc Szöllősi - - Patric R. J. Östergård - doi: "10.37236/8565" - url: https://www.erdosproblems.com/132 - github_repo: lyfar/erdos-132-moment-obstruction-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.Erdos132N14.erdos132_for_fourteen_of_published_inputs - main_results: - - declaration: LeanPool.Erdos132N14.hopfPannwitzLowMultiplicityDistance14 - informal: >- - Every injectively labelled fourteen-point planar configuration has - a realized distance represented by at most fourteen unordered pairs; - the proof chooses that distance to be a diameter. - source_ref: >- - The planar diameter bound, conventionally associated with - Hopf--Pannwitz; Perles's clockwise-extreme endpoint proof. - - declaration: LeanPool.Erdos132N14.erdos132_for_fourteen_of_published_inputs - informal: >- - Assuming precisely the published planar few-distance cardinality - and Szöllősi--Östergård classification interfaces, every injectively - labelled fourteen-point planar configuration realizes at least two - distances whose unordered-pair multiplicities are at most fourteen. - source_ref: >- - Erdős Problem 132; Hopf--Pannwitz's planar diameter bound; Wei, - "A Proof of Erdős--Fishburn's Conjecture for g(6)=13," EJC 19(4) - P38 (2012), doi:10.37236/2917; Szöllősi--Östergård, EJC 27(1) - P1.23 (2020), Table 1 and Theorem 16, doi:10.37236/8565. - tags: - - discrete-geometry - - few-distance-sets - - erdos-problems - - planar-configurations - msc: - - "52C10" - - "05D99" - - slug: erdos346 - title: "Erdős Problem 346: Ratio Limit Forces the Golden Ratio" - summary: >- - Formalizes the limit-exists interpretation of Erdős Problem 346: for a - strictly increasing positive integer sequence with a uniform ratio gap - whose finite deletions are complete but whose infinite deletions are not, - if the consecutive quotient sequence converges to some real limit greater - than one, then that limit must equal the golden ratio and the quotient - sequence tends to it. - branch: number theory - entry_module: LeanPool.Erdos346 - authors: - - KitaKen1 - source: - title: "Erdős Problem #346" - authors: - - KitaKen1 - url: "https://www.erdosproblems.com/346" - github_repo: KitaKen1/erdos346-ratio-limit-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Erdos346.intended_problem_if_limit_exists - main_results: - - declaration: Erdos346.intended_problem_if_limit_exists - informal: >- - Existence of any limit greater than one for the quotient sequence forces - convergence to the golden ratio, the intended Erdős–Graham conclusion. - - declaration: Erdos346.ratio_limit_eq_goldenRatio - informal: >- - If the quotient sequence converges to a limit greater than one under the - completeness and incompleteness deletion hypotheses, that limit equals - the golden ratio. - - declaration: Erdos346.ratio_tendsto_goldenRatio_of_limit_exists - informal: >- - Under the same hypotheses, the quotient sequence converges to the golden - ratio itself. - tags: - - number-theory - - golden-ratio - - erdos-problems - msc: - - "11B05" - - "11J70" - - slug: channel-capacity - title: Uniqueness of Shannon Capacity-Achieving Priors - summary: >- - Formalizes uniqueness of the capacity-achieving input distribution for a - Markov-kernel channel in three aligned layers: a finite-alphabet theorem - (compactness, continuity, and strict concavity of mutual information on - Mathlib's weak topology for `ProbabilityMeasure`), a concrete discharged - theorem for compact-Polish channels with a common strictly positive - jointly continuous density, and a general measure-theoretic packaging that - isolates the abstract well-conditioning hypotheses, plus a counterexample - showing pairwise-distinct rows do not give an injective prior pushforward - (the non-degeneracy required for uniqueness). - branch: information theory - entry_module: LeanPool.ChannelCapacity - authors: - - Adam Benenson - source: - title: Elements of Information Theory - authors: - - Thomas M. Cover - - Joy A. Thomas - doi: "10.1002/047174882X" - github_repo: abenenson/channel-capacity - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - ChannelCapacity.exists_unique_capacity_achieving_prior_of_finite - main_results: - - declaration: ChannelCapacity.exists_unique_capacity_achieving_prior_of_finite - informal: >- - For a Markov kernel between finite alphabets (nonempty input) whose - rows are linearly independent (`Kernel.RowMatrixFullRank`), a - capacity-achieving input prior exists and is unique. - source_ref: "R. G. Gallager, Information Theory and Reliable Communication, Wiley, 1968, Section 4.5." - - declaration: ChannelCapacity.exists_unique_capacity_achieving_prior_discharged - informal: >- - For compact-Polish channels with a common strictly positive jointly - continuous density, given injectivity of the prior-pushforward map and - upper semicontinuity of mutual information (the latter carried as a - field of `Kernel.ContinuousPositiveDensity`, not derived), the - capacity-achieving prior is unique. - - declaration: ChannelCapacity.exists_unique_capacity_achieving_prior - informal: >- - General measure-theoretic packaging: under abstract well-conditioning - hypotheses, the capacity-achieving prior is unique. - tags: - - information-theory - - channel-capacity - - mutual-information - - kullback-leibler - - markov-kernel - msc: - - "94A17" - - "94A15" - - "60A10" - - slug: johnson-lindenstrauss-lean - title: The Johnson–Lindenstrauss Lemma and Quantized JL - summary: >- - Formalizes the Johnson–Lindenstrauss lemma end to end — a Gaussian random - projection preserves pairwise distances of a finite point set up to a - (1 ± ε) factor, so a distortion-ε embedding of any m-point set exists once - k is logarithmic in m — together with the quantized JL (QJL) one-bit sign - sketch: unbiasedness of the asymmetric 1-bit estimator and its sub-Gaussian - distortion bound. - branch: probability - entry_module: LeanPool.JohnsonLindenstraussLean - authors: - - claytomode - source: - title: Extensions of Lipschitz mappings into a Hilbert space - authors: - - William B. Johnson - - Joram Lindenstrauss - doi: "10.1090/conm/026/737400" - github_repo: claytomode/johnson-lindenstrauss-lean - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - JL.johnson_lindenstrauss - - JL.johnson_lindenstrauss_pointset - main_results: - - declaration: JL.johnson_lindenstrauss - informal: The abstract probabilistic-method union bound — if the summed per-pair distortion-failure probability is below one, a distortion-preserving map exists. - - declaration: JL.johnson_lindenstrauss_pointset - informal: The end-to-end Johnson–Lindenstrauss lemma — a Gaussian random projection into k dimensions embeds any m-point set with (1 ± ε) distortion once k is logarithmic in m. - source_ref: "S. Dasgupta, A. Gupta, An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structures and Algorithms 22 (2003), 60-65, doi:10.1002/rsa.10073." - - declaration: JL.qjlEstimator_unbiased - informal: >- - The QJL asymmetric 1-bit estimator is unbiased for the normalized - inner product between key and query. - source_ref: "A. Zandieh, M. Daliri, I. Han, QJL: 1-Bit Quantized JL Transform for KV Cache Quantization with Zero Overhead, AAAI 2025, arXiv:2406.03482." - - declaration: JL.qjlEstimator_concentration_exp - informal: >- - Sub-Gaussian distortion bound for the QJL 1-bit estimator: deviation - at least ε has probability at most 2·exp(-m·ε²/(π·‖q‖²)). - source_ref: "A. Zandieh, M. Daliri, I. Han, QJL: 1-Bit Quantized JL Transform for KV Cache Quantization with Zero Overhead, AAAI 2025, arXiv:2406.03482." - tags: - - dimensionality-reduction - - random-projection - - johnson-lindenstrauss - - gaussian - - quantization - msc: - - "68W20" - - "60G15" - - "68P05" - - slug: cut-and-project - title: Period Lengths of Rational Cut-and-Project Gap Sequences - summary: >- - Formalizes the closed-form minimal periods of one-dimensional rational - cut-and-project gap sequences at the level of the paper's residue model, plus a - fully geometric aperiodicity theorem for irrational slopes. For coprime α, β ∈ ℕ - and window parameter ω ≥ 0, write N = ⌊ωα⌋ + ⌊ωβ⌋ + 1 and D = α² + β². The - strip-to-residue reduction of the paper — sorting the hits of the length-N - progression twisted by the geometric multiplier −αβ⁻¹ (mod D) — is taken as the - definition of the gap sequence rather than derived from the lattice; for it, the - multiset convention has minimal period N/D when D ∣ N and N otherwise, and the - set convention (multiplicities flattened to {0,1}) has minimal period 1 when - N ≥ D and N otherwise. The irrational case is geometric from first principles: - for positive irrational slope and ω > 0, the accepted points of ℤ² sorted by the - physical projection have a gap sequence with no finite period, via injectivity - of the projection and Kronecker density of the internal image. - branch: number theory - entry_module: LeanPool.CutAndProject - authors: - - Dirk Kunert - source: - title: "Period lengths of one-dimensional rational cut-and-project sequences" - authors: - - Dirk Kunert - url: "https://github.com/dkunert/cut-and-project/blob/main/LaTeX/rational_cut_and_project_gap_periods.tex" - github_repo: dkunert/cut-and-project - license: MIT - status: verified - provenance: AI - main_declarations: - - CutAndProject.main_theorem_geometric_concrete - main_results: - - declaration: CutAndProject.main_theorem_geometric_concrete - informal: >- - The gap sequence of the residue model of the rational cut-and-project - multiset (sorted hits of the length-N progression under the multiplier - −αβ⁻¹ mod D) has minimal period N/D if D divides N and N otherwise, where - N = ⌊ωα⌋ + ⌊ωβ⌋ + 1 and D = α² + β². - - declaration: CutAndProject.set_main_theorem_geometric_concrete - informal: >- - The set-convention residue-model gap sequence (multiplicities flattened to - {0,1}) has minimal period 1 if N ≥ D and N otherwise. - - declaration: CutAndProject.Irrational.prop_irrational - informal: >- - For positive irrational slope and window parameter ω > 0, the projected gap - sequence has no finite period. - tags: - - number-theory - - aperiodic-order - - cut-and-project - - quasicrystals - msc: - - "52C23" - - "11K06" - - slug: sundogcert - title: Sundog certificates - summary: >- - A referee-free, axiom-clean Lean development of coding-theory certificates: syndrome - decoding soundness and lossiness bounds for GF(2) parity-check schemes (with a - Reed–Solomon evaluation-dual variant), and a fully machine-checked correctness proof of - the Karp reduction chain 3SAT ≤ 3DM ≤ X3C ≤ bounded-weight syndrome decoding. Only the - logical iff of each reduction is formalized; polynomial-time bounds, NP classes, and - Cook–Levin are unformalized external inputs, so NP-hardness of decoding is conditional - on those standard facts. - branch: complexity theory - entry_module: LeanPool.Sundogcert - authors: - - Humiliati - source: - title: "Sundog certificates" - authors: - - Humiliati - url: "https://github.com/humiliati/sundogcert" - github_repo: humiliati/sundogcert - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Sundog.SATReductionMain.sat_iff_decodes - - Sundog.Certificate.accept_sound - main_results: - - declaration: Sundog.SATReductionMain.sat_iff_decodes - informal: >- - The end-to-end reduction correctness: a formula is satisfiable iff its image under the - composed 3SAT-to-decoding reduction is a yes-instance of bounded-weight syndrome decoding. - - declaration: Sundog.MatchingNPHard.threeDM_iff_X3C - informal: >- - The 3-dimensional matching instance has a perfect matching iff the reduced - exact-cover-by-3-sets instance has an exact cover, both directions machine-checked. - - declaration: Sundog.Certificate.accept_sound - informal: >- - Soundness of the cheap syndrome-decoding verifier when it accepts: an exhibited light - same-syndrome witness certifies the body is safe within the decoding radius. - - declaration: Sundog.Certificate.reject_sound - informal: >- - Soundness of the verifier when it rejects: a sound column-weight lower bound on the - minimum same-syndrome weight rules out every light witness, so the body is genuinely unsafe. - tags: - - complexity - - coding-theory - - np-hardness - msc: - - "68Q17" - - "94B35" - - slug: runge-kutta-order-conditions - title: Certified Runge-Kutta Order Conditions - summary: Certifies Runge-Kutta order conditions over planar rooted trees, including RK4 order four, Dormand-Prince order five, and Gauss-Legendre order six. - branch: numerical analysis - entry_module: LeanPool.RungeKuttaOrderConditions - authors: - - Carles Marín - source: - title: What Order a Method Knows - authors: - - Carles Marín - doi: 10.5281/zenodo.20787666 - github_repo: karlesmarin/runge-kutta-order-conditions-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - RungeKuttaOrderConditions.Butcher.gauss_order6_rat - main_results: - - declaration: RungeKuttaOrderConditions.Butcher.satisfiesOrderConditions_iff - informal: Reduces order-p Runge-Kutta conditions to a finite catalogue check over rooted trees. - - declaration: RungeKuttaOrderConditions.Butcher.rk4_order4 - informal: Certifies that classical RK4 satisfies all planar rooted-tree order conditions through order four. - - declaration: RungeKuttaOrderConditions.Butcher.rk4_not_ord5 - informal: Shows that RK4 fails a fifth-order condition, pinning its order exactly. - - declaration: RungeKuttaOrderConditions.Butcher.orderCond_node_perm - informal: Proves order conditions are invariant under permutations of a node's children. - - declaration: RungeKuttaOrderConditions.Butcher.dp_order5 - informal: Certifies the Dormand-Prince RK45 fifth-order tableau through order five. - - declaration: RungeKuttaOrderConditions.Butcher.gauss_order4 - informal: Certifies the Gauss-Legendre three-stage tableau through order four over Q(sqrt(15)). - - declaration: RungeKuttaOrderConditions.Butcher.gauss_order6 - informal: Certifies the denominator-cleared (360-scaled) integer Z(sqrt(15)) Gauss-Legendre tableau through order six, the scaled certificate feeding the homogeneity bridge. - - declaration: RungeKuttaOrderConditions.Butcher.gauss_order6_rat - informal: Certifies the Gauss-Legendre s=3 tableau through order six over Q(sqrt(15)), via the formalized homogeneity bridge from the 360-scaled integer certificate. - tags: - - numerical-analysis - - runge-kutta - - rooted-trees - - order-conditions - msc: - - "65L06" - - "65L05" - - "05C05" - - slug: erdos-moser-distinct-subset-sums - title: Leo Moser's finite distinct-subset-sums inequality - summary: >- - Formalizes the exact finite sum-of-squares inequality that Guy's 1982 - account states as Theorem 2 and attributes to Leo Moser: a finite set A - of natural numbers with distinct subset sums satisfies - 4^|A| - 1 <= 3 * sum(a^2). It then derives direct and square-root lower - bounds for the largest element of A. This does not claim the stronger - historical asymptotic bound attributed jointly to Erdős and Moser, and - Erdős Problem 1 remains open. - branch: additive combinatorics - entry_module: LeanPool.ErdosMoser - authors: - - Egor Lyfar - source: - title: Sets of Integers Whose Subsets Have Distinct Sums - authors: - - Richard K. Guy - doi: "10.1016/S0304-0208(08)73500-X" - url: https://www.erdosproblems.com/1 - github_repo: lyfar/erdos-moser-distinct-subset-sums-lean - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.ErdosMoser.leoMoserVarianceBound - main_results: - - declaration: LeanPool.ErdosMoser.leoMoserVarianceBound - informal: >- - If a finite set A of natural numbers has distinct subset sums, then - 4^|A| - 1 is at most three times the sum of the squares of the - elements of A. - source_ref: >- - Richard K. Guy, "Sets of Integers Whose Subsets Have Distinct - Sums," Theorem 2 (credited there to Leo Moser), Annals of Discrete - Mathematics 12 (1982), 141-154, - doi:10.1016/S0304-0208(08)73500-X. - - declaration: LeanPool.ErdosMoser.erdosMoserMaxBound - informal: >- - For a nonempty finite set A of natural numbers with distinct subset - sums, 4^|A| - 1 is at most 3|A| times the square of its largest - element. - - declaration: LeanPool.ErdosMoser.erdosMoserMaxSqrt - informal: >- - The equivalent square-root lower bound on the largest element: - sqrt((4^|A| - 1)/(3|A|)) is at most max(A). - tags: - - additive-combinatorics - - number-theory - - distinct-subset-sums - - erdos-problems - msc: - - "11B13" - - "11B75" - - slug: odlyzko-bound - title: Poitou's Explicit Odlyzko Bound for Root Discriminants - summary: >- - Formalizes the unconditional explicit-formula argument for a lower bound - on root discriminants of totally complex number fields. Develops the - completed Dedekind zeta function and its functional equation, Poitou's - explicit formula, the Tartar test function, and a certified numerical - estimate proving that degree at least 18 forces root discriminant at - least 8.25. - branch: algebraic number theory - entry_module: LeanPool.Odlyzko - authors: - - The FLT Project - - √2 - source: - title: Sur les petits discriminants - authors: - - Georges Poitou - url: https://www.numdam.org/item/SDPP_1976-1977__18_1_A6_0/ - github_repo: ImperialCollegeLondon/FLT - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - NumberField.Odlyzko.odlyzkoBound - main_results: - - declaration: NumberField.Odlyzko.odlyzkoBound - informal: >- - Proves that every totally complex number field of degree at least 18 - has absolute discriminant at least 8.25 raised to its degree. - source_ref: >- - Georges Poitou, Sur les petits discriminants, equation (26) and the - table on p. 17; Lean source adapted from ImperialCollegeLondon/FLT - commit 01370b013e55c83f1994af814cd31cb8fb2ef653. - tags: - - number-theory - - discriminants - - number-fields - - explicit-formula - msc: - - "11R29" - - "11R42" - - slug: vlasov-mean-field - title: Mean-field derivation and well-posedness of the Vlasov equation - summary: Formalizes the mean-field limit, Dobrushin stability, and forward-in-time well-posedness for the Vlasov equation via Wasserstein-1 optimal transport. - branch: kinetic theory - entry_module: LeanPool.Vlasov - authors: - - Joseph K. Miller - source: - title: A Formalization of the Mean-Field Derivation of the Vlasov Equation - authors: - - Joseph K. Miller - arxiv: "2607.08986" - github_repo: Hydrodynamical/Vlasov_Meanfield_Formalization - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Vlasov.vlasovWellPosedness - - Vlasov.dobrushin - - Vlasov.meanFieldLimit - main_results: - - declaration: Vlasov.vlasovWellPosedness - informal: Forward-in-time existence of a Lagrangian solution to the Vlasov equation on any finite horizon, for interaction forces that are Lipschitz gradients of an even pair potential (class AssW) — with no smallness restriction on the Lipschitz constant. - - declaration: Vlasov.dobrushin - informal: A Wasserstein-1 form of Dobrushin's 1979 stability estimate (the original uses a truncated metric) - two Lagrangian Vlasov solutions with finite first moments at every time separate at most at an exponential Gronwall rate. - - declaration: Vlasov.meanFieldLimit - informal: Conditional mean-field limit - given Wasserstein-1 convergence of the initial empirical data and a uniform Dobrushin stability estimate for the empirical curves, the N-particle empirical measures converge to the Vlasov solution uniformly on any finite horizon. - - declaration: Vlasov.weak_isLagrangianVlasovSolutionOn - informal: On a time window with L * T^2 < 1, a weak solution with the standing regularity (even C^2 pair potential AssW2, finite first moments, narrow continuity, time-continuity of the convolution force at each point, continuity of the convolution-derivative field, and a uniform spatial first-moment bound on the window) is Lagrangian - the pushforward of its initial datum along the characteristic flow. - tags: - - kinetic-theory - - mean-field-limit - - optimal-transport - - wasserstein-distance - - pde - msc: - - "35Q83" - - "82C22" - - "49Q22" - - slug: poincare-three-body-nonintegrability - title: Poincaré's Nonintegrability Theorem for the Restricted Three-Body Problem - summary: >- - Formalizes Poincaré's classical parameter-analytic nonintegrability - theorem for the planar circular restricted three-body problem. The - development defines the rotating-frame Hamiltonian on its collision-free - domain, proves logarithmic blow-up of the resonant disturbing average at - an apoapsis collision boundary, and analytically propagates the resulting - dense resonant obstruction to exclude an additional analytic first - integral. The final statement is the fixed-coordinate, global - uniform-domain special case of the planar part of Yagasaki's Theorem 1.1; - his local meromorphic resonant-orbit obstruction is Theorem 3.1. - branch: dynamical systems - entry_module: LeanPool.PoincareThreeBody - authors: - - Gershon Bialer - source: - title: A new proof of Poincaré's result on the restricted three-body problem - authors: - - Kazuyuki Yagasaki - arxiv: "2111.11031" - doi: "10.1063/5.0266087" - url: https://arxiv.org/abs/2111.11031 - github_repo: gersh/lean-pool - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.PoincareThreeBody.nonintegrability_of_collisionBand - main_results: - - declaration: LeanPool.PoincareThreeBody.nonintegrability_of_collisionBand - informal: >- - No jointly real-analytic family near zero mass can be an additional - first integral of the planar circular restricted three-body - Hamiltonian with phase differential independent from the Hamiltonian - differential anywhere on the uniform collision-free domain. - source_ref: >- - Kazuyuki Yagasaki, "A new proof of Poincaré's result on the - restricted three-body problem," Theorem 1.1 (p. 2), planar case; - compare the precise local meromorphic obstruction near resonant - unperturbed elliptic orbits in Theorem 3.1 (p. 8). The Lean result is - the fixed-coordinate global uniform-domain special case: a global - real-analytic family restricts and complexifies on the local - neighborhoods used in the source theorem. - - declaration: LeanPool.PoincareThreeBody.delaunayHomological_obstruction - informal: >- - If the first homological equation holds and its perturbing Fourier - coefficient is nonzero at every positive rational Kepler resonance, - then the leading differential of a putative analytic first integral - is nowhere independent of the unperturbed Hamiltonian frequency. - - declaration: LeanPool.PoincareThreeBody.delaunayDenseResonance_obstruction - informal: >- - For the rotating Kepler frequency in Delaunay actions, orthogonality - of a continuous candidate differential to every positive rational - resonance vector forces dependence with the Hamiltonian frequency - at every positive action. - - declaration: LeanPool.PoincareThreeBody.homologicalEquation_obstruction - informal: >- - At a resonant action in two degrees of freedom, a nonzero perturbing - Fourier mode and the first homological equation force the - unperturbed frequency and the differential of a putative leading - first-integral coefficient to be linearly dependent. - - declaration: LeanPool.PoincareThreeBody.denseResonance_obstruction - informal: >- - If the resonant obstruction holds on a dense set of actions and the - frequency and differential vary continuously, then they are linearly - dependent at every action. - tags: - - dynamical-systems - - celestial-mechanics - - hamiltonian-systems - - nonintegrability - msc: - - "70F07" - - "37J30" - - "37J40" - - slug: jacobian-diffgeo - title: The Jacobian of a Compact Riemann Surface - summary: >- - Constructs the Jacobian of a compact Riemann surface as the period-lattice - quotient of the dual of holomorphic 1-forms, and proves the surrounding - theory it needs: the genus as the dimension of the space of holomorphic - 1-forms, discreteness and full rank 2g of the period lattice, the - Abel-Jacobi map and its injectivity in positive genus, functoriality - (pushforward, pullback, degree, - and the projection formula), Riemann-Roch, Serre duality via Laurent - tails, the residue theorem, the Dolbeault-Cech comparison, and finite - dimensionality of the first cohomology by a Schwartz/Montel argument. The - headline no-hack guard is that the genus vanishes exactly when the surface - is homeomorphic to the sphere. - branch: complex analysis and geometry - entry_module: LeanPool.JacobianDiffgeo - authors: - - Rado Kirov - source: - title: >- - Jacobians of compact Riemann surfaces (AI challenge) - authors: - - Kevin Buzzard - url: "https://gist.github.com/kbuzzard/778bc714030b3e974ab5f4038783d1a9" - github_repo: rkirov/jacobian-fable - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - genus_eq_zero_iff_homeo - - Jacobian.ofCurve_inj - main_results: - - declaration: genus_eq_zero_iff_homeo - informal: >- - A compact connected Riemann surface has genus zero if and only if it - is homeomorphic to the 2-sphere. - - declaration: Jacobian.ofCurve_inj - informal: >- - The Abel-Jacobi map of a compact Riemann surface of positive genus is - injective: two points with the same image coincide. - - declaration: Jacobian.pushforward_pullback - informal: >- - For a non-constant holomorphic map between compact Riemann surfaces, - the pushforward composed with the pullback on Jacobians is - multiplication by the degree of the map. - tags: - - riemann-surfaces - - complex-geometry - - abel-jacobi - - riemann-roch - - serre-duality - msc: - - "14H40" - - "30F30" - - "32G20" - - slug: classification-of-surfaces - title: Classification of Compact Surfaces - summary: >- - Formalizes the classification of compact connected Hausdorff topological - 2-manifolds with boundary as the sphere, an orientable normal-form - quotient, or a non-orientable normal-form quotient, using triangulation - and finite cyclic polygonal presentations. - branch: geometric and low-dimensional topology - entry_module: LeanPool.ClassificationOfSurfaces - authors: - - Ryan McCorvie - - Jack McCarthy - - SF Lean Meetup contributors - source: - title: Classification of Compact Surfaces - url: "https://github.com/mccorvie/classification-of-surfaces/tree/master/ClassificationOfSurfaces" - github_repo: mccorvie/classification-of-surfaces - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanEval.Topology.ClassificationOfSurfaces.classification_of_surfaces - main_results: - - declaration: LeanEval.Topology.ClassificationOfSurfaces.classification_of_surfaces - informal: >- - Every compact connected Hausdorff topological 2-manifold with - boundary is homeomorphic to the sphere, an orientable normal-form - quotient, or a non-orientable normal-form quotient. - tags: - - geometric-topology - - low-dimensional-topology - - surfaces - - manifolds - - triangulation - msc: - - "57K20" - - "57N05" - - slug: erdos132-three-chain - title: Planar Point Sets Whose Non-Diameter Distances Form a Geometric 3-Chain - summary: >- - Formalizes a support theorem about planar point sets whose non-diameter squared - distances are confined to a geometric progression of ratio three. The chain-quadruple - theorem shows that no planar quadruple with all six squared distances in such a chain - can span two chain steps, and classifies the seven labelled patterns that do occur: the - rhombus of two glued equilateral triangles, in five positions of its long edge, and the - equilateral triangle with its centroid, in two. A Gram obstruction then rules out any - five-point chain configuration. The lift to n at least thirteen runs through the - diameter graph: the Caro--Wei independence bound, proved here, produces an independent - five-set once the diameter graph is known to have at most as many edges as the set has - points. That edge bound is the Hopf--Pannwitz theorem of 1934, and it already exists in - the pool; this project reproves it in its own model as a supporting lemma, claiming no - novelty for it, and the overlap is set out in full below. The result is - unconditional. - - There is no prior write-up behind this development. The restriction to ratio 3 is ad hoc: - the 3-chain statement was produced by the author's automated research pipeline in August - 2026, while the cited source poses the ambient Erdos problem and does not state this - restriction or result. This theorem does not resolve Erdos 132, whose distance spectra need - not be geometric 3-chains. The descriptive labels used in the Lean docstrings are internal - to this development and refer to no numbered result elsewhere. - - Overlap disclosure. Erdos132ThreeChain.hopfPannwitz is NOT offered as a new result. The - pooled project Erdos132N14 already contains the same diameter bound, and the version - here exists only so that this project is self-contained. Its statement, verbatim from - LeanPool/Erdos132N14/HopfPannwitz.lean, is - - theorem diameterEdges_card_le - (P : Configuration ι) (S : Finset ι) {d : ℝ} - (hd : 0 < d) - (hmax : ∀ f ∈ pairs S, P.pairDistance f ≤ d) : - (diameterEdges P S d).card ≤ S.card - - where, from LeanPool/Erdos132N14/Basic.lean, `Configuration ι` is a structure carrying - an injective `point : ι → ℂ`, `pairs S = (S ×ˢ S).filter fun e ↦ e.1 < e.2` for a - `LinearOrder ι`, `pairDistance P e = dist (P.point e.1) (P.point e.2)`, and - `diameterEdges P S d = (pairs S).filter fun e ↦ P.pairDistance e = d`. The corresponding - statement here is - - theorem hopfPannwitz {X : Finset Point} {D : ℝ} (hD : IsSqDiameter X D) : - ∑ v ∈ X, degree (DiameterAdj D) X v ≤ 2 * X.card - - with `Point = ℝ × ℝ`, `sqDist` the squared Euclidean distance, - `DiameterAdj D p q = (p ≠ q ∧ sqDist p q = D)`, `degree adj X v = (X.filter (adj v)).card`, - and `IsSqDiameter X D` asserting that `D` is realised by two distinct points of `X` and - dominates every squared distance inside `X`. The two are the same mathematical theorem — - a positive maximal distance is realised at most as often as there are points — carried - by different models: an injective map from a linearly ordered index type into the - complex plane, counting a Finset of unordered pairs, against a Finset of points of the - real plane, in squared distances and degree-sum form. Neither statement is an instance - of the other without a model bridge: a LinearOrder on ℝ × ℝ, a dist-to-sqDist transfer, - and a handshake lemma. The proofs differ too, Perles charging there against pendant - deletion driven by Cramer's rule here. No project in the pool imports another, and - CONTRIBUTING.md records that independence as the property the fast per-project build - relies on, so this project reproves the bound rather than importing it. It is a - supporting lemma of this development, not one of its results, and it is therefore not - listed among the main results below. - - Two supporting identities also exist elsewhere in the pool. From - LeanPool/Erdos97ConvexOctagon/CayleyMenger.lean, - - theorem cm4_sqDist_eq_zero (a b c d : Plane) : - cm4 (sqDist a b) (sqDist a c) (sqDist a d) - (sqDist b c) (sqDist b d) (sqDist c d) = 0 - - over `Plane = EuclideanSpace ℝ (Fin 2)` is the same four-point Cayley--Menger vanishing - as this project's - - theorem gramDet_eq_zero (a b c d : Point) : - gramDet (sqDist a b) (sqDist a c) (sqDist a d) (sqDist b c) (sqDist b d) (sqDist c d) = 0 - - over `Point = ℝ × ℝ`; both are one-line `ring` identities in the coordinates. From - LeanPool/DistanceGeometry/SchoenbergHard.lean, - - theorem schoenberg {D : Matrix (Fin n) (Fin n) ℝ} (hD : IsPreDistMatrix D) : - EmbedsIn D k ↔ (centeredGram D).PosSemidef ∧ (centeredGram D).rank ≤ k - - characterizes Euclidean distance matrices in general dimension. This project never - invokes it: the four-point condition is used directly as a `ring` identity and the - five-point obstruction is a sum-of-squares contradiction, so no positive-semidefiniteness - criterion is needed. - branch: discrete geometry - entry_module: LeanPool.Erdos132ThreeChain - authors: - - Egor Lyfar - source: - title: "Erdős Problem #132" - authors: - - Paul Erdős - - János Pach - url: https://www.erdosproblems.com/132 - github_repo: lyfar/lean-pool - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Erdos132ThreeChain.nonDiameterSqDists_ne_chain - main_results: - - declaration: Erdos132ThreeChain.nonDiameterSqDists_ne_chain - informal: >- - Let n be at least thirteen, let a be positive, and let X be a finite set of points of - the real plane with exactly n elements. Suppose D is the squared diameter of X, - meaning that D is realised by two distinct points of X and dominates every squared - distance inside X. Then for every natural number h, the set of squared distances - realised by distinct points of X and different from D is not equal to the geometric - 3-chain consisting of a times 3^j for j below h. - - declaration: Erdos132ThreeChain.four_chain_adjacent - informal: >- - Let a be positive and let S be a four-element set of points of the real plane such - that every squared distance between distinct points of S equals a times a - nonnegative power of three. Then there is a positive r, itself equal to a times a - nonnegative power of three, for which every squared distance between distinct points - of S equals r or 3r. - - declaration: Erdos132ThreeChain.pair_classification - informal: >- - Let c be positive and let A, B, C, D be points of the real plane with the squared - distance from A to B equal to c and every other pairwise squared distance equal to c - times a nonnegative power of three. Then each of the four squared distances from A - and B to C and to D equals c or 3c, at least one of the two squared distances from A - and B to C equals c, likewise for D, the squared distance from C to D equals c or 3c, - the squared distance from A to C equals c or that from B to D equals c, the squared - distance from B to C equals c or that from A to D equals c, and the whole labelling - is one of exactly seven explicit patterns: five with a single edge of squared length - 3c, and two in which the three edges of squared length 3c form a triangle. - - declaration: Erdos132ThreeChain.no_five_chain_finset - informal: >- - Let a be positive. There is no five-element set of points of the real plane in which - every squared distance between distinct points equals a times a nonnegative power of - three. - - declaration: Erdos132ThreeChain.no_four_on_circle - informal: >- - Let c be positive. There are no four points of the real plane, each at squared - distance c from a common point, whose six pairwise squared distances all equal c or - 3c. - - declaration: Erdos132ThreeChain.carowei - informal: >- - Let adj be a symmetric irreflexive relation with decidable adjacency on a type, and - let V be a finite set of vertices. Then V has a subset S, no two elements of which - are adjacent, whose cardinality is at least the sum over v in V of one divided by - one plus the number of neighbours of v inside V. - - declaration: Erdos132ThreeChain.exists_independent_five - informal: >- - Let adj be a symmetric irreflexive relation with decidable adjacency on a type, and - let V be a finite set of at least thirteen vertices whose degrees inside V sum to at - most twice the cardinality of V. Then V has a five-element subset no two elements of - which are adjacent. - - declaration: Erdos132ThreeChain.chainTwo_nonDiameter_support - informal: >- - There is a five-element set of points of the real plane whose squared diameter is 4 - and whose set of non-diameter squared distances is exactly the two-term geometric - 3-chain consisting of 1 and 3. The configuration that the headline forbids for at - least thirteen points is therefore realisable at five points with chain length two. - tags: - - discrete-geometry - - distance-geometry - - erdos-problems - msc: - - "52C10" - - "52C35" - - "05C69" - - slug: erdos81-paper-iii-sum-zero-triangles - title: Near-perfect triangle packings from sum-zero triples - summary: >- - Records a reusable formalization byproduct accompanying Paper III: pairwise edge-disjoint - triangles in any finite complete graph such that every vertex is incident to at most three - uncovered edges, using a cyclic-group sum-zero triple system, with a bridge to Mathlib's - SimpleGraph API. - branch: extremal and design combinatorics - entry_module: LeanPool.Erdos81PaperIIIContrib - authors: - - Juan Pablo Traverso Gianini - - Aristotle - - Vasily Ilin - source: - title: >- - SumZeroTriangles.lean, a reusable formalization byproduct accompanying Linear-Error Clique - Partitions of Split Graphs via Structured Triangle Packing - authors: - - Aristotle - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/main/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Contrib/SumZeroTriangles.lean - github_repo: jtraverso/erdos-81-chordal-clique-partitions - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - SumZeroTriangles.exists_simpleGraph_triangle_packing - main_results: - - declaration: SumZeroTriangles.exists_simpleGraph_triangle_packing - informal: >- - For every finite vertex type, its complete simple graph has a family of pairwise - edge-disjoint triangle graphs whose uncovered graph has degree at most three at every - vertex. - - declaration: SumZeroTriangles.exists_triangle_packing_clique - informal: >- - Every finite complete graph has a family of pairwise edge-disjoint triangles leaving - uncovered degree at most three at every vertex. - tags: - - extremal-combinatorics - - triangle-packing - - cyclic-groups - msc: - - "05B07" - - "05C70" - - slug: matching-logic-completeness - title: Global completeness of one-sorted definedness-free matching logic - summary: >- - Formalizes the one-sorted case of Chen and Roșu's global-completeness - argument and independently discharges its local-completeness oracle by - mechanizing the canonical-model construction from their 2019 Matching - μ-Logic technical report (Theorem 73, Lemmas 80–81, and strong local - completeness Theorem 83). The resulting theorem supplies both local - completeness and soundness for countably infinite variable types. - branch: mathematical logic and formal methods - entry_module: LeanPool.MatchingLogic - authors: - - Aurélien Eveil - - Anthropic - - OpenAI - source: - title: >- - Completeness and incompleteness of basic matching logic; Matching μ-Logic - (2019 technical report) - authors: - - Xiaohong Chen - - Grigore Roșu - arxiv: "2608.13306" - url: https://hdl.handle.net/2142/102281 - github_repo: eveil-labs/matching-logic-lean - commit: 4628a26bb2db117de1a11cbedcb620db9124cdb8 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - MatchingLogic.global_completeness_entryIII - main_results: - - declaration: MatchingLogic.locality - informal: >- - For a backward-closed subset of a matching-logic model, two valuations - that agree inside the subset and both stay outside it otherwise give - identical denotations after restriction to that subset. - - declaration: MatchingLogic.two_copies - informal: >- - The double-cover model over a nonempty backward-closed proper subset has - exactly the denotation predicted by each of its two projections back to - the original model. - - declaration: MatchingLogic.semantic_localization - informal: >- - For a closed theory and closed conclusion, global semantic consequence - is equivalent to local consequence from the localized theory. - - declaration: MatchingLogic.strongLocalCompleteness - informal: >- - One-sorted basic matching logic has strong local completeness over an - arbitrary one-sorted finitary signature when the element-variable type - is countably infinite. - - declaration: MatchingLogic.global_completeness_entryIII - informal: >- - For a one-sorted finitary signature, closed premises and conclusion, - and a countably infinite element-variable type, global semantic - consequence is equivalent to derivability, with both soundness and - local completeness discharged. - tags: - - matching-logic - - mathematical-logic - - modal-logic - - completeness - - formal-methods - msc: - - "03B45" - - "03B70" - - slug: finite-graph-fundamental-group - title: Fundamental groups of finite connected graphs - summary: >- - Constructs the standard interval-cell realization of a finite connected directed - multigraph and proves that its topological fundamental group is a free group of rank - E - V + 1. The proof uses an explicit spanning-tree basis and develops a verified - path-lifting cover, a contraction of the covering tree, root-change equivalences, the - tree/triviality criterion, and the corresponding free-abelian result. - branch: algebraic topology and graph theory - entry_module: LeanPool.FiniteGraphFundamentalGroup - authors: - - Arthur Freitas Ramos - - David Hulak - - Ruy de Queiroz - source: - title: "Algebraic Topology, Additional Topic 1.A: Graphs and Free Groups" - authors: - - Allen Hatcher - url: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf - github_repo: Arthur742Ramos/finite-graph-fundamental-group - commit: dd57e3ab8bc5a7042fcc5498e778b008d87ac032 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - FiniteGraphFreeGroup.graphTopologicalFundamentalGroupEquiv - main_results: - - declaration: FiniteGraphFreeGroup.graphTopologicalFundamentalGroupEquiv - informal: >- - For a finite weakly connected directed multigraph, the topological fundamental group - of its interval-cell quotient realization at any vertex is explicitly isomorphic to - the free group on E + 1 - V generators. - - declaration: FiniteGraphFreeGroup.graphCombinatorialToTopologicalEquiv - informal: >- - The endomorphism group at a vertex in the graph's free groupoid is explicitly - isomorphic to the topological fundamental group of the interval-cell realization. - - declaration: FiniteGraphFreeGroup.graphCoverProjection_isCovering - informal: >- - The canonical path-lifting projection from rooted free-groupoid paths to graph - vertices is a quiver covering, with bijective maps on every incoming and outgoing - edge star. - - declaration: FiniteGraphFreeGroup.graphRealization_tree_simplyConnected - informal: >- - The interval-cell realization of a directed arborescence is simply connected, via an - explicit continuous contraction to its root. - - declaration: FiniteGraphFreeGroup.graphFundamentalGroup_subsingleton_iff_cycleRank_zero - informal: >- - The combinatorial fundamental group is trivial exactly when the graph's cycle rank - E + 1 - V is zero. - - declaration: FiniteGraphFreeGroup.graphFundamentalGroupFreeAbelianizationEquiv - informal: >- - The additive abelianization of the graph fundamental group is the free abelian group - on E + 1 - V generators. - tags: - - algebraic-topology - - graph-theory - - fundamental-groups - - free-groups - - covering-spaces - msc: - - "05C25" - - "55Q05" - - "20F65" - - slug: composition-algebras - title: Hurwitz's Classification of Euclidean Composition Algebras - summary: >- - Formalizes Hurwitz's classification of nontrivial finite-dimensional real - Euclidean composition algebras: every such algebra has dimension 1, 2, 4, - or 8 and is real-linearly equivalent to the real numbers, complex numbers, - quaternions, or octonions through a map preserving the unit, multiplication, - and norm form. The development also constructs the octonions from an explicit Fano multiplication table, - proves Degen's eight-square identity, and shows - that Cayley--Dickson doubling preserves the composition law for - associative inputs. - branch: nonassociative algebra - entry_module: LeanPool.CompositionAlgebras - authors: - - Bryan Ehrlich - source: - title: Über die Composition der quadratischen Formen von beliebig vielen Variablen - authors: - - Adolf Hurwitz - url: https://eudml.org/doc/58420 - github_repo: ehrlich-b/composition-algebras - commit: e37e22b0571a170ba18a0d8db29fe2a857906b92 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - CompositionAlgebra.hurwitz_classification - main_results: - - declaration: CompositionAlgebra.hurwitz_classification - informal: >- - Every nontrivial finite-dimensional real Euclidean composition algebra - is linearly equivalent to the real numbers, complex numbers, - quaternions, or octonions through a map preserving the unit, - multiplication, and norm form. - - declaration: CompositionAlgebra.finrank_eq_one_or_two_or_four_or_eight - informal: >- - The real dimension of a nontrivial finite-dimensional Euclidean - composition algebra is exactly 1, 2, 4, or 8. - - declaration: CompositionAlgebra.CD.instCompositionAlgebra - informal: >- - The Cayley--Dickson double of a nontrivial associative Euclidean - composition algebra again satisfies the composition law. - - declaration: Octonion.nucleus_real - informal: >- - Any octonion that associates in the third slot with every pair has - zero imaginary coordinates and hence lies on the real line. - tags: - - nonassociative-algebra - - composition-algebras - - octonions - - hurwitz-theorem - - cayley-dickson - msc: - - "17A75" - - "17A35" - - "11E88" - - slug: ulms-theorem - title: Ulm's theorem for countable reduced abelian p-groups - summary: >- - Formalizes Ulm's classification theorem for countable reduced abelian - p-groups in arbitrary Lean universes. The development constructs the - transfinite Ulm filtration and invariants, proves their invariance under - additive equivalence, establishes Kaplansky's one-generator extension - lemma, and completes the countable back-and-forth classification. A - finite-group regression distinguishes Z/4 x Z/4 from Z/2 x Z/8 at the - zeroth Ulm invariant. - branch: abelian group theory - entry_module: LeanPool.UlmsTheorem - authors: - - Elan Roth - source: - title: Ulm's theorem for countable reduced abelian p-groups - authors: - - Elan Roth - url: https://github.com/elanroth/UlmsTheorem - github_repo: elanroth/UlmsTheorem - commit: 3bb8a10ae12eb957a0f05db248d3becaf6a7e222 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - UlmsTheorem.ulm_theorem - main_results: - - declaration: UlmsTheorem.ulm_theorem - informal: >- - For any prime p and any two countable reduced abelian p-groups in the - same arbitrary Lean universe, the groups are additively isomorphic if - and only if all of their classical Ulm invariants, indexed by - universe-zero ordinals, agree. - - declaration: UlmsTheorem.iso_of_ulm_invariants - informal: >- - Equal Ulm invariants at every universe-zero ordinal imply an additive - equivalence between countable reduced abelian p-groups, via finite - height-preserving partial isomorphisms and a back-and-forth limit. - - declaration: UlmsTheorem.ulmInvariant_iso_invariant - informal: >- - An additive equivalence preserves every classical Ulm invariant. - - declaration: UlmsTheorem.UlmDifferentialRegression.no_iso_44_28 - informal: >- - The groups Z/4 x Z/4 and Z/2 x Z/8 are not additively isomorphic; the - proof exercises the classification theorem and independently detects - their different zeroth Ulm invariants. - tags: - - abelian-groups - - p-groups - - classification-theorems - - ordinal-filtrations - - ulm-invariants - msc: - - "20K10" - - slug: local-complex-geometry - title: Foundational local complex-analytic geometry - summary: >- - Develops a common Lean foundation for several classical local theorems in - several complex variables: complex-analytic Weierstrass preparation with - full germ uniqueness, Noetherianity of holomorphic-germ rings, finite - projection for hypersurface germs, the holomorphic constant-rank normal - form, and the finite-family local analytic Nullstellensatz. The imported - development also exposes the arbitrary-ideal Nullstellensatz and regular - level-set, submersion, and immersion consequences. - branch: several complex variables and local analytic geometry - entry_module: LeanPool.LocalComplexGeometry - authors: - - BochaoKong - source: - title: Foundational theorems in local complex-analytic geometry - authors: - - BochaoKong - url: https://github.com/BochaoKong/nullstellensatz - github_repo: BochaoKong/nullstellensatz - commit: 029697242294aea7989bf5d391199696a43490df - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LocalComplexGeometry.localAnalyticNullstellensatz - main_results: - - declaration: LocalComplexGeometry.localAnalyticNullstellensatz - informal: >- - If a finite family of functions analytic at the origin has common - zero set locally contained in the zero set of another analytic - function, then a positive power of the latter is locally an analytic - linear combination of the family. - - declaration: LocalComplexGeometry.holomorphicGerm_isNoetherian - informal: >- - For every finite complex dimension, including dimension zero, the - local ring of holomorphic germs at the origin is Noetherian. - - declaration: LocalComplexGeometry.hypersurface_finiteProjection - informal: >- - A nonzero hypersurface germ vanishing at the origin admits a linear - coordinate change whose local quotient is finite free over the base - germ ring, together with an explicit prepared equation and a proper, - surjective finite-fiber projection of the local zero set. - - declaration: LocalComplexGeometry.holomorphic_constantRank_normalForm - informal: >- - An analytic map of locally constant complex rank is locally equivalent, - through analytic source and target coordinates, to the standard rank - projection. - - declaration: ClassicalComplexWPT.classicalComplexWeierstrassPreparation - informal: >- - A complex-analytic germ of exact order in one distinguished variable - factors uniquely into a nonvanishing analytic unit and a monic - Weierstrass polynomial whose analytic coefficient germs vanish at the - origin. - source_ref: >- - BochaoKong/classical-complex-wpt, commit - b4a7273fe5c9752753c52e10494097569089642d (2026-08-28). - - declaration: LocalComplexGeometry.analyticNullstellensatz_ideal - informal: >- - For every ideal of the local holomorphic-germ ring, the ideal of germs - vanishing on its local zero set equals its radical. - - declaration: LocalComplexGeometry.holomorphic_regularLevelSet_parametrization - informal: >- - Near a regular point, the level set of a holomorphic map admits an - analytic parametrization with an analytic local inverse on the level - set. - tags: - - complex-analysis - - analytic-geometry - - nullstellensatz - - weierstrass-preparation - msc: - - "32A05" - - "32A10" - - "32B05" - - "32C25" - - "13E05" - - slug: rado-theorem - title: Radó's theorem for Riemann surfaces - summary: >- - Proves that every connected Hausdorff one-dimensional complex manifold - is second countable. The development constructs a nonconstant harmonic - function by Perron's method, builds the étale space of its - harmonic-conjugate germs, and applies a formalized Poincaré-Volterra - lemma to its discrete-fiber evaluation map. - branch: Riemann surfaces and general topology - entry_module: LeanPool.Rado - authors: - - Rado Kirov - source: - title: Radó's theorem for Riemann surfaces - authors: - - Rado Kirov - url: https://github.com/rkirov/jordan_pick - github_repo: rkirov/jordan_pick - commit: b3c9b7cf7358bf81a077d78ad67e6e8247869ddd - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - rado_riemannSurface - main_results: - - declaration: rado_riemannSurface - informal: >- - Every connected Hausdorff topological space carrying the structure - of a one-dimensional complex manifold has a second-countable - topology. - - declaration: Rado.poincare_volterra - informal: >- - A connected Hausdorff, locally compact, locally connected and - locally second-countable space is second countable whenever it - admits a continuous, locally injective map into a - second-countable Hausdorff space. - - declaration: Rado.exists_harmonic_extension - informal: >- - Every continuous real-valued function on a Euclidean circle extends - continuously to its closed disk and harmonically to the open disk; - unlike the existing LeanComplexAnalysis Poisson theorems, this - constructs the harmonic function rather than assuming one. - - declaration: Rado.IsPerronFamily.surfaceHarmonicOn_perronSup - informal: >- - The pointwise upper envelope of a Perron family of surface - subharmonic functions is harmonic on its open domain. - tags: - - riemann-surfaces - - second-countability - - harmonic-functions - - perron-method - msc: - - "30F15" - - "30F20" - - "54D65" - - slug: hanson-wright - title: The Hanson-Wright inequality for sub-Gaussian quadratic forms - summary: >- - Proves the finite-dimensional Hanson-Wright concentration inequality for - quadratic forms of independent mean-zero sub-Gaussian real random variables - whose maximum least global-MGF sub-Gaussian scale is positive, for matrices - with positive Frobenius and operator norms. The bound uses the fixed universal - coefficient 1 / (256 * exp(1)^2). This is the MGF normalization of the usual - result; no formal identification with an Orlicz psi-2 norm is claimed. - The development derives the required moment-generating-function certificate - instead of assuming it, and contributes reusable scalar Bernstein, Chernoff, - cumulant, sub-Gaussian-process, and finite-maximum infrastructure. - branch: high-dimensional probability and concentration inequalities - entry_module: LeanPool.HansonWright - authors: - - Yuanhe Zhang - - Jason D. Lee - - Fanghui Liu - source: - title: High-Dimensional Probability (second edition) - authors: - - Roman Vershynin - doi: 10.1017/9781009490672 - url: https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html - github_repo: Lean-MoDS/StatsMLlib - commit: 37286c3d2c7e17642fbe26988be40f3ff982f944 - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - LeanPool.HansonWright.hanson_wright_inequality_hdp - main_results: - - declaration: LeanPool.HansonWright.hanson_wright_inequality_hdp - informal: >- - For independent mean-zero real random variables with finite least - global-MGF sub-Gaussian scales and positive maximum coordinate MGF scale, - and a matrix with positive Frobenius and operator norms, the centered - quadratic form satisfies the Hanson-Wright two-regime exponential tail - bound with the fixed universal coefficient 1 / (256 * exp(1)^2). - - declaration: LeanPool.HansonWright.hasHansonWrightMGF_of_subgaussian - informal: >- - Given positive K and operator norm and C at least 64 * exp(1)^2, - independent coordinates with global-MGF sub-Gaussian parameter K^2 - satisfy the local quadratic cumulant-generating-function estimate needed - for Hanson-Wright. - - declaration: LeanPool.bernstein_inequality - informal: >- - A finite family of independent, centered, uniformly bounded real random - variables satisfies the classical scalar Bernstein tail inequality. - tags: - - probability-theory - - concentration-inequalities - - high-dimensional-probability - - random-matrices - msc: - - "60E15" - - "60B20" - - "62H12" - - slug: isograph-canonical-labelling - title: Verified canonical labelling of finite graphs - summary: >- - Implements an individualisation-refinement canonical-labelling algorithm - in the style of nauty and proves that its canonical adjacency form is - invariant under vertex relabelling and classifies finite graphs up to - isomorphism. The imported core includes the executable search, refinement, - orbit-pruning and backjumping machinery together with its equivariance, - termination, soundness and optimality proofs. - branch: graph theory and verified algorithms - entry_module: LeanPool.IsoGraph - authors: - - Alex Meiburg - source: - title: IsoGraph - authors: - - Alex Meiburg - url: https://github.com/Timeroot/IsoGraph - github_repo: Timeroot/IsoGraph - commit: bbfcefd15087d1dcb63ebdad015a087e1d6890a1 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - IsoGraph.Canon.canonAdj_eq_iff - - IsoGraph.Canon.canonAdj_relabel - main_results: - - declaration: IsoGraph.Canon.canonAdj_eq_iff - informal: >- - Two Boolean adjacency functions on Fin n have equal canonical forms - exactly when one is a relabelling of the other by a vertex permutation. - - declaration: IsoGraph.Canon.canonAdj_relabel - informal: >- - Relabelling a finite graph by any permutation leaves its canonical - adjacency form unchanged. - - declaration: IsoGraph.Canon.exists_relabel_of_canonAdj_eq - informal: >- - Equality of canonical adjacency forms yields an explicit permutation - relating the original adjacency functions. - tags: - - graph-theory - - graph-isomorphism - - canonical-labelling - - verified-algorithms - msc: - - "05C60" - - "68R10" - - slug: permanent-formula-lower-bound - title: Quartic-over-logarithmic lower bound for rational permanent formulas - summary: >- - Proves that every valid rational arithmetic formula over the complex - numbers computing the n-by-n permanent, for n at least 32, has at least - n^4 / (192 log_2 n) variable leaves. The development derives the bound - through coefficient-field transcendence degree, Jacobian minors, - matching-polynomial identities, and a cyclic block decomposition. - branch: algebraic complexity theory - entry_module: LeanPool.PermanentFormulaLowerBound - authors: - - OpenAI - - Dean Cureton - source: - title: Ten Advances in Mathematics and Theoretical Computer Science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - PermanentFormulaLowerBound.permanent_rational_formula_logarithmic_lower_bound - main_results: - - declaration: PermanentFormulaLowerBound.permanent_rational_formula_logarithmic_lower_bound - informal: >- - For n at least 32, every valid rational formula over the complex - numbers that computes the n-by-n permanent has at least - n^4 / (192 log_2 n) variable leaves. - - declaration: PermanentFormulaLowerBound.permanent_rational_formula_lower_bound - informal: >- - The same quartic-over-logarithmic lower bound controls variable - leaves, total leaves, and vertices, and half that bound controls - internal gates. - tags: - - algebraic-complexity - - arithmetic-formulas - - permanent - - lower-bounds - - transcendence-degree - msc: - - "68Q17" - - "68Q25" - - "15A15" - - slug: flt-regular - title: Fermat's Last Theorem for regular primes - summary: >- - Formalizes Kummer's proof of Fermat's Last Theorem for every odd regular - prime exponent. The development proves both cases of the theorem and - includes the required theory of cyclotomic integers, relative units, - Hilbert 90 and 92, ramification, class groups, and Kummer's lemma. - branch: algebraic number theory - entry_module: LeanPool.FltRegular - authors: - - FltRegular contributors - source: - title: Fermat's Last Theorem for regular primes - authors: - - Riccardo Brasca - - Ruben Van de Velde - - Andrew Yang - - Chris Birkbeck - - Alex J. Best - - Eric Rodriguez - url: https://github.com/leanprover-community/flt-regular - github_repo: leanprover-community/flt-regular - commit: 94a1d78956a348568c80975ec9005dc5762ab10a - license: Apache-2.0 - status: verified - provenance: human - main_declarations: - - flt_regular - main_results: - - declaration: flt_regular - informal: >- - Fermat's Last Theorem holds for every odd regular prime exponent. - - declaration: FltRegular.caseI - informal: >- - The first case of Fermat's Last Theorem holds for every regular prime: - no primitive solution exists when the exponent does not divide the - product of the three terms. - - declaration: FltRegular.caseII - informal: >- - The second case of Fermat's Last Theorem holds for every odd regular - prime: no primitive nonzero solution exists when the exponent divides - the product of the three terms. - - declaration: eq_pow_prime_of_unit_of_congruent - informal: >- - A unit of the cyclotomic field of an odd regular prime that is - congruent to an integer modulo the prime is itself a prime-th power. - tags: - - algebraic-number-theory - - fermats-last-theorem - - cyclotomic-fields - - class-groups - - regular-primes - msc: - - "11D41" - - "11R18" - - "11R29" - - slug: feige - title: Feige's sharp unit-slack inequality - summary: >- - For small-universe probability spaces, proves the sharp fixed-dimensional - unit-slack lower bound for sums of independent nonnegative random - variables: if each coordinate has mean at most one, then the probability - that the sum lies below its mean plus one is at least (n/(n+1))^n, and - this constant is optimal. The supporting library formalizes the - Vlassis--Thomas exact distribution-free calibration theorem, Grünbaum's - centroid-halfspace inequality and sharp simplex example, and the - normalized-exponential bridge joining them. - branch: probability theory and convex geometry - entry_module: LeanPool.Feige - authors: - - Zhengqing Zhou - source: - title: >- - Sharp Small-Deviation Inequalities for Sums of Independent - Nonnegative Random Variables - authors: - - Weibo Fu - - Yanjun Han - - Guanyang Wang - - Jun Yan - - Peng Zhang - - Zhengqing Zhou - arxiv: "2607.23980" - github_repo: pengzhang91/Feige - commit: 98ab466e74280ae9d40622c19dc7f24f01b60864 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Feige.sharp_unit_slack_feige_complete - main_results: - - declaration: Feige.sharp_unit_slack_feige_complete - informal: >- - For every positive dimension n and every small-universe probability - space, the sharp fixed-dimensional lower bound for the strict event - that a sum of independent nonnegative random variables is below its - expectation plus one is (n/(n+1))^n, and an explicit finite family - proves optimality. - - declaration: VlassisThomas.exactCalibration - informal: >- - On every small-universe probability space, the Dirichlet statistic - used in the proof is super-uniform for every independent family of - measurable, integrable, nonnegative random variables whose coordinate - means are at most one. - - declaration: Grunbaum.grunbaum_centroid_halfspace - informal: >- - Every closed halfspace containing the centroid of a positive-volume - convex body in finite-dimensional Euclidean space contains at least - the sharp Grünbaum fraction of the body's volume. - tags: - - probability - - concentration-inequalities - - convex-geometry - - sharp-constants - msc: - - "60E15" - - "52A20" - - slug: kasami-cyclic-additive - title: Carlet's Kasami cyclic-additive conjecture - summary: >- - Proves Carlet's cyclic-additive conjecture for the Kasami power function: - over every finite field of characteristic two with cardinality 2^n and - every k coprime to n, the normalized derivative image has the exact - coefficient triple count 2^(2n-3) for all distinct nonzero slopes. The - development combines the Müller--Cohen--Matthews permutation theorem, - multiplicative character sums, Fourier inversion, and a Fermat-cubic - Frobenius-kernel argument. - branch: finite fields and coding theory - entry_module: LeanPool.KasamiCyclicAdditive - authors: - - D.S. McNeil - - Gábor P. Nagy - - Attila Vajda - source: - title: On a conjecture on the Kasami APN function - authors: - - Gábor P. Nagy - - Attila Vajda - arxiv: "2608.18584" - url: https://github.com/dsm054/kasami_cyclic_additive - github_repo: dsm054/kasami_cyclic_additive - commit: c8ab49ebd8e9e53f5aa7585f1d43acf1ff68d10f - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - KasamiCyclicAdditive.carlet_kasami_cyclic_additive_literature - main_results: - - declaration: KasamiCyclicAdditive.carlet_kasami_cyclic_additive_literature - informal: >- - For every finite field K of characteristic two with |K| = 2^n, - every k coprime to n, and all distinct nonzero v₁ and v₂ in K, the - coefficient triple count of the normalized Kasami derivative image is - exactly 2^(2n-3). - - declaration: KasamiCyclicAdditive.carlet_kasami_cyclic_additive_core - informal: >- - In the normalized range 1 ≤ k < n, the same coefficient triple count - is 2^(2n-3), assuming characteristic two, |K| = 2^n, coprimality, - and distinct nonzero coefficients. - - declaration: KasamiCyclicAdditive.kasami_half_size - informal: >- - Under the normalized admissibility hypotheses, the image of the - Kasami derivative has exactly half as many elements as the field. - tags: - - finite-fields - - coding-theory - - APN-functions - - difference-sets - - character-sums - msc: - - "11T06" - - "11T71" - - "94A60" - - slug: erdos81-paper-ii-chordal-heredity - title: "Chordal graphs: minimal separators and simplicial vertices" - summary: >- - A reusable formalization byproduct accompanying Paper II: defines chordal graphs through - the cycle-and-chord characterization, proves induced-subgraph heredity and the clique - property for finite minimal separators of fixed vertex pairs, proves Dirac's - simplicial-vertex theorem, and proves the connected non-complete case of the two-vertex - conclusion. - branch: graph theory - entry_module: LeanPool.Erdos81PaperIIContrib - authors: - - Juan Pablo Traverso Gianini - source: - title: Complete-Split Extremizers for a Fractional Triangle-Cover Functional on Chordal Graphs - authors: - - Juan Pablo Traverso Gianini - url: https://github.com/jtraverso/lean-pool/blob/822738f3089d92c5e65a6bfb8c88c992e4699167/LeanPool/Erdos81PaperIIContrib/Chordal.lean - github_repo: jtraverso/lean-pool - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - SimpleGraph.IsChordal.exists_two_nonadj_isSimplicial - main_results: - - declaration: SimpleGraph.IsChordal.comap - informal: >- - A graph that embeds as an induced subgraph of a chordal graph is itself chordal. - - declaration: SimpleGraph.IsChordal.induce - informal: >- - Every induced subgraph of a chordal graph is chordal. - - declaration: SimpleGraph.IsChordal.minimalSeparator_isClique - informal: >- - Every finite minimal separator of two fixed vertices in a chordal graph is a clique. - - declaration: SimpleGraph.IsChordal.exists_isSimplicial - informal: >- - Every nonempty finite chordal graph has a simplicial vertex. - - declaration: SimpleGraph.IsChordal.exists_two_nonadj_isSimplicial - informal: >- - Every connected non-complete finite chordal graph has two non-adjacent simplicial - vertices. - tags: - - graph-theory - - chordal-graphs - - induced-subgraphs - - minimal-separators - - simplicial-vertices - msc: - - "05C75" - - slug: polynomial-freiman-ruzsa - title: The polynomial Freiman–Ruzsa conjecture - summary: >- - Formalizes Marton's polynomial Freiman–Ruzsa conjecture by the entropy method, including - the characteristic-two bounds with exponents twelve and nine, the bounded-torsion theorem - with an explicit exponent, and quantitative consequences for affine dimension and - approximate homomorphisms. The supporting development builds entropic Ruzsa distance, - multidistance, conditional entropy, and additive-energy machinery. This optimized import - reuses the pooled Zhang–Yeung entropy and kernel library in place of duplicate upstream - modules while preserving all six closed headline statements. - branch: additive combinatorics and information theory - entry_module: LeanPool.PFR - authors: - - PFR contributors - contributors: - - Aaron Anderson - - Mantas Bakšys - - Jonas Bayer - - Mauricio Collares - - Rémy Degenne - - Yaël Dillies - - Bhavik Mehta - - Ella Yu - - Ben Eltschig - - Sébastien Gouëzel - - Kalle Kytölä - - Rob Lewis - - Paul Lezeau - - Lorenzo Luccioli - - Heather Macbeth - - Patrick Massot - - Arend Mellendijk - - Kyle Miller - - Pietro Monticone - - Kim Morrison - - Oliver Nash - - Utensil Song - - Terence Tao - - Floris van Doorn - - Sky Wilshaw - - Lawrence Wu - source: - title: Marton's conjecture (the polynomial Freiman–Ruzsa conjecture) - authors: - - W. T. Gowers - - Ben Green - - Freddie Manners - - Terence Tao - arxiv: 2311.05762 - url: https://github.com/teorth/pfr - github_repo: teorth/pfr - commit: 07839bddb1395c8808fa47db821ce2e4fff9362e - upstream_dependencies: - - title: AddCombi - github_repo: leanprover-community/add-combi - commit: 2d061f437c1ba00ba9b16d7cba1c7603ba4807a4 - license: Apache-2.0 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - Marton.pfr_conjecture - main_results: - - declaration: Marton.pfr_conjecture - informal: >- - A nonempty finite set A of doubling at most K in an abelian group of exponent two is - covered by fewer than 2 K^12 cosets of a finite subgroup no larger than A. - - declaration: Marton.pfr_conjecture_nine - informal: >- - Strengthens the characteristic-two covering theorem to fewer than 2 K^9 cosets of a - finite subgroup no larger than A. - - declaration: Marton.torsion_pfr_conjecture - informal: >- - In an abelian group of exponent m at least two, a nonempty finite set of doubling at - most K is covered by fewer than m K^(256 m^3 + 1) cosets of a subgroup no larger than A. - - declaration: Marton.weak_pfr_int - informal: >- - A nonempty finite set of doubling at most K in a finite-rank free abelian group has a - subset of relative size at least K^(-34) and affine dimension at most - (80 / log 2) log K. - - declaration: Marton.homomorphism_pfr - informal: >- - A map between finite abelian groups of exponent two whose additive defects lie in S - differs from a homomorphism by a map taking at most |S|^10 values. - - declaration: Marton.approx_hom_pfr - informal: >- - A map between finite abelian groups of exponent two that is additive on at least a - 1/K fraction of pairs agrees with a homomorphism on at least - (|G| / (2^144 K^122) - 1) / 2 points. - tags: - - additive-combinatorics - - polynomial-freiman-ruzsa - - entropy - - approximate-homomorphisms - msc: - - "11B30" - - "11P70" - - "20K01" - - "94A17" - - slug: infinitary-logic - title: Infinitary logic and countable model theory - summary: >- - Formalizes the syntax and semantics of infinitary logic and a broad - collection of its classical theorems: Scott sentences and ranks, Karp's - theorem, model existence, the sharp Morley--Hanf bound, Craig, Lyndon, - and Malitz interpolation, López--Escobar definability, undefinability of - well-ordering, the Silver--Burgess and Morley counting dichotomies, and - an honest coded-fragment interface whose HF compactness theorem is - derived from first-order compactness. - branch: mathematical logic and model theory - entry_module: LeanPool.InfinitaryLogic - authors: - - Cameron Freer - source: - title: Infinitary Logic in Lean 4 - authors: - - Cameron Freer - url: https://github.com/cameronfreer/infinitary-logic - github_repo: cameronfreer/infinitary-logic - commit: 4d0c43c5288952267e7d5478d79302cae17945a9 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - FirstOrder.Language.scottSentence_characterizes - main_results: - - declaration: FirstOrder.Language.scottSentence - informal: >- - Every countable structure in a countable relational language has an - associated infinitary sentence. - - declaration: FirstOrder.Language.scottSentence_characterizes - informal: >- - A countable structure satisfies the Scott sentence of another - countable structure exactly when the two structures are isomorphic. - - declaration: FirstOrder.Language.scottRank - informal: >- - The Scott rank of a countable structure is the supremum of its - element ranks plus one. - - declaration: FirstOrder.Language.scottRank_lt_omega1 - informal: >- - The Scott rank of every countable structure in a countable - relational language is a countable ordinal. - - declaration: FirstOrder.Language.karp_theorem_w - informal: >- - Two relational structures admit a nonempty potential isomorphism - exactly when they agree on every infinitary sentence over every - branching carrier in their shared universe. - - declaration: FirstOrder.Language.model_existence - informal: >- - Every countable set of infinitary sentences in an equality-respecting - consistency property over a countable language has a countable model. - - declaration: FirstOrder.Language.morley_hanf - informal: >- - Beth omega-one is a Hanf bound for every infinitary sentence. - - declaration: FirstOrder.Language.Lomega1omegaHanfNumber_eq_beth_omega1 - informal: >- - The Hanf number of countable-conjunction infinitary logic is exactly - beth omega-one. - - declaration: FirstOrder.Language.exists_small_model_of_hasArbLargeModels - informal: >- - A sentence with arbitrarily large models has, at every infinite - cardinality, a model realizing only countably many complete types. - - declaration: FirstOrder.Language.exists_complete_sentence_of_lomega1omegaSmall - informal: >- - Every infinitary-small model belongs to a complete infinitary - subclass contained in any prescribed sentence it satisfies. - - declaration: FirstOrder.Language.exists_complete_kCategorical_of_hasArbLargeModels - informal: >- - A categorical sentence with arbitrarily large models has a complete - categorical refinement at the same infinite cardinality. - - declaration: FirstOrder.Language.craig_interpolation - informal: >- - Every sentence entailment has an interpolant using only the function - and relation symbols shared by its two endpoints. - - declaration: FirstOrder.Language.craig_pcSeparation - informal: >- - Disjoint projective infinitary classes can be separated by a single - infinitary sentence. - - declaration: FirstOrder.Language.lyndon_interpolation - informal: >- - Every sentence entailment has an interpolant whose positive and - negative relation occurrences lie in the corresponding shared - occurrence sets. - - declaration: FirstOrder.Language.malitz_interpolation - informal: >- - An entailment with universal consequent in a relational language has - a universal interpolant over the shared vocabulary. - - declaration: FirstOrder.Language.exists_model_relPreserving - informal: >- - If a sentence has chains of every countable length, some model - carries a relation-preserving map from the rationals. - - declaration: FirstOrder.Language.wellOrder_type_boundedness - informal: >- - The order types of all well-ordered models of one infinitary sentence - are bounded below omega-one. - - declaration: FirstOrder.Language.wellOrdering_undefinable - informal: >- - No single infinitary sentence defines exactly the well-orders. - - declaration: FirstOrder.Language.lopezEscobar_iff - informal: >- - A class of coded countable structures is Borel and - isomorphism-invariant exactly when it is the model class of one - infinitary sentence. - - declaration: FirstOrder.Language.lopezEscobar_action_iff - informal: >- - The López--Escobar characterization equivalently uses invariance - under the natural permutation action on coded structures. - - declaration: FirstOrder.Language.wellOrderClass_not_measurableSet - informal: >- - The class of coded well-orders is not a Borel set in the logic space. - - declaration: gandy_harrington_for_relation - informal: >- - A Borel equivalence relation on a Polish space with uncountably many - classes has a continuous Cantor injection of pairwise inequivalent - points. - - declaration: FirstOrder.Language.silverBurgessDichotomy - informal: >- - A standard-Borel quotient has at most countably many classes or - exactly continuum many classes. - - declaration: FirstOrder.Language.morley_counting - informal: >- - Assuming the Silver--Burgess dichotomy, the number of isomorphism - classes of coded countable models of an infinitary sentence is at - most aleph-one or exactly the continuum. - - declaration: FirstOrder.Language.iso_borel_of_bounded_scottHeight - informal: >- - Isomorphism among models of an infinitary sentence is Borel whenever - their Scott heights have a common countable bound. - - declaration: FirstOrder.Language.FamilyPresentation - informal: >- - A coded-family presentation packages codes with their indexed - decoding and a certificate for admissible infinitary formation. - - declaration: FirstOrder.Language.CodedFamily - informal: >- - A coded family packages a certified family code with an explicitly - indexed decoding, made unique by the presentation's decoding law. - - declaration: FirstOrder.Language.AdmissibleFragment - informal: >- - An admissible fragment is closed under exactly the coded infinitary - families certified by its presentation. - - declaration: FirstOrder.Language.hfFragment - informal: >- - The hereditarily finite fragment consists of the infinitary formulas - whose syntax contains no infinitary connective. - - declaration: FirstOrder.Language.hfAdmissibleFragment - informal: >- - The hereditarily finite fragment forms an admissible coded fragment - without assuming compactness data. - - declaration: FirstOrder.Language.finitaryFragment_compact - informal: >- - Every finitely satisfiable theory contained in the first-order image - fragment has a model. - tags: - - mathematical-logic - - infinitary-logic - - model-theory - - descriptive-set-theory - msc: - - "03C75" - - "03E15" - - "03C30" - - slug: nash-williams - title: Nash-Williams fronts and 2-better-quasi-orders - summary: >- - Formalizes Nash-Williams fronts through their well-founded prefix trees and ordinal ranks, - proves the finite-color Nash-Williams theorem, and derives the infinite Ramsey theorem for - arbitrary finite arity. The order-theoretic development proves that 2-better-quasi-orders - are well-quasi-orders and establishes closure under finite products, lexicographic sums, - domination of subsets, and pointwise embedding of sequences. Throughout, TwoBQO r is the - good-triple property of an arbitrary binary relation r (every pair-sequence has m < n < l - with f(m,n) related to f(n,l)); it assumes no reflexivity or transitivity of r. - branch: infinitary combinatorics and order theory - entry_module: LeanPool.NashWilliams - authors: - - Yann Pequignot - source: - title: "Towards better: A motivated introduction to better-quasi-orders" - authors: - - Yann Pequignot - doi: 10.4171/EMSS/4-2-2 - url: https://github.com/yannpequignot/TwoBQO - github_repo: yannpequignot/TwoBQO - commit: 8adea92d7fa41700372ba16b9ff816dc7500579b - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - Front.IsFront.nash_williams - - TwoBQO.wellQuasiOrdered - main_results: - - declaration: Front.IsFront.nash_williams - informal: >- - Every two-coloring of a Nash-Williams front on an infinite set has an infinite - restriction whose front is monochromatic. - - declaration: Front.IsFront.nash_williams_fin - informal: >- - Every finite coloring of a Nash-Williams front has an infinite restriction whose - front is monochromatic. - - declaration: infinite_ramsey - informal: >- - Every finite coloring of the r-element subsets of an infinite set of natural numbers - is constant on all r-element subsets of some infinite subset. - - declaration: TwoBQO.wellQuasiOrdered - informal: >- - Every 2-better-quasi-order is a well-quasi-order. - - declaration: TwoBQO.lexSigmaQO - informal: >- - The 2-BQO good-triple property is preserved by lexicographic sums: if the index - relation and every fibre relation have it, so does the lexicographic-sum relation on - the sigma type. This asserts nothing about reflexivity or transitivity of the sum; - those are proved separately (LexSumRelQO.refl and LexSumRelQO.trans) under the - corresponding order hypotheses on the index and fibres. - - declaration: TwoBQO.embedForAll_wqo - informal: >- - If a preorder is a 2-better-quasi-order, then its infinite sequences are - well-quasi-ordered by the relation admitting a strictly monotone reindexing along - which corresponding terms are related. - tags: - - infinitary-combinatorics - - ramsey-theory - - better-quasi-orders - - well-quasi-orders - - ordinal-ranks - msc: - - "03E05" - - "05D10" - - "06A07" - - slug: ehrhart-volume-inequality - title: Ehrhart's sharp volume inequality - summary: >- - Proves Ehrhart's sharp upper bound for the volume of a centered - full-dimensional convex body whose origin is its unique interior lattice - point. The development builds the weighted complex-analytic and variational - machinery needed for the bound and verifies sharpness using the centered - simplex, including its barycenter and exact volume. - branch: convex geometry and geometry of numbers - entry_module: LeanPool.EhrhartVolumeInequality - authors: - - OpenAI - - Dean Cureton - source: - title: Ten Advances in Mathematics and Theoretical Computer Science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - Ehrhart.Volume.ehrhart_volume_inequality_for_sets - main_results: - - declaration: Ehrhart.Volume.ehrhart_volume_inequality_for_sets - informal: >- - In every positive dimension, a compact full-dimensional convex set - with barycenter zero and origin as its unique interior lattice point - has volume at most (n + 1)^n / n!. - - declaration: Ehrhart.SimplexVolume.normalizedVolume_centeredSimplex - informal: >- - The centered dilation of the standard n-simplex has normalized volume - exactly (n + 1)^n / n!, so the volume bound is sharp. - - declaration: Ehrhart.SimplexVolume.barycenter_centeredSimplex - informal: >- - The centered dilation of the standard simplex has barycenter zero. - - declaration: Ehrhart.SimplexVolume.exists_centeredBody_sharp - informal: >- - In every positive dimension there exists a centered convex body with - a unique interior lattice point that attains the sharp volume bound. - tags: - - convex-geometry - - geometry-of-numbers - - lattice-points - - ehrhart-theory - msc: - - "52A40" - - "11H06" - - slug: sphere-packing - title: Sharp asymptotic upper bounds for sphere packing - summary: >- - Formalizes the sharp high-dimensional asymptotics of the unrestricted - Cohn--Elkies linear program and bridges them to sphere-packing density. - The development constructs radial test functions through Mellin, - saddle-point, and harmonic analysis, proves radialization preserves the - unrestricted optimum, and certifies the exact natural- and base-two - exponential rates. - branch: discrete geometry and harmonic analysis - entry_module: LeanPool.SpherePacking - authors: - - OpenAI - - Sidharth Hariharan - - Gareth Ma - - Dean Cureton - source: - title: Ten Advances in Mathematics and Theoretical Computer Science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs/tree/94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - PackingBounds.sharpFullCohnElkiesManuscriptConclusions - main_results: - - declaration: PackingBounds.FullMain.exact_limit - informal: >- - The d-th root of the unrestricted Cohn--Elkies linear-program bound - converges to the sharp constant sqrt(e / (2 pi)). - - declaration: PackingBounds.FullMain.exact_binary_exponent - informal: >- - The base-two logarithm of the unrestricted linear-program bound, - divided by dimension, converges to - -1/2 log_2(2 pi / e). - - declaration: PackingBounds.PackingBridge.sphere_packing_sharp_asymptotic_upper - informal: >- - There is a vanishing error sequence such that, in every positive - dimension, sphere-packing density is at most - (sqrt(e / (2 pi)) plus that error) to the dimension. - - declaration: PackingBounds.sharpFullCohnElkiesManuscriptConclusions - informal: >- - Packages the complete sharp asymptotic conclusions for the - unrestricted program, including quotient-root, natural-logarithmic, - and base-two limits, vanishing-error formulas, a universal bound for - all admissible functions, and a certified decimal exponent interval. - tags: - - sphere-packing - - discrete-geometry - - harmonic-analysis - - linear-programming-bounds - msc: - - "52C17" - - "41A60" - - "42A38" - - slug: bollobas-nikiforov - title: The Bollobás–Nikiforov conjecture - summary: 'Proves the Bollobás–Nikiforov conjecture (2007): every finite noncomplete simple - graph G on at least two vertices satisfies λ₁(G)² + λ₂(G)² ≤ 2 (1 - 1/ω(G)) |E(G)|, where - λ₁ ≥ λ₂ are the two largest adjacency eigenvalues and ω(G) is the clique number. The - inequality is derived from a weighted spectral inequality for symmetric entrywise - nonnegative matrices supported on the edges of G, which in turn rests on a complete - positivity theorem for planar Gram matrices and a rank-two Gram inequality. The rank-two - Gram inequality also identifies the conic parameter of Coutinho, Spier and Zhang with the - clique number. The supporting library develops the completely positive cone and its - closedness, the Motzkin–Straus inequality in completely positive form, total - nonnegativity of truncated-square and convex kernels via Cauchy–Binet, weighted Laplacian - updates, Schur complements, Perron eigenvectors, the Schur product theorem and Cauchy - interlacing over Mathlib.' - branch: spectral graph theory - entry_module: LeanPool.BollobasNikiforov - authors: - - Gabriel Coutinho - - Yinchen Liu - - Thomás Jung Spier - - Quanyu Tang - - Shengtong Zhang - source: - title: The Bollobás–Nikiforov inequality for nonnegative edge weights - authors: - - Gabriel Coutinho - - Yinchen Liu - - Thomás Jung Spier - - Quanyu Tang - - Shengtong Zhang - url: https://github.com/ShengtongZhang-alt/BN - github_repo: ShengtongZhang-alt/BN - commit: edb5259dfd055ea31b4c46ac9ea4d33a758c2b99 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - BollobasNikiforov.lambda1_sq_add_lambda2_sq_le - main_results: - - declaration: BollobasNikiforov.lambda1_sq_add_lambda2_sq_le - informal: For every finite simple graph G on at least two vertices that is not complete, - the two largest adjacency eigenvalues satisfy λ₁(G)² + λ₂(G)² ≤ 2 (1 - 1/ω(G)) |E(G)|, - with ω(G) the clique number and |E(G)| the number of edges. - - declaration: BollobasNikiforov.weighted - informal: If B is a real symmetric matrix, indexed by a finite type with at least two - elements, that is entrywise nonnegative, has zero diagonal and vanishes off the edges - of a graph G on that type, then the sum of the squares of its two largest positive - eigenvalues is at most (1 - 1/ω(G)) times its squared Frobenius norm. - - declaration: BollobasNikiforov.matrix_theorem - informal: If planar vectors z₁, …, zₙ lie in a closed half-plane through the origin and - X is their Gram matrix, then X ∘ X plus the weighted Laplacian with weights X_ij² on - the pairs i < j with X_ij < 0 is completely positive. - - declaration: BollobasNikiforov.gram_le - informal: For every real positive semidefinite matrix X of rank at most two, - ∑_{i,j} (A_G)_ij (X_ij)₊² ≤ (1 - 1/ω(G)) ‖X‖_F², where (·)₊ is the positive part. - - declaration: BollobasNikiforov.chiVec3_eq_cliqueNum - informal: For every graph G on a nonempty finite vertex type, the rank-two conic - parameter χ''_{vec,3}(G) of Coutinho, Spier and Zhang, defined as the supremum of - ‖X‖_F² over positive semidefinite X of rank at most two with nonnegative entries on - the edges and unit off-edge mass, equals the clique number ω(G). - - declaration: BollobasNikiforov.motzkinStraus - informal: Motzkin–Straus inequality; for every entrywise nonnegative vector y, - yᵀ A_G y ≤ (1 - 1/ω(G)) (∑ᵢ yᵢ)². - - declaration: BollobasNikiforov.posSemidef_hadamard - informal: Schur product theorem over the reals; the Hadamard product of two positive - semidefinite matrices is positive semidefinite. - - declaration: BollobasNikiforov.isClosed_isCompletelyPositive - informal: The cone of completely positive real matrices is closed. - - declaration: BollobasNikiforov.IsTotallyNonneg.mul - informal: Cauchy–Binet; the product of two totally nonnegative real matrices is totally - nonnegative. - - declaration: BollobasNikiforov.exists_nonneg_eigenvector_lambdaMax - informal: Perron; a real symmetric entrywise nonnegative matrix on a nonempty finite - index type has an entrywise nonnegative unit eigenvector for its largest eigenvalue. - tags: - - spectral-graph-theory - - clique-number - - completely-positive-matrices - - motzkin-straus - msc: - - "05C50" - - "15B48" - - "05C35" - - "15A18" - - slug: minimum-degree-matchings - title: BKLO simultaneous perfect matchings and spread matchings - summary: >- - Formalizes the r = 2 specialization of the simultaneous factor-selection step in BKLO - Lemma 10.7: under quantitative parity, minimum-degree, codegree, and incidence hypotheses, - a linearly sized, overlapping family of neighbourhood graphs admits perfect matchings whose - edge sets are pairwise disjoint. The proof replaces the published randomized greedy process - by a deterministic pessimistic-estimator sweep. Supporting results develop perfect, - near-perfect, edge-disjoint, and quantitatively weighted spread matchings from sharp - minimum-degree hypotheses. - branch: graph theory - entry_module: LeanPool.MinimumDegreeMatching - authors: - - Juan Pablo Traverso Gianini - - Aristotle - source: - title: >- - Edge-decompositions of graphs with high minimum degree - authors: - - Ben Barber - - Daniela Kühn - - Allan Lo - - Deryk Osthus - doi: 10.1016/j.aim.2015.09.032 - arxiv: "1410.5750" - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/tree/6736c816e1f9cd105c2295a8e4716ad55609ccb0/preprints/PAPER_III/05_formalization/lean_v1.4_freeze - github_repo: jtraverso/erdos-81-chordal-clique-partitions - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - BKLOK2.lemma107K2_holds - main_results: - - declaration: BKLOK2.lemma107K2_holds - informal: >- - Specializes BKLO Lemma 10.7 to r = 2 in its apex-neighbourhood configuration: for every - sufficiently large admissible configuration, all indexed neighbourhood graphs admit - perfect matchings whose edge sets are pairwise disjoint. The proof is deterministic. - - declaration: SimpleGraph.exists_isPerfectMatching_of_minDegree - informal: >- - Every finite simple graph of even order whose minimum degree is at least half its - order has a perfect matching. - - declaration: SimpleGraph.exists_isMatching_compl_singleton_of_minDegree - informal: >- - Every finite simple graph of odd order n and minimum degree at least (n - 1) / 2 - has a matching whose vertex set is the complement of a singleton. - - declaration: SimpleGraph.exists_involutions_pairwise_ne - informal: >- - If every vertex of a finite even vertex set has at least half the set size plus t - neighbors inside the set, then the graph contains t + 1 pairwise edge-disjoint - perfect matchings of that set, represented by an indexed family of partner involutions. - These involutions are pairwise distinct when the vertex set is nonempty. - - declaration: SimpleGraph.exists_spread_involution - informal: >- - Under the same minimum-degree condition with slack t, every nonnegative edge-weight - function admits a perfect matching whose weight is at most 1 / (t + 1) times the - total weight over ordered pairs in the vertex set. - tags: - - graph-theory - - perfect-matchings - - edge-disjoint-matchings - - pseudorandomness - msc: - - "05C70" - - slug: non-sofic-group - title: Existence of a finitely presented non-sofic group - summary: >- - Constructs a finitely presented group that is not sofic, resolving the - question whether every group admits arbitrarily accurate finite - permutation approximations. The development combines an explicit - elementary group with property T, a finite-table construction of a - finitely presented group, finite-model expansion, and a local obstruction - to sofic approximation. - branch: geometric and combinatorial group theory - entry_module: LeanPool.NonSoficGroup - authors: - - OpenAI - - Dean Cureton - source: - title: Ten Advances in Mathematics and Theoretical Computer Science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - SoficGroups.exists_finitelyPresented_nonsofic_group - main_results: - - declaration: SoficGroups.exists_finitelyPresented_nonsofic_group - informal: >- - There exists a group admitting a finite presentation but no sofic - approximation. - tags: - - group-theory - - sofic-groups - - property-t - - finite-approximations - msc: - - "20F65" - - "20E22" - - slug: infinite-connes-rigidity - title: Infinite counterexamples to Connes's rigidity conjecture - summary: >- - Constructs one finitely generated property-(T) ICC group and an infinite - family of finitely generated property-(T) ICC groups that are pairwise - nonisomorphic and are each nonisomorphic to the first group, while all of - their tracial group von Neumann algebras are normally star-isomorphic. - This unconditional infinite-family result is independent of the pooled - conditional two-group Connes-rigidity formalization. - branch: operator algebras and geometric group theory - entry_module: LeanPool.InfiniteConnesRigidity - authors: - - OpenAI - - Dean Cureton - source: - title: Ten advances in mathematics and theoretical computer science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - ConnesRigidity.infiniteConnesRigidity - main_results: - - declaration: ConnesRigidity.infiniteConnesRigidity - informal: >- - Registry-friendly alias of the exact canonical infinite-family - Connes-rigidity counterexample theorem below. - - declaration: ConnesRigidity.exists_infinite_pairwise_nonisomorphic_propertyT_icc_groups_with_isomorphic_factors - informal: >- - There exist a finitely generated property-(T) ICC group and an - infinite sequence of finitely generated property-(T) ICC groups such - that all their tracial group von Neumann algebras are normally - star-isomorphic, although the sequence is pairwise nonisomorphic and - none of its groups is isomorphic to the distinguished group. - tags: - - operator-algebras - - von-neumann-algebras - - property-t - - group-theory - - connes-rigidity - msc: - - "46L10" - - "20F65" - - "22D10" - - slug: metric-codes - title: Improved asymptotic bounds for binary and spherical codes - summary: >- - Formalizes asymptotic coding bounds obtained from multilevel harmonic - analysis, branching rules, and spectral decompositions. The development - proves a binary-code exponent strictly below the MRRW rate, constructs a - strict hierarchy of spherical-code bounds, and derives the published - 0.39661 asymptotic kissing-number exponent. This import incorporates - substantial proof and performance optimizations by Dean Cureton. - branch: coding theory and discrete geometry - entry_module: LeanPool.MetricCodes - authors: - - OpenAI - - Dean Cureton - source: - title: Ten Advances in Mathematics and Theoretical Computer Science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs/tree/94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - MetricCodes.Johnson.main_binary_theorem - main_results: - - declaration: MetricCodes.Johnson.main_binary_theorem - informal: >- - For every relative distance strictly between zero and one half, the - combined variational upper bound on the asymptotic binary-code rate is - valid and is strictly smaller than the MRRW rate. - - declaration: MetricCodes.Spherical.HigherHierarchy.main_general - informal: >- - Every feasible finite interlacing certificate gives an eventual - exponential upper bound for spherical codes, while the spherical-code - rate is bounded by the closed hierarchy's variational rate. - - declaration: MetricCodes.Spherical.HigherHierarchy.strict_hierarchy - informal: >- - At every angle parameter strictly between zero and one, successive - levels of both the global and localized hierarchies improve strictly, - with the localized hierarchy also strictly improving the row and - classical bounds. - - declaration: MetricCodes.Spherical.HigherHierarchy.NumericalMaximum.eventually_kissingNumber_lt_published - informal: >- - In all sufficiently large dimensions, the kissing number is at most - 2 raised to 0.39661 times the dimension. - tags: - - coding-theory - - spherical-codes - - kissing-number - - harmonic-analysis - - asymptotic-bounds - msc: - - "94B65" - - "52C17" - - "41A60" - - slug: quantum-parallel-repetition - title: Quantum parallel repetition - summary: >- - Proves a quantitative parallel-repetition theorem for finite two-player - quantum games: whenever the entangled value is below one, the entangled - values of parallel repetitions decay exponentially, with an explicit - universal exponent in the game gap and answer-alphabet sizes. - branch: quantum information theory - entry_module: LeanPool.QuantumParallelRepetition - authors: - - OpenAI - - Dean Cureton - source: - title: Ten Advances in Mathematics and Theoretical Computer Science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - QuantumParallelRepetition.distributionUniformExponential - main_results: - - declaration: QuantumParallelRepetition.distributionUniformExponential - informal: >- - There is a universal positive constant giving an explicit exponential - upper bound on the repeated entangled value of every finite game with - positive gap and nonempty answer alphabets. - - declaration: QuantumParallelRepetition.standardQuantumParallelRepetition - informal: >- - Every finite two-player game whose entangled value is below one has - exponentially decaying entangled value under parallel repetition. - tags: - - quantum-information - - nonlocal-games - - parallel-repetition - - theoretical-computer-science - msc: - - "81P68" - - "68Q12" - - slug: gap-cvp - title: Polynomial-factor hardness of the closest vector problem - summary: >- - Formalizes polynomial-factor NP-hardness of the closest vector problem - through an explicit reduction from 3-SAT. The development also derives - NP-hardness of binary nearest codeword, binary syndrome decoding, and - finite p-norm variants of GapCVP. It includes machine-level polynomial-time - bounds, binary encodings, affine systems, Reed--Solomon-style interpolation, - and the lattice and coding-theory reductions needed by the final results. - branch: computational complexity and lattice problems - entry_module: LeanPool.GapCVP - authors: - - OpenAI - - Dean Cureton - source: - title: Ten Advances in Mathematics and Theoretical Computer Science - authors: - - OpenAI - url: https://github.com/openai/ten-proofs - github_repo: openai/ten-proofs - commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - GapCVP.Comparator.gapCVP400IsNPHard - main_results: - - declaration: GapCVP.Comparator.gapCVP400IsNPHard - informal: >- - GapCVP in the Euclidean norm with approximation factor n^(1/400), - where n is the lattice dimension, is NP-hard as a promise problem: - every NP language reduces in polynomial time to the promise problem - on encoded well-formed instances (positive dimension, nonsingular - integer basis, positive rational radius) with an integer target, - whose YES instances have a lattice vector within the radius of the - target and whose NO instances have every lattice vector at distance - greater than n^(1/400) times the radius. - - declaration: GapCVP.Comparator.binaryNearestCodewordIsNPHard - informal: >- - Binary nearest codeword is NP-hard as a promise problem. - - declaration: GapCVP.Comparator.binarySyndromeDecodingIsNPHard - informal: >- - Binary syndrome decoding is NP-hard as a promise problem. - - declaration: GapCVP.Comparator.finitePNormGapCVPIsNPHard - informal: >- - For every rational p at least one, the finite p-norm version of - GapCVP is NP-hard as a promise problem. - tags: - - computational-complexity - - np-hardness - - lattice-problems - - coding-theory - msc: - - "68Q17" - - "68Q25" - - "11H06" - - "94B35" - - slug: navier-stokes-and-euler - title: Finite-time breakdown for Navier–Stokes and Euler - summary: Formalizes forced three-dimensional Navier–Stokes breakdown on Euclidean space and the - periodic torus for every positive viscosity, and finite-time singularity formation for unforced - Euler from smooth, compactly supported, divergence-free initial data. The Euler result specifies - maximal Sobolev lifespan, divergence of the velocity C¹ norm, and divergence of the time integral - of the vorticity norm. - branch: partial differential equations and mathematical fluid dynamics - entry_module: LeanPool.NavierStokesAndEuler - authors: - - OpenAI - source: - title: Finite time blowup for Navier–Stokes and Finite time blowup for the Euler equation - authors: - - OpenAI - url: https://github.com/openai/NavierStokesAndEuler - github_repo: openai/NavierStokesAndEuler - commit: 8937a8f4cbc7abaab5e9e97d1cc7f5d2319d9538 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - NavierStokes.Comparator.navier_stokes_breakdown_R3 - - Euler.euler_breakdown_R3 - main_results: - - declaration: NavierStokes.Comparator.navier_stokes_breakdown_R3 - informal: For every positive viscosity, there are smooth divergence-free initial data and smooth - forcing on ℝ³, with the prescribed spatial and temporal decay, admitting no global smooth - Navier–Stokes solution with uniformly bounded kinetic energy. - - declaration: NavierStokes.Comparator.navier_stokes_breakdown_periodic - informal: For every positive viscosity, there are smooth periodic divergence-free initial data - and smooth periodic forcing with the prescribed temporal decay, admitting no global smooth - periodic Navier–Stokes solution. - - declaration: Euler.euler_breakdown_R3 - informal: There exist smooth divergence-free initial data on ℝ³ whose spatial derivatives decay - faster than every polynomial and which admit no global smooth unforced Euler solution with - uniformly bounded kinetic energy. - - declaration: Euler.exists_compact_smooth_euler_singularity - informal: There exist nonzero smooth compactly supported divergence-free initial data, a maximal - time 0 < Tstar ≤ 1, and an unforced Euler solution on [0, Tstar) in the all-order Sobolev - class with bounded kinetic energy. Smooth Sobolev solutions on [0, T] exist exactly when T - < Tstar. The velocity C¹ norm has infinite left limsup at Tstar, and the time integral of - the spatial supremum of the vorticity norm diverges. - tags: - - navier-stokes - - euler-equations - - finite-time-blowup - - fluid-dynamics - - sobolev-spaces - msc: - - 35Q30 - - 35Q31 - - 35B44 - - slug: exponential-chaos - title: Chaos and the Julia set of the complex exponential - summary: >- - Formalizes the dynamics of z ↦ exp(z): escaping points, points with dense - forward orbit, and repelling periodic points are dense in the complex - plane; the map is chaotic in Devaney's sense and its Julia set is the - whole plane. The development proves eventual covering of every compact - set omitting zero, constructs contracting inverse branches, and rules - out every locally uniform spherical subsequential limit on nonempty - open sets. It includes general normal-sequence restriction lemmas, - quantitative open mapping, and a two-target obstruction for arbitrary - separated uniform codomains. The Lean proofs were primarily generated - by AI under Lasse Rempe's direction. The initial six-lemma architecture - adapts John Harrison's HOL Light formalization, credited with its - retained BSD notice in the entry module. - branch: complex dynamics - entry_module: LeanPool.ExpChaotic - authors: - - Lasse Rempe - source: - title: 'The Exponential Map is Chaotic: An Invitation to Transcendental Dynamics' - authors: - - Zhaiming Shen - - Lasse Rempe-Gillen - arxiv: '1408.1129' - doi: 10.4169/amer.math.monthly.122.10.919 - url: https://github.com/LR-UK/exp-chaotic - github_repo: LR-UK/exp-chaotic - commit: 303b2d3d1121ebc3a86399de8f1e71c17a8b6dd8 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - ExpChaotic.devaney_chaotic - - ExpChaotic.juliaSet_exp - main_results: - - declaration: ExpChaotic.devaney_chaotic - informal: >- - The complex exponential is continuous, has dense periodic points, - and the monoid action generated by its iterates is topologically - transitive on the complex plane. - - declaration: ExpChaotic.juliaSet_exp - informal: >- - The Julia set of the complex exponential is the whole plane, where - the Fatou set is defined by local normality of sphere-valued - iterates with arbitrary locally uniform sphere-valued limits. - - declaration: ExpChaotic.dense_repellingPeriodicSet - informal: >- - Points p with a positive period n and derivative norm - greater than one for the nth exponential iterate are dense in - the complex plane. - - declaration: ExpChaotic.eventually_covers_compact - informal: >- - For every nonempty open set U in the complex plane and every - compact set K omitting zero, all sufficiently large iterates of - the exponential map send U onto a set containing K. - - declaration: ExpChaotic.dense_escapingSet - informal: >- - Points whose forward exponential orbit eventually leaves every - centered Euclidean ball are dense in the complex plane. - - declaration: ExpChaotic.not_countable_denseOrbitSet - informal: >- - The set of complex points with dense forward exponential orbit - is uncountable. - - declaration: ExpChaotic.not_tendstoLocallyUniformly_sphericalExpIterate - informal: >- - On every nonempty open set in the complex plane, no strictly - increasing subsequence of exponential iterates converges locally - uniformly to any function into the Riemann sphere. - tags: - - complex-dynamics - - exponential-map - - julia-sets - - chaos - - holomorphic-functions - msc: - - 37F10 - - 30D05 - - slug: chvatal-conjecture - title: Chvátal's conjecture and sharp Boolean correlation - summary: >- - Formalizes Chang, Liu, and Liu's proof that a hereditary family on a - finite nonempty ground set has a largest intersecting subfamily that is - a star. Develops the sharp Boolean Fourier correlation inequality, - its antipodal and quadratic forms, sharpness examples, and the - weighted star theorem for nonnegative decreasing weights. - branch: extremal combinatorics - entry_module: LeanPool.Chvatal - authors: - - Boon Suan Ho - - Chvatal formalization contributors - source: - title: A proof of Chvátal's conjecture via a sharp correlation inequality - authors: - - Fan Chang - - Hong Liu - - Miao Liu - arxiv: '2609.19123v1' - url: https://github.com/boonsuan/chvatal - github_repo: boonsuan/chvatal - commit: c19ed3aaac9e42d446f963a862d39d8a09eddbf9 - announcement_date: '2026-09-17T03:58:22Z' - announcement_url: https://github.com/boonsuan/chvatal/commit/c46f715af9d400523eb0d4c9c0d2335abbdc9fdd - notes: >- - GPT-6 generated the formalization under Boon Suan Ho's direction; - the mathematical proof is due to Chang, Liu, and Liu. No human expert - review or source-author endorsement is claimed. The upstream MIT - copyright and permission notice is retained in every imported file. - license: MIT - status: verified - provenance: AI - main_declarations: - - Chvatal.chvatal - - Chvatal.sharp_correlation - main_results: - - declaration: Chvatal.chvatal - informal: >- - Every intersecting subfamily A of a hereditary finite family D - on a finite nonempty ground type has cardinality at most that - of some star of D. - - declaration: Chvatal.exists_largest_intersecting_star - informal: >- - One star of D is itself intersecting and simultaneously dominates - all intersecting subfamilies of the hereditary family D in size, - on a finite nonempty ground type. - - declaration: Chvatal.sharp_correlation - informal: >- - For increasing Boolean functions f and g on any finite cube, the - sum of squared Fourier coefficients of g weighted by the largest - influence of f on each nonempty index is at most the harmonic - mean of Cov(f,g) and Cov(f,g*), with zero-denominator value zero. - - declaration: Chvatal.two_spectral_le_quadratic_covariance - informal: >- - For increasing Boolean f and g and every real t, twice their - squared Fourier energy over index pairs with odd intersection is - bounded by t^2 Cov(f,g) + (1-t)^2 Cov(f,g*). - - declaration: Chvatal.antipodal_correlation - informal: >- - If g is also self-dual and the finite coordinate type is nonempty, - Cov(f,g) is at least one quarter of f's minimum coordinate influence. - - declaration: Chvatal.andFunction_sharp - informal: >- - The AND function attains equality in the sharp correlation bound - for every increasing Boolean second function, including constants. - - declaration: Chvatal.exists_largest_weighted_intersecting_star - informal: >- - For any nonnegative antitone real weight on subsets of a finite - nonempty ground type, one full-cube star is intersecting and its - total weight dominates every intersecting family. - tags: - - extremal-combinatorics - - intersecting-families - - Fourier-analysis - msc: - - 05D05 - - 60E15 - - slug: asymptotic-moore-bound - title: Asymptotically attaining the Moore bound - summary: >- - Proves that, for each fixed positive diameter k, the largest order of a - finite simple graph with maximum degree at most d and extended diameter - at most k is asymptotic to d^k as d tends to infinity. Constructs regular - halved-flag graphs over every finite field, with exact order and diameter, - a sharp degree cap, and matching asymptotics through all prime powers. - Also proves the corresponding lower-limit bound for the edge extremum - defined using line-graph diameter. Reusable infrastructure includes common - bases for complete flags, odd-even sorting routes, exact flag enumeration, - projective-line completion counts, and bipartite graph expansion. The - analytic bridge retains the Wiener-Ikehara and prime-counting dependency - closure from the Apache-2.0 PrimeNumberTheoremAnd project. The paper - describes AI-assisted formalization supervised by the human authors. - branch: extremal graph theory - entry_module: LeanPool.MooreBound - authors: - - Wouter Cames van Batenburg - - Samuel Korsky - - PrimeNumberTheoremAnd contributors - source: - title: Asymptotically attaining the Moore bound - authors: - - Wouter Cames van Batenburg - - Samuel Korsky - arxiv: '2608.03965' - url: https://github.com/woutercvb/wewantmoore - github_repo: woutercvb/wewantmoore - commit: d59bd80ea93fabb9faf769e790ab47692645e022 - announcement: https://arxiv.org/abs/2608.03965v1 - announcement_date: '2026-08-04T17:30:09Z' - release: https://github.com/woutercvb/wewantmoore/releases/tag/v1.0 - release_date: '2026-08-05T01:19:30Z' - third_party_url: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd - third_party_commit: 0c7abf7be7765dc5ffd21afc1c37b018199ec3c9 - third_party_license: Apache-2.0 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - MooreBound.DegreeDiameter.theorem_1_1 - main_results: - - declaration: MooreBound.DegreeDiameter.theorem_1_1 - informal: >- - For each positive natural k, n(k,d)/d^k tends to 1 as the natural - degree bound d tends to infinity. Here n(k,d) is the maximum number - of vertices of a finite simple graph with maximum degree at most d - and extended diameter at most k. - - declaration: MooreBound.DegreeDiameter.corollary_1_2 - informal: >- - For each natural ell at least 2, the lower limit of h(ell,d)/d^ell - is at least 1. Here h(ell,d) is one plus the maximum number of edges - of a finite simple graph with maximum degree at most d whose line - graph has extended diameter at most ell. - - declaration: MooreBound.DegreeDiameter.proposition_3_1 - informal: >- - For every positive k and prime power q, the concrete halved-flag graph - is regular, has order [2k+1]_q!/(q+1)^k, degree at most - (q+1)^k((q+1)^k-1), and extended diameter exactly k. Along all prime - powers q tending to infinity, its actual degree divided by this cap - and its order divided by the kth power of its actual degree both - tend to 1. - - declaration: MooreBound.DegreeDiameter.alternating_route - informal: >- - Any two complete flags in an n-dimensional vector space over a - division ring admit an n-step route that changes only ranks of the - prescribed alternating parity at each step. - - declaration: MooreBound.prime_between - informal: >- - For every positive real epsilon, every sufficiently large real x - admits a prime strictly between x and (1+epsilon)x. - tags: - - graph-theory - - finite-geometry - - asymptotics - - prime-number-theorem - msc: - - '05C35' - - '05B25' - - '11N05' - - slug: language-generation-characterization - title: Finite witnesses and the width hierarchy for language generation - summary: >- - Characterizes generation in the limit for arbitrary families of infinite - languages over any countably infinite universe by finite positive witnesses. - Ordered-history generators admit a single set-driven normalization, chosen - before the target language. The separation-width invariant takes every value - 0, 1, 2, ..., omega, omega+1; generation is equivalent to width at most omega. - Reusable infrastructure includes locking witnesses, positive separation, - bounded diagonal capture over arbitrary universes, transport under equivalences, - and finite-query normalization on natural numbers. The formalized scope is - the semantic characterization and hierarchy, not the paper's additional - density-one or Turing-computability results in Appendices D and E. - branch: formal language theory and learning theory - entry_module: LeanPool.LanguageGeneration - authors: - - Xiaoyu Li - - Andi Han - - Jiaojiao Jiang - - Junbin Gao - - Shuangping Li - - Peng Zhang - source: - title: 'Characterizing Language Generation in the Limit: Finite Witnesses and a Separation-Width Hierarchy' - authors: - - Xiaoyu Li - - Andi Han - - Jiaojiao Jiang - - Junbin Gao - arxiv: '2609.10525' - url: https://github.com/xiaoyulics/language-generation-characterization - github_repo: xiaoyulics/language-generation-characterization - commit: 4f7d3f2148017ae15135ba39ddc4d655bb175ab6 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - GenLimit.FiniteWitness.Simplified.ordinary_iff_finiteWitnesses - main_results: - - declaration: GenLimit.FiniteWitness.Simplified.ordinary_iff_finiteWitnesses - informal: >- - For any family H of infinite languages over a countably infinite universe, - there exists one ordered-history generator that eventually outputs an - unseen target element on every exact presentation of every target in H - if and only if H admits a finite positive witness assignment whose - nonempty active subfamilies always have infinite common intersection. - - declaration: GenLimit.FiniteWitness.Simplified.universal_normalization - informal: >- - Every ordered-history generator over a countably infinite universe has - one set-driven normalization that locks on every infinite target on - which the original generator succeeds on all exact presentations. - The normalization is chosen before the target. - - declaration: GenLimit.FiniteWitness.ordinary_iff_width - informal: >- - A family of infinite languages over a countably infinite universe is - generatable in the limit exactly when its separation width is at most omega. - - declaration: GenLimit.FiniteWitness.full_range - informal: >- - Over every countably infinite universe, every value in - WithTop (WithTop Nat), representing 0, 1, 2, ..., omega, omega+1, - is the separation width of some family of infinite languages. - - declaration: GenLimit.FiniteWitness.bounded_capture - informal: >- - Over an arbitrary universe, every sequence of uniformly bounded finite - sets has infinitely many terms contained in one set D such that D - contains none of a prescribed countable list of infinite cores. - - declaration: GenLimit.FiniteWitness.executable_universal_normalization - informal: >- - The finite-search normalization on natural numbers locks on every - infinite language on which the supplied generator eventually succeeds - on every exact presentation. This is a semantic correctness theorem - for the defined procedure, without a formal Turing-computability claim. - tags: - - language-generation - - learning-theory - - finite-witnesses - - combinatorics - msc: - - 68Q32 - - 68Q45 - - slug: serre-mass-formula - title: Serre's mass formula for totally ramified local-field extensions - summary: >- - Proves both of Serre's mass formulas for totally ramified separable - extensions of a nonarchimedean local field of any positive degree, - together with the discriminant bound, tame-ramification criterion, - finiteness dichotomy, convergence, and automorphism-weighted orbit count. - The proof develops Eisenstein monogenicity, quantitative Newton-Hensel - lifting, local constancy of root counts, and Haar measures of lattices - over discrete valuation rings. - branch: local fields and ramification theory - entry_module: LeanPool.MassFormula - authors: - - Hyeon Seung-Hyeon - source: - title: Une «formule de masse» pour les extensions totalement ramifiées de degré donné d'un corps local - authors: - - Jean-Pierre Serre - url: https://gallica.bnf.fr/ark:/12148/bpt6k6234149b/f323.item - github_repo: 0stellensatz/MassFormula - commit: 7fa41a621f724f110183904d3fd19c6730a932ad - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - MassFormula.tsum_one_div_q_pow_c - main_results: - - declaration: MassFormula.tsum_one_div_q_pow_c - informal: >- - For every nonarchimedean local field K with residue-field cardinality q - and every positive integer n, the sum of q^(-c(L)) over the totally - ramified degree-n subextensions L of a separable closure of K is n, - where c(L) is the discriminant exponent minus n plus one. - - declaration: MassFormula.tsum_one_div_w_mul_q_pow_c - informal: >- - For every nonarchimedean local field K, positive n, and set of - representatives of K-isomorphism classes of totally ramified separable - degree-n extensions, the sum of 1/(|Aut_K(L)| q^c(L)) is one. - - declaration: MassFormula.sigma_infinite_iff - informal: >- - For positive n, the set of totally ramified separable degree-n subextensions is infinite - exactly when K and its residue field have the same characteristic p - and p divides n. - - declaration: MassFormula.sub_one_le_d - informal: For positive n, the discriminant exponent of a totally ramified degree-n extension is at least n-1. - - declaration: MassFormula.c_eq_zero_iff - informal: >- - For positive n and a totally ramified degree-n extension, the wild discriminant exponent - vanishes exactly when the residue characteristic does not divide n. - - declaration: MassFormula.summable_one_div_q_pow_c - informal: For positive n, the real-valued series of the unweighted mass terms is summable. - - declaration: MassFormula.ncard_isomorphic_mul_w - informal: >- - For a nonarchimedean local field K and a totally ramified separable - extension L of positive degree n, the number of K-isomorphic copies of L - inside a separable closure, multiplied by |Aut_K(L)|, is n. - tags: - - number-theory - - local-fields - - ramification - - discriminants - - haar-measure - msc: - - 11S15 - - 11S05 - - slug: schoenflies - title: Jordan–Schönflies theorem - summary: Formalizes the Jordan curve theorem and the extension of every homeomorphism between Jordan curves to a plane homeomorphism. - branch: geometric topology - entry_module: LeanPool.Schoenflies - authors: - - Álvaro Begué - source: - title: The Jordan–Schönflies theorem in Lean - url: https://github.com/alonamaloh/schoenflies-lean - github_repo: alonamaloh/schoenflies-lean - commit: 05a43d29cde026618777db3d4e4316204ccca237 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Schoenflies.jordan_schoenflies_of_homeomorph - main_results: - - declaration: Schoenflies.jordan_schoenflies_of_homeomorph - informal: Every homeomorphism between two Jordan curves in the plane extends to a self-homeomorphism of the plane. - - declaration: Schoenflies.jordan_schoenflies - informal: A homeomorphism between Jordan curves extends to a global homeomorphism agreeing with it on the source curve. - - declaration: Schoenflies.forall_isSeparating_of_isJordanCurve - informal: Every Jordan curve separates the plane into an inside and an outside region. - tags: - - jordan-curve - - schoenflies-theorem - - geometric-topology - - homeomorphism - msc: - - "57K10" - - "54C25" - - slug: pdl-craig-interpolation - title: Craig interpolation and Beth definability for propositional dynamic logic - summary: >- - Proves Craig interpolation for full propositional dynamic logic, including - tests and Kleene star, by constructing interpolants from uniform cyclic - tableaux. Interpolants use only atomic propositions and atomic programs - common to the two formulas. The development also proves the local Beth - definability property for a propositional letter, and supplies reusable - syntax, Kripke semantics, Fischer-Ladner closure, tableau soundness and - completeness, finite reachability and game-theoretic infrastructure. - The results assert existence; no executable extraction algorithm or - complexity bound is claimed. Human contributors and Aristotle jointly - developed the formalization. The completeness modules reuse the finite - game determinacy theorem already pooled in `LeanPool.Lean4GlCoalgebras` - instead of duplicating it. - branch: modal logic - entry_module: LeanPool.PDL - authors: - - PDL formalization contributors - contributors: - - Malvin Gattinger - - Madeleine Gignoux - - Noam Cohen - - Haitian Wang - - Wietse Bosman - - Xiaoshuang Yang - - Amos Nicodemus - - Andrés Goens - - DdosSantosGomes - - Eshel Yaron - - Dionysis Yusofy - source: - title: 'Propositional Dynamic Logic has Craig Interpolation: a tableau-based proof' - authors: - - Manfred Borzechowski - - Malvin Gattinger - - Helle Hvid Hansen - - Revantha Ramanayake - - Francisco Trucco Dalmas - - Yde Venema - arxiv: '2503.13276' - url: https://github.com/m4lvin/lean4-pdl - github_repo: m4lvin/lean4-pdl - commit: ec3b050e59cd829a350e92c512951a381a80cddd - announcement: https://leanprover.zulipchat.com/#narrow/channel/579630-Project-announcements/near/622297905 - announcement_date: '2026-09-07T14:27:11Z' - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - PDL.interpolation - - PDL.beth - main_results: - - declaration: PDL.interpolation - informal: >- - If the implication from PDL formula phi to psi is valid in every - Kripke model, there exists a PDL formula theta whose atomic proposition - and program vocabulary is contained in the intersection of theirs, - such that phi implies theta and theta implies psi in every model. - - declaration: PDL.beth - informal: >- - If a PDL formula phi implicitly determines a propositional letter p - (all two-letter substitutions agree wherever both substituted formulas - hold), there is a formula psi using only phi's vocabulary other than p - such that phi entails that p is equivalent to psi. - - declaration: PDL.consIffSat - informal: >- - A free sequent is tableau-consistent if and only if it is satisfiable - in a Kripke model. - tags: - - modal-logic - - propositional-dynamic-logic - - craig-interpolation - - beth-definability - - tableaux - msc: - - 03B45 - - 03B70 - - 03F03 - - slug: nivat-conjecture - title: Nivat's conjecture - summary: Formalizes the proof of Nivat's conjecture found by GPT-6 Pro. A configuration - on the integer lattice over a finite alphabet whose number of distinct patterns on - some m-by-n rectangle is at most m times n has a nonzero period. Develops rational - alphabet labelings, the Laurent-polynomial shift action and its annihilator ideals, - the exact line ideal of a half-plane-supported difference of orbit-closure points, - complexity descent to a smaller rectangle, and Szabados's two-factor theorem for - finite-range configurations. - branch: symbolic dynamics - entry_module: LeanPool.Nivat - authors: - - Boon Suan Ho - source: - title: Pattern complexity and Nivat's conjecture - authors: - - GPT-6 Pro - - Boon Suan Ho - url: https://github.com/boonsuan/nivat - github_repo: boonsuan/nivat - commit: 84fe839635bdebb7d5e80c209b4f578a0c767fcf - announcement_date: '2026-09-14T04:54:56Z' - announcement_url: https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-14-000003&version=1 - palomar: PALOMAR-2026-09-14-000003 - notes: The proof and manuscript were produced by GPT-6 Pro and the Lean development - by GPT-6 Astra in Codex, both prompted and directed by Boon Suan Ho, who claims no - human mathematical authorship. The manuscript ships in the upstream repository as - paper/nivat.tex. The Lean sources are identical to the commit archived by Palomar - on 2026-09-14 (0d94547e); later upstream commits only touched the README and the - paper's abstract. No human expert review is claimed upstream; Bryna Kra's guest - post on Terence Tao's blog (2026-09-13) discusses the arrival of AI proofs of the - conjecture. The upstream MIT copyright and permission notice is retained in every - imported file. - license: MIT - status: verified - provenance: AI - main_declarations: - - Nivat.nivat - - NivatSubmission.nivat - main_results: - - declaration: Nivat.nivat - informal: For every finite alphabet A and configuration c from the integer lattice - to A, if for some positive m and n the number of distinct patterns of c on the - m-by-n rectangle, counted over all integer translations, is at most m times n, - then there is a nonzero lattice vector h with c(z + h) = c(z) for every z. - - declaration: NivatSubmission.nivat - informal: The same statement written with Mathlib vocabulary only, as registered - with Palomar. The pattern set is the range of the restriction of translates of - c to the product of the integer intervals [0, m) and [0, n), its natural - cardinality is at most m times n, and the conclusion is a lattice vector h - different from (0, 0) with c(z + h) = c(z) for all z. - - declaration: Nivat.nivat_rational - informal: A finite-range rational-valued configuration whose pattern count on some - m-by-n rectangle with m, n positive is at most m times n is periodic. This is - proved by strong induction on the area of the rectangle. - - declaration: Nivat.two_factors - informal: If a finite-range rational configuration has pattern count at most m - times n on an m-by-n rectangle with m, n positive and is annihilated by the - product of two difference operators in nonzero directions h and t, then it is - periodic. - - declaration: Nivat.Descent.exists_smaller_low_complexity_rectangle - informal: If two pointwise limits x and y of translates of a finite-range rational - configuration c have nonzero difference whose Laurent annihilator ideal is - generated by a nonzero polynomial, and c has pattern count at most m times n on - an m-by-n rectangle, then applying that polynomial to c yields a configuration - with pattern count at most its area on some rectangle of strictly smaller - positive area. - - declaration: Nivat.Algebra.exact_line_in_basis - informal: A nonzero rational configuration that vanishes on the open lower - half-plane of a lattice basis and is periodic with positive period q along the - first basis direction has Laurent annihilator ideal generated by a single - line-supported polynomial, obtained from a monic divisor of X^q - 1 of positive - degree. - tags: - - symbolic-dynamics - - pattern-complexity - - periodicity - - Nivat-conjecture - - Laurent-polynomials - msc: - - 37B10 - - 37B51 - - 68R15 - - slug: erdos548 - title: "The Erdős–Sós conjecture (Erdős Problem 548)" - summary: >- - Proves the Erdős–Sós conjecture (1962), Erdős Problem 548: every graph on n ≥ k+1 vertices - with at least (k−1)n/2 + 1 edges contains every tree on k+1 vertices as a (not necessarily - induced) subgraph. The headline statement is the FrontierMath Erdős benchmark statement, - reviewed by Thomas Bloom and carried verbatim by Google DeepMind's Formal Conjectures; the - proof was found autonomously by GPT-6 Astra. It is a finite double counting over permutation - words of the host vertices: rotating the first qualifying prefix of a cut word gives a gluing - inequality for two rooted branches, and reversing the two blocks of a cut moves the root to a - newly attached leaf. A strong induction on the order t of the tree bounds the - adjacency-marked cut words by the rooted-copy count plus (t−2)·n!, so a tree-free host on n - vertices has at most (t−2)n/2 edges, the sharp classical bound. - branch: extremal graph theory - entry_module: LeanPool.Erdos548 - authors: - - Tom Adamczewski - source: - title: "Erdős problem #548 (Erdős–Sós conjecture): proof" - authors: - - Tom Adamczewski - url: https://github.com/tadamcz/erdos548 - github_repo: tadamcz/erdos548 - commit: c02708a6956d7b953c2e889c8bc24a0c97796313 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Erdos548.erdos_548 - main_results: - - declaration: Erdos548.erdos_548 - informal: >- - Let n ≥ k+1. Every simple graph on n vertices with at least (k−1)n/2 + 1 edges contains - every tree on k+1 vertices as a subgraph (an injective graph homomorphism). This is - Erdős Problem 548, the Erdős–Sós conjecture, in the benchmark phrasing. - - declaration: Erdos548.tree_free_edge_bound - informal: >- - If a tree T on t ≥ 2 vertices is not contained in a graph G on n ≥ 1 vertices, then - 2·|E(G)| ≤ (t−2)·n. This is the classical sharp form of the Erdős–Sós bound. - - declaration: Erdos548.rooted_word_tree_bound - informal: >- - For a tree T on t ≥ 2 vertices rooted at r, a host graph G and a nonempty - repetition-free host word of length n, the number of cut permutation words of that - word whose cut ends at a neighbour of the first letter is at most the number supporting - a rooted copy of T on the prefix plus (t−2)·n!. - - declaration: Erdos548.marked_word_gluing_count - informal: >- - For a permutation-closed family W of repetition-free words, all of the same length m, - and families A, B, C of letter sets with C ⊆ A and A R ∧ B X → C (R ∪ X) for disjoint - R, X, the marked cuts (l, k) with l ∈ W and k ≤ m qualifying for A plus those qualifying - for B number at most the allowed cuts plus those qualifying for C. - tags: - - extremal-graph-theory - - erdos-problems - - trees - - combinatorics - msc: - - "05C35" - - "05C05" - - slug: finite-max-flow-min-cut - title: Finite max-flow / min-cut - summary: >- - Develops finite networks with real nonnegative capacities, proves cut-flow weak duality and - maximum-flow attainment, and derives max-flow/min-cut equality by an augmenting-path argument. - branch: combinatorial optimization and graph theory - entry_module: LeanPool.MaxFlowMinCut - authors: - - Juan Pablo Traverso Gianini - - Aristotle - source: - title: >- - Reusable max-flow/min-cut component accompanying Linear-Error Clique Partitions of Split - Graphs via Structured Triangle Packing - authors: - - Juan Pablo Traverso Gianini - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/b3423f3e8c8db7c2d1b279673293ef3079faf903/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Contrib/MaxFlowMinCut.lean - github_repo: jtraverso/erdos-81-chordal-clique-partitions - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Contrib.MaxFlowMinCut.maxflow_eq_mincut - main_results: - - declaration: Contrib.MaxFlowMinCut.maxflow_eq_mincut - informal: >- - Every finite network with real nonnegative capacities and distinct source and sink admits - a feasible flow and a source-sink cut whose flow value equals the cut capacity. - - declaration: Contrib.MaxFlowMinCut.value_le_capacity - informal: >- - The value of every feasible flow is at most the capacity of every source-sink cut. - - declaration: Contrib.MaxFlowMinCut.max_flow_exists - informal: >- - Every finite network with real nonnegative capacities admits a maximum-value feasible - flow. - tags: - - combinatorial-optimization - - graph-theory - - network-flows - msc: - - "05C21" - - "90C35" - - slug: even-graph-cycle-decomposition - title: Even graphs are edge-disjoint unions of cycles - summary: >- - Decomposes every finite loopless edge set with even degree at each vertex into edge-disjoint - simple cycles and derives a simultaneous fixed-point-free pairing of incident edges. - branch: graph theory and combinatorics - entry_module: LeanPool.EvenGraphCycles - authors: - - Juan Pablo Traverso Gianini - - Aristotle - source: - title: >- - Reusable even-graph cycle decomposition accompanying Linear-Error Clique Partitions of - Split Graphs via Structured Triangle Packing - authors: - - Juan Pablo Traverso Gianini - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/b3423f3e8c8db7c2d1b279673293ef3079faf903/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Contrib/EvenGraphCycles.lean - github_repo: jtraverso/erdos-81-chordal-clique-partitions - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Finset.exists_cycleDecomp - - Finset.exists_edge_pairing_of_even - main_results: - - declaration: Finset.exists_cycleDecomp - informal: >- - Every finite loopless edge set whose degree is even at every vertex is the edge-disjoint - union of simple cycles of length at least three. - - declaration: Finset.exists_edge_pairing_of_even - informal: >- - If every vertex has even degree, then the incident edges at every vertex admit a - simultaneous fixed-point-free involutive pairing. - tags: - - graph-theory - - eulerian-graphs - - cycle-decompositions - msc: - - "05C45" - - slug: naslund-counterexample - title: Square-difference-free polynomials beyond Naslund's conjectured bound - summary: >- - Constructs square-difference-free subsets of polynomials over the field with three elements - that refute Naslund's Conjecture 13 at every degree bound divisible by four and at least - eight. An explicit degree-eight lift multiplies cardinality by 810 and yields a lower - bound of 16/21 for the lower limiting normalized logarithmic growth rate. - branch: additive combinatorics and number theory - entry_module: LeanPool.NaslundCounterexample - authors: - - JD Jones - source: - title: Square-difference-free sets in F_3[T] past the conjectured bound - authors: - - JD Jones - url: https://github.com/JD-Jones-ASES/ns-lean - github_repo: JD-Jones-ASES/ns-lean - commit: 035e9b0c147630e35631e4401433660f695d1fba - license: MIT - status: verified - provenance: AI - main_declarations: - - NaslundCounterexample.conjecture_fails - - NaslundCounterexample.liminf_ge - main_results: - - declaration: NaslundCounterexample.exists_card_810 - informal: >- - There is a square-difference-free set of 810 polynomials over F_3 with degree less - than eight. - - declaration: NaslundCounterexample.conjecture_fails_at_eight - informal: >- - The upper bound 3^6 = 729 in Naslund's Conjecture 13 fails for square differences - over F_3 at degree bound eight. - - declaration: NaslundCounterexample.exists_card_pow - informal: >- - For every positive integer e, there is a square-difference-free set of 810^e - polynomials over F_3 with degree less than 8e. - - declaration: NaslundCounterexample.exists_card_twenty_seven_mul_pow - informal: >- - For every natural number e, there is a square-difference-free set of 27 times 810^e - polynomials over F_3 with degree less than 8e + 4. - - declaration: NaslundCounterexample.conjecture_fails - informal: >- - For every n divisible by four with n at least eight, there is a square-difference-free - set of polynomials over F_3 of degree less than n whose cardinality exceeds 3^(3n/4). - - declaration: NaslundCounterexample.liminf_ge - informal: >- - If D(n) is the largest cardinality of a square-difference-free set of polynomials over - F_3 of degree less than n, then the lower limit of log D(n) divided by n log 3 is at - least 16/21. - tags: - - additive-combinatorics - - square-difference-free - - function-fields - - polynomial-constructions - msc: - - "11B75" - - "05D05" - - slug: dilatations - title: Dilatations of categories and commutative rings - summary: >- - Constructs categorical dilatations from sieves of permitted numerators - and denominator morphisms, and proves their universal property for - Sigma-regular functors satisfying the sieve containment condition. - Develops restriction and union of centers, dual codilatations, and - ordinary localization as a special case. The composition isomorphism - retains the explicit Sigma-regularity hypothesis left open in the - original paper. For commutative rings, an exponent-profile center - recovers the ring dilatation; the naive single-index comparison is - refuted over Z[X]. A finite category example refutes a general - intermediate-category characterization. Independent ring modules - provide fraction arithmetic, a universal property, reindexing, and - the unique isomorphism for two-stage dilatations. - branch: category theory and commutative algebra - entry_module: LeanPool.Dilatations - authors: - - Arnaud Mayeux - - Jujian Zhang - source: - title: Dilatations of categories, via their Lean formalization - authors: - - Arnaud Mayeux - arxiv: '2608.09305' - url: https://github.com/rndmx/DilCat - github_repo: rndmx/DilCat - commit: 604559654c948566675da3f7709b8ad3126bd487 - announced: '2026-08-10T08:52:54Z' - prior_formalization: - url: https://github.com/ProjConstruction/Proj - commit: 6bfe4ef61d50c20cb3b91f4ef58649c9490b594f - authors: - - Arnaud Mayeux - - Jujian Zhang - arxiv: '2606.01438' - license: Apache-2.0 - note: >- - The upstream single-file development incorporates the ring - construction from Project/Dilatation/lemma.lean, Family.lean, - and Multicenter.lean, introduced by Jujian Zhang on 2025-03-03; - both prior authors are credited here. - license: Apache-2.0 - status: verified - provenance: mix - provenance_note: >- - Mayeux's paper discloses Claude assistance under the author's direction; - it does not quantify the share of generated proofs. The development - incorporates Mayeux and Zhang's earlier ring formalization. This port - includes substantial AI proof repair, optimization, and documentation. - main_declarations: - - CategoryTheory.Dilatations.Dila_universal_property - main_results: - - declaration: CategoryTheory.Dilatations.Dila_universal_property - informal: >- - For a nonempty indexed center in a category C and a functor F from - C to D, assume that localization of D at the images of the chosen - denominators is faithful and that each pushed-forward numerator - sieve lies in the sieve generated by its mapped denominator. - There is a unique functor from the dilatation to D whose composite - with the canonical functor from C is F. - source_ref: Section 5; original Theorem 3.10. - - declaration: CategoryTheory.Dilatations.Iso315 - informal: >- - The successive dilatation at Z and the pushforward of W is - isomorphic to the dilatation at the combined center Z.sum W, - assuming that the composite canonical functor to the successive - dilatation is Sigma-regular for Z.sum W. This regularity is an - explicit hypothesis, not a theorem of the development. - source_ref: Section 8.2 and Appendix B; conditional original Proposition 3.15. - - declaration: CategoryTheory.Dilatations.Iso318 - informal: >- - For two families of numerator sieves with the same denominator - morphisms, dilating at their combined two-copy center is isomorphic - to dilating at the pointwise union of the sieves. - source_ref: Section 8.3; original Proposition 3.18. - - declaration: CategoryTheory.Dilatations.Prop51.Iso51 - informal: >- - For any multicenter in a commutative ring A, the categorical - dilatation of the multiplicative one-object category of A at - all finitely supported exponent profiles is isomorphic to the - one-object category of the ring dilatation A[M]. At profile nu, - the denominator is the product of a_i raised to nu_i and the - numerator ideal is the product of (M_i + (a_i)) raised to nu_i. - source_ref: Theorem 10.1; corrected original Proposition 5.1. - - declaration: CategoryTheory.Dilatations.NaiveCenterCounterexample.no_C_compatible_equiv - informal: >- - For A = Z[X], ideal (X), and denominator 2, there is no equivalence - from the categorical dilatation at the naive one-index center to - the one-object category of the ring dilatation that commutes - strictly with the canonical functors from the base category. - source_ref: Section 10.2 and Appendix B. - - declaration: CategoryTheory.Dilatations.Fact52.no_realizing_center - informal: >- - The constructed category with three arrows from X to Y admits - no equivalence to a dilatation of the base category with two such - arrows that commutes strictly with the canonical base functors. - This is an explicit counterexample, not a classification of - intermediate categories up to arbitrary equivalence. - source_ref: Theorem 11.1. - - declaration: CategoryTheory.Dilatations.iteratedDilatationEquiv_existsUnique - informal: >- - For a multicenter M in any commutative ring A and any subset K of - its indices, first dilate along K and then along the complementary - ideals and denominators mapped to the first-stage ring. There is - exactly one isomorphism of algebras over the first-stage ring from - this two-stage dilatation to A[M], compatible with its canonical map. - source_ref: Appendix C, Proposition C.1 and Lemma C.2. - tags: - - categories - - localization - - dilatations - - commutative-algebra - - universal-properties - msc: - - 18A30 - - 13B30 - - slug: gaussian-moments-counterexamples - title: Gaussian Moments Conjecture counterexamples - summary: >- - Proves explicit complex polynomial counterexamples to the Gaussian Moments - Conjecture in every dimension at least three, using genuine expectations - under finite products of standard real Gaussian measures. Retains reusable - polynomial integrability, Stein identities, complex contractions, coefficient - formulas, injective-coordinate marginals, and related moment results. - branch: probability and polynomial algebra - entry_module: LeanPool.GaussianMomentsCounterexamples - authors: - - Christopher D. Long - source: - arxiv: '2607.18186' - url: https://github.com/long-mathematics/gaussian-moments-counterexamples - github_repo: long-mathematics/gaussian-moments-counterexamples - commit: c31bb63aaa7cfccf1191893f8e7d04fa71bae025 - announcement: https://github.com/long-mathematics/gaussian-moments-counterexamples/pull/2 - completion_date: '2026-09-17T15:47:31Z' - notes: >- - Imports the completed Gaussian core and related moment infrastructure. - The July manuscript predates the September Lean proof release. Five and - six term counts refer to the natural complex coordinates; total degrees - four and three also hold in the original real coordinates. The separate - Jacobian-map and route-arithmetic files are excluded. No general inversion, - Jacobian-reduction route, or resolution of the two-dimensional conjecture - is claimed. Upstream MIT notice is retained in the project entry module. - Upstream Lean authorship is inferred from the public - codex/lean-formalization branch; its exact proof-authoring model and extent - of human editing are unreported. This port and optimization are Codex-authored. - license: MIT - status: verified - provenance: AI - main_declarations: - - GaussianMomentsCounterexamples.not_GMC_of_three_le - main_results: - - declaration: GaussianMomentsCounterexamples.not_GMC_of_three_le - informal: >- - For every n at least three, the Gaussian Moments Conjecture fails: - some complex polynomial has all positive pure Gaussian moments zero, - while a polynomial multiplier prevents eventual mixed-moment vanishing. - - declaration: GaussianMomentsCounterexamples.master_three - informal: >- - For every complex univariate polynomial A and positive m, the expectation - of A(Q3) times P3 to power m equals m! times the degree-m coefficient of - A(X) times (1+X) to power m-1. The explicit P3 has degree four and five - terms in its natural complex coordinates. - - declaration: GaussianMomentsCounterexamples.master_four - informal: >- - The analogous arbitrary-A coefficient identity for the explicit - four-variable polynomial P4, of degree three and six terms in its - natural complex coordinates, gives an independent counterexample. - tags: - - gaussian-measure - - polynomial-moments - - counterexamples - - formal-power-series - msc: - - '60E05' - - '13F20' - - slug: centered-maximal-constant - title: Bounds for the centered Hardy-Littlewood maximal constant - summary: >- - Proves that the weak type (1,1) constant c_2 of the centered maximal - operator over axis-parallel squares is at least the explicit radical - phi, approximately 1.68551, using a weighted periodic lattice and - integrable approximations. Also proves c_d <= 2^d in every natural - dimension and 1.685 < phi < 1.686. The operator acts on real-valued - integrable functions, with Lebesgue outer measure of strict level - sets. Includes the maximal-function API, a Vitali covering argument, - lattice witnesses, smearing estimates, and the limiting lower bound. - No optimality of phi is claimed. - branch: harmonic analysis - entry_module: LeanPool.CenteredMaximal - authors: - - Yongxi Lin - source: - title: A lower bound 1.6855 for the planar centred maximal constant over squares - authors: - - Yongxi Lin - url: https://github.com/CoolRmal/centered-maximal-constant - github_repo: CoolRmal/centered-maximal-constant - commit: c6a8cb29e8ecce9ac4614a8c7e4bf6938366f986 - notes: >- - First completed publicly on 2026-09-18 at 22:15:07 UTC in commit - 06a726f113241d445903f0c0eae1ebe40d8b539f. - The lower bound is documented and proved in upstream docs/PROOF.md; - the weighted smearing argument adapts Aldaz (2000), Lemma 1.1, - and the covering upper bound follows Tao's 245A Notes 5, Exercise 42. - Upstream formalization.yaml and docs/HISTORY.md disclose AI generation - of the configuration and proofs under Yongxi Lin's direction. - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.CenteredMaximal.ofReal_phi_le_weakTypeConstant_two - main_results: - - declaration: LeanPool.CenteredMaximal.ofReal_phi_le_weakTypeConstant_two - informal: >- - The explicit radical phi is a lower bound for c_2, the infimum of - all weak type (1,1) bounds for the centered maximal operator over - axis-parallel squares in the real plane. - - declaration: LeanPool.CenteredMaximal.weakTypeConstant_le_two_pow - informal: >- - For every natural number d, including zero, the centered maximal - operator over axis-parallel cubes in real d-space has weak type - (1,1) constant c_d at most 2^d. - - declaration: LeanPool.CenteredMaximal.phi_mem_Ioo - informal: The explicit radical constant phi lies strictly between 1.685 and 1.686. - tags: - - harmonic-analysis - - maximal-functions - - weak-type-inequalities - - lattice-constructions - msc: - - 42B25 - - 26D15 - - slug: pentagonal-number-theorem-analytic - title: Euler's pentagonal number theorem, two independent proofs, and the Jacobi triple product - summary: >- - Two independent proofs of Euler's pentagonal number theorem that share no lemmas, so each - confirms the result on its own. The primary development is combinatorial: Franklin's sign-reversing involution on partitions into distinct parts, - giving p_e(n) - p_o(n) = (-1)^|k| when n is the generalized pentagonal number k(3k-1)/2 - and 0 otherwise, stated under a single integer index, together with the identification of - that signed count with the coefficient of X^n in the truncated product over ℤ⟦X⟧. It - builds on Mathlib's Nat.Partition.genFun. The second route develops q-Pochhammer symbols, - the Gaussian binomial coefficient, Euler's two identities and the Cauchy identity, proves - the Jacobi triple product both as a formal power series identity in - (LaurentPolynomial ℂ)⟦X⟧ and analytically over ℂ for ‖q‖ < 1, and reaches the same - theorem for the infinite product by specialization. The Lean proofs in both routes were largely - generated by Aristotle; the Franklin route additionally has an author-written blueprint - that the formalization follows lemma by lemma, and carries substantially more manual - intervention and review, whereas the q-series route is machine-generated blueprint and - all. The Franklin route and the analytic pentagonal number theorem overlap the pooled - `pentagonal-number-theorem` development, which states its analytic form over RCLike fields - rather than only over the complex numbers; the q-Pochhammer, Gaussian binomial and Euler - identity definitions overlap arXiv:2607.01544. The project's distinct contribution is the - complex-analytic layer: an analytic Jacobi triple product over the complex numbers for - norm q < 1, its extension to the punctured disc, the analytic Cauchy identity, and the - summability and locally-uniform-convergence machinery they require. Neither the pooled - entry nor the non-archimedean development of arXiv:2607.01544 covers that setting. - branch: number theory and enumerative combinatorics - entry_module: LeanPool.PentagonalNumberTheoremAnalytic - authors: - - Jonathan Conrad - - Paula Muermann - - Maryna Viazovska - source: - title: Pentagonal Number Theorem - authors: - - Jonathan Conrad - - Paula Muermann - - Maryna Viazovska - url: https://viazovska.github.io/PentagonalNumberTheorem - github_repo: viazovska/PentagonalNumberTheorem - revision: "5c3413265fcb44f74e67ebb5abddcac4b82e611b" - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - PentagonalNumberTheorem.Franklin.euler_pentagonal_number_theorem_packaged - main_results: - - declaration: PentagonalNumberTheorem.Franklin.euler_pentagonal_number_theorem_packaged - informal: >- - The primary result. For every n, either n is a generalized pentagonal number - k(3k-1)/2 for some integer k, in which case the number of partitions of n into an - even number of distinct parts exceeds the number with an odd number of distinct - parts by exactly (-1)^|k|; or n is not of that form, in which case the two counts - are equal. Proved by Franklin's sign-reversing involution. - - declaration: PentagonalNumberTheorem.Franklin.coeff_prod_eq_pe_sub_po - informal: >- - The coefficient of X^n in the truncated product ∏_{k=1}^{n}(1 - X^k) over ℤ⟦X⟧ equals - p_e(n) - p_o(n), the signed count of partitions of n into distinct parts. Truncating - at n is harmless because the omitted factors are ≡ 1 mod X^{n+1}. - - declaration: QSeries.jacobiTripleProduct - informal: >- - For ‖q‖ < 1, ‖z‖ < 1 and z ≠ 0, the product (q;q)_∞ (-z;q)_∞ (-q/z;q)_∞ equals the - bilateral theta series ∑_{k ∈ ℤ} z^k q^{k(k-1)/2}. - - declaration: QSeries.euler_pentagonal_number - informal: >- - For ‖q‖ < 1, the infinite product (q;q)_∞ equals ∑_{k ∈ ℤ} (-1)^k q^{ω(k)} over the - generalized pentagonal numbers ω(k) = k(3k-1)/2, written as two one-sided sums. - Obtained from the Jacobi triple product by specialization. - - declaration: QSeries.FormalPowerSeries.jacobiTripleProduct - informal: >- - The Jacobi triple product as an identity of formal power series in - (LaurentPolynomial ℂ)-coefficient power series, with no convergence hypothesis: the - product of q-Pochhammer symbols equals the bilateral theta series coefficientwise. - tags: - - number-theory - - partitions - - q-series - - pentagonal-number-theorem - - jacobi-triple-product - msc: - - "11P81" - - "11P84" - - "05A17" - - "05A19" - - "33D15" - - slug: acmax-conjecture - title: Kolokolnikov's ACMAX conjecture - summary: >- - Proves that every simple graph on n ≥ 4 vertices with 2(n−2) edges has algebraic - connectivity at most 2, attained by the complete bipartite graph K₂,ₙ₋₂. Develops - Laplacian test-vector bounds, sparse cut certificates, and non-backtracking walk - estimates used in the proof. - branch: spectral and extremal graph theory - entry_module: LeanPool.ACMax - authors: - - Zeru Zhu - - Jinzheng Li - - Yuanjie Ren - source: - title: Maximizing algebraic connectivity for certain families of graphs, Conjecture 1.5 - authors: - - Theodore Kolokolnikov - arxiv: '1412.6147' - doi: 10.1016/j.laa.2014.12.023 - url: https://github.com/MerLeanProver/ACMaxConjecture/tree/78736ca2f5c29a4d5ad7dfdee4ac715bbfcde770 - github_repo: MerLeanProver/ACMaxConjecture - commit: 78736ca2f5c29a4d5ad7dfdee4ac715bbfcde770 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - ACMax.acmax_conjecture - main_results: - - declaration: ACMax.acmax_conjecture - informal: >- - For n ≥ 4, the algebraic connectivity of K₂,ₙ₋₂ is 2, and every simple graph - on Fin n with exactly 2(n−2) edges has algebraic connectivity at most 2. - Algebraic connectivity is the second-smallest eigenvalue of the combinatorial - Laplacian, defined using Mathlib's ordered spectrum. - - declaration: ACMax.algConn_mul_sq_le - informal: >- - For a graph on any nonempty finite vertex type and any real vector whose - coordinates sum to zero, algebraic connectivity times the squared norm is at - most the Laplacian quadratic form. - - declaration: ACMax.algConn_le_two_of_testvector - informal: >- - A nonzero real vector with coordinate sum zero and Laplacian quadratic form - at most twice its squared norm certifies algebraic connectivity at most 2. - - declaration: ACMax.algConn_completeBipartite_two - informal: >- - For n ≥ 4, the complete bipartite graph with parts of sizes 2 and n−2 has - algebraic connectivity exactly 2. - tags: - - spectral-graph-theory - - extremal-graph-theory - - laplacian - - non-backtracking-walks - msc: - - '05C50' - - '05C35' - - slug: hopf-problem - title: A complex structure on the six-sphere - summary: >- - Formalizes Levent Alpöge's construction of a complex-analytic atlas on - the standard six-sphere. The development builds a compact complex - threefold from torus fibrations with cusp and elliptic fibres, computes - its homology and fundamental group, proves that its underlying - topological space is homeomorphic to the six-sphere, and transports the - complex atlas across that homeomorphism. It adapts Lean developments by Yury G. - Kudryashov and Sebastian Kumar, together with the statement from the - Formal Conjectures Authors. - branch: complex geometry and differential topology - entry_module: LeanPool.HopfProblem - authors: - - Boris Alexeev - - Yury G. Kudryashov - - Sebastian Kumar - - The Formal Conjectures Authors - source: - title: "HopfProblem: formalization of Alpöge's six-sphere construction" - authors: - - Boris Alexeev - url: https://github.com/plby/HopfProblem/tree/9ac8a456b526527837d7082ff775213ca8bc9809 - github_repo: plby/HopfProblem - commit: 9ac8a456b526527837d7082ff775213ca8bc9809 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Mathoverflow1973.mathoverflow_1973 - main_results: - - declaration: Mathoverflow1973.SixSphereComplexAtlas.exists_complex_analytic_atlas - informal: >- - The standard unit six-sphere admits a complex three-dimensional - charted-space structure making it a complex-analytic manifold. - - declaration: Mathoverflow1973.SixSphereComplexAtlas.exists_complex_atlas - informal: >- - The standard unit six-sphere admits a complex three-dimensional - charted-space structure making it a continuously differentiable - complex manifold. - - declaration: Mathoverflow1973.mathoverflow_1973 - informal: >- - The formalized statement of MathOverflow question 1973: the standard - six-sphere carries a complex three-dimensional manifold atlas. - tags: - - complex-geometry - - differential-topology - - complex-manifolds - - six-sphere - - torus-fibrations - msc: - - "32Q55" - - "57R15" - - slug: goemans-cost-counterexample - title: Counterexample to Goemans' unsplittable-flow cost conjecture - summary: >- - Refutes Goemans' cost conjecture for single-source unsplittable flow, - Conjecture 1.3 of Traub, Vargas Koch, and Zenklusen's paper. A simple, - loopless, acyclic graph with seven vertices and nine arcs admits a - splittable flow of cost 58, but every unsplittable routing within the - additive maximum-demand capacity bound costs at least 60, attained - by an explicit routing. The development proves completeness of the - walk enumeration and transfers the refutation from integers to any - linearly ordered commutative ring, including the rationals and reals. - Jason Hickey directed Claude's formal verification of the counterexample - discovered by Dmitry Rybin with GPT-5.6 Pro; upstream also acknowledges - Katherine Schlitz. Planarity is not part of the formalized claim. - branch: combinatorial optimization - entry_module: LeanPool.GoemansFlow - authors: - - Jason Hickey - source: - title: A verified counterexample to Goemans' cost conjecture - authors: - - Jason Hickey - url: https://github.com/jyh/dinitz-verify/tree/ffba3523f0edd14be3460d039f22a6b98c02fd9e - github_repo: jyh/dinitz-verify - commit: ffba3523f0edd14be3460d039f22a6b98c02fd9e - arxiv: '2308.02651' - announcement: https://leanprover.zulipchat.com/#narrow/channel/583339-AI-authored-projects/topic/Counterexample.20to.20Dinitz-Garg-Goemans.20conjecture/near/612748998 - announced_at: '2026-07-25T17:18:11Z' - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - GoemansFlow.goemans_cost_conjecture_false - main_results: - - declaration: GoemansFlow.goemans_cost_conjecture_false - informal: >- - The rational form of Goemans' cost conjecture is false, even for - simple, loopless, acyclic finite digraphs with explicit capacities, - positive demands, and distinct terminals different from the source. - source_ref: Conjecture 1.3 of arXiv:2308.02651v1 - - declaration: GoemansFlow.not_GoemansCostConjectureFull - informal: >- - The same cost conjecture is false over every linearly ordered - commutative ring, by transfer of the integer counterexample. - - declaration: GoemansFlow.optimum_is_sixty - informal: >- - On the explicit seven-vertex, nine-arc instance, the minimum cost - among unsplittable routings satisfying the additive maximum-demand - capacity bound is exactly 60. - - declaration: GoemansFlow.fractional_cost - informal: The given splittable flow on the counterexample instance costs exactly 58. - - declaration: GoemansFlow.path_complete - informal: >- - For each of the three terminals, the lists of arcs giving a walk - from the source are exactly the two explicitly exhibited paths. - tags: - - network-flows - - combinatorial-optimization - - counterexamples - msc: - - '90C35' - - '05C21' - - slug: bicausalot-palomar - title: "BicausalOT: measurable selection and coupling strategies" - summary: Develops Kuratowski–Ryll-Nardzewski selection, Jankov–von Neumann uniformization, - and finite-horizon Bellman identities for locally feasible coupling strategies. Exact Borel - measurable optimal strategies and initial couplings are obtained on Polish Borel spaces - for weakly continuous probability kernels and nonnegative extended-real lower semicontinuous - costs. Equivalence with optimization over bicausal path laws is not established. - branch: probability - entry_module: LeanPool.BicausalOT - authors: - - KT. Wu - source: - url: https://github.com/maxwellapexlab/bicausalot-palomar - github_repo: maxwellapexlab/bicausalot-palomar - commit: 3fe500ff64a75d7ab71b474d0c656d5471f62992 - license: Apache-2.0 - status: verified - main_declarations: - - MeasurableSelection.exists_measurable_selection - main_results: - - declaration: MeasurableSelection.exists_measurable_selection - informal: A weakly measurable multifunction from an arbitrary measurable space to a Polish - space with nonempty closed values admits an everywhere-defined Borel-measurable selector. - tags: - - probability - msc: - - 28B20 - - 54C65 - - 54H05 - - 03E15 - - 28A20 - - 68V20 - provenance: AI - - slug: asymptotic-triangle-packing - title: Finite hypergraph rounding and the asymptotic triangle-packing gap - summary: >- - Formalizes a finite ceiling-carrying nibble theorem for uniform hypergraphs that are - near-regular outside a small exceptional set and have suitably bounded codegrees, and - derives the uniform subquadratic gap between fractional and integral triangle packing in - finite graphs, together with its fractional-cover formulation. - branch: hypergraph matchings and extremal combinatorics - entry_module: LeanPool.AsymptoticTrianglePacking - authors: - - Juan Pablo Traverso Gianini - - Aristotle - source: - title: >- - Asymptotic triangle cover--packing development accompanying Linear-Error Clique Partitions - of Split Graphs via Structured Triangle Packing - authors: - - Juan Pablo Traverso Gianini - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/tree/main/preprints/PAPER_III - github_repo: jtraverso/erdos-81-chordal-clique-partitions - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - LeanPool.AsymptoticTrianglePacking.trianglePackingGap - main_results: - - declaration: LeanPool.AsymptoticTrianglePacking.nearRegularNibbleTheorem - informal: >- - For every uniformity and positive error tolerance, sufficiently large finite uniform - hypergraphs with bounded codegrees, near-regular degrees outside a small exceptional - set, and a global degree ceiling admit a matching covering the prescribed fraction of - their vertices. - source_ref: Finite near-regular hypergraph nibble theorem used in the Paper III development. - - declaration: LeanPool.AsymptoticTrianglePacking.trianglePackingGap - informal: >- - For every positive error tolerance, all sufficiently large finite graphs have - fractional triangle-packing optimum at most their integral triangle-packing optimum - plus the error tolerance times the square of the number of vertices. - source_ref: Paper III AX1, fractional-integral triangle-packing gap. - - declaration: LeanPool.AsymptoticTrianglePacking.triangleCoverPackingGap - informal: >- - For every positive error tolerance, all sufficiently large finite graphs have - fractional triangle-cover optimum at most their integral triangle-packing optimum - plus the error tolerance times the square of the number of vertices. - source_ref: Paper III AX1, cover-side formulation. - tags: - - hypergraph-matching - - nibble-method - - extremal-combinatorics - msc: - - "05C70" - - "05C35" - - - slug: dross-fractional-triangle-decomposition - title: Dross's fractional triangle decomposition theorem - summary: >- - Formalizes the exact nine-tenths minimum-degree threshold for fractional - triangle decomposition of finite graphs. The proof uses a finite - max-flow/min-cut argument and induction that peels high-degree triangles. - branch: extremal graph theory and fractional decompositions - entry_module: LeanPool.DrossFractionalTriangleDecomposition - authors: - - Juan Pablo Traverso Gianini - - Aristotle - source: - title: Fractional Triangle Decompositions in Graphs with Large Minimum Degree - authors: - - François Dross - doi: 10.1137/15M1014310 - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/tree/dec9f4750c2261681e376f2f67db9eda16a4eca0/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Ax2 - github_repo: jtraverso/erdos-81-chordal-clique-partitions - commit: dec9f4750c2261681e376f2f67db9eda16a4eca0 - license_grant: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/dec9f4750c2261681e376f2f67db9eda16a4eca0/LEAN_SOURCE_APACHE_2_0_GRANT.md - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - LeanPool.DrossFractionalTriangleDecomposition.dross_fractional_flow_exact - main_results: - - declaration: LeanPool.DrossFractionalTriangleDecomposition.dross_fractional_flow_exact - informal: >- - Every finite graph whose minimum degree is at least nine tenths - of its number of vertices has a fractional triangle decomposition: - nonnegative triangle weights sum to one at each edge. - source_ref: Dross (2016), fractional triangle decomposition theorem. - tags: - - extremal-graph-theory - - fractional-decomposition - - max-flow-min-cut - msc: - - "05C70" - - "05C35" - - - slug: caffarelli-kohn-nirenberg - title: Caffarelli–Kohn–Nirenberg partial regularity - summary: >- - Partial regularity for three-dimensional incompressible Navier–Stokes - suitable weak solutions with locally L^q forcing, q > 5/2: quantitative - epsilon regularity and vanishing parabolic one-dimensional Hausdorff - measure of the singular set. Includes the supporting Sobolev, singular - integral, pressure, heat-potential and parabolic Morrey infrastructure, - equivalence of suitable-solution formulations, and a nonzero shear-flow - example. Includes the completed Leray, pressure, and linear regularity support needed - by further Navier–Stokes regularity projects. - branch: Analysis - entry_module: LeanPool.CaffarelliKohnNirenberg - authors: - - Scott Armstrong - - Vlad Vicol - source: - github_repo: scottnarmstrong/CaffarelliKohnNirenberg - url: https://github.com/scottnarmstrong/CaffarelliKohnNirenberg/tree/381d658ead0f03a18361965cc0427ce3fa5844ab - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - CKN.caffarelliKohnNirenberg - - CKN.epsilonRegularityL3 - main_results: - - declaration: CKN.epsilonRegularityL3 - informal: >- - For q > 5/2, sufficiently small unit-cylinder L^3 velocity, L^(3/2) - pressure and L^q force masses give an almost-everywhere equal - quantitatively bounded parabolic Holder representative on the closed - half-cylinder and regularity at each point of its open interior. - - declaration: CKN.epsilonRegularityGradient - informal: >- - A sufficiently small extended nonnegative limsup of the normalized - squared weak-gradient mass implies regularity at an interior point - of a suitable weak solution. - - declaration: CKN.caffarelliKohnNirenberg - informal: >- - The singular set of a suitable weak solution with locally L^q - forcing, q > 5/2, has zero one-dimensional Hausdorff measure in the - parabolic metric. - - declaration: CKN.isSuitableWeakSolution_iff_integrable - informal: >- - The manuscript's suitable weak-solution predicate is equivalent to - the predicate that explicitly requires integrability of all tested - weak-equation and local-energy integrands. - - declaration: CKN.isSuitableWeakSolutionIntegrable_shearFlow - informal: >- - The nonzero decaying sinusoidal shear flow with zero pressure and - force is a suitable weak solution on all space and times in (0,1). - tags: - - Navier–Stokes - - partial differential equations - - harmonic analysis - msc: - - 35Q30 - - 35B65 - - 42B20 - - 28A78 - - slug: hasse-minkowski - title: The Hasse–Minkowski theorem over ℚ - summary: >- - Proves the local–global principle for quadratic forms over the rationals: - a form is isotropic over ℚ exactly when it is isotropic over ℝ and over - every p-adic completion, together with Meyer's theorem that an indefinite - rational form in at least five variables is isotropic. The development - builds the Hilbert symbol at every place of ℚ with Serre's explicit - formulas (including the 2-adic case), proves Hilbert reciprocity and the - existence theorem for prescribed local symbols, the Hasse invariant and - the rank-two and rank-three local representability criteria, Legendre's - ternary descent, and the rank-four and rank-≥5 inductions. - branch: number theory - entry_module: LeanPool.HasseMinkowski - authors: - - jayyswan - - Nirvana Coppola - - María Inés de Frutos-Fernández - source: - title: The Hasse–Minkowski theorem over ℚ - url: https://github.com/jayyswan/hasse-minkowski - github_repo: jayyswan/hasse-minkowski - commit: 47f0e9b5cba6e6354b882bc5856a11962ba61f4a - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - HasseMinkowski.hasseMinkowski - - HasseMinkowski.meyer - main_results: - - declaration: HasseMinkowski.hasseMinkowski - informal: >- - A quadratic form over ℚ is isotropic if and only if it is isotropic - over ℝ and over every p-adic completion ℚ_[p] (the Hasse–Minkowski - local–global principle, in isotropy form). - - declaration: HasseMinkowski.meyer - informal: >- - Every indefinite quadratic form over ℚ in at least five variables is - isotropic (Meyer's theorem). - tags: - - number-theory - - quadratic-forms - - local-global - - p-adic - msc: - - '11E12' - - '11E08' - - '11E88' - - - slug: puiseux - title: Puiseux's theorem - summary: >- - Constructs Puiseux series as the bounded-denominator subfield of rational-exponent - Hahn series. Proves that this field is algebraic over Laurent series over any - coefficient field, and is an algebraic closure when the coefficient field is - algebraically closed of characteristic zero. The Newton-Puiseux proof includes - reusable Hensel splitting and polynomial translation lemmas. Dean Cureton directed - the AI formalization and subsequently simplified its proofs. - branch: algebra - entry_module: LeanPool.Puiseux - authors: - - Dean Cureton - source: - title: Puiseux's theorem in Lean 4 - url: https://github.com/deancureton/Puiseux/tree/73e78681d7bb78ddb202e29cb92ab3c16b723126 - github_repo: deancureton/Puiseux - commit: 73e78681d7bb78ddb202e29cb92ab3c16b723126 - announcement: https://x.com/deancureton/status/2082886228517564682 - announced_at: '2026-07-30T17:49:09Z' - provenance_evidence: https://x.com/deancureton/status/2082886230266487252 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - PuiseuxSeries.isAlgClosure - main_results: - - declaration: PuiseuxSeries.isAlgClosure - informal: >- - For an algebraically closed field K of characteristic zero, the field of - Puiseux series over K is an algebraic closure of the Laurent series field K((t)). - - declaration: PuiseuxSeries.isAlgebraic - informal: >- - Over any field K, the Puiseux series field is algebraic over the Laurent series field. - - declaration: PuiseuxSeries.mem_subfield_iff_support - informal: >- - A rational-exponent Hahn series belongs to the Puiseux subfield if and only if - its support lies in (1/n) times the integers for some positive integer n. - - declaration: Polynomial.Monic.exists_factorization_rootMultiplicity - informal: >- - A monic degree-m polynomial over K[[t]] whose reduction has a root of multiplicity - k, with 0 < k < m, factors into monic polynomials of degrees k and m-k. - tags: - - field-theory - - puiseux-series - - algebraic-closure - - hensel-lemma - msc: - - '12F05' - - '13F25' - - - slug: lean-spectral-theory - title: lean-spectral-theory - summary: The spectral theorem for possibly unbounded self-adjoint operators on complex Hilbert - spaces, formalized through the Cayley transform and real projection-valued measures. The - selected spectral theorem constructs a PVM whose scalar measures give the exact finite-second-moment - domain and first-moment quadratic form, with measurable-set uniqueness, and is proved - equivalent to equality with the unbounded coordinate spectral integral. The selected Stone - theorem proves both that strongly continuous one-parameter unitary groups have self-adjoint - generators and that every self-adjoint operator generates such a group. - branch: functional analysis - entry_module: LeanPool.SpectralTheory - authors: - - Ezzeri Esa - source: - url: https://github.com/savarin/lean-spectral-theory - github_repo: savarin/lean-spectral-theory - commit: d19e279f0ab244657f02ce678e8d67fa69e2f407 - license: Apache-2.0 - status: verified - main_declarations: - - PalomarSpectralStone.spectral_theorem_intrinsic - main_results: - - declaration: PalomarSpectralStone.spectral_theorem_intrinsic - informal: Every self-adjoint partial linear operator on a complex Hilbert space has a - real projection-valued measure representing its domain and scalar moments, unique on - Borel sets. - - declaration: PalomarSpectralStone.stone_theorem_intrinsic - informal: Every strongly continuous one-parameter unitary group has a self-adjoint - generator, and every self-adjoint operator generates such a group. - tags: - - functional-analysis - msc: - - 47A10 - - 47B15 - provenance: AI - - slug: nemytskii-lebesgue - title: Nemytskii operators on Lebesgue spaces - summary: >- - Develops superposition operators induced by Caratheodory functions: - measurability, Lp-to-Lq growth bounds and sequential continuity, - Lipschitz estimates, monotonicity and coercivity, and Frechet remainder - estimates. For Lebesgue measure on the real line and finite-dimensional - real fibers, characterizes when every Lp input has an Lq output by the - standard integrable-envelope growth condition, for 1 <= p,q < infinity. - First-order remainder estimates on arbitrary measure spaces assume a - Caratheodory derivative with the stated Holder-compatible growth. - The integral-functional result bounds the integrated remainder and - proves integrability of the linearized integrand; it does not assert - integrability of the original integrand without its separate growth - hypotheses. Caratheodory here requires strongly measurable sections - and almost-everywhere continuity in the fiber variable. - branch: nonlinear functional analysis - entry_module: LeanPool.NemytskiiLebesgue - authors: - - Petr Girg - - Petr Nečesal - - Martin Dvořák - - Jakub Psutka - source: - title: Nemytskii operators on Lebesgue spaces - url: https://github.com/madvorak/nemytskii-lebesgue/tree/d276c27a1e3ebbd9eb09f25243b5e9808ca168d7 - github_repo: madvorak/nemytskii-lebesgue - commit: d276c27a1e3ebbd9eb09f25243b5e9808ca168d7 - completion_commit: 12e914c8b05694f2ea339a27c4af3d0bdf068922 - release: https://github.com/madvorak/nemytskii-lebesgue/releases/tag/v4.33.1 - completion_date: '2026-09-21T13:49:28Z' - dependencies: - - url: https://github.com/madvorak/utilities-zcu/tree/49f60eacb1401f82358b7d66ee447ea7f2fdebf1 - license: Apache-2.0 - scope: UtilitiesZCU/Basic.lean - provenance_note: >- - Upstream credits the four listed authors. Its README and available - commit history contain no AI disclosure; this is evidence under the - repository's human-provenance rubric, not proof of tool non-use. - The imported version includes substantial AI-assisted Mathlib porting - and proof refactoring by OpenAI Codex, so it is recorded as mixed. - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - NemytskiiLebesgue.IsCaratheodory.memLp_nemytskii_volume_iff_growth - main_results: - - declaration: NemytskiiLebesgue.IsCaratheodory.memLp_nemytskii_volume_iff_growth - informal: >- - For f from R times R^m to R^k with strongly measurable sections and - almost-everywhere continuous fibers, and 1 <= p,q < infinity, - every Lp input gives an Lq output if and only if there exist an Lq - real envelope a and a nonnegative constant C such that almost - everywhere, simultaneously for every fiber vector xi, - norm(f(x,xi)) <= abs(a(x)) + C * norm(xi)^(p/q). - source_ref: Upstream NemytskiiLebesgue/ConverseImplication.lean. - - declaration: NemytskiiLebesgue.IsCaratheodory.nemytskii_seqContinuous_of_memLp_of_growth_real - informal: >- - On any measure space, the same Caratheodory and growth assumptions - imply that a sequence of Lp functions converging in Lp to an Lp - function has superposition images converging in Lq, for finite - exponents p,q at least one and finite-dimensional real fibers. - source_ref: Upstream NemytskiiLebesgue/Continuity.lean. - - declaration: NemytskiiLebesgue.IsCaratheodory.eLpNorm_frechet_remainder_le_ofReal_mul_eLpNorm - informal: >- - On a measure space, suppose f and its fiber derivative f' - are Caratheodory, f' is the Frechet derivative at every fiber point - almost everywhere, p,r >= 1 satisfy 1/q = 1/r + 1/p, and f' is - bounded by b + C * norm(xi)^(p/r) with nonnegative b in Lr and C > 0. - At each Lp function u, the Lq integral quantity (a quasi-norm when - q < 1) of the first-order remainder is at most epsilon times the - Lp norm of l whenever l is in Lp and - its Lp norm is sufficiently small. - source_ref: >- - Upstream NemytskiiLebesgue/Frechet.lean; generalized in this port - by removing an unused finite-measure assumption. - - declaration: NemytskiiLebesgue.IsCaratheodory.integralFunctional_frechet_deriv - informal: >- - On a measure space with finite-dimensional real inputs, - under Caratheodory assumptions on f and its fiber derivative f', - conjugate exponents p,q, and derivative growth b + C*norm(xi)^(p-1) - with nonnegative b in Lq and C > 0, the linearized integrand is - integrable for every Lp perturbation. The norm of the integral of - the first-order remainder is at most epsilon times the Lp norm of - the perturbation when that norm is sufficiently small. - source_ref: >- - Upstream NemytskiiLebesgue/IntegralFunctional.lean; generalized in - this port by removing an unused finite-measure assumption. - tags: - - nonlinear-analysis - - nemytskii-operators - - lebesgue-spaces - - differentiation - msc: - - 47H30 - - 46E30 - - 47J05 - - - slug: besicovitchs-1-2 - title: A machine-checked bound of 0.6934 for Besicovitch's 1/2-problem - summary: >- - Proves the planar Besicovitch threshold is at most 6934/10000, improving the published - 7/10 record. The development formalizes the finite six-point reduction, thirty rational Gram - certificates, the endpoint isolation argument, and Besicovitch's classical lower-bound example. - branch: geometric measure theory - entry_module: LeanPool.Besicovitch - authors: - - Yongxi Lin - source: - title: The bound sigma_1(R^2) <= 6934/10000 for Besicovitch's 1/2-problem - authors: - - Yongxi Lin - url: https://github.com/CoolRmal/Besicovitchs-1-2 - github_repo: CoolRmal/Besicovitchs-1-2 - commit: 17c9597b2a901bf386d5bb8382d0a74d2069f25b - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.Besicovitch.sigmaOne_plane_le_barS - main_results: - - declaration: LeanPool.Besicovitch.sigmaOne_plane_le_barS - informal: >- - The planar one-dimensional rectifiability threshold is at most 6934/10000, improving the - published upper bound 7/10. - - declaration: LeanPool.Besicovitch.forcesOneRectifiability_plane_of_barS_lt - informal: >- - Every threshold strictly above 6934/10000 forces countable one-rectifiability in the plane. - - declaration: LeanPool.Besicovitch.Example.one_half_le_sigmaOne_plane - informal: >- - The planar threshold is at least one half, witnessed by Besicovitch's purely unrectifiable - example. - - declaration: LeanPool.Besicovitch.weightedPairScore_le_of_separated - informal: >- - Thirty exact rational Gram certificates prove the weighted six-point score is at most - -1/2000 for every separated admissible pair in the unit ball. - tags: - - besicovitch-problem - - measure-theory - - rectifiability - - finite-certificates - - gram-matrices - msc: - - "28A75" - - "28A78" - - "49Q15" - - "68V20" - - "90C05" - - - slug: stallings-folding - title: Verified Stallings folding for finitely generated subgroups of F2 - summary: Constructs a finite inverse automaton recognizing the subgroup generated by any - finite list of signed words in the rank-two free group. Proves exact subgroup preservation - and correctness of canonical reduced-word membership testing. Reusable infrastructure - includes inverse-automaton walks, reduction-compatible traversal, finite fold congruences, - coset potentials, flower graphs, and basis roses. Folding uses exhaustive finite search - over Boolean relations; no complexity bound, core pruning, subgroup basis, intersection, - or index algorithm is claimed. - branch: computational group theory - entry_module: LeanPool.StallingsFolding - authors: - - Arthur Freitas Ramos - source: - title: Verified finite automata for finitely generated subgroups of F2 - url: https://github.com/Arthur742Ramos/stallings-folding/tree/117dde0a8415d3da1787e15ad22a6223a744432d - github_repo: Arthur742Ramos/stallings-folding - commit: 117dde0a8415d3da1787e15ad22a6223a744432d - completion_commit: 0291588d2737f10d4fb17be947277680cd3c6462 - completion_date: '2026-09-23T04:05:29Z' - announcement: https://palomar-registry.org/entry?id=PALOMAR-2026-09-23-000004&version=1 - announcement_date: '2026-09-23T05:20:13Z' - mathematical_source: John R. Stallings, Topology of Finite Graphs, Inventiones Mathematicae - 71 (1983), 551-565, doi:10.1007/BF02095993. This is the rank-two finite-recognizer special - case, implemented through least congruences rather than a sequence of local folds. - background: Ilya Kapovich and Alexei Myasnikov, Stallings Foldings and Subgroups of Free - Groups, Journal of Algebra 248 (2002), 608-668, doi:10.1006/jabr.2001.9033. - scope: All four Stallings library modules and the proved finite-recognizer endpoint from - Solution.lean. The standalone challenge placeholder, statement-audit tooling, and CI - files are omitted. Mathlib is the only dependency. - port_extensions: >- - Flower vertices now contain exactly the internal positions of each word, - instead of padding all words to a common maximum length. The new Restriction - module proves preservation when folding an edge-closed finite vertex set - containing the basepoint; its potential is required only on the selected set. - These AI-assisted port extensions are not attributed to the upstream release. - provenance_note: Upstream formalization.yaml records human-directed GPT-6 Codex proof - development. Lean Pool porting, cleanup, and optimization were also performed by OpenAI - Codex. Author attribution follows upstream. - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Stallings.folded_recognizer_exists - main_results: - - declaration: Stallings.folded_recognizer_exists - informal: For every finite list S of signed words on two generators, there exist a finite - type of states, a base state, and a partial transition table with inverse-labelled reverse - edges, such that traversing the canonical reduced word of any g in FreeGroup (Fin 2) - returns to the base exactly when g lies in the subgroup generated by the word values - in S. The witness is the computed flower-graph fold. - source_ref: Upstream Solution.lean, completeStatement and folded_recognizer_exists. - - declaration: Stallings.foldWords_loopSubgroup_eq_generatorSubgroup - informal: For every finite input list of signed words in the rank-two free group, the - based-loop subgroup of the computed folded flower graph equals the subgroup generated - by the input words. - source_ref: Upstream Stallings/Flower.lean. - - declaration: Stallings.foldedMembershipTest_eq_true_iff - informal: For any finite generating list S and any rank-two free-group element g, the - executable folded-automaton test returns true if and only if g belongs to the subgroup - generated by S. There is no efficiency guarantee for constructing the automaton. - source_ref: Upstream Stallings/Flower.lean. - tags: - - free-groups - - stallings-folding - - finite-automata - - verified-algorithms - msc: - - 20E05 - - 68Q45 - - 03B35 - - - slug: boolean-multiplication - title: Exact Boolean multiplicative complexity of four-term polynomial multiplication - summary: >- - Proves that multiplying two binary polynomials with n coefficients needs - exactly 0, 1, 3, 6, and 9 AND gates for n = 0, 1, 2, 3, and 4, with XOR - and constants free and arbitrary reuse of earlier nonlinear wires. - The main lower bound excludes every eight-gate circuit for n = 4. - Develops Boolean algebraic normal forms equivalent to all Boolean - functions, a general circuit model, dimension bounds, circuit rewiring, - a Reed-Muller minimum-weight theorem, and exterior-algebra and jet - arguments controlling nonlinear feedback. - branch: computational complexity - entry_module: LeanPool.BooleanMultiplication - authors: - - Gregory Morse - source: - title: >- - Unrestricted Boolean Multiplicative Complexity of Four-Term Binary - Polynomial Multiplication: Rational Places, Hasse Jets, and the - Failure of Nonlinear Feedback - arxiv: '2608.30238v1' - url: https://github.com/GregoryMorse/unrestricted-boolean-mul/tree/79a801760668f52bb00a9e1a406e841375c58a6d - github_repo: GregoryMorse/unrestricted-boolean-mul - commit: 79a801760668f52bb00a9e1a406e841375c58a6d - announcement_date: '2026-08-31T04:33:48Z' - announcement_url: https://github.com/GregoryMorse/unrestricted-boolean-mul/releases/tag/n4-arxiv-v2 - notes: >- - The complete Lean release predates the arXiv v1 submission of - 2026-08-31T04:45:12Z. The selected September 22 commit corrects - release metadata; eligibility rests on the complete August 31 Lean release at - 2ebc0cf42039e4b5de84cb7f9d97cad4791c669d. The August 29 release - contained computational artifacts, not the Lean formalization. - The author discloses GPT-5.6 Sol for original research and proof - development, Opus 5 as a critical referee, and Codex for release - preparation. Gregory Morse directed and reviewed the work and - assumes responsibility. AI provenance records the proof-producing - agent workflow; human direction alone is not classified as mixed - formalization. The complete upstream MIT notice is retained in - LeanPool/BooleanMultiplication.lean. Only the published n <= 4 - dependency closure is imported; no n = 5 or n = 6 claim is made. - license: MIT - status: verified - provenance: AI - main_declarations: - - UnrestrictedBooleanMul.N4.mc_mul_four - main_results: - - declaration: UnrestrictedBooleanMul.mc_mul_zero - informal: The empty polynomial multiplication map needs zero AND gates. - source_ref: Proposition 2.3; the n = 0 target has no output coordinates. - - declaration: UnrestrictedBooleanMul.mc_mul_one - informal: Multiplying two one-coefficient binary polynomials needs exactly one AND gate. - source_ref: Proposition 2.3. - - declaration: UnrestrictedBooleanMul.mc_mul_two - informal: Multiplying two two-coefficient binary polynomials needs exactly three AND gates. - source_ref: Proposition 2.3. - - declaration: UnrestrictedBooleanMul.mc_mul_three - informal: Multiplying two three-coefficient binary polynomials needs exactly six AND gates. - source_ref: Proposition 2.4; the lower bound is proved internally. - - declaration: UnrestrictedBooleanMul.N4.mc_mul_four - informal: >- - Multiplying two four-coefficient binary polynomials needs exactly - nine AND gates, allowing free XOR, constants, and unrestricted - nonlinear reuse. - source_ref: Theorem 9.1. - - declaration: UnrestrictedBooleanMul.N4.no_eight_gate_circuit - informal: >- - No eight-AND circuit with free XOR, constants, and arbitrary - nonlinear reuse computes all seven output coefficients of the - product of two four-coefficient binary polynomials. - source_ref: Theorem 9.1, lower bound. - tags: - - computational-complexity - - boolean-circuits - - polynomial-multiplication - - algebraic-normal-form - msc: - - '68Q06' - - '68Q17' - - '68W30' - - '15A75' - - '94D10' - - slug: champernowne-normality - title: Champernowne normality in every base - summary: >- - Proves that concatenating the base-b digits of the positive integers - gives a normal digit sequence for every integer base b at least 2: - every nonempty length-k word of digits below b has limiting frequency - b^(-k). This is a discrete, all-base generalization of D. G. - Champernowne's 1933 base-ten result, without a real-number expansion - bridge. Includes reusable substring-counting inequalities, a - digit-string equivalence, exact cohort counts, and quantitative prefix - bounds used to establish the frequency limit. - branch: number theory and asymptotic analysis - entry_module: LeanPool.Champernowne - authors: - - Arthur Champernowne - source: - title: Champernowne Sequence Normality Proof - authors: - - Arthur Champernowne - url: https://github.com/OldMathematician/ChampernowneNormality - github_repo: OldMathematician/ChampernowneNormality - commit: 31a76dbde2727745beb6fb81af4cab1ce6a15fda - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Champernowne.champernowne_normal - main_results: - - declaration: Champernowne.champernowne_normal - informal: >- - For every integer base b at least 2, the sequence formed by - concatenating the base-b digits of 1, 2, 3, and so on is normal: - each nonempty word w of digits below b has limiting occurrence - frequency 1 / b^(length w), counting overlapping occurrences. - - declaration: Champernowne.champernowne_normal_ten - informal: >- - The decimal digit sequence 123456789101112... is normal in base ten. - - declaration: Champernowne.sum_countOccurrences_cohort - informal: >- - For a nonempty length-k word of base-b digits with nonzero first - digit and k at most m, its total occurrences inside all m-digit - positive integers equal b^(m-k) + (m-k)(b-1)b^(m-k-1), with natural - subtraction in the exponents. - - declaration: Champernowne.sum_countOccurrences_cohort_zero - informal: >- - For a nonempty length-k word of base-b digits starting with zero - and k at most m, its total occurrences inside all m-digit positive - integers equal (m-k)(b-1)b^(m-k-1), with natural subtraction in the - exponent. - tags: - - normal-numbers - - digit-expansions - - combinatorics-on-words - - asymptotic-frequency - msc: - - 11K16 - - 68R15 - - slug: chip-firing-with-lean - title: Chip-Firing with Lean 4 - summary: A Lean 4 / Mathlib formalization of Baker and Norine's Riemann-Roch theorem for - finite graphs. Chip configurations on a finite graph, modulo chip-firing moves, form an - abelian group analogous to the group of line bundles on a compact Riemann surface. Each - class can be assigned a rank, analogous to the dimension of a linear series. These ranks - then satisfy a formula of the form r(D) - r(K-D) = deg D + 1 - g, in exact analogy to - Riemann surfaces. This formula is formalized, along with the analog of Clifford's theorem - for finite graphs. - branch: combinatorics - entry_module: LeanPool.ChipFiring - authors: - - Dhyey Dharmendrakumar Mavani - - Nathan Pflueger - source: - url: https://github.com/dhyeymavani2003/chip-firing-with-lean - github_repo: dhyeymavani2003/chip-firing-with-lean - commit: 5b422d855c02af60591121ba7c11b5e8242542b8 - license: Apache-2.0 - status: verified - main_declarations: - - ChipFiring.Propositions.riemann_roch - - ChipFiring.Propositions.clifford - main_results: - - declaration: ChipFiring.Propositions.riemann_roch - informal: For a finite connected graph, the ranks of a divisor and its canonical complement - satisfy the graph Riemann–Roch formula. - - declaration: ChipFiring.Propositions.clifford - informal: For a special divisor on a finite connected graph, its rank is at most half - its degree. - tags: - - combinatorics - msc: - - 05C57 - - 14T20 - provenance: mix - - slug: densityhalesjewett - title: The density Hales–Jewett theorem and Szemerédi's theorem - summary: 'A Lean 4 formalization of the density Hales–Jewett theorem: for every finite alphabet - α and every density δ > 0, all sufficiently large n have the property that any set of - at least δ·|α|^n words of length n over α contains a combinatorial line. The basis for - the formalization is the combinatorial proof of Dodos, Kanellopoulos, and Tyros. As a - consequence, we also formalize Szemerédi''s theorem on the integers: for every k ≥ 3 and - δ > 0, the following holds for all sufficiently large n. Any subset of {0, …, n−1} of - size at least δ·n contains a nonconstant arithmetic progression of length k.' - branch: combinatorics - entry_module: LeanPool.DensityHalesJewett - authors: - - Gabriel Dahia - source: - url: https://github.com/gdahia/densityhalesjewett - github_repo: gdahia/densityhalesjewett - commit: 1c2f5aa1b99caa8e2a87beacf80f07f066c309d3 - license: Apache-2.0 - status: verified - main_declarations: - - Combinatorics.Line.exists_of_density_atTop - main_results: - - declaration: Combinatorics.Line.exists_of_density_atTop - informal: Every positive-density subset of sufficiently long words over a finite nontrivial - alphabet contains a combinatorial line. - - declaration: Combinatorics.ArithmeticProgression.exists_of_density_nat_atTop - informal: Every positive-density subset of a sufficiently long initial interval of the - natural numbers contains an arithmetic progression of any prescribed length at least - three with nonzero common difference. - tags: - - combinatorics - msc: - - 05D10 - - 05A05 - - 11B75 - - 68R15 - provenance: AI - - slug: markovprocess - title: MarkovProcess - summary: A Lean formalization of continuous-path Markov processes associated with conservative - Feller transition semigroups on nonempty locally compact Polish spaces. A uniform Kolmogorov - displacement-moment bound yields a unique measurable kernel from each starting point to - continuous-path space, with the prescribed finite-dimensional distributions. The library - also develops strong Markov properties, generators and resolvents, stopping and killing, - and Brownian motion as an example. - branch: probability - entry_module: LeanPool.MarkovProcess - authors: - - Scott Armstrong - source: - url: https://github.com/scottnarmstrong/markovprocess - github_repo: scottnarmstrong/markovprocess - commit: dbade0d12c4f84179441a9df7d7147df7a8bcaa0 - license: Apache-2.0 - status: verified - main_declarations: - - MarkovProcess.existsUnique_continuousProcess - main_results: - - declaration: MarkovProcess.existsUnique_continuousProcess - informal: A conservative Feller transition semigroup on a nonempty locally compact Polish - space satisfying the Kolmogorov moment criterion determines a unique continuous-path - Markov kernel with its prescribed finite-dimensional distributions. - tags: - - probability - msc: - - 60J25 - - 60J35 - - 60G17 - provenance: AI - - slug: nagatafactoriality - title: Nagata's prime-generated factoriality theorem - summary: Nagata’s factoriality criterion for noetherian domains localized at prime-generated - multiplicative sets, with concrete and abstract localization interfaces and applications - to polynomial rings through Laurent and fraction-field localizations. - branch: commutative algebra - entry_module: LeanPool.NagataFactoriality - authors: - - Arthur F. Ramos - - Ruy J. G. B. de Queiroz - - Anjolina G. de Oliveira - source: - url: https://github.com/arthur742ramos/nagatafactoriality - github_repo: arthur742ramos/nagatafactoriality - commit: 2efdf7f86085972bd758449ca43c76a2e7bdf4af - license: Apache-2.0 - status: verified - provenance: AI - provenance_note: >- - Inferred from explicit Copilot coauthor trailers in the pinned upstream history: - 83.5% of surviving noncomment Lean code lines, including 94.8% of the core - localization lemmas and 96.6% of the theorem module, trace to those commits. - Commits 5b1698879b05fc76597723347544cb0b02c268c4 and - ef73ee27e4b0a8d1260350801c6915bfb56b6658 add substantial new proofs, including - prime-generated descent and the fraction-field polynomial application. - Coauthorship records support this classification but do not identify who - wrote each individual line. - main_declarations: - - NagataFactoriality.nagata_theorem - main_results: - - declaration: NagataFactoriality.nagata_theorem - informal: A noetherian integral domain is a unique factorization domain when localization - at a prime-generated multiplicative set is a unique factorization domain. - tags: - - commutative-algebra - msc: - - 13F15 - - slug: salem-theorem - title: Every Pisot number is a two-sided limit of Salem numbers - summary: Proves Salem’s two-sided approximation theorem for Pisot numbers, preserving both the polynomial - reciprocal-family construction and the exceptional quadratic-unit construction. - branch: algebraic number theory - entry_module: LeanPool.SalemTheorem - authors: - - Stephanie Alexander - source: - url: https://github.com/stalex444/salem-theorem - github_repo: stalex444/salem-theorem - commit: 887be765bf24c6a645600dfc7bd0a1055a1761dd - license: MIT - status: verified - provenance: AI - main_declarations: - - SalemTheorem.salem_theorem - main_results: - - declaration: SalemTheorem.salem_theorem - informal: Every Pisot number is approached by Salem numbers from both below and above. - - declaration: SalemTheorem.salem_construction_two_sided - informal: For the nondegenerate Pisot polynomial pattern, reciprocal-family polynomials have Salem - roots arbitrarily near the Pisot root from either side. - - declaration: SalemTheorem.salem_quadratic_unit - informal: The quadratic-unit family supplies Salem roots approaching each reciprocal quadratic Pisot - unit from either side. - tags: - - number-theory - - pisot-numbers - - salem-numbers - msc: - - 11R06 - - 11R09 - - slug: rr-equals-nonmeager-paper - title: The nonmeagre lower bound for the rearrangement number - summary: Formalizes the inequality non(M) ≤ rr in ZFC via relational systems, category and slalom bounds, - finite-vector constructions, and rearrangements of conditionally convergent real series. The reverse - inequality from the accompanying manuscript is outside the upstream Lean formalization. - branch: set theory - entry_module: LeanPool.RearrangementNumber - authors: - - Vinicius de Oliveira Rodrigues - source: - url: https://github.com/vo-rodrigues/rr-equals-nonmeager-paper - github_repo: vo-rodrigues/rr-equals-nonmeager-paper - commit: a8e74a91fe0b9f3534194910172349c9926e3b3b - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - NonMRR.nonM_le_rr - main_results: - - declaration: NonMRR.nonM_le_rr - informal: The uniformity of the meagre ideal is at most the rearrangement number. - tags: - - set-theory - - cardinal-characteristics - - series - msc: - - 03E17 - - 40A05 - - slug: li-criterion-rh-equivalence-lean - title: Li's criterion for the Riemann Hypothesis - summary: 'Li''s criterion for the Riemann zeta function, formalized in Lean 4 over - Mathlib: Mathlib''s `RiemannHypothesis` holds if and only if every Li-Keiper coefficient - of the completed xi function has nonnegative real part. The zero-indexed analytic - coefficient at n is Li''s classical lambda_(n+1), the n-th Taylor coefficient - at 0 of the logarithmic derivative of s |-> xi(1/(1-s)). A second compared theorem - identifies this analytic coefficient with Li''s arithmetic coefficient, the symmetrized - sum over the nontrivial zeros counted with multiplicity, and proves that symmetrized - family summable so the sum is the genuine one. Both results are unconditional: - the biconditional, the coefficient identification, the genus-1 Hadamard factorization - of xi (with multiplicity), genus-1 summability of 1/|rho|^2, and the growth bound - order(xi) <= 1 are all proved. This is a formalization of a classically known - equivalence (X.-J. Li, 1997); it is not a proof of the Riemann Hypothesis, nor - of the positivity of the coefficients.' - branch: analytic number theory - entry_module: LeanPool.LiCriterion - authors: - - Nicholas Bulka - source: - url: https://github.com/nicholasbulka/li-criterion-rh-equivalence-lean - github_repo: nicholasbulka/li-criterion-rh-equivalence-lean - commit: 35df682f3b709ffe5fbcfdd452dfa964bd622b87 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - li_criterion - - li_coefficients_eq_zero_sum - main_results: - - declaration: li_criterion - informal: The Riemann hypothesis is equivalent to nonnegativity of every Li–Keiper - coefficient. - - declaration: li_coefficients_eq_zero_sum - informal: The analytic Li–Keiper coefficients equal the symmetrically summed arithmetic - coefficients over the nontrivial zeros. - tags: - - number-theory - - riemann-hypothesis - - complex-analysis - msc: - - 11M26 - - slug: hs-virial-nbc-identity - title: Hard-sphere Mayer NBC identity and fork-packing volume bound - summary: A kernel-checked Lean formalization of the exact no-broken-circuit volume identity - for hard-sphere Mayer cluster integrals and the stronger dimension-three fork-packing - upper bound for each tree-owned NBC region. The development proves finite broken-circuit - cancellation, the measurable finite-region decomposition, the anchored Euclidean product-measure - theorem, and its explicit flat-coordinate transport. Its geometric result is the attachment-aware - estimate Vol(NBC_T) <= (17/32)^hardSphereNu T * hardSphereKappa^(k-1), where hardSphereNu - T is the maximum cardinality of a packing of pairwise vertex-disjoint, order-compatible - forks in T. A formal fork is an ordered triple (a,b,c) with a < b < c, center a, leaves - b,c, and tree edges {a,b} and {a,c}. The central contribution is the complete pointwise-to-integral - bridge together with the orientation-aware measure-preserving factorization that realizes - the per-fork volume suppression. - branch: mathematical physics - entry_module: LeanPool.HardSphereNBC - authors: - - ukiyois - - OpenCode agent sessions - source: - url: https://github.com/ukiyois/hs-virial-nbc-identity - github_repo: ukiyois/hs-virial-nbc-identity - commit: fd0a1e73c336574fa9655f9ab3c0cd9ce5c4fcce - license: Apache-2.0 - status: verified - main_declarations: - - PalomarHS.main_result - - PalomarHS.nbc_region_real_volume_le_fork_factor - main_results: - - declaration: PalomarHS.main_result - informal: The absolute hard-sphere Mayer coefficient, multiplied by the factorial, equals - the sum of nonnegative no-broken-circuit region volumes over spanning trees. - - declaration: PalomarHS.nbc_region_real_volume_le_fork_factor - informal: For a spanning tree in three dimensions, its no-broken-circuit region has volume - at most (17/32) to the maximum disjoint compatible-fork packing size times the unit-ball - volume to the number of tree edges. - tags: - - mathematical-physics - msc: - - 05C15 - - 82B05 - provenance: AI - - slug: lean-scv - title: Classical several complex variables - summary: Formalizes Cauchy and Taylor theory, Hartogs phenomena, removable singularities, analytic - sets and germs, Weierstrass theory, holomorphic convexity, Cartan–Thullen and Bochner tube theorems, - plurisubharmonic functions, Levi convexity, and elementary Runge-domain theory. Weierstrass preparation - and coordinate-origin germ Noetherianity overlap LocalComplexGeometry; this development also supplies - germ unique factorization and global extension and convexity results. - branch: complex analysis - entry_module: LeanPool.SeveralComplexVariables - authors: - - Bastiaan J Braams - source: - url: https://github.com/bjbraams/lean-scv - github_repo: bjbraams/lean-scv - commit: caef1ae776ff79933718312357980d46628d3702 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - SeveralComplexVariables.analyticOnNhd_of_separately_analytic_locally_bounded - main_results: - - declaration: SCV.cauchy_formula_polydisc - informal: Holomorphic functions on a polydisc satisfy the iterated Cauchy integral formula. - - declaration: SeveralComplexVariables.analyticOnNhd_of_separately_analytic_locally_bounded - informal: On an open subset of a finite complex coordinate space, a Banach-valued separately analytic - map that is locally bounded is jointly analytic. - source_ref: Boas (2013), Section 2.7, Theorem 7 (the classical two-variable scalar case). - - declaration: SCV.identity_theorem - informal: Holomorphic functions on a connected domain that agree on a nonempty open subset agree - throughout the domain. - tags: - - complex-analysis - - several-complex-variables - - holomorphic-functions - msc: - - 32A10 - - 32D05 - - 32E10 - - slug: egz-formal-proof - title: Convex geometry and the Erdős–Ginzburg–Ziv problem - summary: For every fixed positive dimension d, the Erdős–Ginzburg–Ziv constant of - (ZMod p)^d equals p times its hollow-family constant plus o(p) as p tends to infinity - through primes. Includes the convex-flag Helly and centerpoint theorems, flag - decomposition, balanced combinations, relative expansion, and the polynomial hollow-family - bound. - branch: additive combinatorics - entry_module: LeanPool.ErdosGinzburgZiv - authors: - - Dmitrii Zakharov - source: - doi: 10.19086/da.165216 - arxiv: '2002.09892' - url: https://github.com/zakharov2k/egz-formal-proof - github_repo: zakharov2k/egz-formal-proof - commit: 7c82267c43a51dd5cb3e440412780ff82aff7500 - license: MIT - status: verified - provenance: AI - main_declarations: - - EGZ.theorem_1_2 - - EGZ.main_upper_bound - main_results: - - declaration: EGZ.theorem_1_2 - informal: For each positive dimension d, s((F_p)^d) = p w((F_p)^d) + o(p) as p - tends to infinity through primes. - - declaration: EGZ.main_upper_bound - informal: For each positive d and ε > 0, sufficiently large primes satisfy s((F_p)^d) - ≤ (w((F_p)^d) + ε) p. - tags: - - additive-combinatorics - - zero-sum - - convex-geometry - msc: - - 11B30 - - 11H06 - - slug: lean-snumbers - title: Pietsch s-numbers and the maximal difference theorem - summary: Formalizes the five classical s-number sequences, their comparison inequalities, the maximal - difference theorem, compact-operator singular value decomposition, John ellipsoids, entropy numbers, - and compactness characterizations. - branch: functional analysis - entry_module: LeanPool.SNumbers - authors: - - Mario Ullrich - source: - url: https://github.com/mario-ullrich/lean-snumbers - github_repo: mario-ullrich/lean-snumbers - commit: d984f39b53eb1579920fe987de10382dcf553f4e - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - SNumbers.approximationNumber_le_e_mul_hilbertNumber - main_results: - - declaration: SNumbers.approximationNumber_le_e_mul_hilbertNumber - informal: For a bounded operator, its nth approximation number is at most e times (n+1) times its - nth Hilbert number. - - declaration: SNumbers.allSNumbers_eq_on_HilbertSpace - informal: Every s-number sequence agrees with the approximation numbers on operators between Hilbert - spaces. - - declaration: John.john_decomposition - informal: In John position, the identity decomposes into a positive weighted sum of rank-one operators - from contact points. - tags: - - functional-analysis - - operator-theory - - s-numbers - msc: - - 47B06 - - 46B20 - - slug: lean-superorthogonality - title: Square-function estimates from type IV superorthogonality - summary: Proves the type IV superorthogonality square-function estimate for finite and countable index - sets, including the pointwise combinatorial estimate and unconditional convergence in L^(2r). - branch: harmonic analysis - entry_module: LeanPool.Superorthogonality - authors: - - Joris Roos - source: - url: https://github.com/roos-j/lean-superorthogonality - github_repo: roos-j/lean-superorthogonality - commit: 6d3a3645c1e6f6a3c50bfdc12c4a51250e63bc31 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Superorthogonal.sqfct_estimate_of_type_iv_superorthogonal - main_results: - - declaration: Superorthogonal.sqfct_estimate_of_type_iv_superorthogonal - informal: For an integer r >= 1, a countable type IV superorthogonal family in L^(2r), with - almost-everywhere summable squared norms and square function in L^(2r), converges unconditionally - in L^(2r) and satisfies the square-function bound with constant C(r). - tags: - - harmonic-analysis - - superorthogonality - - square-functions - msc: - - 42B25 - - 46E30 - - slug: bkarforestformula - title: The BKAR forest interpolation formula - summary: Proves the Brydges–Kennedy–Abdesselam–Rivasseau forest interpolation formula for smooth functions - on a finite complete graph’s edge-coupling space, with unordered forest-cube contributions, ordered - sector variants, and choice independence. - branch: analysis and mathematical physics - entry_module: LeanPool.BKARForestFormula - authors: - - Scott Armstrong - source: - url: https://github.com/scottnarmstrong/bkarforestformula - github_repo: scottnarmstrong/bkarforestformula - commit: a07f44a534240fe6339558951dd78a19e4ef7c51 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - BKAR.bkar_formula_forestIndex_cube_contributions - main_results: - - declaration: BKAR.bkar_formula_forestIndex_cube_contributions - informal: A smooth function evaluated at the all-one configuration is the sum over forests of cube - integrals of mixed partial derivatives at path-minimum interpolation points. - tags: - - analysis - - graph-theory - - mathematical-physics - msc: - - 81T08 - - 05C05 - - slug: nikodym - title: The sharp finite-field Nikodym exponent - summary: Proves a lower bound for Nikodym sets over arbitrary finite fields and a matching exponent - upper construction over sufficiently large prime fields, using interpolation, algebraic degree bounds, - and multiquadratic constructions. - branch: additive combinatorics - entry_module: LeanPool.Nikodym - authors: - - Ting-Wei Chao - - Zach Hunter - - Cosmin Pohoata - - Hung-Hsun Hans Yu - - Shengtong Zhang - source: - url: https://github.com/shengtongzhang-alt/nikodym - github_repo: shengtongzhang-alt/nikodym - commit: 38d16dcdbaef04f6f85e5dee92905b7db2eba930 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Nikodym.card_ge_pow_sub - - Nikodym.exists_isNikodym_card_le - main_results: - - declaration: Nikodym.card_ge_pow_sub - informal: Every Nikodym set in dimension d ≥ 2 over a finite field of size q has size at least q^d - − (8d²+1)q^(d−2^(1−d)). - - declaration: Nikodym.exists_isNikodym_card_le - informal: For every positive ε and dimension d ≥ 2, sufficiently large prime fields admit a Nikodym - set of size at most q^d − q^(d−2^(1−d)−ε). - tags: - - combinatorics - - finite-fields - - nikodym-sets - msc: - - 05B25 - - 11T99 - - slug: lean-malliavin - title: Malliavin calculus and the Clark–Ocone representation - summary: Constructs the Malliavin derivative, Gaussian integration by parts, closability, - Wiener chaos, Brownian Itô integration and martingale representation. The Clark–Ocone - endpoint uses a generating pre-Brownian process with continuous linear coordinates - on a separable real Gaussian Banach space, its natural filtration, and pairs in the - closed Malliavin graph; the conditional-expectation sections agree almost everywhere - at positive times. - branch: stochastic analysis - entry_module: LeanPool.Malliavin - authors: - - Ezzeri Esa - source: - url: https://github.com/savarin/lean-malliavin - github_repo: savarin/lean-malliavin - commit: a1a6ccc6eedbdf15fc3aebf9146d1fddfa0c8272 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - PalomarClarkOcone.generated_clark_ocone - main_results: - - declaration: PalomarClarkOcone.generated_clark_ocone - informal: For a generating pre-Brownian process on a separable real Gaussian Banach - space whose coordinates are pointwise represented by continuous linear functionals, - constructs a time realization of the Malliavin derivative and an Itô isometry for - the natural filtration, with martingale representation for every terminal L² variable. - For pairs in the closed Malliavin graph, the Clark–Ocone formula identifies the - derivative's predictable projection with its conditional-expectation sections for - almost every positive time. - tags: - - probability - - malliavin-calculus - - stochastic-analysis - msc: - - 60H07 - - 60H05 - - slug: 1mfld - title: The classification of compact 1-manifolds - summary: 'A Lean 4 (mathlib) proof of the classification of 1-manifolds, specifically that - every compact, connected, Hausdorff topological space charted on the half-line [0,∞) is - homeomorphic to the circle S¹ or to the closed unit interval [0,1]. The proof follows - David Gale''s outline: shrink to a finite atlas of interval charts, then induct on the - atlas size, merging two overlapping charts at each step; a connected overlap glues two - charts into one (two boundary charts glue onto [0, 1]), while a disconnected overlap has - exactly two components and yields the circle.' - branch: topology - entry_module: LeanPool.OneManifold - authors: - - Jim Fowler - - Dennis Sweeney - source: - url: https://github.com/sweeneyde/1mfld - github_repo: sweeneyde/1mfld - commit: 83e961ba28dd8182e7bae844756aba614eb1512e - license: Apache-2.0 - status: verified - main_declarations: - - OneMfld.homeomorph_circle_or_unitInterval - main_results: - - declaration: OneMfld.homeomorph_circle_or_unitInterval - informal: Every compact connected Hausdorff space charted on the nonnegative real half-line - is homeomorphic to the circle or the closed unit interval. - tags: - - topology - msc: - - 57N99 - - 54F65 - - 68V20 - provenance: mix - - slug: inflation-termination - title: Inflation termination for classical pair-source networks - summary: 'For finite binary pair-source networks without isolated vertices, classifies - when the inflation hierarchy characterizes compatibility with finite latent alphabets: - precisely forests of double stars, where order two suffices. Also proves triangle - nontermination for arbitrary measurable latent probability spaces, quantitative - obstructions, finite and measurable Finner inequalities, and graph witnesses.' - branch: causal inference and probability - entry_module: LeanPool.InflationTermination - authors: - - William Blair - source: - url: https://github.com/williamjblair/inflation-termination - github_repo: williamjblair/inflation-termination - commit: 2aa1f05ce932fdeef3896c83d287abd77cd8befb - license: MIT - status: verified - provenance: AI - main_declarations: - - TriangleInflation.Graph.classification_NW - main_results: - - declaration: TriangleInflation.Graph.classification_NW - informal: For finite binary pair-source graphs without isolated vertices, some - finite order of the Navascués–Wolfe hierarchy characterizes compatibility with - finite latent alphabets exactly for double-star forests; order two suffices there. - - declaration: TriangleInflation.no_finite_characterizing_orderM - informal: At every positive inflation order, a triangle law passes both the AI - and NW tests but has no compatible model over any measurable latent probability spaces. - tags: - - causal-inference - - probability - - inflation-hierarchy - - graph-theory - msc: - - 60A05 - - 05C05 - - slug: lean-scottish-book-155 - title: A counterexample to Scottish Book Problem 155 - summary: A Lean proof that there are real Banach spaces X and Y and a bijection U from X - to Y which preserves every pairwise distance on every closed ball of radius one quarter - while failing to be a global isometry. This gives a counterexample to Scottish Book Problem - 155. - branch: functional analysis - entry_module: LeanPool.ScottishBook155 - authors: - - Yoshito Ishiki - source: - url: https://github.com/yoshito-ishiki-math/lean-scottish-book-155 - github_repo: yoshito-ishiki-math/lean-scottish-book-155 - commit: 233eff4af7901eadee7927a1eb1f814ad0f3c118 - license: MIT - status: verified - main_declarations: - - ScottishBook155.claim14 - main_results: - - declaration: ScottishBook155.claim14 - informal: There exist real Banach spaces and a bijection between them that preserves all - pairwise distances on each closed ball of radius one quarter but is not a global isometry. - tags: - - functional-analysis - msc: - - 46B20 - - 54E40 - provenance: AI - - slug: marshallhalltheorem - title: Marshall Hall's theorem through finite cores - summary: Formalizes Marshall Hall's finite-index free-factor theorem for finitely generated - subgroups of finite-rank free groups. The proof extracts the finitely many suffix states - of a subgroup generating set, completes the resulting partial generator actions to permutations - of a finite state set, identifies the connected orbit with a finite core, and extends - a geodesic spanning tree to obtain an inclusion-compatible free-product decomposition. - It also exposes an explicit finite permutation separator for every finitely generated - subgroup and element outside it, yielding the LERF consequence. The repository also proves - binary and finite-indexed free-group instances of Grushko--Neumann rank additivity, and - the arbitrary-factor binary theorem. The arbitrary-factor proof is carried by a finite - labelled-graph fold, source-unfold, and monochromatic-vertex contraction reduction. - branch: group theory - entry_module: LeanPool.MarshallHall - authors: - - Arthur Freitas Ramos - - David Barros Hulak - - Ruy J. G. B. de Queiroz - source: - url: https://github.com/arthur742ramos/marshallhalltheorem - github_repo: arthur742ramos/marshallhalltheorem - commit: 99f419644cebf61a5f78eac334fcc1f19b0ccc73 - license: Apache-2.0 - status: verified - main_declarations: - - MarshallHall.marshallHall - - MarshallHall.rank_freeProduct_eq_add - main_results: - - declaration: MarshallHall.marshallHall - informal: Every finitely generated subgroup of a finite-rank free group is an inclusion-compatible - free factor of a finite-index subgroup. - - declaration: LERFChallenge.freeGroup_finite_permutation_separator - informal: Every element outside a finitely generated subgroup of a finite-rank free group - can be separated by an explicit finite permutation action. - - declaration: MarshallHall.rank_freeProduct_eq_add - informal: The rank of a free product of finitely generated groups is the sum of their - ranks. - tags: - - group-theory - msc: - - 20E07 - - 20F65 - - 05C25 - provenance: AI - - title: Carlson special functions and complex Dirichlet averages - summary: Formalizes Carlson special functions through complex Dirichlet averages, with simplex measures, - Gamma-regularized analytic continuation, Euler inversion, Euler–Poisson equations, and quadratic transformations. - Includes the complete supporting several-complex-variable and special-function libraries. - branch: complex analysis - main_declarations: - - PalomarSnapshot.r_first_quadratic - - PalomarSnapshot.joint_average_continuation - main_results: - - declaration: PalomarSnapshot.r_first_quadratic - informal: The Gamma-regularized Carlson R function satisfies the first quadratic transformation - for complex exponent and Dirichlet parameters with positive-real-part unsquared variables. - - declaration: PalomarSnapshot.joint_average_continuation - informal: For a holomorphic scalar function on a convex open domain D, its - Gamma-regularized Dirichlet average extends jointly holomorphically to all complex - Dirichlet parameters and all node vectors in D. The extension agrees with the - regularized simplex integral when every parameter has positive real part; the - ordinary normalized average differs by the factor Gamma of the parameter sum. - tags: - - special-functions - - complex-analysis - - dirichlet-averages - msc: - - 33C65 - - 32A10 - provenance: AI - slug: lean-codes - entry_module: LeanPool.CarlsonFunctions - authors: - - Bastiaan J Braams - source: - url: https://github.com/bjbraams/lean-codes - github_repo: bjbraams/lean-codes - commit: fcc2be9a086c1bdd572db91f005e868b80a8d644 - license: Apache-2.0 - status: verified - - title: Optimal fully dynamic matching on the line - summary: Constructs a measurable hierarchical policy for fully dynamic matching of uniformly arriving - demand and replenishment on the unit interval. Proves logarithmic-over-inventory bounds for long-run - root-mean-square distance from every initial inventory and for initialized finite-horizon expected - average cost. - branch: probability and optimization - main_declarations: - - FD1D.V5.Palomar.optimalDynamicMatchingUpperBound - main_results: - - declaration: FD1D.V5.Palomar.optimalDynamicMatchingUpperBound - informal: For every inventory size m at least two, an online policy has long-run RMS cost at most - (72000/log 2) log(m+1)/m from every initial inventory, and the same expected-average-cost bound - after initialization for horizons at least 2m². - tags: - - online-matching - - probability - - stochastic-optimization - msc: - - 60J20 - - 90B15 - provenance: AI - slug: fd1d-lean - entry_module: LeanPool.FullyDynamicMatching - authors: - - Yash Kanoria - source: - url: https://github.com/ykanoria/fd1d-lean - github_repo: ykanoria/fd1d-lean - commit: 26a5b4e2fd83fa86053239e09e92af370c62f18c - license: Apache-2.0 - status: verified - - slug: euclidean-jordan - title: Euclidean Jordan algebras and the frame Peirce decomposition - summary: 'The frame Peirce decomposition: relative to a Jordan frame, a Euclidean Jordan - algebra is the internal direct sum of the blocks V_ij over unordered pairs of frame indices, - and in finite dimension the diagonal blocks are lines, dim V_ii = 1 - the Faraut-Koranyi - table entry. The frame is carried as data, so frame conjugacy is sidestepped and nothing - here claims anything about the rank of J. Machine checked, not peer reviewed, no priority - claim.' - branch: nonassociative algebra - entry_module: LeanPool.EuclideanJordan - authors: - - Bryan Ehrlich - source: - url: https://github.com/ehrlich-b/euclidean-jordan - github_repo: ehrlich-b/euclidean-jordan - commit: a83407e4dbf73c77c31a2291c02f20b37a5ebbaf - license: Apache-2.0 - status: verified - main_declarations: - - EuclideanJordan.frameBlock_isInternal - main_results: - - declaration: EuclideanJordan.frameBlock_isInternal - informal: A Euclidean Jordan algebra is the internal direct sum of the Peirce blocks associated - with a Jordan frame. - - declaration: EuclideanJordan.finrank_frameBlock_diag - informal: In finite dimension, every diagonal block of a Jordan frame has dimension one. - tags: - - nonassociative-algebra - msc: - - 17C20 - - 17C27 - - 17C37 - - 17C65 - - 17A15 - - 46L70 - provenance: AI - - slug: kaltonpeck - title: Rank parity and complex structures on the Kalton–Peck space - summary: Develops symplectic Fredholm theory and the real Kalton–Peck space to prove rank parity of - finite-rank perturbations and rule out a complex structure on a hyperplane. - branch: functional analysis - entry_module: LeanPool.KaltonPeck - authors: - - Avik Das - source: - url: https://github.com/adas1236/KaltonPeck - github_repo: adas1236/KaltonPeck - commit: 752dec47b173d05df3eeb579c1cd392335b87b60 - license: MIT - status: verified - provenance: human - main_declarations: - - KaltonPeck.noHyperplaneComplexStructure - main_results: - - declaration: KaltonPeck.rankParityGeneral - informal: The abstract symplectic Fredholm construction yields a rank-parity theorem. - - declaration: KaltonPeck.rankParityZ2 - informal: On the real Kalton–Peck space, T² + 1 has even rank whenever its rank is finite. - - declaration: KaltonPeck.noHyperplaneComplexStructure - informal: A hyperplane of the real Kalton–Peck space cannot carry a complex structure. - tags: - - functional-analysis - - banach-spaces - - fredholm-theory - msc: - - 46B03 - - 47A53 - - slug: quantum-query-foundations - title: Adversary bounds and the polynomial method for quantum queries - summary: Develops finite quantum query algorithms, real adversary primal and dual programs, strong - duality, exact Boolean block composition, explicit constant-factor bounded-error query characterizations, - and degree bounds for output-probability polynomials. Query models include idle and blank sectors; - bounds count queries, not gate complexity or running time. Uniform algorithm extraction assumes - a supplied dual certificate. - branch: quantum query complexity - entry_module: LeanPool.QuantumQuery - authors: - - Troy Lee - source: - arxiv: 1011.3020v2 - url: https://github.com/troyjlee/quantum-query-complexity/tree/517ba85eb211232771429804d8583e3f52b9d541 - github_repo: troyjlee/quantum-query-complexity - commit: 517ba85eb211232771429804d8583e3f52b9d541 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - QuantumQueryComplexity.boundedErrorQQuery_characterized_by_advPM - main_results: - - declaration: QuantumQueryComplexity.advDual_eq_advPM - informal: For every total Boolean function on a finite Boolean input cube, the all-pairs real - adversary dual infimum equals the real symmetric primal supremum. - - declaration: QuantumQueryComplexity.advPM_composeFun_eq - informal: The negative-weight adversary value of total Boolean functions composed on disjoint - blocks equals the product of their adversary values, including constant functions and empty - index types. - - declaration: QuantumQueryComplexity.boundedErrorQQuery_characterized_by_advPM - informal: For a total Boolean function at error one third, quantum value-query complexity lies - between ADV±/36 and 2^14 ADV±. The value oracle swaps a blank answer with the queried letter - and includes an idle query index. - - declaration: QuantumQueryComplexity.xorQQuery_characterized_by_advPM - informal: For the Boolean XOR oracle with idle and blank sectors, query complexity at error one - third lies between ADV±/72 and 2^15 ADV±. - - declaration: QuantumQueryComplexity.qQueryOn_third_le_of_hasDualOn_uniform - informal: A supplied all-pairs real dual certificate of nonnegative cost c gives a finite quantum - algorithm using at most 8192(1+c) value queries at error one third, for any finite promise domain - and nonempty decidable output type. - - declaration: QuantumQueryComplexity.QAlg.exists_probability_polynomial - informal: Each output probability of a q-query value-oracle algorithm on Boolean inputs agrees - on the Boolean cube with a real polynomial of total degree at most 2q. - - declaration: QuantumQueryComplexity.exists_xor_probability_polynomial - informal: Each output probability of a q-query Boolean XOR algorithm agrees on the Boolean cube - with a real polynomial of total degree at most 2q. - tags: - - quantum-computing - - query-complexity - - adversary-method - - polynomial-method - msc: - - 68Q12 - - 81P68 - - title: Beyond the Bethe approximation of the permanent - summary: Constructs a deterministic polynomial-time approximation algorithm for the permanent of - every nonnegative rational matrix, with exponential base strictly smaller than the square root - of two. The development includes stable-polynomial inequalities, rational convex optimization, - finite-precision arithmetic, and a Turing-machine complexity proof. - branch: theoretical computer science - provenance: AI - main_declarations: - - BeyondBethe.theoremOne - main_results: - - declaration: BeyondBethe.theoremOne - informal: There is a fixed rational algorithm approximating the permanent of every nonnegative - rational n-by-n matrix within c^n for an absolute c below sqrt(2), with a deterministic polynomial-time - Turing-machine implementation. - tags: - - permanent - - approximation-algorithms - - computational-complexity - - stable-polynomials - msc: - - 68W25 - - 15A15 - slug: formalization-beyond-bethe - entry_module: LeanPool.BeyondBethe - authors: - - Nima Anari - - Samuel Schlesinger - - Bolton Bailey - - Christian Reitwiessner - source: - url: https://github.com/nimaanari/formalization-beyond-bethe - github_repo: nimaanari/formalization-beyond-bethe - commit: 325cda6d2118870f7f121a9a986b7ea9ffdd7a26 - license: Apache-2.0 - status: verified - - title: Conway refinement for omnific integers - summary: 'Proves multiplicative refinement for omnific integers: every equality ab = cd admits a - two-by-two factor refinement. Develops factorization and refinement of generalized power-series - integer parts and connects these to surreal-number and standalone Mathlib formulations.' - branch: ordered algebra - main_declarations: - - ConwayRefinement.Palomar.conwayRefinement - main_results: - - declaration: ConwayRefinement.Palomar.conwayRefinement - informal: If four omnific integers satisfy ab = cd, there are omnific integers e, f, g, h with - a = ef, b = gh, c = eg, and d = fh, including degenerate cases. - tags: - - surreal-numbers - - ordered-algebra - - factorization - msc: - - 06A05 - - 13F25 - provenance: AI - slug: conway-refinement - entry_module: LeanPool.ConwayRefinement - authors: - - Dan Abramov - source: - url: https://github.com/gaearon/conway-refinement - github_repo: gaearon/conway-refinement - commit: 264445c93b78554c408e99e4e7f663693b4e91ab - license: Apache-2.0 - status: verified - - slug: lean-operator-theory - title: Unitary dilation and the Crouzeix–Palencia bound - summary: Proves unitary power dilation for Hilbert-space contractions, von Neumann’s polynomial inequality, - and the Crouzeix–Palencia polynomial spectral-set estimate for the closed numerical range. - branch: operator theory - entry_module: LeanPool.OperatorTheory - authors: - - Ezzeri Esa - source: - url: https://github.com/savarin/lean-operator-theory - github_repo: savarin/lean-operator-theory - commit: 6f90bb7e6b056b54f56005e0416d77d6d7c2a478 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - PalomarCrouzeixPalencia.crouzeix_palencia - main_results: - - declaration: PalomarCrouzeixPalencia.exists_unitary_power_dilation - informal: Every contraction on a complex Hilbert space admits a unitary power dilation. - - declaration: PalomarCrouzeixPalencia.vonNeumann_inequality - informal: Polynomial evaluation on a contraction is bounded by the polynomial supremum on the closed - unit disk. - - declaration: PalomarCrouzeixPalencia.crouzeix_palencia - informal: The closure of the numerical range is a polynomial spectral set with constant 1 + √2. - tags: - - operator-theory - - functional-analysis - - spectral-theory - msc: - - 47A12 - - 47A20 - - 47A25 - - slug: nrr-aak - title: Equal-area and equal-perimeter convex partitions - summary: Formalizes the Akopyan–Avvakumov–Karasev theorem that every planar convex body can be partitioned - into any prescribed positive number of convex pieces with equal area and equal perimeter. Area - is Lebesgue measure and perimeter is Hausdorff measure of the boundary. - branch: convex geometry - main_declarations: - - HumanVerification.equalAreaEqualPerimeterPartition - main_results: - - declaration: HumanVerification.equalAreaEqualPerimeterPartition - informal: Every compact convex planar set with nonempty interior admits a convex partition into - any positive number of pieces whose areas and perimeters are all equal. - tags: - - convex-geometry - - equivariant-topology - - partitions - msc: - - 52A10 - - 55M20 - provenance: mix - entry_module: LeanPool.NandakumarRamanaRao - authors: - - Arseniy Akopyan - source: - url: https://github.com/akopjan/nrr-aak - github_repo: akopjan/nrr-aak - commit: 9e57be63320f29cbb8d2da7c4a8ad6afdef681eb - license: Apache-2.0 - status: verified - - slug: brill-noether-graphs - title: Brill–Noether theory and gonality of finite graphs - summary: Formalizes Brill–Noether existence through genus five, chains of marked banana graphs and - loops, the treewidth lower bound for gonality, and a tricycle example whose gonality drops under - regular subdivision. Includes divisor and transmission infrastructure and the Demazure-product - support used by vertex gluing. - branch: combinatorics and tropical geometry - main_declarations: - - Highlights.brill_noether_existence_through_five - main_results: - - declaration: Highlights.brill_noether_existence_through_five - informal: Every finite connected loopless multigraph of genus g at most five has, - for integers r and d, a divisor of degree d and rank at least r whenever - (r + 1) * (g - d + r) ≤ g. - - declaration: Highlights.treewidth_le_gonality - informal: On a finite connected loopless multigraph, for every natural number k, - any degree-k divisor of rank at least one bounds the treewidth by k; this is the - witness formulation of the treewidth lower bound for divisorial gonality. - - declaration: Highlights.gonality_drops_under_regular_subdivision - informal: There is a graph whose divisorial gonality strictly decreases under a regular subdivision. - tags: - - brill-noether-theory - - graph-theory - - chip-firing - - tropical-geometry - msc: - - 14T15 - - 05C50 - provenance: AI - entry_module: LeanPool.BrillNoetherGraphs - authors: - - Nathan Pflueger - source: - url: https://github.com/npflueger/brill-noether-graphs - github_repo: npflueger/brill-noether-graphs - commit: ce62dd82fd499fcec9b5a4110a5312befd6e96e2 - license: Apache-2.0 - status: verified - - slug: regts-sevenster - title: A proof of the Regts–Sevenster conjecture, formalized - summary: A Lean 4 proof of the Regts–Sevenster conjecture. A complex-valued graph - parameter, normalized at the empty graph and invariant under isomorphism, has - exponentially bounded edge-connection rank if and only if it is a mixed partition - function. For a natural rank base R, a representing model can be chosen with k - even and 2ℓ odd colours satisfying k + 2ℓ ≤ R. The development also proves the - even connection-rank growth limit, minimum total colour dimension, minimal parity - formulas and prescribed-dimension criterion. Schrijver's factorial argument supplies - nilpotent-trace vanishing; semisimplicity and Deligne's fibre-functor theorem - give the model; a polynomial commutant estimate gives the sharp colour budget. - Deligne's theorem and the required symmetric-group character theory are proved - within the development. The appendix's trace-zeta results give rationality and - degree bounds, with an independent proof of nilpotent-trace vanishing. The theorems - of record use only propext, Classical.choice and Quot.sound, checked by pinned - audits. - branch: algebraic graph theory - entry_module: LeanPool.RegtsSevenster - authors: - - William Whistler - source: - url: https://github.com/willwhistler/regts-sevenster - github_repo: willwhistler/regts-sevenster - commit: bb0a60543d48afac3f36938104b5c85c7c0396c7 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - RS.regts_sevenster_characterisation - main_results: - - declaration: RS.regts_sevenster_characterisation - informal: Normalized isomorphism-invariant complex graph parameters with exponentially - bounded edge-connection ranks are precisely mixed partition functions. - - declaration: RS.regts_sevenster_total - informal: A natural rank bound R admits a representing mixed model with total - color dimension at most R. - - declaration: RS.deligne_theorem - informal: The development proves its stated Deligne fiber-functor theorem for - the symmetric tensor categories used in the representation. - tags: - - graph-parameters - - tensor-categories - - partition-functions - msc: - - 05C50 - - 18M20 - - slug: nashembedding - title: Nash's smooth isometric embedding theorem - summary: 'A Lean 4 / Mathlib formalization of Nash''s C^∞ isometric embedding theorem - for closed manifolds: every compact smooth manifold without boundary, equipped - with a smooth Riemannian metric, admits a smooth injective map into some Euclidean - space ℝ^q whose differential pulls the Euclidean inner product back to the given - metric. The proof formalized is Günther''s (1989), which replaces the Nash-Moser - iteration by an elliptic fixed-point argument, following the presentation in A. - J. Wassermann''s Cambridge Part III lecture notes as the primary source. The development - has two layers. The first proves the theorem for the flat torus: a self-contained - Sobolev toolkit on 𝕋ⁿ (weighted ℓ² Sobolev spaces on ℤⁿ, Rellich compactness, - the Sobolev sup bound, three multiplication theorems, mollifiers, periodization, - Fourier inversion and Parseval), the approximation of smooth metrics by realizable - ones, Günther''s perturbation theorem proved by a contraction in Hˢ, and their - assembly into the statement that every smooth 2π-periodic positive-definite matrix - field on ℝⁿ is the pullback metric of a smooth periodic immersion that is injective - modulo 2πℤⁿ. The second layer reduces the closed-manifold case to the torus: a - Whitney embedding from Mathlib is extended to the ambient space, the metric is - extended by a pseudo-inverse construction, cut off, symmetrized and periodized, - and the torus theorem is applied. No bound on the embedding dimension is claimed. - Concrete instances (round spheres of every dimension, products of spheres, the - flat torus 𝕊¹ × 𝕊¹) are derived, and the flat-torus instance is shown to reproduce - the torus theorem''s conclusion, tying the two layers together. Nothing here is - claimed to be new.' - branch: differential geometry - entry_module: LeanPool.NashEmbedding - authors: - - David Wiygul - source: - url: https://github.com/langlerangle/nashembedding - github_repo: langlerangle/nashembedding - commit: 515a8245c51c952db59d81ef840a146ed853e4ea - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - NashEmbeddingTheorem.nash_isometric_embedding - main_results: - - declaration: NashEmbeddingTheorem.nash_isometric_embedding - informal: Every compact smooth boundaryless manifold with a smooth Riemannian - metric admits a smooth isometric embedding in some finite-dimensional Euclidean - space. - tags: - - differential-geometry - - isometric-embedding - - sobolev-spaces - msc: - - 53C24 - - slug: tutte-path - title: Tutte’s path theorem for matroids - summary: Proves the modular-cut path theorem for finite matroids, with reusable rank-partition, - hyperplane-separation, relative-complement, indecomposable-cover, and diamond constructions. - branch: matroid theory - entry_module: LeanPool.TuttePath - authors: - - Tutte formalization contributors - source: - arxiv: 2601.02582 - url: https://github.com/mbaker386/tutte-homotopy-lean - github_repo: mbaker386/tutte-homotopy-lean - commit: ab33c23369865b5fbd30b7a19710a527aa7ebab6 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - TutteFormalization.path_theorem - main_results: - - declaration: TutteFormalization.path_theorem - informal: In a finite connected matroid, any two hyperplanes containing a proper indecomposable - flat and lying outside a modular cut are joined by a finite Tutte path all of whose vertices - contain that flat and avoid the cut. Consecutive vertices are distinct hyperplanes with - indecomposable corank-two intersection; repeated nonconsecutive vertices are permitted. - - declaration: TutteFormalization.path_theorem_of_indecomposable - informal: In any finite matroid, two hyperplanes containing an indecomposable flat F and - lying outside a modular cut are joined by a Tutte path whose vertices contain F and avoid - the cut. This strengthens the cited path theorem by requiring only connectedness of the - contraction by F; properness of F follows from containment in an endpoint hyperplane. - - declaration: TutteFormalization.exists_indecomposable_complement - informal: If an indecomposable flat S and a flat U span the ground, every indecomposable flat - T contained in both extends inside S to an indecomposable flat R spanning the ground with U, - with corank R equal to rank U minus rank T. - - declaration: TutteFormalization.exists_relative_complement - informal: For nested flats T contained in S, there is a flat U containing T whose intersection - with S is T, whose join with S is the ground, and whose corank equals rank S minus rank T. - tags: - - matroids - - combinatorics - - modular-cuts - msc: - - 05B35 - - slug: parameter_free_gradient_p - title: Parameter-Free Gradient Minimization in l_p Geometry - summary: Exact value-gradient query-complexity results for convex differentiable objectives - on finite-dimensional real l_p spaces with attained minima and a positive global primal-to-dual - Lipschitz-gradient bound. For finite real p > 1, the development proves strict-local scale-identification - impossibility at LR/epsilon = 4 and a deterministic method family with three post-secant-initialization - query-rate regimes. For p > 2 it also proves deterministic known-parameter upper and lower - query exponents p/(p+2), with explicit horizon and dimension restrictions on the lower bound. - branch: optimization - entry_module: LeanPool.ParameterFreeGradient - authors: - - Yuning Yang - source: - url: https://github.com/yuningyang19/parameter_free_gradient_p - github_repo: yuningyang19/parameter_free_gradient_p - commit: f71215fa936fe01aff8a765c62b399e131df963b - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - V7.main - main_results: - - declaration: V7.scaleIdentificationImpossibility - informal: For finite real p > 1 and positive epsilon, measurable strict-local exact value-gradient - methods satisfy scale-identification lower bounds on convex smooth one-dimensional objectives - with unique minimizers, even at LR/epsilon = 4. The result includes deterministic and - randomized finite-horizon statements and infinite worst-case expected queried-or-returned - hitting time. - - declaration: V7.knownParameterAboveTwoOptimality - informal: For finite real p > 2 and convex differentiable objectives with attained minima - and a positive global primal-to-dual Lipschitz-gradient bound, deterministic known-parameter - exact value-gradient methods have upper query exponent p/(p+2). The matching lower exponent - concerns the first T queries, requires d >= 2 and 1 <= T <= d, and retains the explicit - min(p, log d) dependence. - - declaration: V7.main - informal: For convex differentiable objectives with attained minima and positive global - L-Lipschitz gradients from l_p to its dual, one deterministic family works for every finite - real p > 1 and positive epsilon after supplied nondegenerate secant initialization with - 0 < M0 <= L. It returns a queried point with dual gradient norm at most epsilon. Writing - R for distance to the minimizer set, the post-initialization oracle-call bound has polynomial - dependence on max(1, LR/epsilon) with exponent 1/2 for 1 < p < 2 and p = 2, and p/(p+2) - for p > 2, plus additive log(e + L/M0) calibration cost. Constants depend only on p, with - a universal constant at p = 2; discovering the secant pair is outside the count. - tags: - - optimization - - oracle-complexity - - gradient-methods - msc: - - 90C25 - - slug: order-closures - title: Order closure, order adherence and Fatou norms - summary: >- - Formalizes order and unbounded-order adherence in vector lattices, - the Gao-Leung characterization of order-continuous norms on - sigma-order-complete Banach lattices, and the Solovay/Stone-space - construction of order-complete C(K) spaces with arbitrarily large - density and norm-closed separable sublattices whose only order-closed - vector-sublattice extension is the whole C(K) space. Constructs solid - sets with a prescribed infinite number kappa of solid generators whose - adherence towers are strict below any ordinal at most kappa's - successor cardinal. Constructs a complete weakly Fatou lattice norm - with constant 2 that is not equivalent to any Fatou lattice norm. - Includes reusable vector-lattice, net-convergence, sublattice, - seminorm, Boolean-algebra and transfinite-iteration infrastructure. - branch: functional analysis - entry_module: LeanPool.OrderClosures - authors: - - Antonio Avilés - - Mitchell A. Taylor - - Pedro Tradacete - - David Muñoz-Lahoz - source: - title: Order closure, order adherence and Fatou norms - arxiv: '2609.06689' - url: https://github.com/pedrotradacete/OrderClosures/tree/6189ba7134f6975e7d8e4c8c093a9869d03a769c - github_repo: pedrotradacete/OrderClosures - commit: 6189ba7134f6975e7d8e4c8c093a9869d03a769c - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - OrderClosures.exists_weaklyFatou_not_equivalent_fatou - main_results: - - declaration: OrderClosures.gaoLeung_orderContinuous_characterization - informal: >- - For a sigma-order-complete Banach lattice X, the following are - equivalent: every vector sublattice has order-closed order - adherence; every vector sublattice has equal order and - unbounded-order adherence; and the norm of X is order continuous. - source_ref: Paper Theorem 1.1, recalling Gao-Leung Theorem 2.7. - - declaration: OrderClosures.gao_counterexample - informal: >- - For every cardinal kappa there is a compact Hausdorff space K - such that C(K, R) is order complete and has density at least - kappa, with a norm-closed separable vector sublattice Y whose - only order-closed vector-sublattice extension is all of C(K, R). - source_ref: Paper Proposition 2.1, via the Solovay Boolean algebra. - - declaration: OrderClosures.solid_sets_require_arbitrarily_many_iterations - informal: >- - For every infinite cardinal kappa and ordinal xi at most the - initial ordinal of kappa's successor cardinal, there is a - vector lattice and a solid set with solid generator number - kappa whose order-adherence tower has a proper step at every - stage below xi. The witness and cardinal/ordinal parameters - are lifted one universe in Lean. - source_ref: >- - Paper Theorem 2.6, with its explicit infinite-cardinal hypothesis. - Lean strengthens the stated size-kappa generating-set conclusion - to minimum solid generator number exactly kappa; this exact-minimum - conclusion is proved in the formalization, not stated in the paper. - - declaration: OrderClosures.exists_weaklyFatou_not_equivalent_fatou - informal: >- - There is a real vector lattice with a complete lattice norm p - satisfying the weak Fatou property with constant 2, such that - no lattice norm q with the Fatou property is equivalent to p. - Norms, completeness and equivalence are represented explicitly - by PaperLatticeNorm, IsCompleteFor and EquivalentNorms. - source_ref: >- - Paper Theorem 3.1; self-contained construction of the result - previously announced by Anthony Wickstead and credited to - Michael Elliott. The paper credits Jaume de Dios Pont for an - earlier Lean formalization of this result. - tags: - - banach-lattices - - order-convergence - - fatou-norms - - transfinite-constructions - msc: - - 46B42 - - 46A40 - - 46A19 - - slug: sendov-conjecture - title: Sendov's conjecture and the Phelps–Rodriguez conjecture - summary: >- - Proves Sendov's conjecture in full generality: if every zero of a complex polynomial of - degree at least two lies in the closed unit disk, then every zero has a critical point - within distance one. Also proves the Phelps–Rodriguez strengthening, with strict - inequality unless the zero lies on the unit circle and the polynomial is a scalar - multiple of z^n - a^n, and recovers Rubinstein's boundary theorem. Formalizes Tao's - digestion of Lech Mazur's proof: the problem is reduced to a numerical inequality in the - degree and one real parameter, settled by exact rational Bernstein certificates for - degrees up to one hundred and by an analytic large-degree reduction beyond. Includes the - top case of Maclaurin's inequality and multiset AM–GM, a defect inequality on the closed - unit disk, a sharp bound on log(sinh h / h), kernel-checked polynomial certificate - infrastructure, and a packed-exponentiation bridge for exact moment integrals. - branch: complex analysis - entry_module: LeanPool.Sendov - authors: - - Terence Tao - source: - title: Sendov's conjecture in Lean - url: https://github.com/teorth/sendov/tree/1ddea92d89f951a0a7cbbffa6c267cf7e6640b1d - github_repo: teorth/sendov - commit: 1ddea92d89f951a0a7cbbffa6c267cf7e6640b1d - announcement_url: https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/ - announced_at: '2026-08-12T23:10:54Z' - mathematical_source: >- - T. Tao, A digestion of the proof of Sendov's conjecture, blog post, 12 August 2026, - digesting Lech Mazur's AI-assisted proof; the boundary case follows Q. Tang and - T. Zhang, arXiv:2508.10341, equation (5.1) and Remark 5.1. - prior_formalization: >- - Lech Mazur's earlier Lean formalization of Sendov's conjecture at ProofAtlas - (https://www.proofatlas.ai/formalizations/sendov-conjecture/), which the upstream - README credits with priority for both the proof and the first formalization. The - present development shares no code with it and additionally states the - Phelps–Rodriguez equality classification. - license: Apache-2.0 - status: verified - provenance: AI - provenance_note: >- - The upstream README states that essentially all Lean source was written by Claude Opus 5 - in Claude Code under the direction and review of Terence Tao, who set the targets, - supplied the informal proof and staged plans, and made the mathematical decisions. - main_declarations: - - Sendov.sendov - - Sendov.phelps_rodriguez - main_results: - - declaration: Sendov.sendov - informal: >- - For every natural number n at least two and every complex polynomial p of degree n - all of whose roots have norm at most one, every root a of p admits a root of the - derivative of p within distance at most one of a. - source_ref: >- - Sendov's conjecture (1962); Tao's blog post of 12 August 2026, Conjecture 1, following - Mazur's proof; upstream Sendov/Conjecture.lean. First publicly announced - 2026-08-12 23:10:54 UTC in that post, with the Lean repository completed the same day. - - declaration: Sendov.phelps_rodriguez - informal: >- - Under the same hypotheses, either some root of the derivative lies at distance - strictly less than one from a, or a has norm one and p equals a nonzero scalar - multiple of X^n - a^n. - source_ref: >- - Phelps–Rodriguez conjecture; Tao's blog post, Conjecture 2; upstream - Sendov/Conjecture.lean. The equality case is Rubinstein's 1968 theorem. - - declaration: SendovConjecture.sendov - informal: >- - The same statement in the Mathlib-only form of the upstream statement of record: - for every complex polynomial p of degree at least two such that every complex w - with p(w) = 0 has norm at most one, every zero a of p admits a root of the - derivative within distance at most one of a. - source_ref: Upstream Solution.lean, the declaration checked by the upstream comparator setup. - - declaration: Sendov.rubinstein_one - informal: >- - If a polynomial of degree n at least two with all roots in the closed unit disk - vanishes at one, then either its derivative has a root within distance strictly less - than one of the point one, or the polynomial equals its leading coefficient times - X^n - 1. - source_ref: >- - Rubinstein's theorem; Tang–Zhang arXiv:2508.10341, equation (5.1) and Remark 5.1, - run to equality; upstream Sendov/Boundary.lean. - - declaration: Sendov.stat_lt_one - informal: >- - For every degree n at least five and every real parameter alpha between zero and - seventeen satisfying the feasibility constraint c^2 ≤ A, the quantity R n alpha of - the blog post's equation stat is strictly less than one. - source_ref: >- - Equation (stat) of Tao's blog post; degrees five to one hundred by Bernstein - certificates (upstream Sendov/FiniteRange), degrees at least 101 by the analytic - reduction (upstream Sendov/LargeDegree). - - declaration: Multiset.esymm_card_pred_le - informal: >- - For a nonempty multiset of nonnegative reals of cardinality m, the (m-1)-st - elementary symmetric function is at most m times the (m-1)-st power of the mean. - source_ref: Maclaurin's inequality, top case; upstream Sendov/Analytic/Maclaurin.lean. - - declaration: Multiset.prod_le_mean_pow - informal: >- - The product of a multiset of nonnegative reals is at most the mean raised to the - cardinality. - source_ref: Multiset AM–GM inequality; upstream Sendov/Analytic/Maclaurin.lean. - - declaration: Sendov.defect - informal: >- - For a multiset of complex numbers in the closed unit disk, the product of the norms - times the sum of the norms of 1/z minus the conjugate of z is at most one minus the - square of that product. - source_ref: The defect lemma; upstream Sendov/Analytic/Defect.lean. - - declaration: Sendov.log_sinh_div_le - informal: >- - For every positive real h, the logarithm of sinh h divided by h is at most the - square root of h^2 + 9 minus three. - source_ref: Upstream Sendov/Common/Sinh.lean; the crux of the beta-bound step. - - declaration: Sendov.pev_pos_of_bern - informal: >- - If the Bernstein expansion of an integer coefficient list on an interval from zero - to p/q equals p^d times a polynomial P and every Bernstein coefficient is positive, - then P is positive on that interval. - source_ref: >- - Lean Pool port infrastructure replacing the upstream per-certificate ring identities; - LeanPool/Sendov/FiniteRange/Certificate.lean. - tags: - - sendov-conjecture - - polynomial-zeros - - critical-points - - bernstein-certificates - msc: - - 30C15 - - 30C10 - - 12D10 - - slug: brooks-subcubic - title: Brooks' theorem for subcubic graphs - summary: >- - Proves that every finite simple graph of maximum degree at most three - with no four-clique admits a three-colouring, without a connectedness - assumption. Includes controlled greedy colouring, colouring across cut - vertices, and a good-triple argument for cubic components. This is the - subcubic case of Brooks' theorem, not the general-degree theorem. - branch: graph theory - entry_module: LeanPool.BrooksSubcubic - authors: - - Juan Pablo Traverso Gianini - source: - title: On colouring the nodes of a network - authors: - - R. L. Brooks - doi: 10.1017/S030500410002168X - url: https://github.com/jtraverso/lean-pool/blob/aced439fd4161d118bf167a1e8d10553f28913fe/LeanPool/BrooksSubcubic/Main.lean - github_repo: jtraverso/lean-pool - commit: aced439fd4161d118bf167a1e8d10553f28913fe - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - BrooksSubcubic.brooks_cubic - main_results: - - declaration: BrooksSubcubic.brooks_cubic - informal: >- - Every finite simple graph with maximum degree at most three and no - clique on four vertices is three-colourable. The graph may be disconnected. - source_ref: >- - The subcubic specialization of Brooks' 1941 colouring theorem; - the linked Apache-2.0 Lean source proves exactly this specialization. - tags: - - graph-theory - - vertex-colouring - - brooks-theorem - - subcubic-graphs - msc: - - 05C15 - - slug: vizing-equitable-edge-colouring - title: Vizing's theorem and equitable edge colourings - summary: >- - Proves the maximum-degree-plus-one edge-colouring bound for finite simple - graphs by fans and Kempe exchanges. A separate component-swapping argument - equitably rebalances any finite proper edge colouring on its original - palette. Together they give equitable proper edge colourings on any - palette of at least maximum degree plus one colours, with every class - bounded by the ceiling of the average class size. Extracted from the - published Paper IV formalization; these are classical colouring results, - not a claim to new mathematics or to the paper's clique-partition bound. - branch: graph theory - entry_module: LeanPool.Vizing - authors: - - Juan Pablo Traverso Gianini - source: - title: Vizing and equitable edge-colouring modules in the published Paper IV freeze - authors: - - Juan Pablo Traverso Gianini - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions - github_repo: jtraverso/erdos-81-chordal-clique-partitions - commit: ff2019f56a50cf300a6531c9f8d4a8fb1a541e3d - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPool.Vizing.exists_equitable_edge_colouring - main_results: - - declaration: LeanPool.Vizing.lineGraph_colorable - informal: >- - The line graph of a finite simple graph G is colourable with - maximum degree of G plus one colours. - source_ref: Vizing's theorem for finite simple graphs; Vizing (1964). - - declaration: SimpleGraph.lineGraph_chromaticNumber_le_maxDegree_succ - informal: >- - The chromatic number of the line graph of a finite simple graph G - is at most maximum degree of G plus one. - - declaration: LeanPool.Vizing.Equitable.exists_equitable_colouring - informal: >- - Every proper edge colouring of a finite simple graph with a finite - palette admits a proper recolouring on the same palette in which - any two colour classes differ in size by at most one. - - declaration: LeanPool.Vizing.exists_equitable_edge_colouring - informal: >- - For every finite simple graph G and natural c at least maximum - degree of G plus one, there is an equitable proper edge colouring - with c colours; each colour class has at most ceiling(|E(G)|/c) edges. - tags: - - graph-theory - - edge-colouring - - vizing-theorem - - equitable-colouring - - kempe-chains - msc: - - 05C15 - - slug: paper-iv-clique-trees - title: Clique trees, subtree representations and chordal treewidth - summary: >- - Develops rooted clique forests with the running-intersection property, constructs - them from perfect elimination orders of finite chordal graphs, characterizes - chordality by their existence, and obtains maximal-clique, separator, and - counting results. Also characterizes chordality under retained-clique gluing - of induced pieces, with exact vertex and edge inclusion-exclusion identities. - The original library was extracted as a reusable byproduct of the published Paper IV - formalization of the chordal-graph case of Erdős problem #81; this library - is not by itself a proof of that paper's partition bound. - Gavril's characterization supplies exact finite-tree subtree representations - and the converse to chordality. Finite subtree Helly places every finite clique - in a bag of any tree decomposition and gives optimal chordal treewidth. - The supplied representation includes an empty connector; no minimal-host - or runtime bound is claimed. - branch: graph theory - entry_module: LeanPool.PaperIVCliqueTree - authors: - - Juan Pablo Traverso Gianini - source: - title: 'Clique partitions with rooted simplicial defect: quantitative stability and sharp eventual bounds' - authors: - - Juan Pablo Traverso Gianini - url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions - github_repo: jtraverso/erdos-81-chordal-clique-partitions - commit: ff2019f56a50cf300a6531c9f8d4a8fb1a541e3d - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - SimpleGraph.isChordal_iff_nonempty_subtreeRepresentation - main_results: - - declaration: SimpleGraph.IsChordal.exists_isPEO - informal: Every finite chordal graph admits a perfect elimination order. - - declaration: SimpleGraph.isChordal_iff_nonempty_cliqueTree - informal: A finite graph is chordal if and only if it has a rooted clique tree. - - declaration: SimpleGraph.IsChordal.exists_maximalCliqueTree_card_le - informal: >- - A nonempty finite chordal graph has a clique tree whose distinct bags are - precisely its maximal cliques, with no more bags than vertices. - - declaration: SimpleGraph.CliqueTree.exists_parent_eq_of_isMinimalSeparator - informal: >- - In a clique tree, each minimal separator of two reachable vertices is - the intersection of two adjacent bags. - - declaration: SimpleGraph.CliqueTree.card_eq_sum_card_bag_sub_sum_card_inter - informal: >- - The total bag size minus the total parent-child bag intersection size - equals the number of graph vertices. - - declaration: SimpleGraph.CliqueTree.toSubtreeRepresentation - informal: >- - A finite rooted clique forest gives nonempty connected vertex sets of - a finite host tree, with adjacency of distinct source vertices exactly - equivalent to intersection of their assigned sets. Empty and disconnected - source graphs are supported through one empty connector node. - source_ref: 'Gavril, JCT B 16 (1974), 47–56, doi:10.1016/0095-8956(74)90094-X; constructor using the PR 579 adapter.' - - declaration: SimpleGraph.isChordal_iff_nonempty_subtreeRepresentation - informal: >- - A finite graph on a vertex type in Type is chordal if and only if it has - an exact representation by nonempty connected vertex sets of a finite tree. - source_ref: 'Gavril, JCT B 16 (1974), 47–56, doi:10.1016/0095-8956(74)90094-X.' - - declaration: SimpleGraph.IsTree.finite_subtree_helly - informal: >- - A finite pairwise-intersecting family of nonempty connected vertex sets - of a finite tree has a common vertex; the empty family is also allowed. - - declaration: Utilities.Treewidth.TreeDecomposition.exists_clique_subset_bag - informal: Every finite clique is contained in one bag of any tree decomposition. - - declaration: SimpleGraph.IsChordal.treewidth_succ_eq_cliqueNum - informal: >- - For every finite nonempty chordal graph on a vertex type in Type, - its natural treewidth plus one equals its clique number. - - declaration: SimpleGraph.IsChordal.exists_optimal_treeDecomposition - informal: >- - A finite nonempty chordal graph admits a tree decomposition whose - width plus one is its clique number. - - declaration: SimpleGraph.CliqueGluing.isChordal_iff - informal: >- - If two induced pieces cover a finite graph, their overlap is a clique, - and no edge joins their exclusive parts, the graph is chordal exactly - when both pieces are chordal. Overlap edges are retained. - source_ref: >- - New Lean Pool extension proved here using the existing Dirac and - simplicial-elimination interfaces; not a claim from the frozen paper. - - declaration: SimpleGraph.CliqueGluing.card_vertices_add_overlap - informal: >- - The vertex count of the graph plus the overlap size equals the sum - of the vertex counts of the two covering pieces. - source_ref: New Lean Pool extension; finite-set inclusion-exclusion. - - declaration: SimpleGraph.CliqueGluing.card_edges_add_choose_overlap - informal: >- - The graph edge count plus the number of pairs in the retained clique - overlap equals the sum of the induced pieces' edge counts. - source_ref: New Lean Pool extension; induced-edge inclusion-exclusion. - tags: - - graph-theory - - chordal-graphs - - clique-trees - - perfect-elimination-orders - - separators - msc: - - 05C75 - - slug: beck-fiala-matrix - title: Strict Beck–Fiala rounding for incidence matrices - summary: >- - Proves the rational integer-making form of Beck–Fiala rounding for finite - zero-one incidence matrices with column degree bounded by a positive t. - Every row has discrepancy strictly less than t, and initially integral - coordinates are preserved. Includes the necessity of the positive-bound - hypothesis. This is distinct from the Komlós discrepancy bound and from - the non-strict hyperedge-selection rounding used in triangle packing. - branch: discrepancy theory - entry_module: LeanPool.BeckFialaMatrix - authors: - - Juan Pablo Traverso Gianini - source: - title: '"Integer-making" theorems' - authors: - - József Beck - - Tibor Fiala - doi: 10.1016/0166-218X(81)90022-6 - url: https://github.com/jtraverso/lean-pool/blob/523a4ae71168258f9142d70bccf650a140355650/LeanPool/BeckFialaMatrix/Basic.lean - github_repo: jtraverso/lean-pool - commit: 523a4ae71168258f9142d70bccf650a140355650 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - BeckFialaMatrix.exists_rounding - main_results: - - declaration: BeckFialaMatrix.exists_rounding - informal: >- - Given a finite zero-one integer matrix with at most t ones per column, - t at least one, and a rational vector in the unit cube, there is a - zero-one integer vector whose discrepancy in every row is strictly - less than t. - source_ref: Rational unit-cube integer-making form with the stated strict discrepancy bound. - - declaration: BeckFialaMatrix.exists_rounding_preserving_integral - informal: >- - The strict rounding can additionally be chosen to leave every initially - integral coordinate unchanged. - source_ref: Strengthened conclusion of the floating-coordinate proof in the pinned source. - - declaration: BeckFialaMatrix.zero_bound_counterexample - informal: >- - The same universally quantified strict rounding claim without the - assumption t at least one is false, already for a one-by-one zero matrix - with t equal to zero. - source_ref: Boundary-condition witness in the pinned formalization. - tags: - - discrepancy-theory - - beck-fiala - - integer-rounding - - incidence-matrices - msc: - - 05D40 - - slug: line-graph-colouring-matching - title: Line-graph colourings and matching decompositions - summary: >- - Connects Mathlib line-graph colourings to palette-indexed partitions of - actual graph edges into Mathlib matching subgraphs. Both conversions - are inverse, preserve each matching exactly, and allow empty colours - without a default colour or a finiteness assumption. An edge equivalence - yields exact edge accounting: natural-number counts require only finite - edges and a finite palette, while indexed cardinal counts allow arbitrary - vertices, edges, and palettes. This is classical representation - interoperability, not a new colouring bound or a claim of priority. - branch: graph theory - entry_module: LeanPool.GraphColouringMatching - authors: - - Juan Pablo Traverso Gianini - source: - title: Original Lean Pool graph-colouring and matching adapters - url: https://github.com/jtraverso/lean-pool/tree/4c1fddb73f9891afc2186f8f11066ea2a08fe4fc/LeanPool/GraphColouringMatching - github_repo: jtraverso/lean-pool - commit: 4c1fddb73f9891afc2186f8f11066ea2a08fe4fc - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - SimpleGraph.lineGraphColoringEquivMatchingDecomposition - main_results: - - declaration: SimpleGraph.lineGraphColoringEquivMatchingDecomposition - informal: >- - Proper colourings of the line graph on any palette are equivalent - to partitions of the original edges into matching subgraphs indexed - by that palette, with empty classes permitted. - source_ref: >- - New adapter proved in GraphColouringMatching/Basic.lean, using - Mathlib's existing lineGraph, Coloring and Subgraph.IsMatching. - - declaration: SimpleGraph.MatchingDecomposition.edgeMatching_lineGraphColoring - informal: >- - Converting a matching decomposition to a colouring and extracting - one colour class returns exactly its original matching subgraph. - - declaration: SimpleGraph.MatchingDecomposition.edgeEquivSigma - informal: >- - Original graph edges are in bijection with the disjoint union of - the edge types of the palette-indexed matching subgraphs. - - declaration: SimpleGraph.MatchingDecomposition.card_edges_eq_sum - informal: >- - For a finite edge set and finite palette, the number of graph edges - equals the sum of the numbers of edges in its matching classes, - even when the vertex set is infinite. - - declaration: SimpleGraph.MatchingDecomposition.cardinal_edges_eq_sum - informal: >- - Without any finiteness assumption on vertices, edges, or colours, - the cardinality of the graph edge set equals the indexed cardinal - sum of the matching-class edge cardinalities, with a universe lift - on the graph edge cardinality when required. - source_ref: >- - Cardinal.lean applies Mathlib's indexed cardinal sum to the existing - edgeEquivSigma; no infinite graph-colouring theorem is assumed. - - declaration: SimpleGraph.lineGraph_colorable_iff_matchingDecomposition - informal: >- - The line graph is k-colourable if and only if the graph edges - decompose into k matchings, allowing unused colours. - tags: - - graph-theory - - line-graphs - - edge-colouring - - matchings - msc: - - 05C15 - - 05C70 - - slug: stafford38-formal - title: Stafford 3.8 - summary: Lean formalization of Stafford's Conjecture 3.8 for every characteristic-zero - field and every Weyl-algebra rank, proving that each nonzero d admits F, R, and - S with 1 = d*R + F*d*S. The development also proves an exact Bernstein-degree - fixed-source strengthening and derived localization, adjoint, and evolutionary - results. - branch: noncommutative algebra - entry_module: LeanPool.Stafford38 - authors: - - Christopher Albert - source: - url: https://github.com/itpplasma/stafford38-formal - github_repo: itpplasma/stafford38-formal - commit: 1e234855eee36d2541dd69580501408288760e21 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - Stafford38.universalStatement - main_results: - - declaration: Stafford38.universalStatement - informal: For every characteristic-zero field, every Weyl algebra rank, and every - nonzero d, there are F, R, S with 1 = d*R + F*d*S. - - declaration: Stafford38.universalFixedSourceStatement - informal: The stated fixed-source strengthening holds with its exact Bernstein-degree - bound. - tags: - - weyl-algebras - - noncommutative-algebra - - bernstein-degree - msc: - - 16S32 - - - slug: elliptic-pde - title: 'Linear elliptic PDE: solvability, regularity and spectral theory' - summary: 'A reusable analytic development of linear uniformly elliptic divergence-form PDE on Euclidean - domains: Hilbert graph models of H¹ and H₀¹, bounded coefficient operators, Poincaré inequalities, - zero-drift/nonnegative-potential Dirichlet existence and uniqueness, Gårding and Fredholm theory - for general drift, interior H² and higher Sobolev regularity, smooth classical representatives, - Sobolev and Hölder embeddings, C¹-domain extension and compactness, variational eigenvalues, - and maximum principles. AI-generated proofs with human mathematical design and transcription; - announced in arXiv:2609.32561 on 2026-09-26. The unfinished boundary-H² program is outside this - completed dependency closure.' - branch: analysis and partial differential equations - entry_module: LeanPool.EllipticPDE - authors: - - Alejandro José Soto Franco - - Kobe Marshall-Stevens - source: - url: https://github.com/alejandro-soto-franco/EllipticPDE - github_repo: alejandro-soto-franco/EllipticPDE - commit: eaf821d31b200bb6ea235f19eecc50cf0f38c294 - arxiv: '2609.32561' - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - EllipticPdes.Regularity.higher_interior_regularity - main_results: - - declaration: EllipticPdes.Sobolev.FullEllipticOp.weak_solution_L2_of_nonneg_zeroth_of_bounded - informal: On a bounded domain in positive dimension, a bounded measurable uniformly elliptic - operator with zero drift and nonnegative zeroth-order coefficient has a unique H₀¹ weak solution - for every L² datum, with a datum-independent norm bound. - - declaration: EllipticPdes.Regularity.interior_H2_estimate - informal: In positive dimension, on an open measurable domain, an H₀¹ weak solution of a - bounded measurable uniformly elliptic divergence-form operator with globally Lipschitz - principal coefficients and an L² datum has interior L² second weak derivatives, - with an estimate whose constant is chosen before the solution and datum. - - declaration: EllipticPdes.Regularity.higher_interior_regularity - informal: In positive dimension, for a uniformly elliptic divergence-form operator with - bounded measurable coefficients, globally Lipschitz principal-coefficient representatives - in W^{k+1,∞}(ℝ^d), and lower-order coefficients in W^{k,∞}(ℝ^d), every H₀¹ weak solution - on an open measurable domain with an H^k datum has interior H^{k+2} regularity. - On each compact subset the iterated L² derivative bounds have a constant chosen before - the solution, datum and its H^k bound. The pointwise Lipschitz hypothesis is separate - from the weak-derivative bundles. - - declaration: EllipticPdes.Sobolev.embL2_isCompact - informal: The inclusion H₀¹(Ω) → L²(Ω) is compact for bounded measurable Euclidean domains, discharging - the compactness input to the Fredholm and spectral results. - - declaration: EllipticPdes.Extension.exists_extW12 - informal: In positive dimension, bounded open Euclidean domains with C¹ boundary admit - a bounded linear H¹ extension operator with compact support control. - tags: - - elliptic-pde - - sobolev-spaces - - regularity - - fredholm-theory - - spectral-theory - msc: - - 35J15 - - 35J25 - - 35B65 - - 46E35 - - slug: qec-certificates - title: Quantum code certificates and hypergraph-product theory - summary: >- - Formalizes general binary row reduction, kernel bases and rank certificates; - quantum-code distance bounds from witnesses and exhaustive low-weight - enumeration; arbitrary-matrix hypergraph-product dimension and X/Z cleaning - bounds; and CSS orthogonality of lifted products over finite groups. - Includes a Pauli expression translation modulo global phase, a Boolean - encoding with proved soundness and model extension, and sound checking - of RUP additions in LRAT certificates, bridged to matrix-level distance bounds. - Concrete code-family datasets, - large certificate replays and measurement protocols are outside this import. - branch: quantum information and coding theory - entry_module: LeanPool.QECCertificates - authors: - - Shuoming An - source: - title: 'Quantum code parameters, checkable by a certificate of provable size' - authors: - - Shuoming An - - Fusheng Yang - arxiv: '2610.03214' - url: https://github.com/QCL-SUAT/QECCertificates - github_repo: QCL-SUAT/QECCertificates - commit: 1ced3ae0317725f80640d4a38b9c03eaba84a9c9 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - QECCertificates.hgp_kunneth - main_results: - - declaration: QECCertificates.hgp_kunneth - informal: >- - For arbitrary finite binary matrices A and B, the number of columns - of their hypergraph product minus its X and Z ranks equals - nullity(A) times nullity(B) plus nullity(transpose A) times - nullity(transpose B), expressed with natural-number rank differences. - source_ref: Upstream GF2/HGPKunneth.lean; the Tillich–Zémor hypergraph-product construction. - - declaration: QECCertificates.hgp_X_distance_ge - informal: >- - If a binary vector is killed by the hypergraph-product X check and - is outside the Z row space, its Hamming weight is at least the - minimum of supplied lower bounds for nonzero vectors in the kernels - of the first input and the transpose of the second input. - - declaration: QECCertificates.hgp_Z_distance_ge - informal: >- - Dually, a vector killed by the Z check and outside the X row space - has weight at least the minimum of supplied lower bounds for the - kernels of the transpose of the first input and the second input. - - declaration: QECCertificates.lp_orthogonal - informal: >- - For arbitrary matrices of binary-valued functions on a finite group, - the left/right regular expansion gives lifted-product X and Z checks - whose every pair of rows has zero binary dot product. - - declaration: QECCertificates.Matrix.rank_eq_length_rowReduce - informal: >- - The rank of a finite binary matrix equals the length of the verified - row-reduction output for its list of rows. - - declaration: QECCertificates.eq_minWeight_of_decide - informal: >- - For two binary matrices, if d is at most the number of columns, - exhaustive enumeration finds no vector of weight below d in the first - kernel outside the second row space, and a supplied vector has those - membership properties and weight d, then the minimum such weight is d. - The empty-set distance convention is the number of columns plus one. - - declaration: QECCertificates.LRAT.unsat_of_checkSteps - informal: >- - Acceptance by the RUP certificate checker implies unsatisfiability of - the original CNF. No correctness assumption on an external SAT solver - is needed; the clause-database interface carries proved soundness laws. - - declaration: QECCertificates.LRAT.buildPair_complete - informal: >- - For a positive number n of input bits and nonempty row supports within - that range, a Boolean assignment satisfying the parity constraints, - odd bilinear pairing and weight bound extends to a model of buildPair - while agreeing on all 2n input variables. The complementary soundness - theorem extracts those constraints from every model. - source_ref: >- - Upstream Reflect/Encode.lean and Reflect/Complete.lean. The import - retains general theorems, not the paper's concrete instance results. - First public paper announcement: 2026-10-02 12:31:45 UTC, arXiv v1. - The upstream paper discloses AI-generated Lean proofs reviewed and - validated by the author. RowspaceKernel is derived from QCL-SUAT/Lean-QEC - at a0855f8a6374c1dda2097bbf07f33ba53760fdcd (Apache-2.0), credited to - Mattias Ehatamm, Yi Lee, Xiaodi Wu and Runzhou Tao, and their paper - End-to-End Formalization of Quantum Error Correction, arXiv:2605.16523. - - declaration: QECCertificates.LRAT.le_minWeight_of_unsat - informal: >- - For a positive number n of columns and k + 1 ≤ n, an unsatisfiable - Boolean encoding gives a matrix-level distance lower bound of k + 1. - Support-row correspondence, nonemptiness, index bounds and duplicate-free - hypotheses are explicit. This composes with kernel-checked RUP replay. - source_ref: >- - General bridge from upstream Reflect/Certified.lean; the minimum-weight - distance wrapper is added during the Lean Pool port. - tags: - - quantum-error-correction - - coding-theory - - hypergraph-products - - certificates - msc: - - 81P45 - - 94B05 - - 03B35 - - slug: quadratic-carleson-palomar-submission - title: Endpoint estimates for Stein's purely quadratic Carleson operator - summary: 'A Lean 4 formalization of four principal endpoint assertions from Fragkos, - Krause, and Lacey''s Endpoint Estimates for Stein''s Purely Quadratic Carleson - Operator: two failures of modular estimates below t log₂ t, a full-modulation - L log L endpoint estimate, and a lacunary L (log₂ L)² log₄ L endpoint estimate.' - branch: harmonic analysis - entry_module: LeanPool.QuadraticCarleson - authors: - - Anastasios Fragkos - source: - url: https://github.com/anastasiosfragkosmath/quadratic-carleson-palomar-submission - github_repo: anastasiosfragkosmath/quadratic-carleson-palomar-submission - commit: 3d3cf4316acac4c23140059a78fb570b1440394b - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - QuadraticCarleson.StatementSurface.PaperTheorems.full_LlogL_endpoint - main_results: - - declaration: QuadraticCarleson.StatementSurface.PaperTheorems.lacunary_sub_log2_modular_failure - informal: Lacunary purely quadratic modulation fails modular estimates below the - specified iterated-logarithmic threshold. - - declaration: QuadraticCarleson.StatementSurface.PaperTheorems.full_sub_log2_modular_failure - informal: Full purely quadratic modulation fails modular estimates below the specified - iterated-logarithmic threshold. - - declaration: QuadraticCarleson.StatementSurface.PaperTheorems.full_LlogL_endpoint - informal: The full-modulation operator satisfies the L log L endpoint estimate. - - declaration: QuadraticCarleson.StatementSurface.PaperTheorems.lacunary_log2_squared_log4_endpoint - informal: The lacunary operator satisfies the L (log₂ L)² log₄ L endpoint estimate. - tags: - - harmonic-analysis - - maximal-operators - - endpoint-estimates - msc: - - 42B20 - - slug: eilenberg-zilber - title: Eilenberg–Zilber theorem for bisimplicial objects - summary: >- - For a bisimplicial object X in an abelian category, constructs a chain homotopy - equivalence between the total complex of the alternating face map double complex - of X and the alternating face map complex of its diagonal. The equivalence is - built from the shuffle map and the Alexander–Whitney map. On normalized complexes, - shuffle followed by Alexander–Whitney is the identity, and the other composite is - homotopic to the identity via the Eilenberg–Mac Lane recursive homotopy. The - result is then transported to the unnormalized complexes along the Moore - normalization comparisons. The development includes the combinatorics of - (p, q)-shuffles, their signs, and the cancellation of diagonal boundary terms. - branch: algebraic topology - entry_module: LeanPool.EilenbergZilber - authors: - - Jeffrey Li - source: - url: https://github.com/jeffrey-dot-li/lean-homology/tree/40c5aae235b0f499ebfc1e68ce574a62b531203c/HomologyLean/SingularHomology/EilenbergZilberTheorem - github_repo: jeffrey-dot-li/lean-homology - commit: 40c5aae235b0f499ebfc1e68ce574a62b531203c - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - LeanPool.EilenbergZilber.BisimplicialObject.eilenbergZilber - main_results: - - declaration: LeanPool.EilenbergZilber.BisimplicialObject.eilenbergZilber - informal: >- - For every bisimplicial object X in an abelian category, the total complex of - the double complex obtained by taking alternating face map complexes in both - simplicial directions is chain homotopy equivalent to the alternating face - map complex of the diagonal simplicial object of X. - source_ref: >- - Generalized Eilenberg–Zilber theorem for bisimplicial objects in an abelian - category: Dold–Puppe, "Homologie nicht-additiver Funktoren. Anwendungen", - Ann. Inst. Fourier 11 (1961), 201–312, Satz 2.9 (Eilenberg–Zilber–Cartier); the - homology-level statement is Weibel, An Introduction to Homological Algebra - (1994), Theorem 8.5.1. The product case X_{p,q} = K_p ⊗ L_q is Eilenberg–Zilber, - "On products of complexes", Amer. J. Math. 75 (1953), 200–204. - - declaration: LeanPool.EilenbergZilber.BisimplicialObject.eilenbergZilberNormalized - informal: >- - The total complex of the bi-normalized Moore double complex of X is chain - homotopy equivalent to the normalized Moore complex of the diagonal, via the - shuffle and Alexander–Whitney maps; shuffle followed by Alexander–Whitney is - the identity. - source_ref: >- - Eilenberg–Mac Lane, "On the groups H(Π, n), II", Ann. of Math. 60 (1954), - 49–139, Theorem 2.1a, stated there for products of complexes and generalized here to - an arbitrary bisimplicial object in an abelian category. - tags: - - eilenberg-zilber - - bisimplicial-objects - - shuffle-map - - alexander-whitney - - chain-homotopy - msc: - - 55U10 - - 18G30 - - 18G35 - - slug: markoff-modp - title: Markoff mod p - summary: Lean 4 formalization of strong approximation for the Markoff surface modulo - sufficiently large primes. The primary source is Bourgain-Gamburd-Sarnak. - branch: number theory - entry_module: LeanPool.MarkoffModP - authors: - - Yuma Mizuno - - Guanghao Li - source: - url: https://github.com/yuma-mizuno/markoff-modp - github_repo: yuma-mizuno/markoff-modp - commit: ac1099fc96b0f1cf6e6099c74a24b783691c21aa - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - BGS.Markoff.reduction_surjective_of_explicitBound - main_results: - - declaration: BGS.Markoff.reduction_surjective_of_explicitBound - informal: For primes exceeding the explicit bound, reduction of integral Markoff - points to the Markoff surface modulo p is surjective as specified by the theorem. - tags: - - markoff-equation - - strong-approximation - - finite-fields - msc: - - 11D25 - - slug: viscosity-solution-theory - title: Viscosity solutions, Aleksandrov differentiability and harmonic Dirichlet problems - summary: Develops second-order semijets, smooth test-function equivalence, Jensen contact sets, - convex Aleksandrov differentiability, the quadratic-penalty Ishii lemma and compact-domain - comparison for degenerate elliptic equations. Perron assembly retains explicit comparison, - barrier and bumping assumptions; these are discharged for classical harmonic Dirichlet - existence on bounded open domains satisfying a uniform exterior sphere condition. - branch: partial differential equations - entry_module: LeanPool.ViscositySolutionTheory - authors: - - William M. Feldman - - AleksandrovDifferentiability contributors - - The Tau Ceti contributors - source: - arxiv: math/9207212 - url: https://github.com/willmfeldman/viscosity-solution-theory/tree/067e2540ad1594408c0b6145c5bde01b4d5791a9 - github_repo: willmfeldman/viscosity-solution-theory - commit: 067e2540ad1594408c0b6145c5bde01b4d5791a9 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - ViscositySolns.dirichlet_harmonic_modulus_of_uniformExteriorSphere - main_results: - - declaration: AleksandrovDifferentiability.convexAleksandrovAE - informal: A convex real-valued function on an open subset of any finite-dimensional real inner - product space is second-order differentiable at Lebesgue-almost every point. - - declaration: ViscositySolns.JensenContactSetPositiveMeasureOnClosedBallTheorem.proof - informal: Under the localized semiconvexity and strict-contact hypotheses, the contact set - arising from sufficiently small linear perturbations has positive Lebesgue measure. - - declaration: ViscositySolns.QuadraticPenaltyIshiiLemmaOn.proof - informal: For an upper semicontinuous function and a lower semicontinuous function on locally - compact subsets of real coordinate space, doubled-variable maxima with the quadratic - penalty yield closed semijets satisfying the Ishii block-matrix bounds. - - declaration: ViscositySolns.comparison_of_constantShift_boundary_proof - informal: A viscosity subsolution lies below a supersolution on a nonempty locally compact - domain with compact closure, assuming boundary comparison, semicontinuity on the closure, - properness, operator continuity, the Ishii structural condition and uniform scalar decrease. - - declaration: ViscositySolns.viscositySubsolution_iff_smoothTestFunctionSubsolution - informal: On an open domain, semijet and smooth C² test-function formulations of viscosity - subsolutions agree for Hessian-continuous, symmetric-part-invariant operators. - - declaration: ViscositySolns.dirichlet_harmonic_modulus_of_uniformExteriorSphere - informal: On a bounded open subset of finite-dimensional Euclidean space with a uniform - exterior sphere condition, Lipschitz data on its closure admit a continuous - extension that agrees on the boundary and is C² and harmonic in the domain, with a - boundary modulus depending only on the domain and Lipschitz constant. - tags: - - pde - - viscosity-solutions - - convex-analysis - - harmonic-functions - - perron-method - msc: - - 35D40 - - 35J60 - - 26B25 - - slug: polylog-integrals - title: Polylogarithms and sixteen log-trigonometric integral identities - summary: >- - Formalizes the sixteen A-series identities from rainrzk's PolyLog - Integrals tutorial, together with reusable real and complex - polylogarithm series, absolute convergence on the closed unit disk, - differentiation, duplication, Euler reflection and Landen identities, - logarithmic moments, Fubini kernels and radial integration machinery. - The formulas involve Catalan's constant, zeta(3), Li_4(1/2) and - Im Li_3((1+i)/2). These are classical identities, not new informal - results. A004 uses the corrected factor pi^2/24; A014 uses a proved - expression entirely inside the unit disk, replacing the tutorial's - invalid series expansion. A015 uses radial differentiation instead - of a half-disc contour integral. The formalization was materially - completed on 2026-10-01 at 00:01:57 UTC. The 2022 informal tutorial - is older; the eligibility date is that of the completed Lean proofs. - branch: analysis - entry_module: LeanPool.PolylogIntegrals - authors: - - rainrzk - source: - title: LeanPolyLog, completed formalization of PolyLog Integrals - url: https://github.com/rainrzk/LeanPolyLog/tree/056b7b42fd1a26f241e2945c67ae821ec5bdc49b - github_repo: rainrzk/LeanPolyLog - commit: 056b7b42fd1a26f241e2945c67ae821ec5bdc49b - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - LeanPolyLog.Proofs.A016 - main_results: - - declaration: LeanPolyLog.Proofs.A001 - informal: >- - Li_2(i) equals -pi^2/48 plus i times Catalan's constant. - source_ref: >- - Upstream Statements.lean and Proofs/A001.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A002 - informal: >- - Li_2(1/(1+i)) equals 5 pi^2/96 - log^2(2)/8 plus i times (pi - log(2)/8 - G). - source_ref: >- - Upstream Statements.lean and Proofs/A002.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A003 - informal: >- - Li_2((sqrt(5)-1)/2) equals pi^2/10 - log^2((1+sqrt(5))/2). - source_ref: >- - Upstream Statements.lean and Proofs/A003.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A004 - informal: >- - The integral from 0 to 1 of log(1+x^2)/x equals pi^2/24, - correcting the tutorial's factor of two. - source_ref: >- - Upstream Statements.lean and Proofs/A004.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A005 - informal: >- - The integral from 0 to 1 of log^2(1+x^2)/x^3 equals pi^2/12 - - log^2(2). - source_ref: >- - Upstream Statements.lean and Proofs/A005.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A006_cos - informal: >- - The integral from 0 to pi/4 of log(cos x) equals G/2 - (pi/4) - log(2). - source_ref: >- - Upstream Statements.lean and Proofs/A006.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A006_sin - informal: >- - The integral from 0 to pi/4 of log(sin x) equals -G/2 - (pi/4) - log(2). - source_ref: >- - Upstream Statements.lean and Proofs/A006.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A007 - informal: >- - The integral from 0 to pi/4 of x log(cos x) equals pi G/8 - pi^2 - log(2)/32 - 21 zeta(3)/128. - source_ref: >- - Upstream Statements.lean and Proofs/A007.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A008 - informal: >- - The integral from 0 to pi/4 of x log(1+tan x) equals pi^2 - log(2)/64 - pi G/8 + 21 zeta(3)/64. - source_ref: >- - Upstream Statements.lean and Proofs/A008.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A009 - informal: >- - The integral from 0 to pi/2 of x^2 cot x equals pi^2 log(2)/4 - 7 - zeta(3)/8. - source_ref: >- - Upstream Statements.lean and Proofs/A009.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A010 - informal: >- - The integral from 0 to pi/4 of x^2 tan^2 x equals -G + pi^2/16 - - pi^3/192 + pi log(2)/4. - source_ref: >- - Upstream Statements.lean and Proofs/A010.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A011 - informal: >- - The integral from 0 to 1 of Li_2(x)/(x sqrt(1-x^2)) equals 3 pi^2 - log(2)/8 - 7 zeta(3)/16. - source_ref: >- - Upstream Statements.lean and Proofs/A011.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A012 - informal: >- - The integral from 0 to 1 of arctan(x) log(x)/(1+x) equals -pi^3/64 - + G log(2)/2. - source_ref: >- - Upstream Statements.lean and Proofs/A012.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A013 - informal: >- - The integral from 0 to 1 of arctan(x) log^2(x)/(1+x) equals -pi^2 - G/24 - pi^3 log(2)/32 + 21 pi zeta(3)/64. - source_ref: >- - Upstream Statements.lean and Proofs/A013.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A014 - informal: >- - The integral from 0 to 1 of log(x) log(1+x)/(1+x^2) equals 11 - pi^3/128 + 3 pi log^2(2)/32 - 2 G log(2) - 3 Im Li_3((1+i)/2). - source_ref: >- - Upstream Statements.lean and Proofs/A014.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A015 - informal: >- - The integral from 0 to pi/2 of x log^2(sin x) equals Li_4(1/2) - - 19 pi^4/2880 + log^4(2)/24 + pi^2 log^2(2)/12. - source_ref: >- - Upstream Statements.lean and Proofs/A015.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Proofs.A016 - informal: >- - The integral from 0 to 1 of Li_3(x)/(x sqrt(1-x^2)) equals - Li_4(1/2)/2 + 41 pi^4/5760 + log^4(2)/48 + pi^2 log^2(2)/6. - source_ref: >- - Upstream Statements.lean and Proofs/A016.lean at the pinned commit; - README prior-appearance table and blueprint/ERRATA.md label classical - sources and corrections. Lean proofs were written with Claude assistance - and reviewed by rainrzk; the informal tutorial is rainrzk (2022). - - declaration: LeanPolyLog.Dilog.hasDerivAt_Li_succ - informal: >- - For every natural s and nonzero complex z in the open unit disk, - Li_(s+1) has derivative Li_s(z)/z at z. - source_ref: Upstream Lemmas/DilogComplex.lean, termwise differentiation. - - declaration: LeanPolyLog.Dilog.Li_two_add_Li_two_div_sub_one - informal: >- - For complex z of norm at most 1 and real part at most 1/2, - Li_2(z) + Li_2(z/(z-1)) = -log^2(1-z)/2, with the principal logarithm. - source_ref: Upstream Lemmas/DilogComplex.lean, complex Landen identity. - - declaration: LeanPolyLog.Arcsin.integral_Lir_div_mul_sqrt - informal: >- - For s at least 2 and a measurable nonnegative weight w on (0,1] - with integrable moments equal to 1/(n+1)^s, the integral from - 0 to 1 of Li_s(x)/(x sqrt(1-x^2)) equals the integral from 0 to - pi/2 of w(sin t) times (pi/2+t). - source_ref: Upstream Lemmas/Arcsin.lean, moment representation and Fubini. - - declaration: LeanPolyLog.Radial.integral_im_logCube - informal: >- - If (log^3((1+r)/2)-log^3((1-r)/2))/r is interval integrable from - 0 to 1, its integral equals the integral from 0 to pi of the - imaginary part of the cube of the principal complex logarithm - of (1+exp(i t))/2. - source_ref: Upstream Lemmas/Radial.lean, radial differentiation and domination. - tags: - - analysis - - polylogarithms - - special-functions - - integration - - complex-analysis - msc: - - 33B30 - - 26A36 - - 30B10 - - slug: anomalous-diffusion-fractal-homogenization - title: Anomalous diffusion by fractal homogenization - summary: >- - For each positive Hölder exponent below one third, constructs a - divergence-free periodic drift on the two-dimensional torus exhibiting - anomalous dissipation for every mean-zero periodic H1 initial datum. - The same drift admits - uniform Hölder time estimates along a sequence of diffusivity intervals - and two distinct vanishing-diffusivity limits for an explicit nonempty - class of smooth data. Includes weak existence and uniqueness, Sobolev - and Fourier/Galerkin infrastructure, flow derivatives, quantitative - homogenization, and the multiscale construction. The no-selection result - is the corrected version documented upstream, not the printed generality - of the original paper's Proposition 5.5. - branch: analysis of partial differential equations - entry_module: LeanPool.AnomalousDiffusion - authors: - - Scott Armstrong - - Vlad Vicol - source: - title: Anomalous diffusion by fractal homogenization, Lean formalization - arxiv: '2305.05048' - url: https://github.com/scottnarmstrong/AVAnomalousDiffusion/tree/fad71d10bff031522cc598a37770238248186964 - github_repo: scottnarmstrong/AVAnomalousDiffusion - commit: fad71d10bff031522cc598a37770238248186964 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - AVenhance.anomalous_dissipation_full - main_results: - - declaration: AVenhance.anomalous_dissipation_full - informal: >- - For every alpha strictly between zero and one third, one periodic, - divergence-free drift has joint spatial and temporal alpha-Hölder - regularity and continuity into the spatial Hölder class. It exhibits - anomalous dissipation with a positive rate depending only on the - initial L2-to-gradient ratio, uniform Hölder-in-time L2 bounds along - diffusivity intervals accumulating at zero, and two distinct - transport limits along separated diffusivity sequences for a smooth - mean-zero data class containing arbitrarily high cosine modes. - source_ref: >- - Upstream Statements/FullTheorem/AnomalousDissipationFull.lean and - CITATION.cff, date-released 2026-10-02; the pinned completion commit is - fad71d10bff031522cc598a37770238248186964, with committer timestamp - 2026-10-02 16:04:04 UTC. Anomalous dissipation is the dimension-two - case of Theorem 1.1 of arXiv:2305.05048; - uniform time regularity and no-selection use upstream's documented - corrections to Sections 5.4 and 5.5. The README explicitly records - predominantly AI-produced proofs under the authors' supervision. - - declaration: AVenhance.anomalous_dissipation - informal: >- - For every positive alpha below one third there exists a periodic, - divergence-free alpha-Hölder drift and a rate function taking values - in (0,1], depending only on the initial L2-to-gradient ratio, such that - the limsup as positive diffusivity tends to zero of integrated gradient - dissipation is at least the squared rate times the initial L2 energy, - for every mean-zero periodic H1 datum and corresponding weak solutions. - - declaration: AVenhance.weak_wellposed - informal: >- - Every bounded measurable periodic drift with positive diffusivity - and periodic-cell L2 initial datum has a weak solution, unique - up to spatial almost-everywhere equality at every time in [0,1]. - tags: - - anomalous-dissipation - - homogenization - - advection-diffusion - msc: - - 35B27 - - 35Q35 - - 76F30 - - title: Davis–Kahan rotation of eigenvectors - summary: Formalizes the sin-Theta, tan-Theta, sin-two-Theta, and tan-two-Theta theorem families - from Section 2 of Davis and Kahan (1970), for real or complex separable Hilbert spaces, including - common-domain unbounded operators and symmetric-norm estimates. The tan-Theta endpoint requires - Rayleigh–Ritz trial data with residual orthogonal to the trial subspace. - branch: operator theory - main_declarations: - - RotationOfEigenvectors.sinTheta_mem_and_bound - main_results: - - declaration: RotationOfEigenvectors.sinTheta_mem_and_bound - informal: With finite symmetric residual norm, a spectral gap gives sine-angle ideal membership - and controls its symmetric norm by the corresponding residual norm. - - declaration: RotationOfEigenvectors.tanTheta - informal: For Rayleigh–Ritz trial data with residual orthogonal to the trial subspace, - a positive directed spectral gap between the trial compression and the unwanted - reducing block gives the symmetric-norm tangent-angle residual estimate and - tangent pole exclusion, assuming the residual has finite symmetric norm. - - declaration: RotationOfEigenvectors.sinTwoTheta_directed_mem_and_bound - informal: Under the printed oriented gap and common-domain self-adjointness hypotheses, finite - symmetric residual norm gives directed sine-double-angle ideal membership, and twice the - residual norm bounds the gap times its norm. - - declaration: RotationOfEigenvectors.sinTwoTheta_ambient_mem_and_bound - informal: Finite symmetric perturbation norm gives ambient sine-double-angle ideal membership - and the whole-space gap bound under the theorem's reducing and perturbed-gap hypotheses. - - declaration: RotationOfEigenvectors.tanTwoTheta - informal: The residual tangent-double-angle estimate holds with the theorem’s source-facing block - hypotheses. - tags: - - operator-theory - - spectral-perturbation - - hilbert-spaces - msc: - - 47A55 - - 47A15 - - 15A42 - provenance: AI - slug: aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization - entry_module: LeanPool.DavisKahan - authors: - - Jon Crall - - Edward Wang - source: - url: https://github.com/aiq-kitware/aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization - github_repo: aiq-kitware/aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization - commit: 38e37da6e147cd0016da1eb05987c3edf7d95b39 - license: Apache-2.0 - status: verified - - slug: nf-weak-partition - title: New Foundations refutes the Weak Partition Principle - summary: >- - Constructs a formal first-order derivation of the negation of the Weak - Partition Principle from New Foundations, with extensionality and the full - integer-stratified comprehension schema. Proves full deductive equivalence - with Metamath's Hailperin-derived finite presentation, including its added - singleton axiom. Replays the source proof through nominal syntax, explicit - freshness certificates, and a de Bruijn calculus. - Includes reusable first-order soundness and completeness, stratification, - lowering, and proof-translation infrastructure. This is a derivability - theorem; it does not assert the consistency of New Foundations. - branch: mathematical logic - entry_module: LeanPool.NFWeakPartition - authors: - - Elliot Glazer - - NFNotWPP contributors - - Jesse Michael Han - - Floris van Doorn - - Ian Klatzco - source: - title: NF proves not-WPP, source-complete Lean formalization - url: https://github.com/elliotglazer/nf-not-wpp/tree/fa58a7a895a73422042e167391c78e9149979913 - github_repo: elliotglazer/nf-not-wpp - commit: fa58a7a895a73422042e167391c78e9149979913 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - NFChoice.Foundation.NFStandard.NF_proves_not_WPP - main_results: - - declaration: NFChoice.Foundation.NFStandard.NF_proves_not_WPP - informal: >- - Standard New Foundations formally derives the negation of WPP. WPP - says that, for all sets x and y, if a function maps y onto x and an - injective function maps y into x, then an injective function maps x - into y. - source_ref: >- - Upstream Proof/NFStandard/NotWPP.lean and the exact source formulation - documented in README.md at the pinned commit. Standard-NF completion: - commit a9f9867ee379835f91a4f05f9c4a352d44ca1775, 2026-08-25 00:16:27 UTC. - Palomar registration - PALOMAR-2026-08-27-000005 is dated 2026-08-27, without a verified time of day. - The responsible public maintainer is Elliot Glazer. Upstream Flypitch4 - credits Ian Klatzco and Claude for its Lean 4 port; this import also - includes AI-assisted porting and proof repairs, hence mixed provenance. - The deterministic Metamath replay is not by itself evidence of AI - mathematical authorship. Independent novelty is not claimed by this card. - - declaration: NFChoice.Foundation.NFStandard.nf_hailperin_deductivelyEquivalent - informal: >- - Extensionality together with every integer-stratified comprehension - instance and Metamath's Hailperin-derived finite first-order presentation, - including its added singleton axiom, derive exactly the same sentences. - source_ref: >- - Upstream Proof/NFStandard/Equivalence.lean at the pinned commit. - Metamath's ax-sn entry credits the singleton addition to SF on - 2015-01-12 and distinguishes it from Hailperin's original presentation: - https://us.metamath.org/nfeuni/ax-sn.html. - tags: - - new-foundations - - weak-partition-principle - - first-order-logic - msc: - - 03E70 - - 03B35 - - slug: conway-soifer-three - title: Conway–Soifer covering conjecture for n = 3 - summary: Proves that ten congruent closed equilateral triangles covering a filled equilateral triangle - of side three have common side at least one, with arbitrary positions, rotations and overlaps. - Consequently ten unit triangles cannot cover a triangle of side 3 + ε for ε > 0. Includes reusable - finite halfspace and ray-endpoint geometry, orientation-independent support and area bounds, - and verified Bernstein polynomial interval checkers. Four derived contact cases and 100 kernel-checked - certificates close the proof without a certificate hypothesis. - branch: discrete geometry - entry_module: LeanPool.ConwaySoifer - authors: - - Vladislav Kuznetsov - source: - title: 'The Conway–Soifer conjecture for n = 3: a Lean formalization' - url: https://github.com/AnanasClassic/conway-soifer-n3-lean/tree/b71b1d22b6f7ebb0f0173fc70a66075f44c86ce6 - github_repo: AnanasClassic/conway-soifer-n3-lean - commit: b71b1d22b6f7ebb0f0173fc70a66075f44c86ce6 - doi: 10.5281/zenodo.22712658 - license: MIT - status: verified - provenance: AI - main_declarations: - - ConwaySoifer.ten_triangle_cover_lower_bound - main_results: - - declaration: ConwaySoifer.ten_triangle_cover_lower_bound - informal: Any ten congruent closed equilateral triangles covering the filled equilateral target - of side three have common side at least one. Positions, rotations and overlaps are unrestricted. - source_ref: Public v1.0.0 release, 2026-09-11T17:12:25Z; upstream README and Geometry/Basic.lean. - - declaration: ConwaySoifer.no_ten_unit_triangle_cover - informal: For every ε > 0, ten closed equilateral triangles of side one cannot cover the filled - equilateral target of side 3 + ε, regardless of position or rotation. - source_ref: Upstream public theorem and its proved scaling corollary. - tags: - - discrete-geometry - - triangle-covering - - conway-soifer - - bernstein-polynomials - msc: - - 52C15 - - 51M16 - - slug: formalizing-gmt - title: Formalizing geometric measure theory - summary: >- - A development of foundations and central results in geometric measure theory, - including Hausdorff content and density bounds, weak convergence and compactness - for Radon measures, tangent-measure machinery, Mattila's Lemmas 14.5--14.7, - and Marstrand's integrality theorem for positive finite densities. - branch: measure theory and geometric analysis - entry_module: LeanPool.FormalizingGMT - authors: - - Theodore Meek - - Ignacio Tejeda - - Annie Cao - - Nathan Pao - source: - url: https://github.com/uw-math-ai/FormalizingGMT - github_repo: uw-math-ai/FormalizingGMT - commit: 0acca38a91c4121b813f8e0b6ff719a4590bf126 - notes: >- - The geometric-measure-theory results follow Mattila, Geometry of Sets and - Measures in Euclidean Spaces, Chapter 14. The weak-convergence, compactness, - covering, and density results follow Evans--Gariepy, Measure Theory and Fine - Properties of Functions, revised edition. - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - mattila_14_10 - - mattila_14_11 - main_results: - - declaration: mattila_14_10 - informal: >- - If a Radon measure on Euclidean n-space has positive finite s-dimensional - density on a set of positive measure, then s is a natural number. - source_ref: Mattila, Theorem 14.10 (Marstrand's theorem). - - declaration: mattila_14_11 - informal: >- - Every n-uniform measure on Euclidean n-space is a constant multiple of - Lebesgue measure. - source_ref: Mattila, Theorem 14.11. - - declaration: MeasureTheory.Measure.weaklyConverges_iff_compactOpenBounds - informal: >- - Vague convergence of Radon measures on Euclidean space is equivalent to - the Portmanteau compact-set upper and open-set lower bounds. - source_ref: Evans--Gariepy, revised edition, Theorem 1.40. - - declaration: MeasureTheory.exists_vaguelyConvergent_subseq_of_compact_bounded - informal: >- - A sequence of Radon measures on Euclidean space that is uniformly bounded - on every compact set has a vaguely convergent subsequence. - source_ref: Evans--Gariepy, revised edition, Theorem 1.41. - tags: - - geometric-measure-theory - - Hausdorff-measure - - tangent-measures - - weak-convergence - msc: - - 28A75 - - 28A78 - - 49Q15 - - slug: civ-axisymmetric - title: Regularity of asymptotically axisymmetric Navier–Stokes solutions - summary: >- - Proves the Constantin–Ignatova–Vicol regularity theorem for suitable - weak solutions of the three-dimensional forced Navier–Stokes equations - with analytic forcing, anisotropic Type II bounds on the angular mean, - and an axisymmetric core at each time. Includes the fully axisymmetric - theorem, meridional smallness, interior spatial analyticity, the - nonanalyticity consequence for singular solutions, and the comparison, - Serrin, and Gustafson–Kang–Tsai results used in their proofs. Reuses Lean - Pool's Caffarelli–Kohn–Nirenberg suitable-solution and analysis library. - branch: Analysis - entry_module: LeanPool.CIVAxisymmetric - authors: - - Scott Armstrong - - Vlad Vicol - source: - arxiv: "2609.20803" - github_repo: scottnarmstrong/CIVAxisymmetric - url: https://github.com/scottnarmstrong/CIVAxisymmetric/tree/9f539dc7bf7ddd414d3ef81df539e1bef229f681 - commit: 9f539dc7bf7ddd414d3ef81df539e1bef229f681 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - CIV.mainTheorem - - CIV.axisymmetricTheorem - main_results: - - declaration: CIV.mainTheorem - informal: >- - A smooth suitable weak solution on the unit parabolic cylinder with - globally finite energy, bounded C^2 spatially analytic force, - anisotropic Type II bounds on its angular mean, and a positive - axisymmetric core at every time is axisymmetric throughout the - cylinder and bounded near the origin up to the terminal time. - - declaration: CIV.axisymmetricTheorem - informal: >- - An axisymmetric smooth suitable weak solution obeying the stated - anisotropic Type II bounds and bounded C^2 force is bounded near the - origin up to the terminal time. - - declaration: CIV.meridionalSmallness - informal: >- - Under the axisymmetric anisotropic hypotheses, the weighted - meridional quantity tends uniformly to zero on a smaller cylinder - as the terminal time approaches. - - declaration: CIV.interiorAnalyticity - informal: >- - Smooth solutions with locally uniformly spatially analytic forcing - are locally uniformly spatially analytic in the interior. - - declaration: CIV.forceUnderCore - informal: >- - For a smooth suitable weak solution with globally finite energy, - bounded C^2 forcing, angular-mean anisotropic bounds, and an - axisymmetric core at every time, failure of boundedness near the - origin implies that the force is not spatially analytic, is nonzero - somewhere in the unit cylinder, and has a nonzero point in every - smaller terminal cylinder. - tags: - - Navier–Stokes - - axisymmetry - - partial differential equations - - regularity - msc: - - 35Q30 - - 35B65 - - 35B33 - - slug: escauriaza-seregin-sverak - title: Escauriaza–Seregin–Šverák and Ladyzhenskaya–Prodi–Serrin regularity - summary: >- - Endpoint and Serrin regularity criteria for three-dimensional incompressible - Navier–Stokes Leray–Hopf solutions: bounded L-infinity-in-time L3 velocity - implies L5 integrability, uniqueness, and a smooth representative on positive - times including the final time. Includes local endpoint regularity, both - Carleman inequalities, spatial unique continuation, and half-space backward - uniqueness. Retains the complete ESS production formalization. - branch: Analysis - entry_module: LeanPool.EscauriazaSereginSverak - authors: - - Scott Armstrong - source: - github_repo: scottnarmstrong/EscauriazaSereginSverak - url: https://github.com/scottnarmstrong/EscauriazaSereginSverak/tree/ae1a0e46086b9b16179dea76b41da10b01845151 - license: Apache-2.0 - status: verified - provenance: AI - main_declarations: - - ESS.essSmooth - - ESS.ladyzhenskayaProdiSerrin - - ESS.serrinCriterion - main_results: - - declaration: ESS.essSmooth - informal: >- - A Leray–Hopf solution whose spatial L3 mass is essentially bounded in - time belongs to space-time L5, is unique among Leray–Hopf solutions with - the same initial datum, and has an almost-everywhere equal smooth - representative on all positive times through the final time. - - declaration: ESS.ladyzhenskayaProdiSerrin - informal: >- - A Leray–Hopf solution satisfying the critical Serrin mixed-norm condition - with spatial exponent greater than three, including the L2-in-time - L-infinity-in-space endpoint, is unique and has a smooth representative - on positive times through the final time. - - declaration: ESS.serrinCriterion - informal: >- - The endpoint L-infinity-in-time L3 criterion and the Serrin criteria for - all spatial exponents greater than three jointly imply uniqueness and - existence of a smooth representative up to the final time. - - declaration: ESS.essLocal - informal: >- - A weak Navier–Stokes solution on the unit cylinder with finite energy, - L3/2 pressure, and velocity essentially bounded in time in spatial L3 - has an almost-everywhere equal parabolically Holder representative on - the closure of the half cylinder. - - declaration: ESS.essGlobal - informal: >- - A Leray–Hopf solution whose spatial L3 mass is essentially bounded in - time has no singular point at an interior positive time. - - declaration: ESS.essL5Unique - informal: >- - A Leray–Hopf solution whose spatial L3 mass is essentially bounded in - time belongs to space-time L5 and is unique among Leray–Hopf solutions - with its initial datum. - - declaration: ESS.uniqueContinuation - informal: >- - A vector field with explicit weak space-time derivatives obeying a - parabolic differential inequality has unique continuation across a - spatial boundary under the stated decay and boundary hypotheses. - - declaration: ESS.backwardUniqueness - informal: >- - A vector field on a three-dimensional half-space satisfying the heat - differential inequality, growth bound, and zero initial trace vanishes. - - declaration: ESS.carlemanGaussian - informal: >- - The Gaussian Carleman inequality bounds weighted field and gradient - energies by the weighted heat-operator energy under its support hypotheses. - - declaration: ESS.carlemanHalfSpace - informal: >- - The half-space Carleman inequality bounds weighted field and gradient - energies by the weighted heat-operator energy under its support hypotheses. - tags: - - Navier–Stokes - - partial differential equations - - regularity - - unique continuation - msc: - - 35Q30 - - 35B65 - - 35B60 - - slug: microwave-networks - title: Microwave network synthesis and power-of-two DFT implementation - summary: >- - Characterizes the complex transmission matrices synthesized from phase - shifters and permutations, with or without hybrid couplers, using series - and parallel composition. Phase-only networks are permuted phase-diagonal - matrices; networks with hybrid couplers are products of permutation and - phase-parametrized block-diagonal layers. Proves that the normalized DFT - of every power-of-two size is implementable, with reusable finite-type - reindexing, independent row and column permutations, and block-diagonal - composition infrastructure. Uses the reciprocal matched network matrix - model of the source paper; no general scattering-physics theorem is claimed. - branch: applied mathematics and signal processing - entry_module: LeanPool.MicrowaveNetworks - authors: - - Matteo Nerini - - Claude AI - source: - title: Toward a Lean Formalization of Analog Computing with Microwaves - arxiv: '2610.04774' - url: https://github.com/matteonerini/formalizing-analog-computing/tree/a89c1e9263658c20a29046f9ed1a8eeb867dd537 - github_repo: matteonerini/formalizing-analog-computing - commit: a89c1e9263658c20a29046f9ed1a8eeb867dd537 - license: Apache-2.0 - status: verified - provenance: mix - main_declarations: - - MiLAC.HybridCouplersPhaseShifters.implementable_dft - main_results: - - declaration: MiLAC.PhaseShifters.implementable_iff_permutedPhaseDiagonal - informal: >- - A finite phase-only network is implementable exactly when its matrix - is a permutation matrix times a diagonal matrix of complex phases. - source_ref: Upstream PhaseShifters.lean. - - declaration: MiLAC.HybridCouplersPhaseShifters.implementable_iff_layered - informal: >- - A finite network is implementable with hybrid couplers and phase - shifters exactly when its matrix is an initial permutation preceded - by finitely many permutation and phase-parametrized block-diagonal layers. - source_ref: Theorem 1 and Section IV of arXiv:2610.04774. - - declaration: MiLAC.HybridCouplersPhaseShifters.implementable_dft - informal: >- - For every natural L, the normalized DFT matrix of size 2^L can be - implemented by hybrid couplers, phase shifters, and permutation networks. - source_ref: Section V of arXiv:2610.04774. - tags: - - applied-mathematics - - signal-processing - - matrix-theory - - fourier-transform - - network-synthesis - msc: - - '94A12' - - '15A23' - - '42A38' - - - slug: small-undecidable-groups - title: Small undecidable groups and effective Adian–Rabin families - summary: Proves the existence of a finitely presented group with three generator - slots and nine relator slots whose word problem is undecidable, and effective - families with undecidable triviality having respectively four generators and eleven - relators, and two generators and ten relators. The dependency closure includes - computable machine-to-Thue reductions, synchronized rewriting, Borisov’s group - simulator, injective HNN and Tietze compression, and general Miller–Tancer and - Gordon constructions. Formalizes the algebraic results of the cited paper; its - four-manifold consequences are outside this project. - branch: combinatorial group theory and computability - entry_module: LeanPool.SmallUndecidableGroups - authors: - - Marc Kegel - - Shana Yunsheng Li - - Qiuyu Ren - source: - title: Small undecidable groups and unrecognizable 4-manifolds - arxiv: '2609.10461' - url: https://github.com/32805433/Adian-Rabin - github_repo: 32805433/Adian-Rabin - commit: fb4a401e8fbdba36e7b35c11fde2baa15592e185 - license: MIT - status: verified - provenance: AI - main_declarations: - - Undecidability.exists_small_undecidable_presentations - main_results: - - declaration: Undecidability.exists_three_generator_nine_relator_group_with_unsolvable_word_problem - informal: There is a finite presentation with three generator slots and nine relator - slots whose word problem is not a computable predicate. - source_ref: arXiv:2609.10461v1, Theorem 1.1. - - declaration: Undecidability.exists_four_generator_eleven_relator_adian_rabin_family - informal: There is a uniformly computable family of finite presentations with - four generator slots and eleven relator slots for which the set of indices presenting - the trivial group is not computable. - source_ref: arXiv:2609.10461v1, Theorem 1.3. - - declaration: Undecidability.exists_two_generator_ten_relator_adian_rabin_family - informal: There is a uniformly computable family of finite presentations with - two generator slots and ten relator slots for which the set of indices presenting - the trivial group is not computable. - source_ref: arXiv:2609.10461v1, Theorem 1.4. - - declaration: Undecidability.exists_small_undecidable_presentations - informal: 'The three small-presentation bounds hold together: an undecidable word - problem with three generators and nine relators, and effective families with - undecidable triviality with four generators and eleven relators and with two - generators and ten relators.' - source_ref: arXiv:2609.10461v1, Theorems 1.1, 1.3, and 1.4. - tags: - - group-theory - - undecidability - - finite-presentations - - hnn-extensions - - rewriting-systems - msc: - - 20F10 - - 03D35 diff --git a/LeanPool/projects/1mfld.yaml b/LeanPool/projects/1mfld.yaml new file mode 100644 index 0000000000..beb52958c8 --- /dev/null +++ b/LeanPool/projects/1mfld.yaml @@ -0,0 +1,33 @@ +slug: 1mfld +title: The classification of compact 1-manifolds +summary: 'A Lean 4 (mathlib) proof of the classification of 1-manifolds, specifically + that every compact, connected, Hausdorff topological space charted on the half-line + [0,∞) is homeomorphic to the circle S¹ or to the closed unit interval [0,1]. The + proof follows David Gale''s outline: shrink to a finite atlas of interval charts, + then induct on the atlas size, merging two overlapping charts at each step; a connected + overlap glues two charts into one (two boundary charts glue onto [0, 1]), while + a disconnected overlap has exactly two components and yields the circle.' +branch: topology +entry_module: LeanPool.OneManifold +authors: +- Jim Fowler +- Dennis Sweeney +source: + url: https://github.com/sweeneyde/1mfld + github_repo: sweeneyde/1mfld + commit: 83e961ba28dd8182e7bae844756aba614eb1512e +license: Apache-2.0 +status: verified +main_declarations: +- OneMfld.homeomorph_circle_or_unitInterval +main_results: +- declaration: OneMfld.homeomorph_circle_or_unitInterval + informal: Every compact connected Hausdorff space charted on the nonnegative real + half-line is homeomorphic to the circle or the closed unit interval. +tags: +- topology +msc: +- 57N99 +- 54F65 +- 68V20 +provenance: mix diff --git a/LeanPool/projects/2-coloring-1-round.yaml b/LeanPool/projects/2-coloring-1-round.yaml new file mode 100644 index 0000000000..6b7e2de52d --- /dev/null +++ b/LeanPool/projects/2-coloring-1-round.yaml @@ -0,0 +1,32 @@ +slug: 2-coloring-1-round +title: 2-Coloring Cycles in One Round +summary: Formalizes certified upper and lower bounds for one-round randomized algorithms + that 2-color cycles. +branch: distributed computing +entry_module: LeanPool.TwoColoringOneRound +authors: +- Jukka Suomela +source: + arxiv: '2603.04235' + github_repo: suomela/2-coloring-1-round +license: MIT +status: verified +provenance: AI +main_declarations: +- Distributed2Coloring.pStar_ge_23879 +- Distributed2Coloring.pStar_lt_24118 +main_results: +- declaration: Distributed2Coloring.pStar_ge_23879 + informal: Proves that every one-round classical algorithm has monochromatic-edge + probability at least 0.23879. +- declaration: Distributed2Coloring.pStar_lt_24118 + informal: Proves that an explicit one-round algorithm attains monochromatic-edge + probability strictly below 0.24118. +tags: +- distributed-computing +- graph-coloring +- randomized-algorithms +- formal-verification +msc: +- 68W15 +- 05C15 diff --git a/LeanPool/projects/a-formalization-of-borel-determinacy-in-lean.yaml b/LeanPool/projects/a-formalization-of-borel-determinacy-in-lean.yaml new file mode 100644 index 0000000000..f77479b868 --- /dev/null +++ b/LeanPool/projects/a-formalization-of-borel-determinacy-in-lean.yaml @@ -0,0 +1,31 @@ +slug: a-formalization-of-borel-determinacy-in-lean +title: A formalization of Borel determinacy in Lean +summary: Formalizes Martin's theorem that every Borel Gale-Stewart game is determined. +branch: descriptive set theory +entry_module: LeanPool.AFormalizationOfBorelDeterminacyInLean +authors: +- Sven Manthe +source: + arxiv: '2502.03432' + github_repo: sven-manthe/A-formalization-of-Borel-determinacy-in-Lean +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- GaleStewartGame.borel_determinacy +- GaleStewartGame.Games.borel_determinacy +main_results: +- declaration: GaleStewartGame.borel_determinacy + informal: Proves that every Borel Gale-Stewart game on a nonempty discrete alphabet + is determined. +- declaration: GaleStewartGame.Games.borel_determinacy + informal: Proves Borel determinacy for the bundled game representation used in the + formalization. +tags: +- descriptive-set-theory +- game-theory +- determinacy +msc: +- 03E15 +- 54H05 +- 91A44 diff --git a/LeanPool/projects/abc-exceptions.yaml b/LeanPool/projects/abc-exceptions.yaml new file mode 100644 index 0000000000..13abd0cfc0 --- /dev/null +++ b/LeanPool/projects/abc-exceptions.yaml @@ -0,0 +1,40 @@ +slug: abc-exceptions +title: Exceptional Set in the abc Conjecture +summary: Formalizes reductions and optimization bounds for a power-saving estimate + on exceptional triples in the abc conjecture. +branch: analytic number theory +entry_module: LeanPool.ABCExceptions +authors: +- Bhavik Mehta +- Arend Mellendijk +source: + title: Bounds on the exceptional set in the abc conjecture + authors: + - Tim Browning + - Jared Duker Lichtman + - Joni Teräväinen + arxiv: '2410.12234' + github_repo: b-mehta/ABC-Exceptions +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- abcConjecture_iff_countTriples +- thm_4_point_3 +main_results: +- declaration: abcConjecture_iff_countTriples + informal: Recasts the abc conjecture as a uniform big-O bound for bounded exceptional + triples. +- declaration: refinedCountTriplesStar_isBigO_B + informal: Bounds the refined exceptional-triple count by the dyadic structured-box + supremum. +- declaration: thm_4_point_3 + informal: Derives the Section 4 optimization bound from the determinant, geometry, + and Fourier hypotheses. +tags: +- number-theory +- analytic-number-theory +- abc-conjecture +msc: +- 11D75 +- 11N37 diff --git a/LeanPool/projects/acmax-conjecture.yaml b/LeanPool/projects/acmax-conjecture.yaml new file mode 100644 index 0000000000..a6d593a5aa --- /dev/null +++ b/LeanPool/projects/acmax-conjecture.yaml @@ -0,0 +1,51 @@ +slug: acmax-conjecture +title: Kolokolnikov's ACMAX conjecture +summary: Proves that every simple graph on n ≥ 4 vertices with 2(n−2) edges has algebraic + connectivity at most 2, attained by the complete bipartite graph K₂,ₙ₋₂. Develops + Laplacian test-vector bounds, sparse cut certificates, and non-backtracking walk + estimates used in the proof. +branch: spectral and extremal graph theory +entry_module: LeanPool.ACMax +authors: +- Zeru Zhu +- Jinzheng Li +- Yuanjie Ren +source: + title: Maximizing algebraic connectivity for certain families of graphs, Conjecture + 1.5 + authors: + - Theodore Kolokolnikov + arxiv: '1412.6147' + doi: 10.1016/j.laa.2014.12.023 + url: https://github.com/MerLeanProver/ACMaxConjecture/tree/78736ca2f5c29a4d5ad7dfdee4ac715bbfcde770 + github_repo: MerLeanProver/ACMaxConjecture + commit: 78736ca2f5c29a4d5ad7dfdee4ac715bbfcde770 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- ACMax.acmax_conjecture +main_results: +- declaration: ACMax.acmax_conjecture + informal: For n ≥ 4, the algebraic connectivity of K₂,ₙ₋₂ is 2, and every simple + graph on Fin n with exactly 2(n−2) edges has algebraic connectivity at most 2. + Algebraic connectivity is the second-smallest eigenvalue of the combinatorial + Laplacian, defined using Mathlib's ordered spectrum. +- declaration: ACMax.algConn_mul_sq_le + informal: For a graph on any nonempty finite vertex type and any real vector whose + coordinates sum to zero, algebraic connectivity times the squared norm is at most + the Laplacian quadratic form. +- declaration: ACMax.algConn_le_two_of_testvector + informal: A nonzero real vector with coordinate sum zero and Laplacian quadratic + form at most twice its squared norm certifies algebraic connectivity at most 2. +- declaration: ACMax.algConn_completeBipartite_two + informal: For n ≥ 4, the complete bipartite graph with parts of sizes 2 and n−2 + has algebraic connectivity exactly 2. +tags: +- spectral-graph-theory +- extremal-graph-theory +- laplacian +- non-backtracking-walks +msc: +- 05C50 +- 05C35 diff --git a/LeanPool/projects/agree-to-disagree.yaml b/LeanPool/projects/agree-to-disagree.yaml new file mode 100644 index 0000000000..8fa269ac03 --- /dev/null +++ b/LeanPool/projects/agree-to-disagree.yaml @@ -0,0 +1,41 @@ +slug: agree-to-disagree +title: Aumann's Agreement Theorem +summary: Formalizes Aumann's agreement theorem for information partitions and a common-prior + setting, together with a p-belief variant bounding posterior disagreement by 2 * + (1 - p). +branch: probability theory +entry_module: LeanPool.AgreeToDisagree +authors: +- Ruiz Chen +- Ben Eltschig +- Ken Ono +- Jujian Zhang +source: + title: 'We Can''t Agree to Disagree, Formally: Aumann''s Theorem and Assumption + Accounting in Lean' + url: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6837298 + github_repo: AxiomMath/AgreeToDisagree +license: MIT +status: verified +provenance: AI +main_declarations: +- AgreeToDisagree.agreeToDisagree +- AgreeToDisagree.agreeToDisagree_beliefs +main_results: +- declaration: AgreeToDisagree.agreeToDisagree + informal: If it is common knowledge at a state that two agents assign fixed posterior + probabilities to an event, and the common-knowledge cell has equal posterior under + their priors, then the two posterior probabilities are equal. +- declaration: AgreeToDisagree.agreeToDisagree' + informal: Generalizes the agreement theorem to a family of agents indexed by an + arbitrary type. +- declaration: AgreeToDisagree.agreeToDisagree_beliefs + informal: Proves that common p-belief of fixed posteriors bounds the difference + between any two agents' real-valued posteriors by 2 * (1 - p). +tags: +- probability +- game-theory +- epistemic-logic +msc: +- 60A10 +- 91A40 diff --git a/LeanPool/projects/aharonikorman.yaml b/LeanPool/projects/aharonikorman.yaml new file mode 100644 index 0000000000..403e8fb769 --- /dev/null +++ b/LeanPool/projects/aharonikorman.yaml @@ -0,0 +1,28 @@ +slug: aharonikorman +title: Disproof of the Aharoni-Korman Conjecture +summary: Formalizes Hollom's counterexample to the Aharoni-Korman fishbone conjecture + for partially ordered sets. +branch: order theory +entry_module: LeanPool.AharoniKorman +authors: +- Bhavik Mehta +source: + arxiv: '2411.16844' + doi: 10.1007/BF00383948 + github_repo: b-mehta/AharoniKorman +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LeanPool.AharoniKorman.aharoni_korman_false +main_results: +- declaration: LeanPool.AharoniKorman.aharoni_korman_false + informal: Proves that not every partially ordered set has either an infinite antichain + or a chain meeting every antichain in a partition. +tags: +- order-theory +- combinatorics +- partial-orders +msc: +- 06A06 +- 06A07 diff --git a/LeanPool/projects/aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization.yaml b/LeanPool/projects/aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization.yaml new file mode 100644 index 0000000000..d2e1b2757f --- /dev/null +++ b/LeanPool/projects/aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization.yaml @@ -0,0 +1,49 @@ +title: Davis–Kahan rotation of eigenvectors +summary: Formalizes the sin-Theta, tan-Theta, sin-two-Theta, and tan-two-Theta theorem + families from Section 2 of Davis and Kahan (1970), for real or complex separable + Hilbert spaces, including common-domain unbounded operators and symmetric-norm estimates. + The tan-Theta endpoint requires Rayleigh–Ritz trial data with residual orthogonal + to the trial subspace. +branch: operator theory +main_declarations: +- RotationOfEigenvectors.sinTheta_mem_and_bound +main_results: +- declaration: RotationOfEigenvectors.sinTheta_mem_and_bound + informal: With finite symmetric residual norm, a spectral gap gives sine-angle ideal + membership and controls its symmetric norm by the corresponding residual norm. +- declaration: RotationOfEigenvectors.tanTheta + informal: For Rayleigh–Ritz trial data with residual orthogonal to the trial subspace, + a positive directed spectral gap between the trial compression and the unwanted + reducing block gives the symmetric-norm tangent-angle residual estimate and tangent + pole exclusion, assuming the residual has finite symmetric norm. +- declaration: RotationOfEigenvectors.sinTwoTheta_directed_mem_and_bound + informal: Under the printed oriented gap and common-domain self-adjointness hypotheses, + finite symmetric residual norm gives directed sine-double-angle ideal membership, + and twice the residual norm bounds the gap times its norm. +- declaration: RotationOfEigenvectors.sinTwoTheta_ambient_mem_and_bound + informal: Finite symmetric perturbation norm gives ambient sine-double-angle ideal + membership and the whole-space gap bound under the theorem's reducing and perturbed-gap + hypotheses. +- declaration: RotationOfEigenvectors.tanTwoTheta + informal: The residual tangent-double-angle estimate holds with the theorem’s source-facing + block hypotheses. +tags: +- operator-theory +- spectral-perturbation +- hilbert-spaces +msc: +- 47A55 +- 47A15 +- 15A42 +provenance: AI +slug: aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization +entry_module: LeanPool.DavisKahan +authors: +- Jon Crall +- Edward Wang +source: + url: https://github.com/aiq-kitware/aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization + github_repo: aiq-kitware/aiq-davis-kahan-1970-rotation-eigenvectgors-perturbation-formalization + commit: 38e37da6e147cd0016da1eb05987c3edf7d95b39 +license: Apache-2.0 +status: verified diff --git a/LeanPool/projects/anderson-conjecture.yaml b/LeanPool/projects/anderson-conjecture.yaml new file mode 100644 index 0000000000..9f4305ad8d --- /dev/null +++ b/LeanPool/projects/anderson-conjecture.yaml @@ -0,0 +1,27 @@ +slug: anderson-conjecture +title: 'The Anderson Conjecture: A Weakly Quasi-Complete Ring Need Not Be Quasi-Complete' +summary: Formalizes a counterexample to Anderson's question, constructing a weakly + quasi-complete Noetherian local ring that is not quasi-complete via the Jensen-Heitmann-Loepp + transfinite construction. +branch: algebra +entry_module: LeanPool.AndersonConjecture +authors: +- FrenzyMath +source: + doi: 10.1007/978-1-4939-0925-4_2 + github_repo: frenzymath/Anderson-Conjecture +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- anderson_main_theorem +main_results: +- declaration: anderson_main_theorem + informal: Proves that there exists a weakly quasi-complete Noetherian local ring + that is not quasi-complete. +tags: +- commutative-algebra +- ring-theory +msc: +- 13B35 +- 13H10 diff --git a/LeanPool/projects/anomalous-diffusion-fractal-homogenization.yaml b/LeanPool/projects/anomalous-diffusion-fractal-homogenization.yaml new file mode 100644 index 0000000000..9f63002058 --- /dev/null +++ b/LeanPool/projects/anomalous-diffusion-fractal-homogenization.yaml @@ -0,0 +1,62 @@ +slug: anomalous-diffusion-fractal-homogenization +title: Anomalous diffusion by fractal homogenization +summary: For each positive Hölder exponent below one third, constructs a divergence-free + periodic drift on the two-dimensional torus exhibiting anomalous dissipation for + every mean-zero periodic H1 initial datum. The same drift admits uniform Hölder + time estimates along a sequence of diffusivity intervals and two distinct vanishing-diffusivity + limits for an explicit nonempty class of smooth data. Includes weak existence and + uniqueness, Sobolev and Fourier/Galerkin infrastructure, flow derivatives, quantitative + homogenization, and the multiscale construction. The no-selection result is the + corrected version documented upstream, not the printed generality of the original + paper's Proposition 5.5. +branch: analysis of partial differential equations +entry_module: LeanPool.AnomalousDiffusion +authors: +- Scott Armstrong +- Vlad Vicol +source: + title: Anomalous diffusion by fractal homogenization, Lean formalization + arxiv: '2305.05048' + url: https://github.com/scottnarmstrong/AVAnomalousDiffusion/tree/fad71d10bff031522cc598a37770238248186964 + github_repo: scottnarmstrong/AVAnomalousDiffusion + commit: fad71d10bff031522cc598a37770238248186964 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- AVenhance.anomalous_dissipation_full +main_results: +- declaration: AVenhance.anomalous_dissipation_full + informal: For every alpha strictly between zero and one third, one periodic, divergence-free + drift has joint spatial and temporal alpha-Hölder regularity and continuity into + the spatial Hölder class. It exhibits anomalous dissipation with a positive rate + depending only on the initial L2-to-gradient ratio, uniform Hölder-in-time L2 + bounds along diffusivity intervals accumulating at zero, and two distinct transport + limits along separated diffusivity sequences for a smooth mean-zero data class + containing arbitrarily high cosine modes. + source_ref: Upstream Statements/FullTheorem/AnomalousDissipationFull.lean and CITATION.cff, + date-released 2026-10-02; the pinned completion commit is fad71d10bff031522cc598a37770238248186964, + with committer timestamp 2026-10-02 16:04:04 UTC. Anomalous dissipation is the + dimension-two case of Theorem 1.1 of arXiv:2305.05048; uniform time regularity + and no-selection use upstream's documented corrections to Sections 5.4 and 5.5. + The README explicitly records predominantly AI-produced proofs under the authors' + supervision. +- declaration: AVenhance.anomalous_dissipation + informal: For every positive alpha below one third there exists a periodic, divergence-free + alpha-Hölder drift and a rate function taking values in (0,1], depending only + on the initial L2-to-gradient ratio, such that the limsup as positive diffusivity + tends to zero of integrated gradient dissipation is at least the squared rate + times the initial L2 energy, for every mean-zero periodic H1 datum and corresponding + weak solutions. +- declaration: AVenhance.weak_wellposed + informal: Every bounded measurable periodic drift with positive diffusivity and + periodic-cell L2 initial datum has a weak solution, unique up to spatial almost-everywhere + equality at every time in [0,1]. +tags: +- anomalous-dissipation +- homogenization +- advection-diffusion +msc: +- 35B27 +- 35Q35 +- 76F30 diff --git a/LeanPool/projects/apportionment.yaml b/LeanPool/projects/apportionment.yaml new file mode 100644 index 0000000000..b24e70c06f --- /dev/null +++ b/LeanPool/projects/apportionment.yaml @@ -0,0 +1,26 @@ +slug: apportionment +title: Apportionmentlib +summary: 'Formalizes apportionment rules and proves the Balinski-Young impossibility + theorem: no anonymous quota rule can be population-monotone.' +branch: social choice theory +entry_module: LeanPool.Apportionment +authors: +- Michał Dobranowski +source: + url: https://archive.org/details/fairrepresentati00bali + github_repo: mdbrnowski/apportionmentlib +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Apportionment.balinski_young +main_results: +- declaration: Apportionment.balinski_young + informal: Proves that no anonymous quota apportionment rule can satisfy population + monotonicity. +tags: +- social-choice-theory +- combinatorics +msc: +- 91B12 +- 91B14 diff --git a/LeanPool/projects/archon-firstproof-results.yaml b/LeanPool/projects/archon-firstproof-results.yaml new file mode 100644 index 0000000000..cdaa148ef6 --- /dev/null +++ b/LeanPool/projects/archon-firstproof-results.yaml @@ -0,0 +1,34 @@ +slug: archon-firstproof-results +title: Archon-FirstProof-Results +summary: Collects formal solutions from the Archon FirstProof benchmark, including + polynomial and graph-theoretic results. +branch: problem-solving mathematics +entry_module: LeanPool.ArchonFirstProofResults +authors: +- FrenzyMath +source: + arxiv: '2602.05192' + github_repo: frenzymath/Archon-FirstProof-Results +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Problem4.harmonic_mean_inequality_full +- Problem6.exists_eps_light_subset +main_results: +- declaration: Problem4.harmonic_mean_inequality_full + informal: Proves the benchmark harmonic-mean inequality for real-rooted polynomial + transforms. +- declaration: Problem6.exists_eps_light_subset + informal: Proves the benchmark light-subset existence theorem in the graph Laplacian + setting. +tags: +- polynomials +- analysis +- combinatorics +- linear-algebra +- graph-theory +msc: +- 26D15 +- 05C50 +- 15A42 diff --git a/LeanPool/projects/artin-wedderburn.yaml b/LeanPool/projects/artin-wedderburn.yaml new file mode 100644 index 0000000000..3c398c3fd5 --- /dev/null +++ b/LeanPool/projects/artin-wedderburn.yaml @@ -0,0 +1,32 @@ +slug: artin-wedderburn +title: Artin-Wedderburn Theorem +summary: Formalizes the Artin-Wedderburn theorem for prime and simple Artinian rings. +branch: ring theory +entry_module: LeanPool.ArtinWedderburn +authors: +- Matevz Miščič +- Maša Žaucer +- Job Petrovčič +source: + title: A First Course in Noncommutative Rings + authors: + - T. Y. Lam + arxiv: '2405.04588' + doi: 10.1007/978-1-4419-8616-0 + github_repo: JobPetrovcic/ArtinWedderburn +license: MIT +status: verified +provenance: human +main_declarations: +- LeanPool.ArtinWedderburn.ArtinWedderburnForPrime +main_results: +- declaration: LeanPool.ArtinWedderburn.ArtinWedderburnForPrime + informal: A nontrivial prime Artinian ring is isomorphic to a matrix ring over a + division ring. +tags: +- ring-theory +- noncommutative-algebra +- artinian-rings +msc: +- 16K20 +- 16P20 diff --git a/LeanPool/projects/asymptotic-moore-bound.yaml b/LeanPool/projects/asymptotic-moore-bound.yaml new file mode 100644 index 0000000000..5315f34636 --- /dev/null +++ b/LeanPool/projects/asymptotic-moore-bound.yaml @@ -0,0 +1,73 @@ +slug: asymptotic-moore-bound +title: Asymptotically attaining the Moore bound +summary: Proves that, for each fixed positive diameter k, the largest order of a finite + simple graph with maximum degree at most d and extended diameter at most k is asymptotic + to d^k as d tends to infinity. Constructs regular halved-flag graphs over every + finite field, with exact order and diameter, a sharp degree cap, and matching asymptotics + through all prime powers. Also proves the corresponding lower-limit bound for the + edge extremum defined using line-graph diameter. Reusable infrastructure includes + common bases for complete flags, odd-even sorting routes, exact flag enumeration, + projective-line completion counts, and bipartite graph expansion. The analytic bridge + retains the Wiener-Ikehara and prime-counting dependency closure from the Apache-2.0 + PrimeNumberTheoremAnd project. The paper describes AI-assisted formalization supervised + by the human authors. +branch: extremal graph theory +entry_module: LeanPool.MooreBound +authors: +- Wouter Cames van Batenburg +- Samuel Korsky +- PrimeNumberTheoremAnd contributors +source: + title: Asymptotically attaining the Moore bound + authors: + - Wouter Cames van Batenburg + - Samuel Korsky + arxiv: '2608.03965' + url: https://github.com/woutercvb/wewantmoore + github_repo: woutercvb/wewantmoore + commit: d59bd80ea93fabb9faf769e790ab47692645e022 + announcement: https://arxiv.org/abs/2608.03965v1 + announcement_date: '2026-08-04T17:30:09Z' + release: https://github.com/woutercvb/wewantmoore/releases/tag/v1.0 + release_date: '2026-08-05T01:19:30Z' + third_party_url: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd + third_party_commit: 0c7abf7be7765dc5ffd21afc1c37b018199ec3c9 + third_party_license: Apache-2.0 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- MooreBound.DegreeDiameter.theorem_1_1 +main_results: +- declaration: MooreBound.DegreeDiameter.theorem_1_1 + informal: For each positive natural k, n(k,d)/d^k tends to 1 as the natural degree + bound d tends to infinity. Here n(k,d) is the maximum number of vertices of a + finite simple graph with maximum degree at most d and extended diameter at most + k. +- declaration: MooreBound.DegreeDiameter.corollary_1_2 + informal: For each natural ell at least 2, the lower limit of h(ell,d)/d^ell is + at least 1. Here h(ell,d) is one plus the maximum number of edges of a finite + simple graph with maximum degree at most d whose line graph has extended diameter + at most ell. +- declaration: MooreBound.DegreeDiameter.proposition_3_1 + informal: For every positive k and prime power q, the concrete halved-flag graph + is regular, has order [2k+1]_q!/(q+1)^k, degree at most (q+1)^k((q+1)^k-1), and + extended diameter exactly k. Along all prime powers q tending to infinity, its + actual degree divided by this cap and its order divided by the kth power of its + actual degree both tend to 1. +- declaration: MooreBound.DegreeDiameter.alternating_route + informal: Any two complete flags in an n-dimensional vector space over a division + ring admit an n-step route that changes only ranks of the prescribed alternating + parity at each step. +- declaration: MooreBound.prime_between + informal: For every positive real epsilon, every sufficiently large real x admits + a prime strictly between x and (1+epsilon)x. +tags: +- graph-theory +- finite-geometry +- asymptotics +- prime-number-theorem +msc: +- 05C35 +- 05B25 +- 11N05 diff --git a/LeanPool/projects/asymptotic-triangle-packing.yaml b/LeanPool/projects/asymptotic-triangle-packing.yaml new file mode 100644 index 0000000000..0fb66bf25f --- /dev/null +++ b/LeanPool/projects/asymptotic-triangle-packing.yaml @@ -0,0 +1,48 @@ +slug: asymptotic-triangle-packing +title: Finite hypergraph rounding and the asymptotic triangle-packing gap +summary: Formalizes a finite ceiling-carrying nibble theorem for uniform hypergraphs + that are near-regular outside a small exceptional set and have suitably bounded + codegrees, and derives the uniform subquadratic gap between fractional and integral + triangle packing in finite graphs, together with its fractional-cover formulation. +branch: hypergraph matchings and extremal combinatorics +entry_module: LeanPool.AsymptoticTrianglePacking +authors: +- Juan Pablo Traverso Gianini +- Aristotle +source: + title: Asymptotic triangle cover--packing development accompanying Linear-Error + Clique Partitions of Split Graphs via Structured Triangle Packing + authors: + - Juan Pablo Traverso Gianini + url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/tree/main/preprints/PAPER_III + github_repo: jtraverso/erdos-81-chordal-clique-partitions +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LeanPool.AsymptoticTrianglePacking.trianglePackingGap +main_results: +- declaration: LeanPool.AsymptoticTrianglePacking.nearRegularNibbleTheorem + informal: For every uniformity and positive error tolerance, sufficiently large + finite uniform hypergraphs with bounded codegrees, near-regular degrees outside + a small exceptional set, and a global degree ceiling admit a matching covering + the prescribed fraction of their vertices. + source_ref: Finite near-regular hypergraph nibble theorem used in the Paper III + development. +- declaration: LeanPool.AsymptoticTrianglePacking.trianglePackingGap + informal: For every positive error tolerance, all sufficiently large finite graphs + have fractional triangle-packing optimum at most their integral triangle-packing + optimum plus the error tolerance times the square of the number of vertices. + source_ref: Paper III AX1, fractional-integral triangle-packing gap. +- declaration: LeanPool.AsymptoticTrianglePacking.triangleCoverPackingGap + informal: For every positive error tolerance, all sufficiently large finite graphs + have fractional triangle-cover optimum at most their integral triangle-packing + optimum plus the error tolerance times the square of the number of vertices. + source_ref: Paper III AX1, cover-side formulation. +tags: +- hypergraph-matching +- nibble-method +- extremal-combinatorics +msc: +- 05C70 +- 05C35 diff --git a/LeanPool/projects/bannai-bannai-stanton.yaml b/LeanPool/projects/bannai-bannai-stanton.yaml new file mode 100644 index 0000000000..2b652d459e --- /dev/null +++ b/LeanPool/projects/bannai-bannai-stanton.yaml @@ -0,0 +1,31 @@ +slug: bannai-bannai-stanton +title: The Bannai-Bannai-Stanton Bound on Distance Sets +summary: 'Formalizes the Bannai-Bannai-Stanton theorem: a distance set `S` in `ℝ^d` + realizing `s` distinct distances has size at most `Nat.choose (d + s) s`. Follows + the short polynomial-method proof of Petrov and Pohoata via a generalized Croot-Lev-Pach + rank lemma, with auxiliary results on the diagonal linear-independence argument + for polynomials, monomial counting of bounded degree, and a rank bound for sums + of matrices.' +branch: combinatorics +entry_module: LeanPool.BannaiBannaiStanton +authors: +- Antoine du Fresne von Hohenesche +source: + github_repo: AntoineduFresne/Bannai-Bannai-Stanton_Theorem + url: https://github.com/AntoineduFresne/Bannai-Bannai-Stanton_Theorem +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- BannaiBannaiStanton.bannai_bannai_stanton_bound +main_results: +- declaration: BannaiBannaiStanton.bannai_bannai_stanton_bound + informal: A distance set `S` in Euclidean space `ℝ^d` realizing `s` distinct distances + satisfies `S.encard ≤ Nat.choose (d + s) s`. +tags: +- combinatorics +- distance-sets +- polynomial-method +msc: +- 52C10 +- 05B30 diff --git a/LeanPool/projects/beck-fiala-matrix.yaml b/LeanPool/projects/beck-fiala-matrix.yaml new file mode 100644 index 0000000000..9689bad4d7 --- /dev/null +++ b/LeanPool/projects/beck-fiala-matrix.yaml @@ -0,0 +1,50 @@ +slug: beck-fiala-matrix +title: Strict Beck–Fiala rounding for incidence matrices +summary: Proves the rational integer-making form of Beck–Fiala rounding for finite + zero-one incidence matrices with column degree bounded by a positive t. Every row + has discrepancy strictly less than t, and initially integral coordinates are preserved. + Includes the necessity of the positive-bound hypothesis. This is distinct from the + Komlós discrepancy bound and from the non-strict hyperedge-selection rounding used + in triangle packing. +branch: discrepancy theory +entry_module: LeanPool.BeckFialaMatrix +authors: +- Juan Pablo Traverso Gianini +source: + title: '"Integer-making" theorems' + authors: + - József Beck + - Tibor Fiala + doi: 10.1016/0166-218X(81)90022-6 + url: https://github.com/jtraverso/lean-pool/blob/523a4ae71168258f9142d70bccf650a140355650/LeanPool/BeckFialaMatrix/Basic.lean + github_repo: jtraverso/lean-pool + commit: 523a4ae71168258f9142d70bccf650a140355650 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- BeckFialaMatrix.exists_rounding +main_results: +- declaration: BeckFialaMatrix.exists_rounding + informal: Given a finite zero-one integer matrix with at most t ones per column, + t at least one, and a rational vector in the unit cube, there is a zero-one integer + vector whose discrepancy in every row is strictly less than t. + source_ref: Rational unit-cube integer-making form with the stated strict discrepancy + bound. +- declaration: BeckFialaMatrix.exists_rounding_preserving_integral + informal: The strict rounding can additionally be chosen to leave every initially + integral coordinate unchanged. + source_ref: Strengthened conclusion of the floating-coordinate proof in the pinned + source. +- declaration: BeckFialaMatrix.zero_bound_counterexample + informal: The same universally quantified strict rounding claim without the assumption + t at least one is false, already for a one-by-one zero matrix with t equal to + zero. + source_ref: Boundary-condition witness in the pinned formalization. +tags: +- discrepancy-theory +- beck-fiala +- integer-rounding +- incidence-matrices +msc: +- 05D40 diff --git a/LeanPool/projects/besicovitchs-1-2.yaml b/LeanPool/projects/besicovitchs-1-2.yaml new file mode 100644 index 0000000000..af7539c134 --- /dev/null +++ b/LeanPool/projects/besicovitchs-1-2.yaml @@ -0,0 +1,47 @@ +slug: besicovitchs-1-2 +title: A machine-checked bound of 0.6934 for Besicovitch's 1/2-problem +summary: Proves the planar Besicovitch threshold is at most 6934/10000, improving + the published 7/10 record. The development formalizes the finite six-point reduction, + thirty rational Gram certificates, the endpoint isolation argument, and Besicovitch's + classical lower-bound example. +branch: geometric measure theory +entry_module: LeanPool.Besicovitch +authors: +- Yongxi Lin +source: + title: The bound sigma_1(R^2) <= 6934/10000 for Besicovitch's 1/2-problem + authors: + - Yongxi Lin + url: https://github.com/CoolRmal/Besicovitchs-1-2 + github_repo: CoolRmal/Besicovitchs-1-2 + commit: 17c9597b2a901bf386d5bb8382d0a74d2069f25b +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.Besicovitch.sigmaOne_plane_le_barS +main_results: +- declaration: LeanPool.Besicovitch.sigmaOne_plane_le_barS + informal: The planar one-dimensional rectifiability threshold is at most 6934/10000, + improving the published upper bound 7/10. +- declaration: LeanPool.Besicovitch.forcesOneRectifiability_plane_of_barS_lt + informal: Every threshold strictly above 6934/10000 forces countable one-rectifiability + in the plane. +- declaration: LeanPool.Besicovitch.Example.one_half_le_sigmaOne_plane + informal: The planar threshold is at least one half, witnessed by Besicovitch's + purely unrectifiable example. +- declaration: LeanPool.Besicovitch.weightedPairScore_le_of_separated + informal: Thirty exact rational Gram certificates prove the weighted six-point score + is at most -1/2000 for every separated admissible pair in the unit ball. +tags: +- besicovitch-problem +- measure-theory +- rectifiability +- finite-certificates +- gram-matrices +msc: +- 28A75 +- 28A78 +- 49Q15 +- 68V20 +- 90C05 diff --git a/LeanPool/projects/bicausalot-palomar.yaml b/LeanPool/projects/bicausalot-palomar.yaml new file mode 100644 index 0000000000..3da99b71c4 --- /dev/null +++ b/LeanPool/projects/bicausalot-palomar.yaml @@ -0,0 +1,35 @@ +slug: bicausalot-palomar +title: 'BicausalOT: measurable selection and coupling strategies' +summary: Develops Kuratowski–Ryll-Nardzewski selection, Jankov–von Neumann uniformization, + and finite-horizon Bellman identities for locally feasible coupling strategies. + Exact Borel measurable optimal strategies and initial couplings are obtained on + Polish Borel spaces for weakly continuous probability kernels and nonnegative extended-real + lower semicontinuous costs. Equivalence with optimization over bicausal path laws + is not established. +branch: probability +entry_module: LeanPool.BicausalOT +authors: +- KT. Wu +source: + url: https://github.com/maxwellapexlab/bicausalot-palomar + github_repo: maxwellapexlab/bicausalot-palomar + commit: 3fe500ff64a75d7ab71b474d0c656d5471f62992 +license: Apache-2.0 +status: verified +main_declarations: +- MeasurableSelection.exists_measurable_selection +main_results: +- declaration: MeasurableSelection.exists_measurable_selection + informal: A weakly measurable multifunction from an arbitrary measurable space to + a Polish space with nonempty closed values admits an everywhere-defined Borel-measurable + selector. +tags: +- probability +msc: +- 28B20 +- 54C65 +- 54H05 +- 03E15 +- 28A20 +- 68V20 +provenance: AI diff --git a/LeanPool/projects/biswal.yaml b/LeanPool/projects/biswal.yaml new file mode 100644 index 0000000000..198e95587a --- /dev/null +++ b/LeanPool/projects/biswal.yaml @@ -0,0 +1,43 @@ +slug: biswal +title: Chebyshev Quotients and Demazure Multiplicities +summary: Formalizes positivity results for the coefficients of Chebyshev-type polynomial + quotients attached to integer partitions, modelling Demazure multiplicities. Proves + that the relevant generating function is a polynomial and that its coefficients + are eventually positive (Theorem 1), gives explicit Cauchy-product and matrix-inverse + formulas for the coefficients (Theorem 2), and establishes their nonnegativity through + a Dyck-path lattice counting model (Theorem 3). +branch: algebraic combinatorics +entry_module: LeanPool.Biswal +authors: +- Jujian Zhang +source: + title: Chebyshev quotients, Demazure multiplicities, and Dyck-path models + authors: + - Evan Chen + - Kenny Lau + - Seewoo Lee + - Ken Ono + - Jujian Zhang + arxiv: '2604.25246' + github_repo: AxiomMath/Biswal +license: MIT +status: verified +provenance: AI +main_declarations: +- Biswal.Theorem23.thm_manifest_coeff_nonneg +main_results: +- declaration: Biswal.Theorem1.genFun_is_polynomial + informal: Proves that the generating function attached to a partition is a polynomial + whenever its parts are bounded by the parameter `m`. +- declaration: Biswal.Theorem1.genFun_coeff_eventually_pos + informal: Proves that the coefficients of the generating function attached to a + partition are eventually strictly positive. +- declaration: Biswal.Theorem23.thm_manifest_coeff_nonneg + informal: Proves that every coefficient of the generating function is nonnegative, + via the explicit Dyck-path lattice counting model. +tags: +- algebraic-combinatorics +- chebyshev +- demazure +msc: +- 05E10 diff --git a/LeanPool/projects/bkarforestformula.yaml b/LeanPool/projects/bkarforestformula.yaml new file mode 100644 index 0000000000..33f2d738f6 --- /dev/null +++ b/LeanPool/projects/bkarforestformula.yaml @@ -0,0 +1,30 @@ +slug: bkarforestformula +title: The BKAR forest interpolation formula +summary: Proves the Brydges–Kennedy–Abdesselam–Rivasseau forest interpolation formula + for smooth functions on a finite complete graph’s edge-coupling space, with unordered + forest-cube contributions, ordered sector variants, and choice independence. +branch: analysis and mathematical physics +entry_module: LeanPool.BKARForestFormula +authors: +- Scott Armstrong +source: + url: https://github.com/scottnarmstrong/bkarforestformula + github_repo: scottnarmstrong/bkarforestformula + commit: a07f44a534240fe6339558951dd78a19e4ef7c51 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- BKAR.bkar_formula_forestIndex_cube_contributions +main_results: +- declaration: BKAR.bkar_formula_forestIndex_cube_contributions + informal: A smooth function evaluated at the all-one configuration is the sum over + forests of cube integrals of mixed partial derivatives at path-minimum interpolation + points. +tags: +- analysis +- graph-theory +- mathematical-physics +msc: +- 81T08 +- 05C05 diff --git a/LeanPool/projects/block-spectral-sensitivity.yaml b/LeanPool/projects/block-spectral-sensitivity.yaml new file mode 100644 index 0000000000..2a5e2ca0c5 --- /dev/null +++ b/LeanPool/projects/block-spectral-sensitivity.yaml @@ -0,0 +1,61 @@ +slug: block-spectral-sensitivity +title: Block sensitivity can exceed spectral sensitivity squared +summary: Constructs a total Boolean function on 14011 blocks of 144 bits with block + sensitivity at least 14011 and squared spectral sensitivity at most U = 7149 + 2 + sqrt(150108). Proves the strict separation lambda(f)^2.12 < bs(f), multiplicativity + of spectral sensitivity under arbitrary block composition, and an exponential lower + bound with factor 14011/U > 1.768 for the self-composed family. Includes reusable + finite product probability, an asymmetric Lovasz local lemma, partial Boolean assignments, + certificate-family spectral bounds, and a Paley doubly regular tournament. The formalized + explicit exponent is 2.12; the exact logarithmic exponent is also retained. +branch: computational complexity +entry_module: LeanPool.BlockSpectralSensitivity +authors: +- Alex Meiburg +source: + title: Block Sensitivity can exceed Spectral Sensitivity Squared + arxiv: '2608.00851' + url: https://github.com/Timeroot/BS_Lam/tree/7bd39a8d41ee7910d3296d0477ad18f8fff9d870 + github_repo: Timeroot/BS_Lam + commit: 7bd39a8d41ee7910d3296d0477ad18f8fff9d870 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- BSLambda.Final.exists_bs_gt_lam_rpow +main_results: +- declaration: BSLambda.Final.exists_bs_gt_lam_rpow + informal: There exists a total Boolean function on ZMod 14011 times Fin 144 whose + block sensitivity strictly exceeds its spectral sensitivity raised to the real + exponent 2.12. + source_ref: Upstream Main.lean; bs_lambda.txt Sections 5-12 and 14. +- declaration: BSLambda.Final.exists_counterexample + informal: There exists a gate labelling whose certificate-union indicator f has + bs(f) at least 14011, lambda(f)^2 at most U = 7149 + 2 sqrt(150108), and lambda(f)^alpha + at most bs(f), where alpha = 2 log(14011)/log(U) > 2. + source_ref: Upstream Main.lean; bs_lambda.txt Sections 5-12 and 14. +- declaration: BSLambda.Final.exists_ratio_blowup + informal: A single gate labelling yields, for every natural m, an m-fold self-composition + F_m satisfying (14011/U)^m * lambda(F_m)^2 <= bs(F_m). The auxiliary theorem ratio_gt + proves 14011/U > 1.768. + source_ref: Upstream Main.lean; bs_lambda.txt Section 14, division-free bound. +- declaration: BSLambda.lam_comp + informal: For any Boolean functions on finite coordinate types, spectral sensitivity + of their block composition equals the product of their spectral sensitivities, + including the degenerate cases. + source_ref: Upstream Spectral/Multiplicative.lean; composition theorem used in Section + 14. +- declaration: BSLambda.LLL.pr_avoid_pos + informal: In a finite uniform product of nonempty alphabets, events supported on + specified coordinate sets have positive probability of all being avoided if each + event probability is at most its weight times the product of one minus the weights + of its dependent neighbours, with every weight strictly between zero and one. + source_ref: Upstream LLL/Asymmetric.lean; asymmetric local lemma in Section 10. +tags: +- boolean-functions +- complexity +- probabilistic-method +msc: +- 68Q17 +- 05D40 +- 05C50 diff --git a/LeanPool/projects/bollobas-nikiforov.yaml b/LeanPool/projects/bollobas-nikiforov.yaml new file mode 100644 index 0000000000..be05fdc188 --- /dev/null +++ b/LeanPool/projects/bollobas-nikiforov.yaml @@ -0,0 +1,84 @@ +slug: bollobas-nikiforov +title: The Bollobás–Nikiforov conjecture +summary: 'Proves the Bollobás–Nikiforov conjecture (2007): every finite noncomplete + simple graph G on at least two vertices satisfies λ₁(G)² + λ₂(G)² ≤ 2 (1 - 1/ω(G)) + |E(G)|, where λ₁ ≥ λ₂ are the two largest adjacency eigenvalues and ω(G) is the + clique number. The inequality is derived from a weighted spectral inequality for + symmetric entrywise nonnegative matrices supported on the edges of G, which in turn + rests on a complete positivity theorem for planar Gram matrices and a rank-two Gram + inequality. The rank-two Gram inequality also identifies the conic parameter of + Coutinho, Spier and Zhang with the clique number. The supporting library develops + the completely positive cone and its closedness, the Motzkin–Straus inequality in + completely positive form, total nonnegativity of truncated-square and convex kernels + via Cauchy–Binet, weighted Laplacian updates, Schur complements, Perron eigenvectors, + the Schur product theorem and Cauchy interlacing over Mathlib.' +branch: spectral graph theory +entry_module: LeanPool.BollobasNikiforov +authors: +- Gabriel Coutinho +- Yinchen Liu +- Thomás Jung Spier +- Quanyu Tang +- Shengtong Zhang +source: + title: The Bollobás–Nikiforov inequality for nonnegative edge weights + authors: + - Gabriel Coutinho + - Yinchen Liu + - Thomás Jung Spier + - Quanyu Tang + - Shengtong Zhang + url: https://github.com/ShengtongZhang-alt/BN + github_repo: ShengtongZhang-alt/BN + commit: edb5259dfd055ea31b4c46ac9ea4d33a758c2b99 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- BollobasNikiforov.lambda1_sq_add_lambda2_sq_le +main_results: +- declaration: BollobasNikiforov.lambda1_sq_add_lambda2_sq_le + informal: For every finite simple graph G on at least two vertices that is not complete, + the two largest adjacency eigenvalues satisfy λ₁(G)² + λ₂(G)² ≤ 2 (1 - 1/ω(G)) + |E(G)|, with ω(G) the clique number and |E(G)| the number of edges. +- declaration: BollobasNikiforov.weighted + informal: If B is a real symmetric matrix, indexed by a finite type with at least + two elements, that is entrywise nonnegative, has zero diagonal and vanishes off + the edges of a graph G on that type, then the sum of the squares of its two largest + positive eigenvalues is at most (1 - 1/ω(G)) times its squared Frobenius norm. +- declaration: BollobasNikiforov.matrix_theorem + informal: If planar vectors z₁, …, zₙ lie in a closed half-plane through the origin + and X is their Gram matrix, then X ∘ X plus the weighted Laplacian with weights + X_ij² on the pairs i < j with X_ij < 0 is completely positive. +- declaration: BollobasNikiforov.gram_le + informal: For every real positive semidefinite matrix X of rank at most two, ∑_{i,j} + (A_G)_ij (X_ij)₊² ≤ (1 - 1/ω(G)) ‖X‖_F², where (·)₊ is the positive part. +- declaration: BollobasNikiforov.chiVec3_eq_cliqueNum + informal: For every graph G on a nonempty finite vertex type, the rank-two conic + parameter χ''_{vec,3}(G) of Coutinho, Spier and Zhang, defined as the supremum + of ‖X‖_F² over positive semidefinite X of rank at most two with nonnegative entries + on the edges and unit off-edge mass, equals the clique number ω(G). +- declaration: BollobasNikiforov.motzkinStraus + informal: Motzkin–Straus inequality; for every entrywise nonnegative vector y, yᵀ + A_G y ≤ (1 - 1/ω(G)) (∑ᵢ yᵢ)². +- declaration: BollobasNikiforov.posSemidef_hadamard + informal: Schur product theorem over the reals; the Hadamard product of two positive + semidefinite matrices is positive semidefinite. +- declaration: BollobasNikiforov.isClosed_isCompletelyPositive + informal: The cone of completely positive real matrices is closed. +- declaration: BollobasNikiforov.IsTotallyNonneg.mul + informal: Cauchy–Binet; the product of two totally nonnegative real matrices is + totally nonnegative. +- declaration: BollobasNikiforov.exists_nonneg_eigenvector_lambdaMax + informal: Perron; a real symmetric entrywise nonnegative matrix on a nonempty finite + index type has an entrywise nonnegative unit eigenvector for its largest eigenvalue. +tags: +- spectral-graph-theory +- clique-number +- completely-positive-matrices +- motzkin-straus +msc: +- 05C50 +- 15B48 +- 05C35 +- 15A18 diff --git a/LeanPool/projects/boolean-isoperimetry.yaml b/LeanPool/projects/boolean-isoperimetry.yaml new file mode 100644 index 0000000000..df78b27573 --- /dev/null +++ b/LeanPool/projects/boolean-isoperimetry.yaml @@ -0,0 +1,45 @@ +slug: boolean-isoperimetry +title: Boolean Isoperimetry and Conway--Guy Coherent Gaps +summary: 'Formalizes Harper''s vertex-isoperimetric theorem for the Boolean cube: + among all families of a fixed cardinality, the simplicial initial segment has the + smallest closed radius-one Hamming neighborhood. Develops the supporting binomial-cascade + identities, Kruskal--Katona shadow bounds, coordinate compressions, and the final + dimension induction. Applies the coherent-gap certificate method to prove normalized + consecutive-gap rigidity for the Conway--Guy sequence from its exact recurrence.' +branch: extremal and additive combinatorics +entry_module: LeanPool.BooleanIsoperimetry +authors: +- Alexey Milovanov +- Egor Lyfar +source: + title: Optimal numberings and isoperimetric problems on graphs; A sum packing problem + of Erdős and the Conway--Guy sequence + authors: + - L. H. Harper + - Tom Bohman + doi: 10.1016/S0021-9800(66)80059-5 + url: https://doi.org/10.1090/S0002-9939-96-03653-2 + github_repo: AlexeyMilovanov/BooleanIsoperimetry +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- BooleanIsoperimetry.harper_theorem +- BooleanIsoperimetry.conwayGuyGapRigidity +main_results: +- declaration: BooleanIsoperimetry.harper_theorem + informal: For every family of vertices in an n-dimensional Boolean cube, the simplicial + initial segment of the same cardinality has a closed radius-one Hamming neighborhood + no larger than that of the family. +- declaration: BooleanIsoperimetry.conwayGuyGapRigidity + informal: In every dimension, any real row whose consecutive subset-sum gaps relative + to the Conway--Guy reference row are at least one dominates that row coordinatewise. +tags: +- additive-combinatorics +- isoperimetry +- boolean-cube +- subset-sums +- coherent-orders +msc: +- 05D05 +- 05C35 diff --git a/LeanPool/projects/boolean-multiplication.yaml b/LeanPool/projects/boolean-multiplication.yaml new file mode 100644 index 0000000000..d7e9c4a60e --- /dev/null +++ b/LeanPool/projects/boolean-multiplication.yaml @@ -0,0 +1,74 @@ +slug: boolean-multiplication +title: Exact Boolean multiplicative complexity of four-term polynomial multiplication +summary: Proves that multiplying two binary polynomials with n coefficients needs + exactly 0, 1, 3, 6, and 9 AND gates for n = 0, 1, 2, 3, and 4, with XOR and constants + free and arbitrary reuse of earlier nonlinear wires. The main lower bound excludes + every eight-gate circuit for n = 4. Develops Boolean algebraic normal forms equivalent + to all Boolean functions, a general circuit model, dimension bounds, circuit rewiring, + a Reed-Muller minimum-weight theorem, and exterior-algebra and jet arguments controlling + nonlinear feedback. +branch: computational complexity +entry_module: LeanPool.BooleanMultiplication +authors: +- Gregory Morse +source: + title: 'Unrestricted Boolean Multiplicative Complexity of Four-Term Binary Polynomial + Multiplication: Rational Places, Hasse Jets, and the Failure of Nonlinear Feedback' + arxiv: 2608.30238v1 + url: https://github.com/GregoryMorse/unrestricted-boolean-mul/tree/79a801760668f52bb00a9e1a406e841375c58a6d + github_repo: GregoryMorse/unrestricted-boolean-mul + commit: 79a801760668f52bb00a9e1a406e841375c58a6d + announcement_date: '2026-08-31T04:33:48Z' + announcement_url: https://github.com/GregoryMorse/unrestricted-boolean-mul/releases/tag/n4-arxiv-v2 + notes: The complete Lean release predates the arXiv v1 submission of 2026-08-31T04:45:12Z. + The selected September 22 commit corrects release metadata; eligibility rests + on the complete August 31 Lean release at 2ebc0cf42039e4b5de84cb7f9d97cad4791c669d. + The August 29 release contained computational artifacts, not the Lean formalization. + The author discloses GPT-5.6 Sol for original research and proof development, + Opus 5 as a critical referee, and Codex for release preparation. Gregory Morse + directed and reviewed the work and assumes responsibility. AI provenance records + the proof-producing agent workflow; human direction alone is not classified as + mixed formalization. The complete upstream MIT notice is retained in LeanPool/BooleanMultiplication.lean. + Only the published n <= 4 dependency closure is imported; no n = 5 or n = 6 claim + is made. +license: MIT +status: verified +provenance: AI +main_declarations: +- UnrestrictedBooleanMul.N4.mc_mul_four +main_results: +- declaration: UnrestrictedBooleanMul.mc_mul_zero + informal: The empty polynomial multiplication map needs zero AND gates. + source_ref: Proposition 2.3; the n = 0 target has no output coordinates. +- declaration: UnrestrictedBooleanMul.mc_mul_one + informal: Multiplying two one-coefficient binary polynomials needs exactly one AND + gate. + source_ref: Proposition 2.3. +- declaration: UnrestrictedBooleanMul.mc_mul_two + informal: Multiplying two two-coefficient binary polynomials needs exactly three + AND gates. + source_ref: Proposition 2.3. +- declaration: UnrestrictedBooleanMul.mc_mul_three + informal: Multiplying two three-coefficient binary polynomials needs exactly six + AND gates. + source_ref: Proposition 2.4; the lower bound is proved internally. +- declaration: UnrestrictedBooleanMul.N4.mc_mul_four + informal: Multiplying two four-coefficient binary polynomials needs exactly nine + AND gates, allowing free XOR, constants, and unrestricted nonlinear reuse. + source_ref: Theorem 9.1. +- declaration: UnrestrictedBooleanMul.N4.no_eight_gate_circuit + informal: No eight-AND circuit with free XOR, constants, and arbitrary nonlinear + reuse computes all seven output coefficients of the product of two four-coefficient + binary polynomials. + source_ref: Theorem 9.1, lower bound. +tags: +- computational-complexity +- boolean-circuits +- polynomial-multiplication +- algebraic-normal-form +msc: +- 68Q06 +- 68Q17 +- 68W30 +- 15A75 +- 94D10 diff --git a/LeanPool/projects/brauergroup_new.yaml b/LeanPool/projects/brauergroup_new.yaml new file mode 100644 index 0000000000..bf64b26f55 --- /dev/null +++ b/LeanPool/projects/brauergroup_new.yaml @@ -0,0 +1,57 @@ +slug: brauergroup_new +title: Brauer Group Core +summary: Formalizes central simple algebra infrastructure, Wedderburn-Artin structure + theorems, and finite-field triviality for Brauer groups. +branch: ring theory +entry_module: LeanPool.BrauerGroupNew +authors: +- Yunzhou Xie +- Yichen Feng +- Jujian Zhang +- Yael Dillies +source: + title: Central Simple Algebras and Galois Cohomology + authors: + - Philippe Gille + - Tamás Szamuely + url: https://doi.org/10.1017/9781316661277 + github_repo: Whysoserioushah/BrauerGroup +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- BrauerGroup.BruaerGroup +- BrauerGroupHom.Br +- WedderburnArtin +main_results: +- declaration: BrauerGroup.BruaerGroup + informal: Constructs the Brauer group of a field from Brauer equivalence classes + of central simple algebras. +- declaration: BrauerGroupHom.Br + informal: Constructs the Brauer group functor from fields to commutative groups + via scalar extension. +- declaration: WedderburnArtin + informal: Proves a Wedderburn-Artin decomposition for simple Artinian rings as matrix + algebras over division rings. +- declaration: trivialBrauer + informal: Proves that every element of the Brauer group of a finite field is trivial. +- declaration: WedderburnArtin_algebra_version + informal: Gives an algebra equivalence form of Wedderburn-Artin for finite-dimensional + central simple algebras. +- declaration: IsCentralSimple.TensorProduct.simple + informal: Proves simplicity of tensor products of central simple algebras over a + field. +- declaration: IsCentralSimple.TensorProduct.isCentral + informal: Proves centrality of tensor products of central algebras over a field. +- declaration: CSA_implies_CSA + informal: Shows that the division algebra factor in a matrix presentation of a central + simple algebra is central over the base field. +tags: +- algebra +- ring-theory +- central-simple-algebras +- brauer-groups +msc: +- 16K20 +- 16K50 +- 16S35 diff --git a/LeanPool/projects/brill-noether-graphs.yaml b/LeanPool/projects/brill-noether-graphs.yaml new file mode 100644 index 0000000000..59f3cdbc6b --- /dev/null +++ b/LeanPool/projects/brill-noether-graphs.yaml @@ -0,0 +1,39 @@ +slug: brill-noether-graphs +title: Brill–Noether theory and gonality of finite graphs +summary: Formalizes Brill–Noether existence through genus five, chains of marked banana + graphs and loops, the treewidth lower bound for gonality, and a tricycle example + whose gonality drops under regular subdivision. Includes divisor and transmission + infrastructure and the Demazure-product support used by vertex gluing. +branch: combinatorics and tropical geometry +main_declarations: +- Highlights.brill_noether_existence_through_five +main_results: +- declaration: Highlights.brill_noether_existence_through_five + informal: Every finite connected loopless multigraph of genus g at most five has, + for integers r and d, a divisor of degree d and rank at least r whenever (r + + 1) * (g - d + r) ≤ g. +- declaration: Highlights.treewidth_le_gonality + informal: On a finite connected loopless multigraph, for every natural number k, + any degree-k divisor of rank at least one bounds the treewidth by k; this is the + witness formulation of the treewidth lower bound for divisorial gonality. +- declaration: Highlights.gonality_drops_under_regular_subdivision + informal: There is a graph whose divisorial gonality strictly decreases under a + regular subdivision. +tags: +- brill-noether-theory +- graph-theory +- chip-firing +- tropical-geometry +msc: +- 14T15 +- 05C50 +provenance: AI +entry_module: LeanPool.BrillNoetherGraphs +authors: +- Nathan Pflueger +source: + url: https://github.com/npflueger/brill-noether-graphs + github_repo: npflueger/brill-noether-graphs + commit: ce62dd82fd499fcec9b5a4110a5312befd6e96e2 +license: Apache-2.0 +status: verified diff --git a/LeanPool/projects/brooks-subcubic.yaml b/LeanPool/projects/brooks-subcubic.yaml new file mode 100644 index 0000000000..e735434471 --- /dev/null +++ b/LeanPool/projects/brooks-subcubic.yaml @@ -0,0 +1,37 @@ +slug: brooks-subcubic +title: Brooks' theorem for subcubic graphs +summary: Proves that every finite simple graph of maximum degree at most three with + no four-clique admits a three-colouring, without a connectedness assumption. Includes + controlled greedy colouring, colouring across cut vertices, and a good-triple argument + for cubic components. This is the subcubic case of Brooks' theorem, not the general-degree + theorem. +branch: graph theory +entry_module: LeanPool.BrooksSubcubic +authors: +- Juan Pablo Traverso Gianini +source: + title: On colouring the nodes of a network + authors: + - R. L. Brooks + doi: 10.1017/S030500410002168X + url: https://github.com/jtraverso/lean-pool/blob/aced439fd4161d118bf167a1e8d10553f28913fe/LeanPool/BrooksSubcubic/Main.lean + github_repo: jtraverso/lean-pool + commit: aced439fd4161d118bf167a1e8d10553f28913fe +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- BrooksSubcubic.brooks_cubic +main_results: +- declaration: BrooksSubcubic.brooks_cubic + informal: Every finite simple graph with maximum degree at most three and no clique + on four vertices is three-colourable. The graph may be disconnected. + source_ref: The subcubic specialization of Brooks' 1941 colouring theorem; the linked + Apache-2.0 Lean source proves exactly this specialization. +tags: +- graph-theory +- vertex-colouring +- brooks-theorem +- subcubic-graphs +msc: +- 05C15 diff --git a/LeanPool/projects/brouwer.yaml b/LeanPool/projects/brouwer.yaml new file mode 100644 index 0000000000..28a8d1072d --- /dev/null +++ b/LeanPool/projects/brouwer.yaml @@ -0,0 +1,59 @@ +slug: brouwer +title: Existence of Nash equilibria via Brouwer's fixed-point theorem +summary: Formalizes Scarf's combinatorial lemma, derives Brouwer's fixed-point theorem + on a simplex and on finite products of simplices, and uses it to prove the existence + of a mixed Nash equilibrium in every finite game. +branch: game theory +entry_module: LeanPool.Brouwer +authors: +- Math_XMUM +source: + doi: 10.1073/pnas.36.1.48 + url: https://github.com/math-xmum/Brouwer + github_repo: math-xmum/Brouwer +license: MIT +status: verified +provenance: human +main_declarations: +- ExistsNashEq +- Brouwer +- Brouwer_Product +- IndexedLOrder.Scarf +- FinGame +main_results: +- declaration: ExistsNashEq + informal: Every finite game has a mixed Nash equilibrium. + source_ref: J. Nash, Equilibrium Points in N-Person Games, Proceedings of the National + Academy of Sciences 36(1):48-49, 1950, doi:10.1073/pnas.36.1.48. +- declaration: Brouwer + informal: Every continuous self-map of a standard simplex has a fixed point. + source_ref: H. Scarf, The Approximation of Fixed Points of a Continuous Mapping, + SIAM Journal on Applied Mathematics 15(5):1328-1343, 1967, doi:10.1137/0115116. +- declaration: Brouwer_Product + informal: Every continuous self-map of a finite product of simplices has a fixed + point. + source_ref: H. Scarf, The Approximation of Fixed Points of a Continuous Mapping, + SIAM Journal on Applied Mathematics 15(5):1328-1343, 1967, doi:10.1137/0115116. +- declaration: IndexedLOrder.Scarf + informal: Scarf's combinatorial lemma producing a colorful simplex. + source_ref: H. Scarf, The Approximation of Fixed Points of a Continuous Mapping, + SIAM Journal on Applied Mathematics 15(5):1328-1343, 1967, doi:10.1137/0115116. +- declaration: FinGame + informal: The structure of a finite game with finitely many players and finite pure + strategy sets. + source_ref: J. Nash, Equilibrium Points in N-Person Games, Proceedings of the National + Academy of Sciences 36(1):48-49, 1950, doi:10.1073/pnas.36.1.48. +- declaration: FinGame.mixedNashEquilibrium + informal: The predicate that a mixed strategy profile is a Nash equilibrium. + source_ref: J. Nash, Equilibrium Points in N-Person Games, Proceedings of the National + Academy of Sciences 36(1):48-49, 1950, doi:10.1073/pnas.36.1.48. +tags: +- game-theory +- nash-equilibrium +- brouwer-fixed-point +- scarf-lemma +- simplex +msc: +- 91A06 +- 91A10 +- 47H10 diff --git a/LeanPool/projects/bruhat-tits.yaml b/LeanPool/projects/bruhat-tits.yaml new file mode 100644 index 0000000000..af9358a1f3 --- /dev/null +++ b/LeanPool/projects/bruhat-tits.yaml @@ -0,0 +1,36 @@ +slug: bruhat-tits +title: Formalisation of the Bruhat-Tits Tree +summary: Formalizes the Bruhat-Tits tree for GL₂ over the fraction field of a discrete + valuation ring. +branch: algebraic geometry +entry_module: LeanPool.BruhatTits +authors: +- Judith Ludwig +- Christian Merten +source: + title: Trees + authors: + - Jean-Pierre Serre + url: https://doi.org/10.1007/978-3-642-61856-7 + github_repo: chrisflav/bruhat-tits +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- BruhatTits.BTtree +- BruhatTits.BTlaplace_surjective +main_results: +- declaration: BruhatTits.BTtree + informal: Proves that the Bruhat-Tits graph is a tree. +- declaration: BruhatTits.btgraph_regular + informal: Proves regularity of the Bruhat-Tits graph when the residue field is finite. +- declaration: BruhatTits.BTlaplace_surjective + informal: Proves surjectivity of the Bruhat-Tits Laplacian on harmonic cochains. +tags: +- algebraic-geometry +- graph-theory +- discrete-valuation-rings +msc: +- 20E42 +- 05C25 +- 13H05 diff --git a/LeanPool/projects/burkholder.yaml b/LeanPool/projects/burkholder.yaml new file mode 100644 index 0000000000..d7d90ed671 --- /dev/null +++ b/LeanPool/projects/burkholder.yaml @@ -0,0 +1,40 @@ +slug: burkholder +title: Burkholder Martingale Transform Inequality +summary: 'Formalizes Burkholder''s sharp Lp inequality for real-valued martingale + transforms: if `f` is a discrete martingale and `w` a predictable multiplier bounded + by `1`, then the Lp norm of the transform `w ⋆ₘ f` is at most `(pStar p - 1)` times + the Lp norm of `f`, where `pStar p` is the maximum of `p` and its conjugate exponent. + The proof builds the Burkholder special function as a majorant of the relevant payoff + in each exponent regime (`1 < p < 2`, `p = 2`, `p > 2`).' +branch: probability +entry_module: LeanPool.Burkholder +authors: +- Daniel Smania +source: + title: Boundary value problems and sharp inequalities for martingale transforms + authors: + - Donald L. Burkholder + doi: 10.1214/aop/1176993220 + github_repo: SmaniaD/Burkholder +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- MeasureTheory.Lp_Burkholder_inequality_martingaleTransform +main_results: +- declaration: MeasureTheory.Lp_Burkholder_inequality_martingaleTransform + informal: 'The sharp Lp Burkholder inequality: for a discrete martingale `f` and + a predictable multiplier `w` with `|w n| ≤ 1`, the Lp norm of the martingale transform + `(w ⋆ₘ f) n` is bounded by `(pStar p - 1)` times the Lp norm of `f n`.' +- declaration: MeasureTheory.martingaleTransform_martingale + informal: The martingale transform of a martingale by a strongly predictable bounded + multiplier is itself a martingale. +- declaration: Majorants.exists_majorant_p_g_1 + informal: For every exponent `p > 1` there exists a Burkholder majorant, the special + function dominating the payoff and driving the inequality. +tags: +- probability +- martingales +- burkholder-inequality +msc: +- 60G42 diff --git a/LeanPool/projects/caffarelli-kohn-nirenberg.yaml b/LeanPool/projects/caffarelli-kohn-nirenberg.yaml new file mode 100644 index 0000000000..c744dafc4b --- /dev/null +++ b/LeanPool/projects/caffarelli-kohn-nirenberg.yaml @@ -0,0 +1,52 @@ +slug: caffarelli-kohn-nirenberg +title: Caffarelli–Kohn–Nirenberg partial regularity +summary: 'Partial regularity for three-dimensional incompressible Navier–Stokes suitable + weak solutions with locally L^q forcing, q > 5/2: quantitative epsilon regularity + and vanishing parabolic one-dimensional Hausdorff measure of the singular set. Includes + the supporting Sobolev, singular integral, pressure, heat-potential and parabolic + Morrey infrastructure, equivalence of suitable-solution formulations, and a nonzero + shear-flow example. Includes the completed Leray, pressure, and linear regularity + support needed by further Navier–Stokes regularity projects.' +branch: Analysis +entry_module: LeanPool.CaffarelliKohnNirenberg +authors: +- Scott Armstrong +- Vlad Vicol +source: + github_repo: scottnarmstrong/CaffarelliKohnNirenberg + url: https://github.com/scottnarmstrong/CaffarelliKohnNirenberg/tree/381d658ead0f03a18361965cc0427ce3fa5844ab +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- CKN.caffarelliKohnNirenberg +- CKN.epsilonRegularityL3 +main_results: +- declaration: CKN.epsilonRegularityL3 + informal: For q > 5/2, sufficiently small unit-cylinder L^3 velocity, L^(3/2) pressure + and L^q force masses give an almost-everywhere equal quantitatively bounded parabolic + Holder representative on the closed half-cylinder and regularity at each point + of its open interior. +- declaration: CKN.epsilonRegularityGradient + informal: A sufficiently small extended nonnegative limsup of the normalized squared + weak-gradient mass implies regularity at an interior point of a suitable weak + solution. +- declaration: CKN.caffarelliKohnNirenberg + informal: The singular set of a suitable weak solution with locally L^q forcing, + q > 5/2, has zero one-dimensional Hausdorff measure in the parabolic metric. +- declaration: CKN.isSuitableWeakSolution_iff_integrable + informal: The manuscript's suitable weak-solution predicate is equivalent to the + predicate that explicitly requires integrability of all tested weak-equation and + local-energy integrands. +- declaration: CKN.isSuitableWeakSolutionIntegrable_shearFlow + informal: The nonzero decaying sinusoidal shear flow with zero pressure and force + is a suitable weak solution on all space and times in (0,1). +tags: +- Navier–Stokes +- partial differential equations +- harmonic analysis +msc: +- 35Q30 +- 35B65 +- 42B20 +- 28A78 diff --git a/LeanPool/projects/cencov-petz.yaml b/LeanPool/projects/cencov-petz.yaml new file mode 100644 index 0000000000..5fe96ed555 --- /dev/null +++ b/LeanPool/projects/cencov-petz.yaml @@ -0,0 +1,46 @@ +slug: cencov-petz +title: Finite Čencov-Petz Uniqueness +summary: Formalizes the finite/discrete Čencov-Petz uniqueness theorem, showing that + any continuous metric family on finite probability simplexes that is monotone under + Markov morphisms is a scalar multiple of the Fisher information metric. +branch: information geometry +entry_module: LeanPool.CencovPetz +authors: +- Adam Benenson +source: + title: Statistical Decision Rules and Optimal Inference + authors: + - N. N. Chentsov + url: https://bookstore.ams.org/mmono-53 + github_repo: abenenson/cencov-petz +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_continuous +main_results: +- declaration: LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_continuous + informal: Proves that every continuous monotone metric family on a finite probability + simplex is a scalar multiple of the Fisher information metric. + source_ref: N. N. Chentsov, Statistical Decision Rules and Optimal Inference, AMS + Translations of Mathematical Monographs 53, 1982. +- declaration: LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_isRational + informal: Establishes the scalar Fisher-form identity at rational simplex points + using split representations. +- declaration: LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_isSplitRepresentable + informal: Proves the scalar Fisher-form identity for simplex points represented + as pushforwards of uniform distributions under splitting maps. +- declaration: LeanPool.CencovPetz.MonotoneMetricFamily.uniformScalar_eq_uniformScalar_two + informal: Shows that the uniform-point scalar is independent of simplex dimension + once the dimension is at least two. +- declaration: LeanPool.CencovPetz.TangentFin.Bilin.B_eq_smul_fisherBilin_uniform + informal: Proves that the permutation-invariant bilinear form at a uniform finite + simplex point is a scalar multiple of the Fisher bilinear form. +tags: +- information-geometry +- fisher-information +- markov-morphisms +- finite-simplex +msc: +- 62B10 +- 53C21 diff --git a/LeanPool/projects/centered-maximal-constant.yaml b/LeanPool/projects/centered-maximal-constant.yaml new file mode 100644 index 0000000000..e3f823fd5a --- /dev/null +++ b/LeanPool/projects/centered-maximal-constant.yaml @@ -0,0 +1,49 @@ +slug: centered-maximal-constant +title: Bounds for the centered Hardy-Littlewood maximal constant +summary: Proves that the weak type (1,1) constant c_2 of the centered maximal operator + over axis-parallel squares is at least the explicit radical phi, approximately 1.68551, + using a weighted periodic lattice and integrable approximations. Also proves c_d + <= 2^d in every natural dimension and 1.685 < phi < 1.686. The operator acts on + real-valued integrable functions, with Lebesgue outer measure of strict level sets. + Includes the maximal-function API, a Vitali covering argument, lattice witnesses, + smearing estimates, and the limiting lower bound. No optimality of phi is claimed. +branch: harmonic analysis +entry_module: LeanPool.CenteredMaximal +authors: +- Yongxi Lin +source: + title: A lower bound 1.6855 for the planar centred maximal constant over squares + authors: + - Yongxi Lin + url: https://github.com/CoolRmal/centered-maximal-constant + github_repo: CoolRmal/centered-maximal-constant + commit: c6a8cb29e8ecce9ac4614a8c7e4bf6938366f986 + notes: First completed publicly on 2026-09-18 at 22:15:07 UTC in commit 06a726f113241d445903f0c0eae1ebe40d8b539f. + The lower bound is documented and proved in upstream docs/PROOF.md; the weighted + smearing argument adapts Aldaz (2000), Lemma 1.1, and the covering upper bound + follows Tao's 245A Notes 5, Exercise 42. Upstream formalization.yaml and docs/HISTORY.md + disclose AI generation of the configuration and proofs under Yongxi Lin's direction. +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.CenteredMaximal.ofReal_phi_le_weakTypeConstant_two +main_results: +- declaration: LeanPool.CenteredMaximal.ofReal_phi_le_weakTypeConstant_two + informal: The explicit radical phi is a lower bound for c_2, the infimum of all + weak type (1,1) bounds for the centered maximal operator over axis-parallel squares + in the real plane. +- declaration: LeanPool.CenteredMaximal.weakTypeConstant_le_two_pow + informal: For every natural number d, including zero, the centered maximal operator + over axis-parallel cubes in real d-space has weak type (1,1) constant c_d at most + 2^d. +- declaration: LeanPool.CenteredMaximal.phi_mem_Ioo + informal: The explicit radical constant phi lies strictly between 1.685 and 1.686. +tags: +- harmonic-analysis +- maximal-functions +- weak-type-inequalities +- lattice-constructions +msc: +- 42B25 +- 26D15 diff --git a/LeanPool/projects/champernowne-normality.yaml b/LeanPool/projects/champernowne-normality.yaml new file mode 100644 index 0000000000..69dc654689 --- /dev/null +++ b/LeanPool/projects/champernowne-normality.yaml @@ -0,0 +1,49 @@ +slug: champernowne-normality +title: Champernowne normality in every base +summary: 'Proves that concatenating the base-b digits of the positive integers gives + a normal digit sequence for every integer base b at least 2: every nonempty length-k + word of digits below b has limiting frequency b^(-k). This is a discrete, all-base + generalization of D. G. Champernowne''s 1933 base-ten result, without a real-number + expansion bridge. Includes reusable substring-counting inequalities, a digit-string + equivalence, exact cohort counts, and quantitative prefix bounds used to establish + the frequency limit.' +branch: number theory and asymptotic analysis +entry_module: LeanPool.Champernowne +authors: +- Arthur Champernowne +source: + title: Champernowne Sequence Normality Proof + authors: + - Arthur Champernowne + url: https://github.com/OldMathematician/ChampernowneNormality + github_repo: OldMathematician/ChampernowneNormality + commit: 31a76dbde2727745beb6fb81af4cab1ce6a15fda +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Champernowne.champernowne_normal +main_results: +- declaration: Champernowne.champernowne_normal + informal: 'For every integer base b at least 2, the sequence formed by concatenating + the base-b digits of 1, 2, 3, and so on is normal: each nonempty word w of digits + below b has limiting occurrence frequency 1 / b^(length w), counting overlapping + occurrences.' +- declaration: Champernowne.champernowne_normal_ten + informal: The decimal digit sequence 123456789101112... is normal in base ten. +- declaration: Champernowne.sum_countOccurrences_cohort + informal: For a nonempty length-k word of base-b digits with nonzero first digit + and k at most m, its total occurrences inside all m-digit positive integers equal + b^(m-k) + (m-k)(b-1)b^(m-k-1), with natural subtraction in the exponents. +- declaration: Champernowne.sum_countOccurrences_cohort_zero + informal: For a nonempty length-k word of base-b digits starting with zero and k + at most m, its total occurrences inside all m-digit positive integers equal (m-k)(b-1)b^(m-k-1), + with natural subtraction in the exponent. +tags: +- normal-numbers +- digit-expansions +- combinatorics-on-words +- asymptotic-frequency +msc: +- 11K16 +- 68R15 diff --git a/LeanPool/projects/channel-capacity.yaml b/LeanPool/projects/channel-capacity.yaml new file mode 100644 index 0000000000..1b456d5494 --- /dev/null +++ b/LeanPool/projects/channel-capacity.yaml @@ -0,0 +1,51 @@ +slug: channel-capacity +title: Uniqueness of Shannon Capacity-Achieving Priors +summary: 'Formalizes uniqueness of the capacity-achieving input distribution for a + Markov-kernel channel in three aligned layers: a finite-alphabet theorem (compactness, + continuity, and strict concavity of mutual information on Mathlib''s weak topology + for `ProbabilityMeasure`), a concrete discharged theorem for compact-Polish channels + with a common strictly positive jointly continuous density, and a general measure-theoretic + packaging that isolates the abstract well-conditioning hypotheses, plus a counterexample + showing pairwise-distinct rows do not give an injective prior pushforward (the non-degeneracy + required for uniqueness).' +branch: information theory +entry_module: LeanPool.ChannelCapacity +authors: +- Adam Benenson +source: + title: Elements of Information Theory + authors: + - Thomas M. Cover + - Joy A. Thomas + doi: 10.1002/047174882X + github_repo: abenenson/channel-capacity +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- ChannelCapacity.exists_unique_capacity_achieving_prior_of_finite +main_results: +- declaration: ChannelCapacity.exists_unique_capacity_achieving_prior_of_finite + informal: For a Markov kernel between finite alphabets (nonempty input) whose rows + are linearly independent (`Kernel.RowMatrixFullRank`), a capacity-achieving input + prior exists and is unique. + source_ref: R. G. Gallager, Information Theory and Reliable Communication, Wiley, + 1968, Section 4.5. +- declaration: ChannelCapacity.exists_unique_capacity_achieving_prior_discharged + informal: For compact-Polish channels with a common strictly positive jointly continuous + density, given injectivity of the prior-pushforward map and upper semicontinuity + of mutual information (the latter carried as a field of `Kernel.ContinuousPositiveDensity`, + not derived), the capacity-achieving prior is unique. +- declaration: ChannelCapacity.exists_unique_capacity_achieving_prior + informal: 'General measure-theoretic packaging: under abstract well-conditioning + hypotheses, the capacity-achieving prior is unique.' +tags: +- information-theory +- channel-capacity +- mutual-information +- kullback-leibler +- markov-kernel +msc: +- 94A17 +- 94A15 +- 60A10 diff --git a/LeanPool/projects/chip-firing-with-lean.yaml b/LeanPool/projects/chip-firing-with-lean.yaml new file mode 100644 index 0000000000..7f0ca8591b --- /dev/null +++ b/LeanPool/projects/chip-firing-with-lean.yaml @@ -0,0 +1,36 @@ +slug: chip-firing-with-lean +title: Chip-Firing with Lean 4 +summary: A Lean 4 / Mathlib formalization of Baker and Norine's Riemann-Roch theorem + for finite graphs. Chip configurations on a finite graph, modulo chip-firing moves, + form an abelian group analogous to the group of line bundles on a compact Riemann + surface. Each class can be assigned a rank, analogous to the dimension of a linear + series. These ranks then satisfy a formula of the form r(D) - r(K-D) = deg D + 1 + - g, in exact analogy to Riemann surfaces. This formula is formalized, along with + the analog of Clifford's theorem for finite graphs. +branch: combinatorics +entry_module: LeanPool.ChipFiring +authors: +- Dhyey Dharmendrakumar Mavani +- Nathan Pflueger +source: + url: https://github.com/dhyeymavani2003/chip-firing-with-lean + github_repo: dhyeymavani2003/chip-firing-with-lean + commit: 5b422d855c02af60591121ba7c11b5e8242542b8 +license: Apache-2.0 +status: verified +main_declarations: +- ChipFiring.Propositions.riemann_roch +- ChipFiring.Propositions.clifford +main_results: +- declaration: ChipFiring.Propositions.riemann_roch + informal: For a finite connected graph, the ranks of a divisor and its canonical + complement satisfy the graph Riemann–Roch formula. +- declaration: ChipFiring.Propositions.clifford + informal: For a special divisor on a finite connected graph, its rank is at most + half its degree. +tags: +- combinatorics +msc: +- 05C57 +- 14T20 +provenance: mix diff --git a/LeanPool/projects/chudnovsky.yaml b/LeanPool/projects/chudnovsky.yaml new file mode 100644 index 0000000000..a1d8776b00 --- /dev/null +++ b/LeanPool/projects/chudnovsky.yaml @@ -0,0 +1,40 @@ +slug: chudnovsky +title: A Detailed Proof of the Chudnovsky Formula +summary: 'Formalizes the Chudnovsky brothers'' rapidly convergent series for 1/π, + proving Mathlib''s `chudnovskySum = π⁻¹`. Follows Milla''s elementary complex-analytic + proof: Weierstrass and Eisenstein theory, Liouville and parallelogram-residue calculus, + the Kummer hypergeometric bridge and the main modular identity relating the j-invariant + to the lattice periods, Baker''s division-value machinery, and the singular-moduli + inputs at the CM point τ = (1 + √-163)/2 — integrality of j(τ) via modular polynomials + and Kronecker''s congruence, and rationality of the Weierstrass s₂ invariant via + Masser''s Appendix A.1.' +branch: number theory +entry_module: LeanPool.Chudnovsky +authors: +- Xuanji Li +source: + title: A Detailed Proof of the Chudnovsky Formula + authors: + - Lorenz Milla + arxiv: '1809.00533' + github_repo: ldct/lean-eval-chudnovsky +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Chudnovsky.chudnovskySum_eq_pi_inv +main_results: +- declaration: Chudnovsky.chudnovskySum_eq_pi_inv + informal: 'Proves the Chudnovsky series identity, i.e. Mathlib''s `chudnovskySum + = π⁻¹`: the series (12 / 640320^(3/2)) · ∑ₙ (-1)ⁿ (6n)! (545140134 n + 13591409) + / ((3n)! (n!)³ 640320^(3n)) sums to 1/π.' +tags: +- number-theory +- pi +- chudnovsky +- modular-forms +- complex-multiplication +msc: +- 11Y60 +- 11F03 +- 33C05 diff --git a/LeanPool/projects/chvatal-conjecture.yaml b/LeanPool/projects/chvatal-conjecture.yaml new file mode 100644 index 0000000000..a556a729dd --- /dev/null +++ b/LeanPool/projects/chvatal-conjecture.yaml @@ -0,0 +1,67 @@ +slug: chvatal-conjecture +title: Chvátal's conjecture and sharp Boolean correlation +summary: Formalizes Chang, Liu, and Liu's proof that a hereditary family on a finite + nonempty ground set has a largest intersecting subfamily that is a star. Develops + the sharp Boolean Fourier correlation inequality, its antipodal and quadratic forms, + sharpness examples, and the weighted star theorem for nonnegative decreasing weights. +branch: extremal combinatorics +entry_module: LeanPool.Chvatal +authors: +- Boon Suan Ho +- Chvatal formalization contributors +source: + title: A proof of Chvátal's conjecture via a sharp correlation inequality + authors: + - Fan Chang + - Hong Liu + - Miao Liu + arxiv: 2609.19123v1 + url: https://github.com/boonsuan/chvatal + github_repo: boonsuan/chvatal + commit: c19ed3aaac9e42d446f963a862d39d8a09eddbf9 + announcement_date: '2026-09-17T03:58:22Z' + announcement_url: https://github.com/boonsuan/chvatal/commit/c46f715af9d400523eb0d4c9c0d2335abbdc9fdd + notes: GPT-6 generated the formalization under Boon Suan Ho's direction; the mathematical + proof is due to Chang, Liu, and Liu. No human expert review or source-author endorsement + is claimed. The upstream MIT copyright and permission notice is retained in every + imported file. +license: MIT +status: verified +provenance: AI +main_declarations: +- Chvatal.chvatal +- Chvatal.sharp_correlation +main_results: +- declaration: Chvatal.chvatal + informal: Every intersecting subfamily A of a hereditary finite family D on a finite + nonempty ground type has cardinality at most that of some star of D. +- declaration: Chvatal.exists_largest_intersecting_star + informal: One star of D is itself intersecting and simultaneously dominates all + intersecting subfamilies of the hereditary family D in size, on a finite nonempty + ground type. +- declaration: Chvatal.sharp_correlation + informal: For increasing Boolean functions f and g on any finite cube, the sum of + squared Fourier coefficients of g weighted by the largest influence of f on each + nonempty index is at most the harmonic mean of Cov(f,g) and Cov(f,g*), with zero-denominator + value zero. +- declaration: Chvatal.two_spectral_le_quadratic_covariance + informal: For increasing Boolean f and g and every real t, twice their squared Fourier + energy over index pairs with odd intersection is bounded by t^2 Cov(f,g) + (1-t)^2 + Cov(f,g*). +- declaration: Chvatal.antipodal_correlation + informal: If g is also self-dual and the finite coordinate type is nonempty, Cov(f,g) + is at least one quarter of f's minimum coordinate influence. +- declaration: Chvatal.andFunction_sharp + informal: The AND function attains equality in the sharp correlation bound for every + increasing Boolean second function, including constants. +- declaration: Chvatal.exists_largest_weighted_intersecting_star + informal: For any nonnegative antitone real weight on subsets of a finite nonempty + ground type, one full-cube star is intersecting and its total weight dominates + every intersecting family. +tags: +- extremal-combinatorics +- intersecting-families +- Fourier-analysis +msc: +- 05D05 +- 60E15 diff --git a/LeanPool/projects/circuit-complexity.yaml b/LeanPool/projects/circuit-complexity.yaml new file mode 100644 index 0000000000..069b2e7a93 --- /dev/null +++ b/LeanPool/projects/circuit-complexity.yaml @@ -0,0 +1,46 @@ +slug: circuit-complexity +title: Circuit Complexity in Lean 4 +summary: A Lean 4 formalization of classical Boolean circuit complexity, including + the Shannon counting lower bound, Schnorr's XOR gate lower bound, a gate-elimination + lower bound, CNF/DNF lower bounds for parity, and nondeterministic quantification + complexity bounds. +branch: boolean circuit complexity lower bounds +entry_module: LeanPool.CircuitComplexity +authors: +- Samuel Schlesinger +source: + url: https://github.com/SamuelSchlesinger/circuit-complexity + github_repo: SamuelSchlesinger/circuit-complexity +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- CircuitComplexity.shannon_lower_bound_circuit +main_results: +- declaration: CircuitComplexity.shannon_lower_bound_circuit + informal: For N at least 6, there exists a Boolean function on N inputs that cannot + be computed by any fan-in-2 AND/OR circuit of size at most 2^N / (5N). +- declaration: CircuitComplexity.schnorr_lower_bound_circuit + informal: Any fan-in-2 AND/OR circuit computing the N-input XOR function (or its + complement) has at least 2(N-1) internal gates. +- declaration: CircuitComplexity.Circuit.gate_elimination_lower_bound + informal: Any circuit over bounded fan-in k AND/OR computing a function with n' + essential inputs has size at least n' / k. +- declaration: CircuitComplexity.DNF.xorBool_complexity_lb + informal: Any DNF formula computing the N-input XOR function requires at least 2^(N-1) + terms. +- declaration: CircuitComplexity.CNF.xorBool_complexity_lb + informal: Any CNF formula computing the N-input XOR function requires at least 2^(N-1) + clauses. +- declaration: CircuitComplexity.existQuantify_complexity_le + informal: If a function f has circuit complexity s, then the existential quantification + of f over a k-bit block has circuit complexity at most 2^k * (s + 1). +tags: +- circuit-complexity +- boolean-functions +- lower-bounds +- shannon-bound +- parity +msc: +- 68Q06 +- 94C11 diff --git a/LeanPool/projects/circuitlib.yaml b/LeanPool/projects/circuitlib.yaml new file mode 100644 index 0000000000..0d7d867149 --- /dev/null +++ b/LeanPool/projects/circuitlib.yaml @@ -0,0 +1,49 @@ +slug: circuitlib +title: 'circuitlib: a circuit verification library for Lean 4' +summary: A categorical semantics framework for digital circuits over Belnap's four-valued + logic, following Ghica-Kaye-Sprunger. Its main theorems prove that combinational + circuits (monotone functions on wire vectors) and sequential circuits (causal monotone + stream functions) form symmetric monoidal categories, with a verified full adder + as demonstration. +branch: categorical circuit semantics +entry_module: LeanPool.Circuitlib +authors: +- Matt Hunzinger +source: + arxiv: '2201.10456' + url: https://github.com/matthunz/circuitlib + github_repo: matthunz/circuitlib +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Circuit.SequentialCircuitCategory.instSymmetricCategory +main_results: +- declaration: Circuit.SequentialCircuitCategory.instSymmetricCategory + informal: The category of sequential circuits, whose morphisms are causal monotone + stream functions as in Ghica-Kaye-Sprunger's denotational semantics, is symmetric + monoidal, with all coherence laws (pentagon, triangle, hexagons, symmetry) proven. +- declaration: Circuit.CombinationalCircuitCategory.instSymmetricCategory + informal: The category of combinational circuits, with wire counts as objects and + monotone wire-vector functions as morphisms, is symmetric monoidal under juxtaposition + of wire bundles. +- declaration: Circuit.SequentialCircuitCategory.tensorHom_causal + informal: Causality is preserved by parallel composition, the key lemma making sequential + circuits closed under the tensor product. +- declaration: Circuit.BelnapLevel.instSemilatticeSup + informal: Belnap's four truth values under the information order form a join-semilattice, + the wire-value semantics in which every circuit denotes a monotone function. +- declaration: Circuit.CombinationalCircuit.adder_def + informal: Correctness of a full adder assembled categorically from two half adders, + gates, braidings, and whiskerings, computing the Boolean sum and carry bits; a + demonstration of the framework. +tags: +- circuits +- hardware +- category-theory +- combinational +- sequential +msc: +- 18M05 +- 68Q60 +- 94C11 diff --git a/LeanPool/projects/civ-axisymmetric.yaml b/LeanPool/projects/civ-axisymmetric.yaml new file mode 100644 index 0000000000..1214efde60 --- /dev/null +++ b/LeanPool/projects/civ-axisymmetric.yaml @@ -0,0 +1,56 @@ +slug: civ-axisymmetric +title: Regularity of asymptotically axisymmetric Navier–Stokes solutions +summary: Proves the Constantin–Ignatova–Vicol regularity theorem for suitable weak + solutions of the three-dimensional forced Navier–Stokes equations with analytic + forcing, anisotropic Type II bounds on the angular mean, and an axisymmetric core + at each time. Includes the fully axisymmetric theorem, meridional smallness, interior + spatial analyticity, the nonanalyticity consequence for singular solutions, and + the comparison, Serrin, and Gustafson–Kang–Tsai results used in their proofs. Reuses + Lean Pool's Caffarelli–Kohn–Nirenberg suitable-solution and analysis library. +branch: Analysis +entry_module: LeanPool.CIVAxisymmetric +authors: +- Scott Armstrong +- Vlad Vicol +source: + arxiv: '2609.20803' + github_repo: scottnarmstrong/CIVAxisymmetric + url: https://github.com/scottnarmstrong/CIVAxisymmetric/tree/9f539dc7bf7ddd414d3ef81df539e1bef229f681 + commit: 9f539dc7bf7ddd414d3ef81df539e1bef229f681 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- CIV.mainTheorem +- CIV.axisymmetricTheorem +main_results: +- declaration: CIV.mainTheorem + informal: A smooth suitable weak solution on the unit parabolic cylinder with globally + finite energy, bounded C^2 spatially analytic force, anisotropic Type II bounds + on its angular mean, and a positive axisymmetric core at every time is axisymmetric + throughout the cylinder and bounded near the origin up to the terminal time. +- declaration: CIV.axisymmetricTheorem + informal: An axisymmetric smooth suitable weak solution obeying the stated anisotropic + Type II bounds and bounded C^2 force is bounded near the origin up to the terminal + time. +- declaration: CIV.meridionalSmallness + informal: Under the axisymmetric anisotropic hypotheses, the weighted meridional + quantity tends uniformly to zero on a smaller cylinder as the terminal time approaches. +- declaration: CIV.interiorAnalyticity + informal: Smooth solutions with locally uniformly spatially analytic forcing are + locally uniformly spatially analytic in the interior. +- declaration: CIV.forceUnderCore + informal: For a smooth suitable weak solution with globally finite energy, bounded + C^2 forcing, angular-mean anisotropic bounds, and an axisymmetric core at every + time, failure of boundedness near the origin implies that the force is not spatially + analytic, is nonzero somewhere in the unit cylinder, and has a nonzero point in + every smaller terminal cylinder. +tags: +- Navier–Stokes +- axisymmetry +- partial differential equations +- regularity +msc: +- 35Q30 +- 35B65 +- 35B33 diff --git a/LeanPool/projects/classification-of-surfaces.yaml b/LeanPool/projects/classification-of-surfaces.yaml new file mode 100644 index 0000000000..f8848a05c4 --- /dev/null +++ b/LeanPool/projects/classification-of-surfaces.yaml @@ -0,0 +1,35 @@ +slug: classification-of-surfaces +title: Classification of Compact Surfaces +summary: Formalizes the classification of compact connected Hausdorff topological + 2-manifolds with boundary as the sphere, an orientable normal-form quotient, or + a non-orientable normal-form quotient, using triangulation and finite cyclic polygonal + presentations. +branch: geometric and low-dimensional topology +entry_module: LeanPool.ClassificationOfSurfaces +authors: +- Ryan McCorvie +- Jack McCarthy +- SF Lean Meetup contributors +source: + title: Classification of Compact Surfaces + url: https://github.com/mccorvie/classification-of-surfaces/tree/master/ClassificationOfSurfaces + github_repo: mccorvie/classification-of-surfaces +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanEval.Topology.ClassificationOfSurfaces.classification_of_surfaces +main_results: +- declaration: LeanEval.Topology.ClassificationOfSurfaces.classification_of_surfaces + informal: Every compact connected Hausdorff topological 2-manifold with boundary + is homeomorphic to the sphere, an orientable normal-form quotient, or a non-orientable + normal-form quotient. +tags: +- geometric-topology +- low-dimensional-topology +- surfaces +- manifolds +- triangulation +msc: +- 57K20 +- 57N05 diff --git a/LeanPool/projects/clawristotle.yaml b/LeanPool/projects/clawristotle.yaml new file mode 100644 index 0000000000..e80d336fdc --- /dev/null +++ b/LeanPool/projects/clawristotle.yaml @@ -0,0 +1,31 @@ +slug: clawristotle +title: 'Clawristotle: Vlasov-Maxwell-Landau steady-state classification' +summary: Formalizes a steady-state classification result for the Vlasov-Maxwell-Landau + system with Coulomb kernel. +branch: kinetic theory +entry_module: LeanPool.Clawristotle +authors: +- Vasily Ilin +source: + arxiv: '2603.15929' + github_repo: Vilin97/Clawristotle +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- VML.Theorem42 +- VML.CoulombConcreteTheorem42 +main_results: +- declaration: VML.Theorem42 + informal: States the abstract steady-state classification theorem for the Vlasov-Maxwell-Landau + framework. +- declaration: VML.CoulombConcreteTheorem42 + informal: Specializes the steady-state classification theorem to the Coulomb kernel. +tags: +- pde +- kinetic-theory +- mathematical-physics +msc: +- 35Q83 +- 82C40 +- 35Q61 diff --git a/LeanPool/projects/common-neighbour-conjecture.yaml b/LeanPool/projects/common-neighbour-conjecture.yaml new file mode 100644 index 0000000000..5958c4f1e5 --- /dev/null +++ b/LeanPool/projects/common-neighbour-conjecture.yaml @@ -0,0 +1,46 @@ +slug: common-neighbour-conjecture +title: Common-neighbour counterexamples for Saxl graphs +summary: Proves that for every base size B at least two and every prescribed degree + bound, there is a finite faithful primitive permutation group of at least that degree + and exact base size B whose generalized Saxl graph contains two nonadjacent vertices + with no common neighbor. The supporting library constructs an explicit infinite + family from binary deleted permutation modules over finite fields and proves the + required base-size, irreducibility, primitivity, and graph-theoretic properties. +branch: permutation groups and algebraic graph theory +entry_module: LeanPool.CommonNeighbourConjecture +authors: +- Aluna Rizzoli +- Adam R. Thomas +source: + title: Common neighbour conjectures for Saxl graphs fail at every base size + authors: + - Aluna Rizzoli + - Adam R. Thomas + doi: 10.5281/zenodo.22231393 + url: https://github.com/alunik/common-neighbour-conjecture + github_repo: alunik/common-neighbour-conjecture + commit: b2c0a8f95aa4dca76a034a0e769a8c8249f38ae2 +license: MIT +status: verified +provenance: human +main_declarations: +- SaxlCounterexamples.MainTheorems.mainTheorem +main_results: +- declaration: SaxlCounterexamples.MainTheorems.mainTheorem + informal: For every natural number B at least two and every degree bound, there + exists a finite faithful primitive permutation group of degree at least the bound + and base size exactly B whose generalized Saxl graph has two nonadjacent vertices + with no common neighbor. +- declaration: SaxlCounterexamples.EveryBase.hq_coreFamily + informal: For every odd d with 3 ≤ d and every tail, the explicit affine wreath-product + family has exact base size tail + 2, acts primitively, and its displayed pair + has no common neighbor in the corresponding generalized Saxl graph. +tags: +- permutation-groups +- group-actions +- algebraic-graph-theory +- saxl-graphs +- counterexamples +msc: +- 20B15 +- 05C25 diff --git a/LeanPool/projects/compact-spectral.yaml b/LeanPool/projects/compact-spectral.yaml new file mode 100644 index 0000000000..aec681ffb1 --- /dev/null +++ b/LeanPool/projects/compact-spectral.yaml @@ -0,0 +1,55 @@ +slug: compact-spectral +title: Spectral theorem for compact self-adjoint operators +summary: 'Formalizes the spectral theorem for compact self-adjoint operators on a + Hilbert space: such an operator admits a Hilbert basis of eigenvectors. The development + passes from weak compactness of Hilbert closed balls and Rayleigh-quotient extrema + to an eigenvector at the operator norm, finiteness of the point spectrum outside + any neighbourhood of zero, finite-rank cutoff approximation, and finally the orthogonal + decomposition into eigenspaces.' +branch: functional analysis +entry_module: LeanPool.CompactSpectral +authors: +- Adam Benenson +source: + title: Advanced Real Analysis + authors: + - Anthony W. Knapp + doi: 10.3792/euclid/9781429799911-2 + github_repo: abenenson/compact-spectral +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- CompactSelfAdjoint.exists_hasEigenvector_iSup_or_iInf_of_isCompactOperator +main_results: +- declaration: CompactSelfAdjoint.exists_hilbertBasis_hasEigenvector_of_isCompactOperator_of_isSelfAdjoint + informal: A compact self-adjoint operator on a Hilbert space admits a Hilbert basis + of eigenvectors. + source_ref: A. W. Knapp, Advanced Real Analysis, Chapter II, Section 2, Theorem + 2.3, doi:10.3792/euclid/9781429799911-2. +- declaration: CompactSelfAdjoint.iSup_eigenspace_orthogonal_eq_bot_of_isCompactOperator_of_isSelfAdjoint + informal: 'The eigenspaces of a compact self-adjoint operator span a dense subspace: + the orthogonal complement of the supremum of all eigenspaces is trivial.' + source_ref: A. W. Knapp, Advanced Real Analysis, Chapter II, Section 2, Theorem + 2.3, doi:10.3792/euclid/9781429799911-2. +- declaration: CompactSelfAdjoint.exists_hasEigenvector_iSup_or_iInf_of_isCompactOperator + informal: A compact self-adjoint operator has an eigenvector whose eigenvalue realizes + either the supremum or the infimum of its Rayleigh quotient. + source_ref: A. W. Knapp, Advanced Real Analysis, Chapter II, Section 2, Theorem + 2.2, doi:10.3792/euclid/9781429799911-2. +- declaration: CompactSelfAdjoint.exists_unit_norm_eq_opNorm_of_isCompactOperator + informal: A compact operator attains its operator norm on the unit sphere of the + Hilbert space. + source_ref: A. W. Knapp, Advanced Real Analysis, Chapter II, Section 1, compactness + preliminaries for compact operators, doi:10.3792/euclid/9781429799911-2. +- declaration: CompactSelfAdjoint.countable_set_hasEigenvalue_ne_zero_of_isCompactOperator_of_isSelfAdjoint + informal: The set of nonzero eigenvalues of a compact self-adjoint operator is countable. + source_ref: A. W. Knapp, Advanced Real Analysis, Chapter II, Section 2, spectral + theorem consequences, doi:10.3792/euclid/9781429799911-2. +tags: +- spectral-theory +- functional-analysis +- compact-operators +msc: +- 47A75 +- 47B07 diff --git a/LeanPool/projects/compactness-and-degeneracy.yaml b/LeanPool/projects/compactness-and-degeneracy.yaml new file mode 100644 index 0000000000..35ccc253aa --- /dev/null +++ b/LeanPool/projects/compactness-and-degeneracy.yaml @@ -0,0 +1,44 @@ +slug: compactness-and-degeneracy +title: Counterexamples to graph compactness and two-degenerate extremal bounds +summary: Formalizes two counterexamples in extremal graph theory. The first gives + a finite family of connected bipartite cyclic graphs whose family extremal number + grows strictly slower than every member's extremal number, disproving the Erdős--Simonovits + compactness conjecture. The second constructs a connected bipartite two-degenerate + graph with extremal number at least c n^(3/2+ε), disproving the proposed universal + n^(3/2) upper bound under the stated bipartition degree condition. +branch: extremal combinatorics and graph theory +entry_module: LeanPool.CompactnessAndDegeneracy +authors: +- OpenAI +- Dean Cureton +source: + title: Ten Advances in Mathematics and Theoretical Computer Science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- CompactnessConjecture.quantitativeCompactnessCounterexample +main_results: +- declaration: CompactnessConjecture.quantitativeCompactnessCounterexample + informal: There is a nonempty finite family of connected bipartite cyclic graphs + such that every member has a uniform positive n^(4/3)-scale extremal lower bound, + while every family-free host on n vertices has its sixteenth power of the edge + count bounded by a constant times n^21. Consequently the family is not compact + and the Erdős--Simonovits compactness conjecture is false. +- declaration: TwoDegenerateGraphs.twoDegenerateExtremalCounterexample + informal: There is a connected bipartite two-degenerate finite graph H such that + each side of every two-coloring has a vertex of degree greater than two, yet ex(n,H) + is eventually at least c n^(3/2+ε) for some positive constants c and ε. +tags: +- extremal-combinatorics +- extremal-graph-theory +- degenerate-graphs +- counterexamples +msc: +- 05C35 +- 05C75 diff --git a/LeanPool/projects/composition-algebras.yaml b/LeanPool/projects/composition-algebras.yaml new file mode 100644 index 0000000000..2f626605ee --- /dev/null +++ b/LeanPool/projects/composition-algebras.yaml @@ -0,0 +1,49 @@ +slug: composition-algebras +title: Hurwitz's Classification of Euclidean Composition Algebras +summary: 'Formalizes Hurwitz''s classification of nontrivial finite-dimensional real + Euclidean composition algebras: every such algebra has dimension 1, 2, 4, or 8 and + is real-linearly equivalent to the real numbers, complex numbers, quaternions, or + octonions through a map preserving the unit, multiplication, and norm form. The + development also constructs the octonions from an explicit Fano multiplication table, + proves Degen''s eight-square identity, and shows that Cayley--Dickson doubling preserves + the composition law for associative inputs.' +branch: nonassociative algebra +entry_module: LeanPool.CompositionAlgebras +authors: +- Bryan Ehrlich +source: + title: Über die Composition der quadratischen Formen von beliebig vielen Variablen + authors: + - Adolf Hurwitz + url: https://eudml.org/doc/58420 + github_repo: ehrlich-b/composition-algebras + commit: e37e22b0571a170ba18a0d8db29fe2a857906b92 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- CompositionAlgebra.hurwitz_classification +main_results: +- declaration: CompositionAlgebra.hurwitz_classification + informal: Every nontrivial finite-dimensional real Euclidean composition algebra + is linearly equivalent to the real numbers, complex numbers, quaternions, or octonions + through a map preserving the unit, multiplication, and norm form. +- declaration: CompositionAlgebra.finrank_eq_one_or_two_or_four_or_eight + informal: The real dimension of a nontrivial finite-dimensional Euclidean composition + algebra is exactly 1, 2, 4, or 8. +- declaration: CompositionAlgebra.CD.instCompositionAlgebra + informal: The Cayley--Dickson double of a nontrivial associative Euclidean composition + algebra again satisfies the composition law. +- declaration: Octonion.nucleus_real + informal: Any octonion that associates in the third slot with every pair has zero + imaginary coordinates and hence lies on the real line. +tags: +- nonassociative-algebra +- composition-algebras +- octonions +- hurwitz-theorem +- cayley-dickson +msc: +- 17A75 +- 17A35 +- 11E88 diff --git a/LeanPool/projects/computability.yaml b/LeanPool/projects/computability.yaml new file mode 100644 index 0000000000..86abab4a24 --- /dev/null +++ b/LeanPool/projects/computability.yaml @@ -0,0 +1,45 @@ +slug: computability +title: Oracle Computability and Turing Degrees +summary: Formalizes oracle-relative computability via partial recursive functions, + Turing reducibility and the Turing degrees as an upper semilattice, a Gödel numbering + with a universal oracle machine, the jump operator, and the arithmetical hierarchy. +branch: computability theory +entry_module: LeanPool.Computability +authors: +- Tanner Duve +- Elan Roth +source: + url: https://github.com/tannerduve/computability + github_repo: tannerduve/computability +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- Computability.RecursiveIn +- Computability.TuringDegree +main_results: +- declaration: Computability.RecursiveIn + informal: 'The inductive relation `RecursiveIn O f` stating that `f : ℕ →. ℕ` is + partial recursive relative to a set of oracle functions `O`.' +- declaration: Computability.exists_code_rel + informal: Universality of the oracle evaluator `evalo`, that is, `RecursiveIn (Set.range + g) f` holds iff there is a code `c` with `evalo g c = f`. +- declaration: Computability.TuringReducible + informal: '`f` is Turing reducible to `g` (`f ≤ᵀ g`) when `f` is recursive in the + singleton oracle set `{g}`.' +- declaration: Computability.TuringDegree + informal: The type of Turing degrees, the antisymmetrization of partial functions + under Turing reducibility. +- declaration: Computability.jump_recIn + informal: The original function is Turing reducible to its jump, `f ≤ᵀ f⌜`. +- declaration: Computability.TuringDegree.instSemilatticeSup + informal: The Turing degrees form an upper semilattice with the Turing join as supremum. +tags: +- computability +- oracle-computability +- turing-degrees +- recursion-theory +- arithmetical-hierarchy +msc: +- 03D30 +- 03D28 diff --git a/LeanPool/projects/computablereal.yaml b/LeanPool/projects/computablereal.yaml new file mode 100644 index 0000000000..da5244e85c --- /dev/null +++ b/LeanPool/projects/computablereal.yaml @@ -0,0 +1,46 @@ +slug: computablereal +title: Verified interval-Cauchy real arithmetic +summary: Represents real numbers as Cauchy sequences of nested rational intervals + quotiented into a field, with executable interval ring arithmetic, proven-convergent + rational bounds for sqrt, exp, and pi, and an IsComputable typeclass that packages + such approximations to drive classical (noncomputable) comparison instances, after + O'Connor's certified exact real computation. +branch: interval arithmetic +entry_module: LeanPool.ComputableReal +authors: +- Alex Meiburg +source: + arxiv: '0805.2438' + doi: 10.1007/978-3-540-71067-7_21 + github_repo: Timeroot/computableReal +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- ComputableℝSeq +- Computableℝ +- IsComputable +main_results: +- declaration: ComputableℝSeq + informal: A sequence of converging rational lower and upper bounds representing + a single real number. +- declaration: Computableℝ + informal: The quotient of interval-Cauchy sequences by equality of value, an ordered + field whose ring operations are executable interval arithmetic. +- declaration: IsComputable + informal: A typeclass packaging an explicit interval-Cauchy approximation of a real + number, used to drive its classical comparison instances. +- declaration: Computableℝ.instField + informal: The field instance on the quotient; inversion extracts a sign witness + classically and is noncomputable. +- declaration: ComputableℝSeq.Sqrt.sqrtq + informal: Computes converging rational interval bounds for the square root of a + rational interval. +tags: +- interval-arithmetic +- real-numbers +- cauchy-sequences +- special-functions +msc: +- 65G40 +- 68V20 diff --git a/LeanPool/projects/concentration-inequalities.yaml b/LeanPool/projects/concentration-inequalities.yaml new file mode 100644 index 0000000000..443a01af57 --- /dev/null +++ b/LeanPool/projects/concentration-inequalities.yaml @@ -0,0 +1,44 @@ +slug: concentration-inequalities +title: Bennett--Bernstein, Freedman, and Hoeffding concentration inequalities +summary: Develops variance-sensitive sub-gamma moment-generating-function estimates, + Bennett and Bernstein bounds, a Freedman--Bernstein upper tail for finite adapted + sums, conditional and independent-variable Hoeffding inequalities. +branch: probability theory and probabilistic combinatorics +entry_module: LeanPool.ConcentrationInequalities +authors: +- Juan Pablo Traverso Gianini +- Aristotle +source: + title: Reusable concentration-inequality components accompanying Linear-Error Clique + Partitions of Split Graphs via Structured Triangle Packing + authors: + - Juan Pablo Traverso Gianini + url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/b3423f3e8c8db7c2d1b279673293ef3079faf903/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Contrib/FreedmanBernstein.lean + github_repo: jtraverso/erdos-81-chordal-clique-partitions +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Contrib.Freedman.subgamma_bernstein_tail +main_results: +- declaration: Contrib.Bennett.hasSubgammaMGF_of_bounded_above + informal: A centered real random variable bounded above by a positive constant, + with bounded second moment, satisfies a Bennett--Bernstein sub-gamma moment-generating-function + estimate. +- declaration: Contrib.Freedman.subgamma_bernstein_tail + informal: A finite adapted sum of conditionally sub-gamma increments satisfies the + corresponding variance-sensitive Freedman--Bernstein upper-tail bound. +- declaration: Contrib.ConditionalHoeffding.hasCondSubgaussianMGF_of_mem_Icc + informal: A bounded random variable with zero conditional expectation satisfies + a conditional Hoeffding sub-Gaussian moment-generating-function estimate. +- declaration: Contrib.Hoeffding.hoeffding_upper + informal: A sum of finitely many independent random variables valued in the unit + interval exceeds its mean by t with probability at most exp(-2 t squared divided + by the number of terms). +tags: +- probability-theory +- concentration-inequalities +- martingales +msc: +- 60E15 +- 60G42 diff --git a/LeanPool/projects/connes-kreimer.yaml b/LeanPool/projects/connes-kreimer.yaml new file mode 100644 index 0000000000..6c0faca6eb --- /dev/null +++ b/LeanPool/projects/connes-kreimer.yaml @@ -0,0 +1,48 @@ +slug: connes-kreimer +title: Connes-Kreimer Hopf algebra of rooted trees +summary: Formalizes the planar Connes-Kreimer/Foissy Hopf algebra of rooted trees, + its cut coproduct, bialgebra and Hopf algebra structures, the commutative quotient, + Adams operators, and the Eulerian idempotent on the quotient. +branch: algebraic combinatorics +entry_module: LeanPool.ConnesKreimer +authors: +- Carles Marín +source: + title: 'How a Tree Remembers Its Cuts: the Connes-Kreimer-Foissy Hopf Algebra of + Rooted Trees, Machine-Checked in Lean 4' + authors: + - Carles Marín + doi: 10.5281/zenodo.20762280 + github_repo: karlesmarin/connes-kreimer-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- CK.instCKHopf +- CK.instHabHopf +- CK.eulerian1_idem_ab +main_results: +- declaration: CK.instCKHopf + informal: Constructs the planar rooted-tree Connes-Kreimer/Foissy Hopf algebra as + a Mathlib `HopfAlgebra`. +- declaration: CK.instHabHopf + informal: Constructs the Hopf algebra structure on the commutative quotient of the + planar rooted-tree algebra. +- declaration: CK.eulerian1_idem_ab + informal: Proves that the first Eulerian idempotent is idempotent on the commutative + quotient. +- declaration: CK.coassoc_map + informal: Lifts the list-level cut coassociativity proof to equality of linear maps + for the coproduct. +- declaration: PowerSeries.logOf_mul + informal: Proves additivity of `PowerSeries.logOf` for power series with constant + coefficient one. +tags: +- hopf-algebras +- rooted-trees +- renormalization +- combinatorics +msc: +- 16T05 +- 05C05 +- 81T15 diff --git a/LeanPool/projects/connes-rigidity.yaml b/LeanPool/projects/connes-rigidity.yaml new file mode 100644 index 0000000000..45fedef6cd --- /dev/null +++ b/LeanPool/projects/connes-rigidity.yaml @@ -0,0 +1,38 @@ +slug: connes-rigidity +title: Conditional formalization of Zhou's Connes-rigidity counterexample +summary: Constructs Zhou's two countable discrete groups and, assuming the project's + universe-polymorphic property-(T) predicate for EL₃(𝔽₂[t]), proves that both satisfy + the same predicate, are ICC and nonisomorphic, while their group von Neumann algebras + are trace-preservingly normally star-isomorphic. +branch: operator algebras and geometric group theory +entry_module: LeanPool.ConnesRigidity +authors: +- Utensil Song +source: + title: ICC property(T) groups without W*-superrigidity + authors: + - Shuoxing Zhou + arxiv: '2608.02327' + url: https://github.com/utensil/connes-rigidity + github_repo: utensil/connes-rigidity +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Connes.theoremA +main_results: +- declaration: Connes.theoremA + informal: At any Hilbert-space universe, assuming property-(T) for EL₃(𝔽₂[t]), constructs + two countable discrete ICC groups satisfying property-(T) at that universe which + are nonisomorphic although their group von Neumann algebras are trace-preservingly + normally star-isomorphic. +tags: +- operator-algebras +- von-neumann-algebras +- property-t +- group-theory +- connes-rigidity +msc: +- 46L10 +- 20F65 +- 22D10 diff --git a/LeanPool/projects/conway-refinement.yaml b/LeanPool/projects/conway-refinement.yaml new file mode 100644 index 0000000000..37adadd0c6 --- /dev/null +++ b/LeanPool/projects/conway-refinement.yaml @@ -0,0 +1,30 @@ +title: Conway refinement for omnific integers +summary: 'Proves multiplicative refinement for omnific integers: every equality ab + = cd admits a two-by-two factor refinement. Develops factorization and refinement + of generalized power-series integer parts and connects these to surreal-number and + standalone Mathlib formulations.' +branch: ordered algebra +main_declarations: +- ConwayRefinement.Palomar.conwayRefinement +main_results: +- declaration: ConwayRefinement.Palomar.conwayRefinement + informal: If four omnific integers satisfy ab = cd, there are omnific integers e, + f, g, h with a = ef, b = gh, c = eg, and d = fh, including degenerate cases. +tags: +- surreal-numbers +- ordered-algebra +- factorization +msc: +- 06A05 +- 13F25 +provenance: AI +slug: conway-refinement +entry_module: LeanPool.ConwayRefinement +authors: +- Dan Abramov +source: + url: https://github.com/gaearon/conway-refinement + github_repo: gaearon/conway-refinement + commit: 264445c93b78554c408e99e4e7f663693b4e91ab +license: Apache-2.0 +status: verified diff --git a/LeanPool/projects/conway-soifer-three.yaml b/LeanPool/projects/conway-soifer-three.yaml new file mode 100644 index 0000000000..c442b15b4a --- /dev/null +++ b/LeanPool/projects/conway-soifer-three.yaml @@ -0,0 +1,42 @@ +slug: conway-soifer-three +title: Conway–Soifer covering conjecture for n = 3 +summary: Proves that ten congruent closed equilateral triangles covering a filled + equilateral triangle of side three have common side at least one, with arbitrary + positions, rotations and overlaps. Consequently ten unit triangles cannot cover + a triangle of side 3 + ε for ε > 0. Includes reusable finite halfspace and ray-endpoint + geometry, orientation-independent support and area bounds, and verified Bernstein + polynomial interval checkers. Four derived contact cases and 100 kernel-checked + certificates close the proof without a certificate hypothesis. +branch: discrete geometry +entry_module: LeanPool.ConwaySoifer +authors: +- Vladislav Kuznetsov +source: + title: 'The Conway–Soifer conjecture for n = 3: a Lean formalization' + url: https://github.com/AnanasClassic/conway-soifer-n3-lean/tree/b71b1d22b6f7ebb0f0173fc70a66075f44c86ce6 + github_repo: AnanasClassic/conway-soifer-n3-lean + commit: b71b1d22b6f7ebb0f0173fc70a66075f44c86ce6 + doi: 10.5281/zenodo.22712658 +license: MIT +status: verified +provenance: AI +main_declarations: +- ConwaySoifer.ten_triangle_cover_lower_bound +main_results: +- declaration: ConwaySoifer.ten_triangle_cover_lower_bound + informal: Any ten congruent closed equilateral triangles covering the filled equilateral + target of side three have common side at least one. Positions, rotations and overlaps + are unrestricted. + source_ref: Public v1.0.0 release, 2026-09-11T17:12:25Z; upstream README and Geometry/Basic.lean. +- declaration: ConwaySoifer.no_ten_unit_triangle_cover + informal: For every ε > 0, ten closed equilateral triangles of side one cannot cover + the filled equilateral target of side 3 + ε, regardless of position or rotation. + source_ref: Upstream public theorem and its proved scaling corollary. +tags: +- discrete-geometry +- triangle-covering +- conway-soifer +- bernstein-polynomials +msc: +- 52C15 +- 51M16 diff --git a/LeanPool/projects/cramer-wold.yaml b/LeanPool/projects/cramer-wold.yaml new file mode 100644 index 0000000000..da24733b23 --- /dev/null +++ b/LeanPool/projects/cramer-wold.yaml @@ -0,0 +1,50 @@ +slug: cramer-wold +title: The Cramer-Wold theorem +summary: Formalizes the finite-dimensional Cramer-Wold theorem for measures on Euclidean + spaces, with Banach-space continuous-dual generalizations, Hilbert-space inner-product + versions, and random-variable law corollaries for identical distributions and `HasLaw`. +branch: probability theory +entry_module: LeanPool.CramerWold +authors: +- Lazar Milikic +source: + title: Some theorems on distribution functions + authors: + - Harald Cramer + - Herman Wold + doi: 10.1112/jlms/s1-11.4.290 + github_repo: Lemmy00/lean-pool +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.CramerWold.euclideanSpace_measure_eq_of_forall_inner_map_eq +main_results: +- declaration: LeanPool.CramerWold.euclideanSpace_measure_eq_of_forall_inner_map_eq + informal: Proves the classical finite-dimensional Cramer-Wold theorem for finite + Borel measures on `EuclideanSpace ℝ (Fin n)`, showing equality when all scalar + projections have equal pushforward measures. +- declaration: LeanPool.CramerWold.measure_eq_of_forall_dual_map_eq + informal: Generalizes the extensionality criterion to finite Borel measures on complete + second-countable real normed spaces, using all continuous-linear real projections. +- declaration: LeanPool.CramerWold.measure_eq_of_forall_inner_map_eq + informal: Specializes Cramer-Wold extensionality to real Hilbert spaces, using scalar + projections of the form `x ↦ ⟪t, x⟫_ℝ`. +- declaration: LeanPool.CramerWold.identDistrib_of_forall_dual_comp_identDistrib + informal: Proves that two finite-measure random variables are identically distributed + when every continuous-linear real projection of them is identically distributed. +- declaration: LeanPool.CramerWold.identDistrib_of_forall_inner_comp_identDistrib + informal: Gives the Hilbert-space random-variable version using all scalar inner-product + projections. +- declaration: LeanPool.CramerWold.hasLaw_of_forall_dual_comp_hasLaw + informal: Proves that a random variable has a target law when every continuous-linear + real projection has the corresponding projected target law. +tags: +- probability +- measure-theory +- characteristic-functions +- cramer-wold +msc: +- 60B11 +- 60E10 +- 28A33 diff --git a/LeanPool/projects/critical-portraits.yaml b/LeanPool/projects/critical-portraits.yaml new file mode 100644 index 0000000000..e3793a2bb1 --- /dev/null +++ b/LeanPool/projects/critical-portraits.yaml @@ -0,0 +1,41 @@ +slug: critical-portraits +title: Counting Critical Portraits +summary: Formalizes the census of degree-d critical portraits, modeled as level-canonical + (d-1)-subsets of the cyclic group ZMod (d*m), and proves that their number equals + C(d*m, d-1)/d for all d, via the Dvoretzky-Motzkin cycle lemma together with an + explicit level-canonicalization bijection (injectivity and surjectivity of the delete-minimum + map). +branch: combinatorics +entry_module: LeanPool.CriticalPortraits +authors: +- Keston Aquino-Michaels +source: + title: Counting critical portraits via the cycle lemma + authors: + - Keston Aquino-Michaels + doi: 10.5281/zenodo.20737896 + github_repo: no-way-labs/lean-critical-portraits +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- CriticalPortraits.card_portraits +- CriticalPortraits.Cycle.cycle_lemma +main_results: +- declaration: CriticalPortraits.card_portraits + informal: The number of degree-d critical portraits at level m equals C(d*m, d-1)/d. +- declaration: CriticalPortraits.Cycle.cycle_lemma + informal: 'The Dvoretzky-Motzkin cycle lemma: a period-n integer sequence with period-sum + 1 has exactly one good cyclic shift.' +- declaration: CriticalPortraits.T_injOn + informal: The delete-minimum map is injective on critical portraits. +- declaration: CriticalPortraits.T_surjOn + informal: The delete-minimum map surjects onto the level-canonical subsets. +tags: +- combinatorics +- cycle-lemma +- critical-portraits +- enumeration +msc: +- 05A15 +- 37F20 diff --git a/LeanPool/projects/cut-and-project.yaml b/LeanPool/projects/cut-and-project.yaml new file mode 100644 index 0000000000..6d9e634a5c --- /dev/null +++ b/LeanPool/projects/cut-and-project.yaml @@ -0,0 +1,50 @@ +slug: cut-and-project +title: Period Lengths of Rational Cut-and-Project Gap Sequences +summary: 'Formalizes the closed-form minimal periods of one-dimensional rational cut-and-project + gap sequences at the level of the paper''s residue model, plus a fully geometric + aperiodicity theorem for irrational slopes. For coprime α, β ∈ ℕ and window parameter + ω ≥ 0, write N = ⌊ωα⌋ + ⌊ωβ⌋ + 1 and D = α² + β². The strip-to-residue reduction + of the paper — sorting the hits of the length-N progression twisted by the geometric + multiplier −αβ⁻¹ (mod D) — is taken as the definition of the gap sequence rather + than derived from the lattice; for it, the multiset convention has minimal period + N/D when D ∣ N and N otherwise, and the set convention (multiplicities flattened + to {0,1}) has minimal period 1 when N ≥ D and N otherwise. The irrational case is + geometric from first principles: for positive irrational slope and ω > 0, the accepted + points of ℤ² sorted by the physical projection have a gap sequence with no finite + period, via injectivity of the projection and Kronecker density of the internal + image.' +branch: number theory +entry_module: LeanPool.CutAndProject +authors: +- Dirk Kunert +source: + title: Period lengths of one-dimensional rational cut-and-project sequences + authors: + - Dirk Kunert + url: https://github.com/dkunert/cut-and-project/blob/main/LaTeX/rational_cut_and_project_gap_periods.tex + github_repo: dkunert/cut-and-project +license: MIT +status: verified +provenance: AI +main_declarations: +- CutAndProject.main_theorem_geometric_concrete +main_results: +- declaration: CutAndProject.main_theorem_geometric_concrete + informal: The gap sequence of the residue model of the rational cut-and-project + multiset (sorted hits of the length-N progression under the multiplier −αβ⁻¹ mod + D) has minimal period N/D if D divides N and N otherwise, where N = ⌊ωα⌋ + ⌊ωβ⌋ + + 1 and D = α² + β². +- declaration: CutAndProject.set_main_theorem_geometric_concrete + informal: The set-convention residue-model gap sequence (multiplicities flattened + to {0,1}) has minimal period 1 if N ≥ D and N otherwise. +- declaration: CutAndProject.Irrational.prop_irrational + informal: For positive irrational slope and window parameter ω > 0, the projected + gap sequence has no finite period. +tags: +- number-theory +- aperiodic-order +- cut-and-project +- quasicrystals +msc: +- 52C23 +- 11K06 diff --git a/LeanPool/projects/dead-ends.yaml b/LeanPool/projects/dead-ends.yaml new file mode 100644 index 0000000000..8218efc740 --- /dev/null +++ b/LeanPool/projects/dead-ends.yaml @@ -0,0 +1,28 @@ +slug: dead-ends +title: Dead Ends in Square-Free Digit Walks +summary: Formalizes the density formula for base-b dead ends in square-free digit + walks. +branch: analytic number theory +entry_module: LeanPool.DeadEnds +authors: +- Evan Chen +- Kenny Lau +- Seewoo Lee +- Ken Ono +- Jujian Zhang +source: + arxiv: '2602.05095' + github_repo: AxiomMath/dead-ends +license: MIT +status: verified +provenance: AI +main_declarations: +- LeanPool.DeadEnds.baseBDeadEnd_density_formula +main_results: +- declaration: LeanPool.DeadEnds.baseBDeadEnd_density_formula + informal: Proves the inclusion-exclusion density formula for base-b dead ends. +tags: +- number-theory +- combinatorics +msc: +- 11N25 diff --git a/LeanPool/projects/demazure-product.yaml b/LeanPool/projects/demazure-product.yaml new file mode 100644 index 0000000000..a470b193fe --- /dev/null +++ b/LeanPool/projects/demazure-product.yaml @@ -0,0 +1,57 @@ +slug: demazure-product +title: Extended Demazure Product on ASP Permutations +summary: Formalizes the extended Demazure product on almost-sign-preserving integer + permutations, including slipface models, residuals, greedy and stingy characterizations, + reduction theorems, adjacent-transposition formulas, and 321-avoiding factorization + results. +branch: algebraic combinatorics +entry_module: LeanPool.DemazureProduct +authors: +- Nathan Pflueger +source: + title: An extended Demazure product on integer permutations via min-plus matrix + multiplication + authors: + - Nathan Pflueger + arxiv: '2206.14227' + github_repo: npflueger/demazure +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LeanPool.DemazureProduct.AspPerm.star +main_results: +- declaration: LeanPool.DemazureProduct.AspPerm.star + informal: Defines the extended Demazure product on almost-sign-preserving permutations. +- declaration: LeanPool.DemazureProduct.AspPerm.star_assoc + informal: Proves associativity of the extended Demazure product on almost-sign-preserving + permutations. +- declaration: LeanPool.DemazureProduct.Reduction.starGreedy + informal: Characterizes the Demazure product as the Bruhat-maximum of ordinary products + of smaller factors. +- declaration: LeanPool.DemazureProduct.Reduction.reduce + informal: Reduces inequalities against a Demazure product to ordinary products of + Bruhat-smaller factors with fixed shift. +- declaration: LeanPool.DemazureProduct.Reduction.lres_stingy + informal: Gives the stingy characterization of the left residual as a Bruhat-minimum + of ordinary products. +- declaration: LeanPool.DemazureProduct.ReducedProducts.star_eq_mul_iff_reducedProduct + informal: Shows that the Demazure product equals ordinary multiplication exactly + for reduced ASP products. +- declaration: LeanPool.DemazureProduct.Transpositions.starSigma + informal: Proves the adjacent-transposition formula for multiplying an ASP permutation + by a no-consecutive-pair involution. +- declaration: LeanPool.DemazureProduct.ASP321a.dprod_eq_iff + informal: Characterizes Demazure factorizations of 321-avoiding ASP permutations + by isolated inversion-set decompositions. +- declaration: LeanPool.DemazureProduct.Tableaux.setValuedTableauEquivHeckeFactorization + informal: Relates fixed-shift set-valued tableaux to Hecke factorizations of 321-avoiding + ASP permutations. +tags: +- algebraic-combinatorics +- demazure-product +- bruhat-order +- permutations +msc: +- 05E05 +- 20F55 diff --git a/LeanPool/projects/demazureoperatorslean.yaml b/LeanPool/projects/demazureoperatorslean.yaml new file mode 100644 index 0000000000..d103459fd5 --- /dev/null +++ b/LeanPool/projects/demazureoperatorslean.yaml @@ -0,0 +1,54 @@ +slug: demazureoperatorslean +title: Demazure Operators and Lean +summary: Formalizes Demazure operators on multivariate polynomials and their nilpotence, + braid, commutation, and symmetric-polynomial linearity relations, with auxiliary + Coxeter-theory support including strong exchange and a conditional Matsumoto theorem. +branch: algebraic combinatorics +entry_module: LeanPool.DemazureOperatorsLean +authors: +- Óscar Álvarez Sánchez +source: + title: Schubert cells, and the cohomology of the spaces G/P + authors: + - I. N. Bernstein + - I. M. Gelfand + - S. I. Gelfand + doi: 10.1070/RM1973v028n03ABEH001557 + github_repo: bolito2/DemazureOperatorsLean +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Demazure.Dem +- CoxeterSystem.strongExchangeProperty +main_results: +- declaration: Demazure.Dem + informal: Defines the Demazure operator as a linear map over the subalgebra of symmetric + polynomials. +- declaration: Demazure.demazure_order_two + informal: Proves that applying the Demazure operator twice gives zero. +- declaration: Demazure.demazure_commutes_adjacent + informal: Proves the braid relation for adjacent Demazure operators. +- declaration: Demazure.demazure_commutes_non_adjacent + informal: Proves that non-adjacent Demazure operators commute. +- declaration: Demazure.demazure_mul_symm + informal: Proves that symmetric polynomials act as scalars with respect to Demazure + operators. +- declaration: CoxeterSystem.strongExchangeProperty + informal: Proves the strong exchange property for Coxeter systems. + source_ref: J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University + Press, 1990. +- declaration: CoxeterSystem.matsumoto_reduced + informal: Auxiliary theorem proving, under the extra MatsumotoCondition hypothesis, + that reduced words for the same Coxeter group element are connected by braid moves. + source_ref: J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University + Press, 1990. +tags: +- algebraic-combinatorics +- demazure-operators +- polynomials +- representation-theory +msc: +- 05E05 +- 13P10 +- 20F55 diff --git a/LeanPool/projects/densityhalesjewett.yaml b/LeanPool/projects/densityhalesjewett.yaml new file mode 100644 index 0000000000..064b3f9adb --- /dev/null +++ b/LeanPool/projects/densityhalesjewett.yaml @@ -0,0 +1,38 @@ +slug: densityhalesjewett +title: The density Hales–Jewett theorem and Szemerédi's theorem +summary: 'A Lean 4 formalization of the density Hales–Jewett theorem: for every finite + alphabet α and every density δ > 0, all sufficiently large n have the property that + any set of at least δ·|α|^n words of length n over α contains a combinatorial line. + The basis for the formalization is the combinatorial proof of Dodos, Kanellopoulos, + and Tyros. As a consequence, we also formalize Szemerédi''s theorem on the integers: + for every k ≥ 3 and δ > 0, the following holds for all sufficiently large n. Any + subset of {0, …, n−1} of size at least δ·n contains a nonconstant arithmetic progression + of length k.' +branch: combinatorics +entry_module: LeanPool.DensityHalesJewett +authors: +- Gabriel Dahia +source: + url: https://github.com/gdahia/densityhalesjewett + github_repo: gdahia/densityhalesjewett + commit: 1c2f5aa1b99caa8e2a87beacf80f07f066c309d3 +license: Apache-2.0 +status: verified +main_declarations: +- Combinatorics.Line.exists_of_density_atTop +main_results: +- declaration: Combinatorics.Line.exists_of_density_atTop + informal: Every positive-density subset of sufficiently long words over a finite + nontrivial alphabet contains a combinatorial line. +- declaration: Combinatorics.ArithmeticProgression.exists_of_density_nat_atTop + informal: Every positive-density subset of a sufficiently long initial interval + of the natural numbers contains an arithmetic progression of any prescribed length + at least three with nonzero common difference. +tags: +- combinatorics +msc: +- 05D10 +- 05A05 +- 11B75 +- 68R15 +provenance: AI diff --git a/LeanPool/projects/desargues.yaml b/LeanPool/projects/desargues.yaml new file mode 100644 index 0000000000..419c535da1 --- /dev/null +++ b/LeanPool/projects/desargues.yaml @@ -0,0 +1,41 @@ +slug: desargues +title: Axiomatic projective geometry (Faure–Frölicher) +summary: Develops, following Faure and Frölicher's Modern Projective Geometry, the + axiomatic theory of projective geometries given by a ternary collinearity relation, + with subspaces and subgeometries, proves that the central projection between two + lines is a bijection, and exhibits every Mathlib projectivization as a projective + geometry. Groundwork from the upstream desargues repository — Desargues' theorem + itself is not formalized. +branch: geometry +entry_module: LeanPool.Desargues +authors: +- Abdullah Uyu +source: + title: Modern Projective Geometry + authors: + - Claude-Alain Faure + - Alfred Frölicher + doi: 10.1007/978-94-015-9590-2 + github_repo: oneofvalts/desargues +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Basic.ProjectiveGeometry +- Basic.centralProjection +- Basic.cen_proj_bij +main_results: +- declaration: Basic.ProjectiveGeometry + informal: Axiomatizes a projective geometry as a typeclass (axioms L₁–L₃, Faure–Frölicher + p. 26) on a ternary collinearity relation. +- declaration: Basic.centralProjection + informal: Defines the central projection of a line onto another from a center point. +- declaration: Basic.cen_proj_bij + informal: The headline theorem — the central projection between two lines is a bijection. +tags: +- projective-geometry +- incidence-geometry +- geometry +msc: +- 51A05 +- 51A30 diff --git a/LeanPool/projects/dilatations.yaml b/LeanPool/projects/dilatations.yaml new file mode 100644 index 0000000000..34c76f74c8 --- /dev/null +++ b/LeanPool/projects/dilatations.yaml @@ -0,0 +1,100 @@ +slug: dilatations +title: Dilatations of categories and commutative rings +summary: Constructs categorical dilatations from sieves of permitted numerators and + denominator morphisms, and proves their universal property for Sigma-regular functors + satisfying the sieve containment condition. Develops restriction and union of centers, + dual codilatations, and ordinary localization as a special case. The composition + isomorphism retains the explicit Sigma-regularity hypothesis left open in the original + paper. For commutative rings, an exponent-profile center recovers the ring dilatation; + the naive single-index comparison is refuted over Z[X]. A finite category example + refutes a general intermediate-category characterization. Independent ring modules + provide fraction arithmetic, a universal property, reindexing, and the unique isomorphism + for two-stage dilatations. +branch: category theory and commutative algebra +entry_module: LeanPool.Dilatations +authors: +- Arnaud Mayeux +- Jujian Zhang +source: + title: Dilatations of categories, via their Lean formalization + authors: + - Arnaud Mayeux + arxiv: '2608.09305' + url: https://github.com/rndmx/DilCat + github_repo: rndmx/DilCat + commit: 604559654c948566675da3f7709b8ad3126bd487 + announced: '2026-08-10T08:52:54Z' + prior_formalization: + url: https://github.com/ProjConstruction/Proj + commit: 6bfe4ef61d50c20cb3b91f4ef58649c9490b594f + authors: + - Arnaud Mayeux + - Jujian Zhang + arxiv: '2606.01438' + license: Apache-2.0 + note: The upstream single-file development incorporates the ring construction + from Project/Dilatation/lemma.lean, Family.lean, and Multicenter.lean, introduced + by Jujian Zhang on 2025-03-03; both prior authors are credited here. +license: Apache-2.0 +status: verified +provenance: mix +provenance_note: Mayeux's paper discloses Claude assistance under the author's direction; + it does not quantify the share of generated proofs. The development incorporates + Mayeux and Zhang's earlier ring formalization. This port includes substantial AI + proof repair, optimization, and documentation. +main_declarations: +- CategoryTheory.Dilatations.Dila_universal_property +main_results: +- declaration: CategoryTheory.Dilatations.Dila_universal_property + informal: For a nonempty indexed center in a category C and a functor F from C to + D, assume that localization of D at the images of the chosen denominators is faithful + and that each pushed-forward numerator sieve lies in the sieve generated by its + mapped denominator. There is a unique functor from the dilatation to D whose composite + with the canonical functor from C is F. + source_ref: Section 5; original Theorem 3.10. +- declaration: CategoryTheory.Dilatations.Iso315 + informal: The successive dilatation at Z and the pushforward of W is isomorphic + to the dilatation at the combined center Z.sum W, assuming that the composite + canonical functor to the successive dilatation is Sigma-regular for Z.sum W. This + regularity is an explicit hypothesis, not a theorem of the development. + source_ref: Section 8.2 and Appendix B; conditional original Proposition 3.15. +- declaration: CategoryTheory.Dilatations.Iso318 + informal: For two families of numerator sieves with the same denominator morphisms, + dilating at their combined two-copy center is isomorphic to dilating at the pointwise + union of the sieves. + source_ref: Section 8.3; original Proposition 3.18. +- declaration: CategoryTheory.Dilatations.Prop51.Iso51 + informal: For any multicenter in a commutative ring A, the categorical dilatation + of the multiplicative one-object category of A at all finitely supported exponent + profiles is isomorphic to the one-object category of the ring dilatation A[M]. + At profile nu, the denominator is the product of a_i raised to nu_i and the numerator + ideal is the product of (M_i + (a_i)) raised to nu_i. + source_ref: Theorem 10.1; corrected original Proposition 5.1. +- declaration: CategoryTheory.Dilatations.NaiveCenterCounterexample.no_C_compatible_equiv + informal: For A = Z[X], ideal (X), and denominator 2, there is no equivalence from + the categorical dilatation at the naive one-index center to the one-object category + of the ring dilatation that commutes strictly with the canonical functors from + the base category. + source_ref: Section 10.2 and Appendix B. +- declaration: CategoryTheory.Dilatations.Fact52.no_realizing_center + informal: The constructed category with three arrows from X to Y admits no equivalence + to a dilatation of the base category with two such arrows that commutes strictly + with the canonical base functors. This is an explicit counterexample, not a classification + of intermediate categories up to arbitrary equivalence. + source_ref: Theorem 11.1. +- declaration: CategoryTheory.Dilatations.iteratedDilatationEquiv_existsUnique + informal: For a multicenter M in any commutative ring A and any subset K of its + indices, first dilate along K and then along the complementary ideals and denominators + mapped to the first-stage ring. There is exactly one isomorphism of algebras over + the first-stage ring from this two-stage dilatation to A[M], compatible with its + canonical map. + source_ref: Appendix C, Proposition C.1 and Lemma C.2. +tags: +- categories +- localization +- dilatations +- commutative-algebra +- universal-properties +msc: +- 18A30 +- 13B30 diff --git a/LeanPool/projects/directed-topology-lean-4.yaml b/LeanPool/projects/directed-topology-lean-4.yaml new file mode 100644 index 0000000000..2312e5f7da --- /dev/null +++ b/LeanPool/projects/directed-topology-lean-4.yaml @@ -0,0 +1,42 @@ +slug: directed-topology-lean-4 +title: Directed Topology in Lean 4 +summary: Foundational library for directed topological spaces, dipaths, dihomotopy, + the fundamental category functor, and a directed Van Kampen pushout theorem. +branch: algebraic topology +entry_module: LeanPool.DirectedTopologyLean4 +authors: +- Dominique Lawson +- Henning Basold +- Peter Bruin +source: + arxiv: '2312.06506' + doi: 10.4230/LIPIcs.ITP.2024.8 + github_repo: Dominique-Lawson/Directed-Topology-Lean-4 +license: MIT +status: verified +provenance: human +main_declarations: +- DirectedSpace +- Dipath +- DirectedVanKampen.directed_van_kampen +main_results: +- declaration: DirectedSpace + informal: Definition of a directed topological space as a topological space with + a class of directed paths closed under constant paths, concatenation, and monotone + reparametrization. +- declaration: Dipath + informal: A directed path in a directed space. +- declaration: Dipath.Dihomotopic + informal: Dihomotopy equivalence relation on dipaths between fixed endpoints. +- declaration: DirectedVanKampen.directed_van_kampen + informal: Proves the directed Van Kampen theorem, showing that the directed fundamental + category functor sends the standard open cover square to a pushout in the category + of categories. +tags: +- directed-topology +- algebraic-topology +- category-theory +msc: +- 55U40 +- 55Q05 +- 18A30 diff --git a/LeanPool/projects/distance-geometry.yaml b/LeanPool/projects/distance-geometry.yaml new file mode 100644 index 0000000000..287db5f23d --- /dev/null +++ b/LeanPool/projects/distance-geometry.yaml @@ -0,0 +1,48 @@ +slug: distance-geometry +title: Euclidean Distance Geometry +summary: Develops the basepoint-centered Gram approach to finite Euclidean distance + geometry. Proves both directions of Schoenberg's characterization, including a rank-controlled + factorization of positive-semidefinite matrices; proves that three affinely independent + centers in Euclidean three-space determine at most two points with prescribed distances; + and verifies the Cayley--Menger determinant for segments and triangles, including + the triangle-area identity underlying Heron's formula. +branch: geometry +entry_module: LeanPool.DistanceGeometry +authors: +- Egor Lyfar +source: + title: Remarks to Maurice Fréchet's article Sur la définition axiomatique d'une + classe d'espace distanciés vectoriellement applicable sur l'espace de Hilbert + authors: + - Isaac Jacob Schoenberg + doi: 10.2307/1968654 + github_repo: lyfar/distance-geometry-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- DistanceGeometry.schoenberg +- DistanceGeometry.trilateration_le_two +main_results: +- declaration: DistanceGeometry.schoenberg + informal: For n at least one, a symmetric hollow n-by-n matrix is a squared-distance + matrix of points in k-dimensional Euclidean space exactly when its basepoint-centered + Gram matrix is positive semidefinite and has rank at most k. +- declaration: DistanceGeometry.encard_setOf_forall_dist_eq_le_two + informal: In a finite-dimensional real inner-product affine space, prescribed distances + to centers whose affine span has codimension one admit at most two solutions. +- declaration: DistanceGeometry.trilateration_le_two + informal: Three affinely independent centers in three-dimensional Euclidean space + admit at most two points with any prescribed triple of distances. +- declaration: DistanceGeometry.cayleyMenger_det_heron + informal: The Cayley--Menger determinant of three planar points equals minus sixteen + times the squared area of their triangle. +tags: +- distance-geometry +- euclidean-geometry +- linear-algebra +- cayley-menger +msc: +- 51K05 +- 52C99 +- 15A18 diff --git a/LeanPool/projects/domain-theory.yaml b/LeanPool/projects/domain-theory.yaml new file mode 100644 index 0000000000..18c32908e9 --- /dev/null +++ b/LeanPool/projects/domain-theory.yaml @@ -0,0 +1,47 @@ +slug: domain-theory +title: DomainTheory +summary: Formalizes Dana Scott's domain theory through Scott information systems and + PRG-19 neighbourhood systems, including approximable maps, products, function spaces, + fixed points, recursive presentations, and initial-algebra constructions for denotational + semantics. +branch: domain theory +entry_module: LeanPool.DomainTheory +authors: +- Catskills Research Company +source: + title: Domains for Denotational Semantics + authors: + - Dana Scott + doi: 10.1007/BFb0012801 + url: https://doi.org/10.1007/BFb0012801 + github_repo: catskillsresearch/domain_theory +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- InfoSys +- Domain.Neighborhood.NeighborhoodSystem +main_results: +- declaration: InfoSys + informal: Defines Scott information systems, with a distinguished bottom token, + finite consistency predicate, entailment relation, and Scott's six axioms. +- declaration: Domain.Neighborhood.NeighborhoodSystem + informal: Defines PRG-19 neighbourhood systems, Scott's neighbourhood presentation + of domains. +- declaration: Domain.Neighborhood.ApproximableMap + informal: Defines approximable maps between neighbourhood systems as relations preserving + finite approximations. +- declaration: Domain.Neighborhood.trianglelefteq_of_isInitial + informal: Proves that an initial algebra for a functor embeds into every other solution + through Scott's subdomain relation. +- declaration: ScottMap.fix_eq + informal: Proves the fixed-point equation for Scott-continuous self maps on complete + lattices. +tags: +- domain-theory +- denotational-semantics +- information-systems +msc: +- 03B70 +- 06B35 +- 68Q55 diff --git a/LeanPool/projects/dross-fractional-triangle-decomposition.yaml b/LeanPool/projects/dross-fractional-triangle-decomposition.yaml new file mode 100644 index 0000000000..19040c11c9 --- /dev/null +++ b/LeanPool/projects/dross-fractional-triangle-decomposition.yaml @@ -0,0 +1,37 @@ +slug: dross-fractional-triangle-decomposition +title: Dross's fractional triangle decomposition theorem +summary: Formalizes the exact nine-tenths minimum-degree threshold for fractional + triangle decomposition of finite graphs. The proof uses a finite max-flow/min-cut + argument and induction that peels high-degree triangles. +branch: extremal graph theory and fractional decompositions +entry_module: LeanPool.DrossFractionalTriangleDecomposition +authors: +- Juan Pablo Traverso Gianini +- Aristotle +source: + title: Fractional Triangle Decompositions in Graphs with Large Minimum Degree + authors: + - François Dross + doi: 10.1137/15M1014310 + url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/tree/dec9f4750c2261681e376f2f67db9eda16a4eca0/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Ax2 + github_repo: jtraverso/erdos-81-chordal-clique-partitions + commit: dec9f4750c2261681e376f2f67db9eda16a4eca0 + license_grant: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/dec9f4750c2261681e376f2f67db9eda16a4eca0/LEAN_SOURCE_APACHE_2_0_GRANT.md +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LeanPool.DrossFractionalTriangleDecomposition.dross_fractional_flow_exact +main_results: +- declaration: LeanPool.DrossFractionalTriangleDecomposition.dross_fractional_flow_exact + informal: 'Every finite graph whose minimum degree is at least nine tenths of its + number of vertices has a fractional triangle decomposition: nonnegative triangle + weights sum to one at each edge.' + source_ref: Dross (2016), fractional triangle decomposition theorem. +tags: +- extremal-graph-theory +- fractional-decomposition +- max-flow-min-cut +msc: +- 05C70 +- 05C35 diff --git a/LeanPool/projects/duality.yaml b/LeanPool/projects/duality.yaml new file mode 100644 index 0000000000..2072fa5daa --- /dev/null +++ b/LeanPool/projects/duality.yaml @@ -0,0 +1,37 @@ +slug: duality +title: Duality theory in linear optimization and its extensions +summary: Formalizes Farkas-like theorems and strong LP duality, including extensions + to a linearly ordered field with bot and top. +branch: linear optimization +entry_module: LeanPool.Duality +authors: +- Martin Dvorak +source: + arxiv: '2409.08119' + doi: 10.46298/afm.14253 + github_repo: madvorak/duality +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- extendedFarkas +- StandardLP.strongDuality +main_results: +- declaration: equalityFarkas + informal: Farkas' lemma for linear equalities over nonnegative variables. +- declaration: inequalityFarkas + informal: Farkas' lemma for linear inequalities over nonnegative variables. +- declaration: extendedFarkas + informal: Extended Farkas theorem for matrices and vectors over a linearly ordered + field with bot and top. +- declaration: ValidELP.strongDuality + informal: Strong duality theorem for extended linear programs. +- declaration: StandardLP.strongDuality + informal: Strong duality theorem for linear programs in standard form. +tags: +- linear-programming +- optimization +- farkas-lemma +msc: +- 90C05 +- 90C46 diff --git a/LeanPool/projects/ec-tate-lean.yaml b/LeanPool/projects/ec-tate-lean.yaml new file mode 100644 index 0000000000..5c52f0669e --- /dev/null +++ b/LeanPool/projects/ec-tate-lean.yaml @@ -0,0 +1,60 @@ +slug: ec-tate-lean +title: Weierstrass models and singular points for Tate's algorithm +summary: Formalizes Weierstrass models of elliptic curves over a commutative ring, + their b- and c-invariants and discriminant, the change-of-coordinates group action, + the discriminant's membership in the Jacobian ideal, and the construction of the + singular point of a singular model over a perfect field, together with the Kronecker + symbol and the Kodaira type enumeration. +branch: number theory +entry_module: LeanPool.EcTateLean +authors: +- Sacha H. +- Sander R. Dahmen +- Anne Baanen +- Alex J. Best +source: + title: Elliptic Curve Handbook + authors: + - Ian Connell + url: https://www.math.rug.nl/~top/ian.pdf + github_repo: KisaraBlue/ec-tate-lean +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Model +- Model.Field.isSingularPoint_singularPoint +main_results: +- declaration: Model + informal: Defines a Weierstrass model of a possibly singular elliptic curve by its + five a-invariants. +- declaration: Model.discr_identity + informal: Proves the identity relating the discriminant to the c-invariants, 1728 + * discr = c4^3 - c6^2. +- declaration: Model.rst_discr + informal: Proves that the discriminant is invariant under the rst change-of-coordinates + transformation. +- declaration: Model.discr_in_jacobian_ideal + informal: Expresses the discriminant as an explicit combination of the Weierstrass + polynomial and its partial derivatives, witnessing membership in the Jacobian + ideal. +- declaration: Model.discr_eq_zero_of_singular + informal: Proves that a model with a singular point has vanishing discriminant. +- declaration: Model.Field.isSingularPoint_singularPoint + informal: Proves that the explicitly constructed singular point of a singular model + over a perfect field is indeed a singular point (Connell, Proposition 1.5.4). + source_ref: Ian Connell, Elliptic Curve Handbook, February 1999, Proposition 1.5.4. +- declaration: Model.Field.move_singular_point_to_origin + informal: Proves that the change of coordinates moving the singular point places + it at the origin. +- declaration: Kodaira + informal: Enumerates the Kodaira reduction types, including the extra types arising + over imperfect residue fields of characteristic 2 or 3. +tags: +- number-theory +- elliptic-curves +- algebraic-geometry +msc: +- 11G05 +- 11G07 +- 14H52 diff --git a/LeanPool/projects/egrs75.yaml b/LeanPool/projects/egrs75.yaml new file mode 100644 index 0000000000..7444be4051 --- /dev/null +++ b/LeanPool/projects/egrs75.yaml @@ -0,0 +1,47 @@ +slug: egrs75 +title: The Erdős–Graham–Ruzsa–Straus two-prime theorem +summary: A complete machine-checked proof that for any two distinct odd primes p and + q there are infinitely many n with binom(2n,n) coprime to pq, including the iterated + carry-controlled digit-clearing step that Bloom and Croot defer to the 1975 paper, + via a per-move lexicographic measure, an elementary staircase lemma, and a Dirichlet-approximation + seed. +branch: number theory +entry_module: LeanPool.Egrs75 +authors: +- Egor Lyfar +source: + title: On the prime factors of (2n choose n) + authors: + - Paul Erdős + - Ronald L. Graham + - Imre Z. Ruzsa + - Ernst G. Straus + doi: 10.1090/S0025-5718-1975-0369288-3 + github_repo: lyfar/egrs75-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Egrs75.MuFinish.egrs_two_prime_mu +- Egrs75.Finish.egrs_two_prime_finish +main_results: +- declaration: Egrs75.MuFinish.egrs_two_prime_mu + informal: Proves that for distinct odd primes p and q there are infinitely many + n with p and q both not dividing the central binomial coefficient binom(2n,n), + via the mu-measure descent. +- declaration: Egrs75.Finish.egrs_two_prime_finish + informal: Reassembles the same theorem via bad-bit-mask strong induction, with the + low-case clearing discharged by Egrs75.MuFinish.egrs_clearing_low_mu. +- declaration: Egrs75.not_dvd_centralBinom_iff_lowDigits + informal: The Kummer bridge, stating that an odd prime p does not divide binom(2n,n) + exactly when every base-p digit of n is at most (p-1)/2. +- declaration: Egrs75.MathlibAPI.irrational_log_div_log + informal: Proves that the ratio of logarithms of two distinct primes is irrational. +tags: +- number-theory +- central-binomial-coefficients +- digit-representations +- erdos-problems +msc: +- 11A63 +- 11B65 diff --git a/LeanPool/projects/egz-formal-proof.yaml b/LeanPool/projects/egz-formal-proof.yaml new file mode 100644 index 0000000000..a8a012c77c --- /dev/null +++ b/LeanPool/projects/egz-formal-proof.yaml @@ -0,0 +1,36 @@ +slug: egz-formal-proof +title: Convex geometry and the Erdős–Ginzburg–Ziv problem +summary: For every fixed positive dimension d, the Erdős–Ginzburg–Ziv constant of + (ZMod p)^d equals p times its hollow-family constant plus o(p) as p tends to infinity + through primes. Includes the convex-flag Helly and centerpoint theorems, flag decomposition, + balanced combinations, relative expansion, and the polynomial hollow-family bound. +branch: additive combinatorics +entry_module: LeanPool.ErdosGinzburgZiv +authors: +- Dmitrii Zakharov +source: + doi: 10.19086/da.165216 + arxiv: '2002.09892' + url: https://github.com/zakharov2k/egz-formal-proof + github_repo: zakharov2k/egz-formal-proof + commit: 7c82267c43a51dd5cb3e440412780ff82aff7500 +license: MIT +status: verified +provenance: AI +main_declarations: +- EGZ.theorem_1_2 +- EGZ.main_upper_bound +main_results: +- declaration: EGZ.theorem_1_2 + informal: For each positive dimension d, s((F_p)^d) = p w((F_p)^d) + o(p) as p tends + to infinity through primes. +- declaration: EGZ.main_upper_bound + informal: For each positive d and ε > 0, sufficiently large primes satisfy s((F_p)^d) + ≤ (w((F_p)^d) + ε) p. +tags: +- additive-combinatorics +- zero-sum +- convex-geometry +msc: +- 11B30 +- 11H06 diff --git a/LeanPool/projects/ehrhart-volume-inequality.yaml b/LeanPool/projects/ehrhart-volume-inequality.yaml new file mode 100644 index 0000000000..d35729cfbd --- /dev/null +++ b/LeanPool/projects/ehrhart-volume-inequality.yaml @@ -0,0 +1,45 @@ +slug: ehrhart-volume-inequality +title: Ehrhart's sharp volume inequality +summary: Proves Ehrhart's sharp upper bound for the volume of a centered full-dimensional + convex body whose origin is its unique interior lattice point. The development builds + the weighted complex-analytic and variational machinery needed for the bound and + verifies sharpness using the centered simplex, including its barycenter and exact + volume. +branch: convex geometry and geometry of numbers +entry_module: LeanPool.EhrhartVolumeInequality +authors: +- OpenAI +- Dean Cureton +source: + title: Ten Advances in Mathematics and Theoretical Computer Science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- Ehrhart.Volume.ehrhart_volume_inequality_for_sets +main_results: +- declaration: Ehrhart.Volume.ehrhart_volume_inequality_for_sets + informal: In every positive dimension, a compact full-dimensional convex set with + barycenter zero and origin as its unique interior lattice point has volume at + most (n + 1)^n / n!. +- declaration: Ehrhart.SimplexVolume.normalizedVolume_centeredSimplex + informal: The centered dilation of the standard n-simplex has normalized volume + exactly (n + 1)^n / n!, so the volume bound is sharp. +- declaration: Ehrhart.SimplexVolume.barycenter_centeredSimplex + informal: The centered dilation of the standard simplex has barycenter zero. +- declaration: Ehrhart.SimplexVolume.exists_centeredBody_sharp + informal: In every positive dimension there exists a centered convex body with a + unique interior lattice point that attains the sharp volume bound. +tags: +- convex-geometry +- geometry-of-numbers +- lattice-points +- ehrhart-theory +msc: +- 52A40 +- 11H06 diff --git a/LeanPool/projects/eilenberg-zilber.yaml b/LeanPool/projects/eilenberg-zilber.yaml new file mode 100644 index 0000000000..59aaa14620 --- /dev/null +++ b/LeanPool/projects/eilenberg-zilber.yaml @@ -0,0 +1,53 @@ +slug: eilenberg-zilber +title: Eilenberg–Zilber theorem for bisimplicial objects +summary: For a bisimplicial object X in an abelian category, constructs a chain homotopy + equivalence between the total complex of the alternating face map double complex + of X and the alternating face map complex of its diagonal. The equivalence is built + from the shuffle map and the Alexander–Whitney map. On normalized complexes, shuffle + followed by Alexander–Whitney is the identity, and the other composite is homotopic + to the identity via the Eilenberg–Mac Lane recursive homotopy. The result is then + transported to the unnormalized complexes along the Moore normalization comparisons. + The development includes the combinatorics of (p, q)-shuffles, their signs, and + the cancellation of diagonal boundary terms. +branch: algebraic topology +entry_module: LeanPool.EilenbergZilber +authors: +- Jeffrey Li +source: + url: https://github.com/jeffrey-dot-li/lean-homology/tree/40c5aae235b0f499ebfc1e68ce574a62b531203c/HomologyLean/SingularHomology/EilenbergZilberTheorem + github_repo: jeffrey-dot-li/lean-homology + commit: 40c5aae235b0f499ebfc1e68ce574a62b531203c +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LeanPool.EilenbergZilber.BisimplicialObject.eilenbergZilber +main_results: +- declaration: LeanPool.EilenbergZilber.BisimplicialObject.eilenbergZilber + informal: For every bisimplicial object X in an abelian category, the total complex + of the double complex obtained by taking alternating face map complexes in both + simplicial directions is chain homotopy equivalent to the alternating face map + complex of the diagonal simplicial object of X. + source_ref: 'Generalized Eilenberg–Zilber theorem for bisimplicial objects in an + abelian category: Dold–Puppe, "Homologie nicht-additiver Funktoren. Anwendungen", + Ann. Inst. Fourier 11 (1961), 201–312, Satz 2.9 (Eilenberg–Zilber–Cartier); the + homology-level statement is Weibel, An Introduction to Homological Algebra (1994), + Theorem 8.5.1. The product case X_{p,q} = K_p ⊗ L_q is Eilenberg–Zilber, "On products + of complexes", Amer. J. Math. 75 (1953), 200–204.' +- declaration: LeanPool.EilenbergZilber.BisimplicialObject.eilenbergZilberNormalized + informal: The total complex of the bi-normalized Moore double complex of X is chain + homotopy equivalent to the normalized Moore complex of the diagonal, via the shuffle + and Alexander–Whitney maps; shuffle followed by Alexander–Whitney is the identity. + source_ref: Eilenberg–Mac Lane, "On the groups H(Π, n), II", Ann. of Math. 60 (1954), + 49–139, Theorem 2.1a, stated there for products of complexes and generalized here + to an arbitrary bisimplicial object in an abelian category. +tags: +- eilenberg-zilber +- bisimplicial-objects +- shuffle-map +- alexander-whitney +- chain-homotopy +msc: +- 55U10 +- 18G30 +- 18G35 diff --git a/LeanPool/projects/elliptic-pde.yaml b/LeanPool/projects/elliptic-pde.yaml new file mode 100644 index 0000000000..da139dc3ec --- /dev/null +++ b/LeanPool/projects/elliptic-pde.yaml @@ -0,0 +1,61 @@ +slug: elliptic-pde +title: 'Linear elliptic PDE: solvability, regularity and spectral theory' +summary: 'A reusable analytic development of linear uniformly elliptic divergence-form + PDE on Euclidean domains: Hilbert graph models of H¹ and H₀¹, bounded coefficient + operators, Poincaré inequalities, zero-drift/nonnegative-potential Dirichlet existence + and uniqueness, Gårding and Fredholm theory for general drift, interior H² and higher + Sobolev regularity, smooth classical representatives, Sobolev and Hölder embeddings, + C¹-domain extension and compactness, variational eigenvalues, and maximum principles. + AI-generated proofs with human mathematical design and transcription; announced + in arXiv:2609.32561 on 2026-09-26. The unfinished boundary-H² program is outside + this completed dependency closure.' +branch: analysis and partial differential equations +entry_module: LeanPool.EllipticPDE +authors: +- Alejandro José Soto Franco +- Kobe Marshall-Stevens +source: + url: https://github.com/alejandro-soto-franco/EllipticPDE + github_repo: alejandro-soto-franco/EllipticPDE + commit: eaf821d31b200bb6ea235f19eecc50cf0f38c294 + arxiv: '2609.32561' +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- EllipticPdes.Regularity.higher_interior_regularity +main_results: +- declaration: EllipticPdes.Sobolev.FullEllipticOp.weak_solution_L2_of_nonneg_zeroth_of_bounded + informal: On a bounded domain in positive dimension, a bounded measurable uniformly + elliptic operator with zero drift and nonnegative zeroth-order coefficient has + a unique H₀¹ weak solution for every L² datum, with a datum-independent norm bound. +- declaration: EllipticPdes.Regularity.interior_H2_estimate + informal: In positive dimension, on an open measurable domain, an H₀¹ weak solution + of a bounded measurable uniformly elliptic divergence-form operator with globally + Lipschitz principal coefficients and an L² datum has interior L² second weak derivatives, + with an estimate whose constant is chosen before the solution and datum. +- declaration: EllipticPdes.Regularity.higher_interior_regularity + informal: In positive dimension, for a uniformly elliptic divergence-form operator + with bounded measurable coefficients, globally Lipschitz principal-coefficient + representatives in W^{k+1,∞}(ℝ^d), and lower-order coefficients in W^{k,∞}(ℝ^d), + every H₀¹ weak solution on an open measurable domain with an H^k datum has interior + H^{k+2} regularity. On each compact subset the iterated L² derivative bounds have + a constant chosen before the solution, datum and its H^k bound. The pointwise + Lipschitz hypothesis is separate from the weak-derivative bundles. +- declaration: EllipticPdes.Sobolev.embL2_isCompact + informal: The inclusion H₀¹(Ω) → L²(Ω) is compact for bounded measurable Euclidean + domains, discharging the compactness input to the Fredholm and spectral results. +- declaration: EllipticPdes.Extension.exists_extW12 + informal: In positive dimension, bounded open Euclidean domains with C¹ boundary + admit a bounded linear H¹ extension operator with compact support control. +tags: +- elliptic-pde +- sobolev-spaces +- regularity +- fredholm-theory +- spectral-theory +msc: +- 35J15 +- 35J25 +- 35B65 +- 46E35 diff --git a/LeanPool/projects/erdos-97-convex-octagon.yaml b/LeanPool/projects/erdos-97-convex-octagon.yaml new file mode 100644 index 0000000000..beccef0afd --- /dev/null +++ b/LeanPool/projects/erdos-97-convex-octagon.yaml @@ -0,0 +1,39 @@ +slug: erdos-97-convex-octagon +title: The Convex-Octagon Case of Erdős Problem 97 +summary: Proves that among eight labelled points in the Euclidean plane in convex + position, at least one point has no four other points at a common distance. The + development reduces a geometric counterexample to a finite incidence model, classifies + normalized row patterns up to symmetry, proves geometric and algebraic obstructions, + and excludes seven canonical first-row masks with a kernel-audited exhaustive search + over the remaining legal row masks and checked geometric obstruction witnesses. + This is the modern four-equidistant formulation; Erdős's 1946 paper posed the stronger + three-equidistant version. This is the convex-octagon case only; the general Erdős + Problem 97 remains open. +branch: discrete geometry +entry_module: LeanPool.Erdos97ConvexOctagon +authors: +- Egor Lyfar +source: + title: On Sets of Distances of n Points + authors: + - Paul Erdős + doi: 10.1080/00029890.1946.11991674 + url: https://www.erdosproblems.com/97 + github_repo: lyfar/erdos-97-octagon-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Erdos97Octagon.erdos97_convex_octagon +main_results: +- declaration: Erdos97Octagon.erdos97_convex_octagon + informal: Among eight labelled points in the Euclidean plane in convex position, + some point has no four other points at a common distance. +tags: +- discrete-geometry +- distance-geometry +- erdos-problems +- convexity +msc: +- 51K05 +- 52A10 diff --git a/LeanPool/projects/erdos-moser-distinct-subset-sums.yaml b/LeanPool/projects/erdos-moser-distinct-subset-sums.yaml new file mode 100644 index 0000000000..1a6f68b9c3 --- /dev/null +++ b/LeanPool/projects/erdos-moser-distinct-subset-sums.yaml @@ -0,0 +1,45 @@ +slug: erdos-moser-distinct-subset-sums +title: Leo Moser's finite distinct-subset-sums inequality +summary: 'Formalizes the exact finite sum-of-squares inequality that Guy''s 1982 account + states as Theorem 2 and attributes to Leo Moser: a finite set A of natural numbers + with distinct subset sums satisfies 4^|A| - 1 <= 3 * sum(a^2). It then derives direct + and square-root lower bounds for the largest element of A. This does not claim the + stronger historical asymptotic bound attributed jointly to Erdős and Moser, and + Erdős Problem 1 remains open.' +branch: additive combinatorics +entry_module: LeanPool.ErdosMoser +authors: +- Egor Lyfar +source: + title: Sets of Integers Whose Subsets Have Distinct Sums + authors: + - Richard K. Guy + doi: 10.1016/S0304-0208(08)73500-X + url: https://www.erdosproblems.com/1 + github_repo: lyfar/erdos-moser-distinct-subset-sums-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.ErdosMoser.leoMoserVarianceBound +main_results: +- declaration: LeanPool.ErdosMoser.leoMoserVarianceBound + informal: If a finite set A of natural numbers has distinct subset sums, then 4^|A| + - 1 is at most three times the sum of the squares of the elements of A. + source_ref: Richard K. Guy, "Sets of Integers Whose Subsets Have Distinct Sums," + Theorem 2 (credited there to Leo Moser), Annals of Discrete Mathematics 12 (1982), + 141-154, doi:10.1016/S0304-0208(08)73500-X. +- declaration: LeanPool.ErdosMoser.erdosMoserMaxBound + informal: For a nonempty finite set A of natural numbers with distinct subset sums, + 4^|A| - 1 is at most 3|A| times the square of its largest element. +- declaration: LeanPool.ErdosMoser.erdosMoserMaxSqrt + informal: 'The equivalent square-root lower bound on the largest element: sqrt((4^|A| + - 1)/(3|A|)) is at most max(A).' +tags: +- additive-combinatorics +- number-theory +- distinct-subset-sums +- erdos-problems +msc: +- 11B13 +- 11B75 diff --git a/LeanPool/projects/erdos-tuza-valtr.yaml b/LeanPool/projects/erdos-tuza-valtr.yaml new file mode 100644 index 0000000000..12397b82e3 --- /dev/null +++ b/LeanPool/projects/erdos-tuza-valtr.yaml @@ -0,0 +1,35 @@ +slug: erdos-tuza-valtr +title: The Erdős–Tuza–Valtr conjecture +summary: Formalizes the Erdős–Tuza–Valtr conjecture, a sharp refinement of the Erdős–Szekeres + happy-ending theorem, proving that any general-position point set of size at least + C(n+2,2)+2 contains the target cup–cap configuration. +branch: combinatorics +entry_module: LeanPool.ErdosTuzaValtr +authors: +- Jineon Baek +source: + title: On the Erdős–Tuza–Valtr Conjecture + arxiv: '2206.04260' + doi: 10.1016/j.ejc.2024.104085 + github_repo: jcpaik/erdos-tuza-valtr +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- ErdosTuzaValtr.main +- Config.main_lemma +main_results: +- declaration: ErdosTuzaValtr.main + informal: Proves the sharp upper bound — a general-position point set with at least + C(n+2,2)+2 points contains the Erdős–Tuza–Valtr cup–cap configuration. +- declaration: Config.main_lemma + informal: The configuration-relative induction lemma that, with the mirror-symmetry + reduction, drives the main proof. +tags: +- combinatorics +- discrete-geometry +- convex-geometry +- ramsey-theory +msc: +- 52C10 +- 05D10 diff --git a/LeanPool/projects/erdos1196.yaml b/LeanPool/projects/erdos1196.yaml new file mode 100644 index 0000000000..814c713ce6 --- /dev/null +++ b/LeanPool/projects/erdos1196.yaml @@ -0,0 +1,29 @@ +slug: erdos1196 +title: Primitive Sets Above x (Erdos Problem 1196) +summary: Formalizes Math Inc.'s proof around primitive sets above x for Erdos Problem + 1196. +branch: analytic number theory +entry_module: LeanPool.Erdos1196 +authors: +- Math Inc. +source: + arxiv: '2605.00301' + github_repo: math-inc/Erdos1196 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- PrimitiveSetsAboveX.mainTheorem +- Erdos1196.erdos_1196 +main_results: +- declaration: PrimitiveSetsAboveX.mainTheorem + informal: Proves the main primitive-sets-above-x theorem. +- declaration: Erdos1196.erdos_1196 + informal: States the formalized version of Erdos Problem 1196. +tags: +- number-theory +- combinatorics +- analysis +msc: +- 11N25 +- 05D05 diff --git a/LeanPool/projects/erdos132-convex-k3.yaml b/LeanPool/projects/erdos132-convex-k3.yaml new file mode 100644 index 0000000000..dd9061bee9 --- /dev/null +++ b/LeanPool/projects/erdos132-convex-k3.yaml @@ -0,0 +1,113 @@ +slug: erdos132-convex-k3 +title: Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures +summary: Proves that every finite cyclically strictly convex planar configuration + has a vertex incident to at most six edges from its three largest distance classes. + The proof selects a maximal-gap vertex and coordinated majorants, closes all exceptional + inner branches, performs the five-row enumeration, and transports all thirteen canonical + cover words to four shared full-two-rung, anti-saturation, terminal-cage, and four-edge-cage + predicates. It also proves exceptional-count rigidity and an explicit heptagon refutation + of a tempting stronger majorant claim, with rational regression witnesses. This + settles the all-n k=3 instance of the paper's suggested degree-2k convex bound; + Morić--Pritchard proved the corresponding top-three edge bound for sufficiently + large n. It does not settle the general-k suggestion or the general Erdős problem + 132. +branch: discrete geometry +entry_module: LeanPool.Erdos132ConvexK3 +authors: +- Egor Lyfar +source: + title: On the Graph of Large Distances + authors: + - Paul Erdős + - László Lovász + - Katalin Vesztergombi + doi: 10.1007/BF02187746 + url: https://www.erdosproblems.com/132 + github_repo: lyfar/erdos-132-convex-k3-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.Erdos132ConvexK3.convex_top_three_degree_six +main_results: +- declaration: LeanPool.Erdos132ConvexK3.convex_top_three_degree_six + informal: For every nonempty finite cyclically strictly convex planar configuration + with three strictly ordered largest squared-distance classes d₁, d₂, and d₃, some + vertex is incident to at most six pairs whose squared distance is one of those + three classes. + source_ref: 'Erdős--Lovász--Vesztergombi, printed p. 542, suggest the stronger convex + bound degree at most 2k. Morić--Pritchard, "Counting large distances in convex + polygons" (arXiv:1103.0412), Theorem 1.4, prove m₁ + m₂ + m₃ ≤ 3n for sufficiently + large n, which gives the same degree-six consequence in that range. The novelty + here is the all-n theorem: it removes the sufficiently-large-n restriction. ErLV89 + Theorem 2.7, printed p. 548, proves only degree at most 3k-1, which specializes + to eight.' +- declaration: LeanPool.Erdos132ConvexK3.convex_top_three_draft_reduction_complete + informal: Cyclic strict convexity and the three-largest-distance-class hypothesis + construct the full maximal-gap, thirteen-word reduction package together with + route-specific closure evidence, without any remaining coordinated-majorant or + geometric-adapter assumption. + source_ref: New proof-producing assembly theorem in this project. It combines the + ErLV89 majorant framework with the project's maximal-gap tail, terminal-color, + five-row, and thirteen-word transport lemmas. +- declaration: LeanPool.Erdos132ConvexK3.global_thirteen_word_closure_complete + informal: Every one of the thirteen canonical maximal-gap cover words transports + to its routed full-two-rung, anti-saturation, terminal-cage, or four-edge-cage + realization, and hence supplies the corresponding degree bound. + source_ref: New geometric transport theorem in this project. The thirteen-word classification + and its four-predicate closure interface are not statements of ErLV89. +- declaration: LeanPool.Erdos132ConvexK3.concrete_word_closures + informal: A thirteen-tag routing interface over four shared geometric realization + predicates maps B:3→2 and D:3→2 to Row1B32WordRealization, B:3→1, BA, D:3→1, and + DC to OnePenultimateWordGeometry, DD to Row4DDWordRealization, and the remaining + six tags to FullTwoRungGeometry. Each inhabited routed predicate supplies a vertex + of degree at most six, with the one-penultimate route proving the stronger bound + five. + source_ref: New derived variant contributed and proved in this Lean Pool project. + The thirteen-word classification and its closures are not statements of ErLV89. +- declaration: LeanPool.Erdos132ConvexK3.fullTwoRung_realization_degree_le_six + informal: Every raw full-two-rung realization has its displayed vertex of degree + at most six. A rigid normalization derives the canonical coordinates, the diameter + branch is impossible, the long branch is counted from four raw boundary arcs, + and the short branch contradicts the exact two-rung identity and metric dichotomy. +- declaration: LeanPool.Erdos132ConvexK3.convex_k3_degree_six_of_reduction + informal: This implication from the abstract reduction interface is retained as + the final assembly map. Its hypothesis is now discharged by convex_top_three_draft_reduction_complete, + yielding the unconditional theorem convex_top_three_degree_six; ErLV89 itself + proves only the degree-eight bound obtained from 3k−1 when k=3. + source_ref: 'Erdős--Lovász--Vesztergombi, printed p. 542: “Perhaps in the convex + case there always exists an x_i such that the degree of x_i is at most 2k.” Alongside + it, Theorem 2.7, printed p. 548, proves only: “If S is the set of vertices of + a convex polygon then the graph G(S, k) has a point of degree at most 3k−1.” Thus + degree six for k=3 is the stronger suggested variant; the proved source bound + is eight, while this project''s global closure proves degree six.' +- declaration: LeanPool.Erdos132ConvexK3.coordinated_exceptional_count_rigidity + informal: Every exceptional jointly minimal pair lies in case (1,2), (2,1), or (2,2). + In (1,2), every first substitute has at least one right move while every second + substitute has exactly two left moves and zero right moves. In (2,1), every first + substitute has exactly two right moves and zero left moves while every second + substitute has at least one left move. Only in (2,2) are those exact substitute + counts proved on both sides. +- declaration: LeanPool.Erdos132ConvexK3.coordinated_exceptional_rank_and_exchange_obstruction + informal: Assuming the three top distance classes and TopThreeAdjacent for both + starting edges z–x and t–u, each forced two-move chain in an exceptional inner-move + case has the rank ladder from the third-largest through the second-largest to + the largest distance, while its proposed intermediate exchange edge remains nonterminal. +- declaration: LeanPool.Erdos132ConvexK3.heptagon_nondiameter_edge_is_majorant + informal: In an explicit rational convex heptagon, a second-largest-distance edge + is already terminal under both permitted cover moves, refuting the stronger claim + that every non-diameter top-three edge admits a longer cover. +- declaration: LeanPool.Erdos132ConvexK3.Witnesses.ninePoint_top_three + informal: An explicit rational nine-point regression configuration realizes squared-distance + classes 855625, 838525, and 811165 as its three largest classes. +- declaration: LeanPool.Erdos132ConvexK3.Witnesses.rationalHexagon_top_three + informal: An explicit rational six-point regression configuration realizes the three + stated rational squared-distance values as its largest classes. +tags: +- discrete-geometry +- distance-graphs +- erdos-problems +- convexity +msc: +- 52C10 +- 05C12 diff --git a/LeanPool/projects/erdos132-n14.yaml b/LeanPool/projects/erdos132-n14.yaml new file mode 100644 index 0000000000..7f58c6223d --- /dev/null +++ b/LeanPool/projects/erdos132-n14.yaml @@ -0,0 +1,50 @@ +slug: erdos132-n14 +title: A Conditional Fourteen-Point Case of Erdős Problem 132 +summary: 'Formalizes the planar Hopf--Pannwitz diameter bound and proves the fourteen-point + low-multiplicity conclusion conditional on exactly two remaining published interfaces: + the known maximum-cardinality bound for planar sets with at most six distances and + the Szöllősi--Östergård classification of thirteen-point planar six-distance sets. + Internally derives the exact failure profile and endpoint deletion, constructs the + three classified templates, eliminates the nonregular cases by multiplicity, and + excludes the regular tridecagon extension by explicit second and fourth moments. + The general Erdős problem remains open.' +branch: discrete geometry +entry_module: LeanPool.Erdos132N14 +authors: +- Egor Lyfar +source: + title: Constructions of maximum few-distance sets in Euclidean spaces + authors: + - Ferenc Szöllősi + - Patric R. J. Östergård + doi: 10.37236/8565 + url: https://www.erdosproblems.com/132 + github_repo: lyfar/erdos-132-moment-obstruction-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.Erdos132N14.erdos132_for_fourteen_of_published_inputs +main_results: +- declaration: LeanPool.Erdos132N14.hopfPannwitzLowMultiplicityDistance14 + informal: Every injectively labelled fourteen-point planar configuration has a realized + distance represented by at most fourteen unordered pairs; the proof chooses that + distance to be a diameter. + source_ref: The planar diameter bound, conventionally associated with Hopf--Pannwitz; + Perles's clockwise-extreme endpoint proof. +- declaration: LeanPool.Erdos132N14.erdos132_for_fourteen_of_published_inputs + informal: Assuming precisely the published planar few-distance cardinality and Szöllősi--Östergård + classification interfaces, every injectively labelled fourteen-point planar configuration + realizes at least two distances whose unordered-pair multiplicities are at most + fourteen. + source_ref: Erdős Problem 132; Hopf--Pannwitz's planar diameter bound; Wei, "A Proof + of Erdős--Fishburn's Conjecture for g(6)=13," EJC 19(4) P38 (2012), doi:10.37236/2917; + Szöllősi--Östergård, EJC 27(1) P1.23 (2020), Table 1 and Theorem 16, doi:10.37236/8565. +tags: +- discrete-geometry +- few-distance-sets +- erdos-problems +- planar-configurations +msc: +- 52C10 +- 05D99 diff --git a/LeanPool/projects/erdos132-three-chain.yaml b/LeanPool/projects/erdos132-three-chain.yaml new file mode 100644 index 0000000000..6be340fd34 --- /dev/null +++ b/LeanPool/projects/erdos132-three-chain.yaml @@ -0,0 +1,136 @@ +slug: erdos132-three-chain +title: Planar Point Sets Whose Non-Diameter Distances Form a Geometric 3-Chain +summary: "Formalizes a support theorem about planar point sets whose non-diameter\ + \ squared distances are confined to a geometric progression of ratio three. The\ + \ chain-quadruple theorem shows that no planar quadruple with all six squared distances\ + \ in such a chain can span two chain steps, and classifies the seven labelled patterns\ + \ that do occur: the rhombus of two glued equilateral triangles, in five positions\ + \ of its long edge, and the equilateral triangle with its centroid, in two. A Gram\ + \ obstruction then rules out any five-point chain configuration. The lift to n at\ + \ least thirteen runs through the diameter graph: the Caro--Wei independence bound,\ + \ proved here, produces an independent five-set once the diameter graph is known\ + \ to have at most as many edges as the set has points. That edge bound is the Hopf--Pannwitz\ + \ theorem of 1934, and it already exists in the pool; this project reproves it in\ + \ its own model as a supporting lemma, claiming no novelty for it, and the overlap\ + \ is set out in full below. The result is unconditional.\nThere is no prior write-up\ + \ behind this development. The restriction to ratio 3 is ad hoc: the 3-chain statement\ + \ was produced by the author's automated research pipeline in August 2026, while\ + \ the cited source poses the ambient Erdos problem and does not state this restriction\ + \ or result. This theorem does not resolve Erdos 132, whose distance spectra need\ + \ not be geometric 3-chains. The descriptive labels used in the Lean docstrings\ + \ are internal to this development and refer to no numbered result elsewhere.\n\ + Overlap disclosure. Erdos132ThreeChain.hopfPannwitz is NOT offered as a new result.\ + \ The pooled project Erdos132N14 already contains the same diameter bound, and the\ + \ version here exists only so that this project is self-contained. Its statement,\ + \ verbatim from LeanPool/Erdos132N14/HopfPannwitz.lean, is\n\n theorem diameterEdges_card_le\n\ + \ (P : Configuration ι) (S : Finset ι) {d : ℝ}\n (hd : 0 < d)\n \ + \ (hmax : ∀ f ∈ pairs S, P.pairDistance f ≤ d) :\n (diameterEdges P\ + \ S d).card ≤ S.card\n\nwhere, from LeanPool/Erdos132N14/Basic.lean, `Configuration\ + \ ι` is a structure carrying an injective `point : ι → ℂ`, `pairs S = (S ×ˢ S).filter\ + \ fun e ↦ e.1 < e.2` for a `LinearOrder ι`, `pairDistance P e = dist (P.point e.1)\ + \ (P.point e.2)`, and `diameterEdges P S d = (pairs S).filter fun e ↦ P.pairDistance\ + \ e = d`. The corresponding statement here is\n\n theorem hopfPannwitz {X : Finset\ + \ Point} {D : ℝ} (hD : IsSqDiameter X D) :\n ∑ v ∈ X, degree (DiameterAdj\ + \ D) X v ≤ 2 * X.card\n\nwith `Point = ℝ × ℝ`, `sqDist` the squared Euclidean distance,\ + \ `DiameterAdj D p q = (p ≠ q ∧ sqDist p q = D)`, `degree adj X v = (X.filter (adj\ + \ v)).card`, and `IsSqDiameter X D` asserting that `D` is realised by two distinct\ + \ points of `X` and dominates every squared distance inside `X`. The two are the\ + \ same mathematical theorem — a positive maximal distance is realised at most as\ + \ often as there are points — carried by different models: an injective map from\ + \ a linearly ordered index type into the complex plane, counting a Finset of unordered\ + \ pairs, against a Finset of points of the real plane, in squared distances and\ + \ degree-sum form. Neither statement is an instance of the other without a model\ + \ bridge: a LinearOrder on ℝ × ℝ, a dist-to-sqDist transfer, and a handshake lemma.\ + \ The proofs differ too, Perles charging there against pendant deletion driven by\ + \ Cramer's rule here. No project in the pool imports another, and CONTRIBUTING.md\ + \ records that independence as the property the fast per-project build relies on,\ + \ so this project reproves the bound rather than importing it. It is a supporting\ + \ lemma of this development, not one of its results, and it is therefore not listed\ + \ among the main results below.\nTwo supporting identities also exist elsewhere\ + \ in the pool. From LeanPool/Erdos97ConvexOctagon/CayleyMenger.lean,\n\n theorem\ + \ cm4_sqDist_eq_zero (a b c d : Plane) :\n cm4 (sqDist a b) (sqDist a c)\ + \ (sqDist a d)\n (sqDist b c) (sqDist b d) (sqDist c d) = 0\n\nover `Plane\ + \ = EuclideanSpace ℝ (Fin 2)` is the same four-point Cayley--Menger vanishing as\ + \ this project's\n\n theorem gramDet_eq_zero (a b c d : Point) :\n gramDet\ + \ (sqDist a b) (sqDist a c) (sqDist a d) (sqDist b c) (sqDist b d) (sqDist c d)\ + \ = 0\n\nover `Point = ℝ × ℝ`; both are one-line `ring` identities in the coordinates.\ + \ From LeanPool/DistanceGeometry/SchoenbergHard.lean,\n\n theorem schoenberg\ + \ {D : Matrix (Fin n) (Fin n) ℝ} (hD : IsPreDistMatrix D) :\n EmbedsIn D\ + \ k ↔ (centeredGram D).PosSemidef ∧ (centeredGram D).rank ≤ k\n\ncharacterizes Euclidean\ + \ distance matrices in general dimension. This project never invokes it: the four-point\ + \ condition is used directly as a `ring` identity and the five-point obstruction\ + \ is a sum-of-squares contradiction, so no positive-semidefiniteness criterion is\ + \ needed." +branch: discrete geometry +entry_module: LeanPool.Erdos132ThreeChain +authors: +- Egor Lyfar +source: + title: 'Erdős Problem #132' + authors: + - Paul Erdős + - János Pach + url: https://www.erdosproblems.com/132 + github_repo: lyfar/lean-pool +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Erdos132ThreeChain.nonDiameterSqDists_ne_chain +main_results: +- declaration: Erdos132ThreeChain.nonDiameterSqDists_ne_chain + informal: Let n be at least thirteen, let a be positive, and let X be a finite set + of points of the real plane with exactly n elements. Suppose D is the squared + diameter of X, meaning that D is realised by two distinct points of X and dominates + every squared distance inside X. Then for every natural number h, the set of squared + distances realised by distinct points of X and different from D is not equal to + the geometric 3-chain consisting of a times 3^j for j below h. +- declaration: Erdos132ThreeChain.four_chain_adjacent + informal: Let a be positive and let S be a four-element set of points of the real + plane such that every squared distance between distinct points of S equals a times + a nonnegative power of three. Then there is a positive r, itself equal to a times + a nonnegative power of three, for which every squared distance between distinct + points of S equals r or 3r. +- declaration: Erdos132ThreeChain.pair_classification + informal: 'Let c be positive and let A, B, C, D be points of the real plane with + the squared distance from A to B equal to c and every other pairwise squared distance + equal to c times a nonnegative power of three. Then each of the four squared distances + from A and B to C and to D equals c or 3c, at least one of the two squared distances + from A and B to C equals c, likewise for D, the squared distance from C to D equals + c or 3c, the squared distance from A to C equals c or that from B to D equals + c, the squared distance from B to C equals c or that from A to D equals c, and + the whole labelling is one of exactly seven explicit patterns: five with a single + edge of squared length 3c, and two in which the three edges of squared length + 3c form a triangle.' +- declaration: Erdos132ThreeChain.no_five_chain_finset + informal: Let a be positive. There is no five-element set of points of the real + plane in which every squared distance between distinct points equals a times a + nonnegative power of three. +- declaration: Erdos132ThreeChain.no_four_on_circle + informal: Let c be positive. There are no four points of the real plane, each at + squared distance c from a common point, whose six pairwise squared distances all + equal c or 3c. +- declaration: Erdos132ThreeChain.carowei + informal: Let adj be a symmetric irreflexive relation with decidable adjacency on + a type, and let V be a finite set of vertices. Then V has a subset S, no two elements + of which are adjacent, whose cardinality is at least the sum over v in V of one + divided by one plus the number of neighbours of v inside V. +- declaration: Erdos132ThreeChain.exists_independent_five + informal: Let adj be a symmetric irreflexive relation with decidable adjacency on + a type, and let V be a finite set of at least thirteen vertices whose degrees + inside V sum to at most twice the cardinality of V. Then V has a five-element + subset no two elements of which are adjacent. +- declaration: Erdos132ThreeChain.chainTwo_nonDiameter_support + informal: There is a five-element set of points of the real plane whose squared + diameter is 4 and whose set of non-diameter squared distances is exactly the two-term + geometric 3-chain consisting of 1 and 3. The configuration that the headline forbids + for at least thirteen points is therefore realisable at five points with chain + length two. +tags: +- discrete-geometry +- distance-geometry +- erdos-problems +msc: +- 52C10 +- 52C35 +- 05C69 diff --git a/LeanPool/projects/erdos137.yaml b/LeanPool/projects/erdos137.yaml new file mode 100644 index 0000000000..8e23bbe338 --- /dev/null +++ b/LeanPool/projects/erdos137.yaml @@ -0,0 +1,44 @@ +slug: erdos137 +title: 'Erdős Problem #137: powerful products of consecutive integers' +summary: 'Formalizes the abc-conditional finiteness results for Erdős Problem #137 + (Erdős–Selfridge): a number N is powerful if every prime dividing N has its square + dividing N, and the question is whether the product F(k,n) = n(n+1)⋯(n+k-1) of k + ≥ 3 consecutive integers can be powerful for infinitely many n. Each route isolates + its abc input (a radical lower bound) as a single explicit hypothesis and proves + everything else outright, concluding that, conditionally, F(k,n) is powerful for + only finitely many n for each fixed k ≥ 3.' +branch: number theory +entry_module: LeanPool.Erdos137 +authors: +- Scott D. Hughes +source: + title: 'Erdős Problem #137: powerful products of consecutive integers' + authors: + - Scott D. Hughes + url: https://www.erdosproblems.com/137 + github_repo: scottdhughes/erdos137 +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Erdos137.erdos137_finite +- Erdos137.erdos137_eventually_not_powerful +main_results: +- declaration: Erdos137.erdos137_finite + informal: Under the Granville–Langevin radical bound RadLB k (an abc-conditional + hypothesis) and k ≥ 3, the set of n ≥ 1 for which F(k,n) is powerful is finite. +- declaration: Erdos137.erdos137_eventually_not_powerful + informal: Under RadLB k and k ≥ 3, there is a threshold beyond which F(k,n) is never + powerful. +- declaration: Erdos137.lemma_star + informal: Unconditionally, the radical squared times the powerful part of m is at + most m squared (rad m ^ 2 * B2 m ≤ m ^ 2). +- declaration: Erdos137.powerful_rad_sq_le + informal: A nonzero powerful number N satisfies rad N ^ 2 ≤ N. +tags: +- number-theory +- powerful-numbers +- erdos-problems +msc: +- 11A51 +- 11N25 diff --git a/LeanPool/projects/erdos346.yaml b/LeanPool/projects/erdos346.yaml new file mode 100644 index 0000000000..ec73004748 --- /dev/null +++ b/LeanPool/projects/erdos346.yaml @@ -0,0 +1,40 @@ +slug: erdos346 +title: 'Erdős Problem 346: Ratio Limit Forces the Golden Ratio' +summary: 'Formalizes the limit-exists interpretation of Erdős Problem 346: for a strictly + increasing positive integer sequence with a uniform ratio gap whose finite deletions + are complete but whose infinite deletions are not, if the consecutive quotient sequence + converges to some real limit greater than one, then that limit must equal the golden + ratio and the quotient sequence tends to it.' +branch: number theory +entry_module: LeanPool.Erdos346 +authors: +- KitaKen1 +source: + title: 'Erdős Problem #346' + authors: + - KitaKen1 + url: https://www.erdosproblems.com/346 + github_repo: KitaKen1/erdos346-ratio-limit-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Erdos346.intended_problem_if_limit_exists +main_results: +- declaration: Erdos346.intended_problem_if_limit_exists + informal: Existence of any limit greater than one for the quotient sequence forces + convergence to the golden ratio, the intended Erdős–Graham conclusion. +- declaration: Erdos346.ratio_limit_eq_goldenRatio + informal: If the quotient sequence converges to a limit greater than one under the + completeness and incompleteness deletion hypotheses, that limit equals the golden + ratio. +- declaration: Erdos346.ratio_tendsto_goldenRatio_of_limit_exists + informal: Under the same hypotheses, the quotient sequence converges to the golden + ratio itself. +tags: +- number-theory +- golden-ratio +- erdos-problems +msc: +- 11B05 +- 11J70 diff --git a/LeanPool/projects/erdos367.yaml b/LeanPool/projects/erdos367.yaml new file mode 100644 index 0000000000..390002a0f5 --- /dev/null +++ b/LeanPool/projects/erdos367.yaml @@ -0,0 +1,55 @@ +slug: erdos367 +title: 'Erdős Problem #367' +summary: Formalizes upper and lower bound results for powerful parts of products of + consecutive integers, including abc-conditional upper bounds, an unconditional odd + r-full lower bound, a Pell-style limsup theorem, and polynomial arithmetic cores. +branch: number theory +entry_module: LeanPool.Erdos367 +authors: +- Scott D. Hughes +source: + title: Powerful parts of consecutive integers and Davenport-Zannier polynomials + authors: + - Scott D. Hughes + url: https://github.com/scottdhughes/erdos367 + github_repo: scottdhughes/erdos367 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Erdos367.erdos_367_k3 +- RFullOdd.erdos367_iv +- Erdos367.erdos367 +main_results: +- declaration: Erdos367.erdos_367_k3 + informal: Proves an abc-conditional k = 3 upper bound for the product of square-full + parts of three consecutive integers. +- declaration: GeneralK.B2_upper_bound + informal: Proves a general-k upper bound for the square-full part of a product of + consecutive integers from an explicit radical lower-bound hypothesis. +- declaration: RFullOdd.erdos367_iv + informal: Gives an unconditional odd r construction where the product of r-full + parts of two consecutive integers is larger than the stated power of n. +- declaration: Erdos367.erdos367 + informal: Proves an unconditional limsup form showing that the product of square-full + parts of three consecutive integers exceeds any fixed multiple of n squared times + log n. +- declaration: Core139.T3 + informal: Proves the integer-form arithmetic core yielding the E3 at least 13/9 + lower-bound construction under explicit prime-supply data. +- declaration: Core4027.IDENT + informal: Verifies the degree-27 polynomial identity used in the Davenport-Zannier + construction. +- declaration: Core4027.T3 + informal: Proves the integer-form arithmetic core yielding the E3 at least 40/27 + lower-bound construction under explicit prime-supply data. +tags: +- number-theory +- erdos-problems +- powerful-numbers +- abc-conjecture +- pell-equations +msc: +- 11D09 +- 11N25 +- 11D45 diff --git a/LeanPool/projects/erdos403.yaml b/LeanPool/projects/erdos403.yaml new file mode 100644 index 0000000000..66b094724f --- /dev/null +++ b/LeanPool/projects/erdos403.yaml @@ -0,0 +1,35 @@ +slug: erdos403 +title: Sums of Distinct Factorials That Are Powers of Two (Erdos Problem 403) +summary: 'Reconstructs the finiteness theorem for Erdos Problem 403: only finitely + many sums of distinct factorials are powers of two, and every such power is at most + 2^7 = 128 (the extremal solution 2! + 3! + 5! = 128). The sharp bound is proved + in the factorial number system via a single fixed modulus 12!, with all computations + kernel-pure.' +branch: number theory +entry_module: LeanPool.Erdos403 +authors: +- Trevor Morris +source: + title: 'Erdos Problem 403: sums of distinct factorials that are powers of two' + authors: + - Trevor Morris + url: https://www.erdosproblems.com/403 + github_repo: gotrevor/erdos-403 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Erdos403.erdos_403_sharp +- Erdos403.erdos_403_finite +main_results: +- declaration: Erdos403.erdos_403_sharp + informal: Every power of two that is a sum of distinct factorials is at most 2^7, + with 128 the largest solution. +- declaration: Erdos403.erdos_403_finite + informal: Only finitely many sums of distinct factorials are powers of two. +tags: +- number-theory +- factorials +- erdos-problems +msc: +- 11B83 diff --git a/LeanPool/projects/erdos548.yaml b/LeanPool/projects/erdos548.yaml new file mode 100644 index 0000000000..d56574081c --- /dev/null +++ b/LeanPool/projects/erdos548.yaml @@ -0,0 +1,57 @@ +slug: erdos548 +title: The Erdős–Sós conjecture (Erdős Problem 548) +summary: 'Proves the Erdős–Sós conjecture (1962), Erdős Problem 548: every graph on + n ≥ k+1 vertices with at least (k−1)n/2 + 1 edges contains every tree on k+1 vertices + as a (not necessarily induced) subgraph. The headline statement is the FrontierMath + Erdős benchmark statement, reviewed by Thomas Bloom and carried verbatim by Google + DeepMind''s Formal Conjectures; the proof was found autonomously by GPT-6 Astra. + It is a finite double counting over permutation words of the host vertices: rotating + the first qualifying prefix of a cut word gives a gluing inequality for two rooted + branches, and reversing the two blocks of a cut moves the root to a newly attached + leaf. A strong induction on the order t of the tree bounds the adjacency-marked + cut words by the rooted-copy count plus (t−2)·n!, so a tree-free host on n vertices + has at most (t−2)n/2 edges, the sharp classical bound.' +branch: extremal graph theory +entry_module: LeanPool.Erdos548 +authors: +- Tom Adamczewski +source: + title: 'Erdős problem #548 (Erdős–Sós conjecture): proof' + authors: + - Tom Adamczewski + url: https://github.com/tadamcz/erdos548 + github_repo: tadamcz/erdos548 + commit: c02708a6956d7b953c2e889c8bc24a0c97796313 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Erdos548.erdos_548 +main_results: +- declaration: Erdos548.erdos_548 + informal: Let n ≥ k+1. Every simple graph on n vertices with at least (k−1)n/2 + + 1 edges contains every tree on k+1 vertices as a subgraph (an injective graph + homomorphism). This is Erdős Problem 548, the Erdős–Sós conjecture, in the benchmark + phrasing. +- declaration: Erdos548.tree_free_edge_bound + informal: If a tree T on t ≥ 2 vertices is not contained in a graph G on n ≥ 1 vertices, + then 2·|E(G)| ≤ (t−2)·n. This is the classical sharp form of the Erdős–Sós bound. +- declaration: Erdos548.rooted_word_tree_bound + informal: For a tree T on t ≥ 2 vertices rooted at r, a host graph G and a nonempty + repetition-free host word of length n, the number of cut permutation words of + that word whose cut ends at a neighbour of the first letter is at most the number + supporting a rooted copy of T on the prefix plus (t−2)·n!. +- declaration: Erdos548.marked_word_gluing_count + informal: For a permutation-closed family W of repetition-free words, all of the + same length m, and families A, B, C of letter sets with C ⊆ A and A R ∧ B X → + C (R ∪ X) for disjoint R, X, the marked cuts (l, k) with l ∈ W and k ≤ m qualifying + for A plus those qualifying for B number at most the allowed cuts plus those qualifying + for C. +tags: +- extremal-graph-theory +- erdos-problems +- trees +- combinatorics +msc: +- 05C35 +- 05C05 diff --git a/LeanPool/projects/erdos81-paper-i-contrib.yaml b/LeanPool/projects/erdos81-paper-i-contrib.yaml new file mode 100644 index 0000000000..bcb5dda145 --- /dev/null +++ b/LeanPool/projects/erdos81-paper-i-contrib.yaml @@ -0,0 +1,43 @@ +slug: erdos81-paper-i-contrib +title: Finitely generated cones and finite LP duality +summary: Proves that finitely generated cones in real normed spaces are closed, including + a conic Carathéodory reduction and PointedCone API forms. Derives a finite-dimensional + inequality form of Farkas' lemma and proves packing attainment with equality to + the covering infimum for finite covering and packing linear programs. The Farkas + and LP statements overlap the pooled Duality development; the generic cone-closedness + results provide the project's distinct reusable API. +branch: convex geometry and optimization +entry_module: LeanPool.Erdos81PaperIContrib +authors: +- Juan Pablo Traverso Gianini +source: + title: Affine Profile Reduction for Fractional Triangle Packings in Split Graphs + authors: + - Juan Pablo Traverso Gianini + url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/tree/main/preprints/PAPER_I + github_repo: jtraverso/erdos-81-chordal-clique-partitions +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LeanPool.Erdos81PaperIContrib.fg_cone_isClosed +main_results: +- declaration: LeanPool.Erdos81PaperIContrib.fg_cone_isClosed + informal: The cone generated by a finite family in a real normed vector space is + a closed set. +- declaration: LeanPool.Erdos81PaperIContrib.farkas_ge + informal: An infeasible finite system of nonnegative linear inequalities admits + a nonnegative separating Farkas certificate. +- declaration: LeanPool.Erdos81PaperIContrib.covering_packing_duality + informal: Given a finite nonnegative matrix with a strictly positive entry in every + column and nonnegative capacities, there is a feasible nonnegative packing whose + unit-objective value is maximal and equals the infimum cost of nonnegative coverings + with unit demand. +tags: +- convex-geometry +- linear-programming +- duality +- farkas-lemma +msc: +- 52A20 +- 90C05 diff --git a/LeanPool/projects/erdos81-paper-ii-chordal-heredity.yaml b/LeanPool/projects/erdos81-paper-ii-chordal-heredity.yaml new file mode 100644 index 0000000000..b874e24953 --- /dev/null +++ b/LeanPool/projects/erdos81-paper-ii-chordal-heredity.yaml @@ -0,0 +1,45 @@ +slug: erdos81-paper-ii-chordal-heredity +title: 'Chordal graphs: minimal separators and simplicial vertices' +summary: 'A reusable formalization byproduct accompanying Paper II: defines chordal + graphs through the cycle-and-chord characterization, proves induced-subgraph heredity + and the clique property for finite minimal separators of fixed vertex pairs, proves + Dirac''s simplicial-vertex theorem, and proves the connected non-complete case of + the two-vertex conclusion.' +branch: graph theory +entry_module: LeanPool.Erdos81PaperIIContrib +authors: +- Juan Pablo Traverso Gianini +source: + title: Complete-Split Extremizers for a Fractional Triangle-Cover Functional on + Chordal Graphs + authors: + - Juan Pablo Traverso Gianini + url: https://github.com/jtraverso/lean-pool/blob/822738f3089d92c5e65a6bfb8c88c992e4699167/LeanPool/Erdos81PaperIIContrib/Chordal.lean + github_repo: jtraverso/lean-pool +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- SimpleGraph.IsChordal.exists_two_nonadj_isSimplicial +main_results: +- declaration: SimpleGraph.IsChordal.comap + informal: A graph that embeds as an induced subgraph of a chordal graph is itself + chordal. +- declaration: SimpleGraph.IsChordal.induce + informal: Every induced subgraph of a chordal graph is chordal. +- declaration: SimpleGraph.IsChordal.minimalSeparator_isClique + informal: Every finite minimal separator of two fixed vertices in a chordal graph + is a clique. +- declaration: SimpleGraph.IsChordal.exists_isSimplicial + informal: Every nonempty finite chordal graph has a simplicial vertex. +- declaration: SimpleGraph.IsChordal.exists_two_nonadj_isSimplicial + informal: Every connected non-complete finite chordal graph has two non-adjacent + simplicial vertices. +tags: +- graph-theory +- chordal-graphs +- induced-subgraphs +- minimal-separators +- simplicial-vertices +msc: +- 05C75 diff --git a/LeanPool/projects/erdos81-paper-iii-sum-zero-triangles.yaml b/LeanPool/projects/erdos81-paper-iii-sum-zero-triangles.yaml new file mode 100644 index 0000000000..660bf8a942 --- /dev/null +++ b/LeanPool/projects/erdos81-paper-iii-sum-zero-triangles.yaml @@ -0,0 +1,39 @@ +slug: erdos81-paper-iii-sum-zero-triangles +title: Near-perfect triangle packings from sum-zero triples +summary: 'Records a reusable formalization byproduct accompanying Paper III: pairwise + edge-disjoint triangles in any finite complete graph such that every vertex is incident + to at most three uncovered edges, using a cyclic-group sum-zero triple system, with + a bridge to Mathlib''s SimpleGraph API.' +branch: extremal and design combinatorics +entry_module: LeanPool.Erdos81PaperIIIContrib +authors: +- Juan Pablo Traverso Gianini +- Aristotle +- Vasily Ilin +source: + title: SumZeroTriangles.lean, a reusable formalization byproduct accompanying Linear-Error + Clique Partitions of Split Graphs via Structured Triangle Packing + authors: + - Aristotle + url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/main/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Contrib/SumZeroTriangles.lean + github_repo: jtraverso/erdos-81-chordal-clique-partitions +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- SumZeroTriangles.exists_simpleGraph_triangle_packing +main_results: +- declaration: SumZeroTriangles.exists_simpleGraph_triangle_packing + informal: For every finite vertex type, its complete simple graph has a family of + pairwise edge-disjoint triangle graphs whose uncovered graph has degree at most + three at every vertex. +- declaration: SumZeroTriangles.exists_triangle_packing_clique + informal: Every finite complete graph has a family of pairwise edge-disjoint triangles + leaving uncovered degree at most three at every vertex. +tags: +- extremal-combinatorics +- triangle-packing +- cyclic-groups +msc: +- 05B07 +- 05C70 diff --git a/LeanPool/projects/erdos865.yaml b/LeanPool/projects/erdos865.yaml new file mode 100644 index 0000000000..d7c73be0b7 --- /dev/null +++ b/LeanPool/projects/erdos865.yaml @@ -0,0 +1,42 @@ +slug: erdos865 +title: A sharp 5/8 bound for Erdős Problem 865 +summary: Formalizes the resolution of Erdős Problem 865 on pairwise sums. For A ⊆ + {1,…,N}, a pairwise-sum triple is a set of distinct a, b, c ∈ A with a+b, a+c, b+c + all in A, and A is triple-free when it has none. The development proves that every + triple-free A ⊆ [1,N] satisfies 8·|A| ≤ 5·N + 53, that any A exceeding this bound + contains a triple, and — via the construction A = [M,2M] ∪ [4M,8M] for N = 8M — + that the constant 5/8 is optimal, so f₃(N) = 5N/8 + O(1). The upper bound runs through + a folded additive lemma over ℤ/mℤ (a four-set union bound) and a folding lemma, + closed by strong induction on N. +branch: additive combinatorics +entry_module: LeanPool.Erdos865 +authors: +- Ricky Cipollini +source: + title: A sharp 5/8 bound for an Erdős–Sós pairwise-sums problem + url: https://github.com/mrricky22/erdos-865-lean + github_repo: mrricky22/erdos-865-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Erdos865.erdos865_upper_bound +- Erdos865.sharpness +main_results: +- declaration: Erdos865.erdos865_upper_bound + informal: Every triple-free set A ⊆ [1,N] satisfies 8·|A| ≤ 5·N + 53, i.e. |A| ≤ + (5/8)·N + O(1). +- declaration: Erdos865.erdos865_contains_triple + informal: Every A ⊆ [1,N] with 5·N + 53 < 8·|A| contains a pairwise-sum triple (the + paper's Theorem 1.1 form). +- declaration: Erdos865.sharpness + informal: For N = 8M with M ≥ 1 there is a triple-free A ⊆ [1,N] with 8·|A| = 5·N + + 16, so the constant 5/8 is optimal. +tags: +- additive-combinatorics +- erdos-problems +- sum-free-sets +- combinatorics +msc: +- 11B75 +- 11B13 diff --git a/LeanPool/projects/escauriaza-seregin-sverak.yaml b/LeanPool/projects/escauriaza-seregin-sverak.yaml new file mode 100644 index 0000000000..27abac311e --- /dev/null +++ b/LeanPool/projects/escauriaza-seregin-sverak.yaml @@ -0,0 +1,70 @@ +slug: escauriaza-seregin-sverak +title: Escauriaza–Seregin–Šverák and Ladyzhenskaya–Prodi–Serrin regularity +summary: 'Endpoint and Serrin regularity criteria for three-dimensional incompressible + Navier–Stokes Leray–Hopf solutions: bounded L-infinity-in-time L3 velocity implies + L5 integrability, uniqueness, and a smooth representative on positive times including + the final time. Includes local endpoint regularity, both Carleman inequalities, + spatial unique continuation, and half-space backward uniqueness. Retains the complete + ESS production formalization.' +branch: Analysis +entry_module: LeanPool.EscauriazaSereginSverak +authors: +- Scott Armstrong +source: + github_repo: scottnarmstrong/EscauriazaSereginSverak + url: https://github.com/scottnarmstrong/EscauriazaSereginSverak/tree/ae1a0e46086b9b16179dea76b41da10b01845151 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- ESS.essSmooth +- ESS.ladyzhenskayaProdiSerrin +- ESS.serrinCriterion +main_results: +- declaration: ESS.essSmooth + informal: A Leray–Hopf solution whose spatial L3 mass is essentially bounded in + time belongs to space-time L5, is unique among Leray–Hopf solutions with the same + initial datum, and has an almost-everywhere equal smooth representative on all + positive times through the final time. +- declaration: ESS.ladyzhenskayaProdiSerrin + informal: A Leray–Hopf solution satisfying the critical Serrin mixed-norm condition + with spatial exponent greater than three, including the L2-in-time L-infinity-in-space + endpoint, is unique and has a smooth representative on positive times through + the final time. +- declaration: ESS.serrinCriterion + informal: The endpoint L-infinity-in-time L3 criterion and the Serrin criteria for + all spatial exponents greater than three jointly imply uniqueness and existence + of a smooth representative up to the final time. +- declaration: ESS.essLocal + informal: A weak Navier–Stokes solution on the unit cylinder with finite energy, + L3/2 pressure, and velocity essentially bounded in time in spatial L3 has an almost-everywhere + equal parabolically Holder representative on the closure of the half cylinder. +- declaration: ESS.essGlobal + informal: A Leray–Hopf solution whose spatial L3 mass is essentially bounded in + time has no singular point at an interior positive time. +- declaration: ESS.essL5Unique + informal: A Leray–Hopf solution whose spatial L3 mass is essentially bounded in + time belongs to space-time L5 and is unique among Leray–Hopf solutions with its + initial datum. +- declaration: ESS.uniqueContinuation + informal: A vector field with explicit weak space-time derivatives obeying a parabolic + differential inequality has unique continuation across a spatial boundary under + the stated decay and boundary hypotheses. +- declaration: ESS.backwardUniqueness + informal: A vector field on a three-dimensional half-space satisfying the heat differential + inequality, growth bound, and zero initial trace vanishes. +- declaration: ESS.carlemanGaussian + informal: The Gaussian Carleman inequality bounds weighted field and gradient energies + by the weighted heat-operator energy under its support hypotheses. +- declaration: ESS.carlemanHalfSpace + informal: The half-space Carleman inequality bounds weighted field and gradient + energies by the weighted heat-operator energy under its support hypotheses. +tags: +- Navier–Stokes +- partial differential equations +- regularity +- unique continuation +msc: +- 35Q30 +- 35B65 +- 35B60 diff --git a/LeanPool/projects/euclidean-jordan.yaml b/LeanPool/projects/euclidean-jordan.yaml new file mode 100644 index 0000000000..d705cdcab8 --- /dev/null +++ b/LeanPool/projects/euclidean-jordan.yaml @@ -0,0 +1,37 @@ +slug: euclidean-jordan +title: Euclidean Jordan algebras and the frame Peirce decomposition +summary: 'The frame Peirce decomposition: relative to a Jordan frame, a Euclidean + Jordan algebra is the internal direct sum of the blocks V_ij over unordered pairs + of frame indices, and in finite dimension the diagonal blocks are lines, dim V_ii + = 1 - the Faraut-Koranyi table entry. The frame is carried as data, so frame conjugacy + is sidestepped and nothing here claims anything about the rank of J. Machine checked, + not peer reviewed, no priority claim.' +branch: nonassociative algebra +entry_module: LeanPool.EuclideanJordan +authors: +- Bryan Ehrlich +source: + url: https://github.com/ehrlich-b/euclidean-jordan + github_repo: ehrlich-b/euclidean-jordan + commit: a83407e4dbf73c77c31a2291c02f20b37a5ebbaf +license: Apache-2.0 +status: verified +main_declarations: +- EuclideanJordan.frameBlock_isInternal +main_results: +- declaration: EuclideanJordan.frameBlock_isInternal + informal: A Euclidean Jordan algebra is the internal direct sum of the Peirce blocks + associated with a Jordan frame. +- declaration: EuclideanJordan.finrank_frameBlock_diag + informal: In finite dimension, every diagonal block of a Jordan frame has dimension + one. +tags: +- nonassociative-algebra +msc: +- 17C20 +- 17C27 +- 17C37 +- 17C65 +- 17A15 +- 46L70 +provenance: AI diff --git a/LeanPool/projects/even-graph-cycle-decomposition.yaml b/LeanPool/projects/even-graph-cycle-decomposition.yaml new file mode 100644 index 0000000000..aba19c85cb --- /dev/null +++ b/LeanPool/projects/even-graph-cycle-decomposition.yaml @@ -0,0 +1,36 @@ +slug: even-graph-cycle-decomposition +title: Even graphs are edge-disjoint unions of cycles +summary: Decomposes every finite loopless edge set with even degree at each vertex + into edge-disjoint simple cycles and derives a simultaneous fixed-point-free pairing + of incident edges. +branch: graph theory and combinatorics +entry_module: LeanPool.EvenGraphCycles +authors: +- Juan Pablo Traverso Gianini +- Aristotle +source: + title: Reusable even-graph cycle decomposition accompanying Linear-Error Clique + Partitions of Split Graphs via Structured Triangle Packing + authors: + - Juan Pablo Traverso Gianini + url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/b3423f3e8c8db7c2d1b279673293ef3079faf903/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Contrib/EvenGraphCycles.lean + github_repo: jtraverso/erdos-81-chordal-clique-partitions +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Finset.exists_cycleDecomp +- Finset.exists_edge_pairing_of_even +main_results: +- declaration: Finset.exists_cycleDecomp + informal: Every finite loopless edge set whose degree is even at every vertex is + the edge-disjoint union of simple cycles of length at least three. +- declaration: Finset.exists_edge_pairing_of_even + informal: If every vertex has even degree, then the incident edges at every vertex + admit a simultaneous fixed-point-free involutive pairing. +tags: +- graph-theory +- eulerian-graphs +- cycle-decompositions +msc: +- 05C45 diff --git a/LeanPool/projects/event-structures.yaml b/LeanPool/projects/event-structures.yaml new file mode 100644 index 0000000000..a476f26c87 --- /dev/null +++ b/LeanPool/projects/event-structures.yaml @@ -0,0 +1,59 @@ +slug: event-structures +title: Event Structures and Causal-Consistent Reversibility +summary: Formalizes event structures with binary conflict, their configurations, computations, + the trace monoid, and causal-consistent reversibility, proving correctness and causal-safety + of rollback, uniqueness of minimal and maximal replays, and conditional replay-existence + characterizations. +branch: concurrency theory +entry_module: LeanPool.EventStructures +authors: +- Vikraman Choudhury +source: + doi: 10.1007/3-540-17906-2_31 + url: https://github.com/vikraman/event-structures + github_repo: vikraman/event-structures +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- EventStructures.EventStructure +- EventStructures.Conf +main_results: +- declaration: EventStructures.EventStructure + informal: Defines an event structure as a partially ordered set of events with an + irreflexive symmetric conflict relation. + source_ref: M. Nielsen, G. Plotkin, G. Winskel, Petri nets, event structures and + domains, part I, Theoretical Computer Science 13 (1981), 85-108, doi:10.1016/0304-3975(81)90112-2. +- declaration: EventStructures.Conf + informal: Defines a configuration as a conflict-free, downward-closed set of events. + source_ref: G. Winskel, Event structures, Advances in Petri Nets 1986, LNCS 255, + Springer, 1987, 325-392, doi:10.1007/3-540-17906-2_31. +- declaration: EventStructures.Rollback.rollback_correctness_finite + informal: Proves that the original configuration is reachable from the rollback + of an event when the configuration is finitely representable. + source_ref: Causal-consistent reversibility in the sense of V. Danos, J. Krivine, + Reversible communicating systems, CONCUR 2004, LNCS 3170, Springer, 2004, 292-307, + doi:10.1007/978-3-540-28644-8_19. +- declaration: EventStructures.Rollback.rollback_causal_safety + informal: Proves causal safety of rollback - events causally after the rolled-back + event are removed. + source_ref: Causal-consistent reversibility in the sense of V. Danos, J. Krivine, + Reversible communicating systems, CONCUR 2004, LNCS 3170, Springer, 2004, 292-307, + doi:10.1007/978-3-540-28644-8_19. +- declaration: EventStructures.Replay.minReplay_unique + informal: Proves uniqueness of the minimal replay of a log - any two minimal replays + compatible with the log reach the same configuration. +- declaration: EventStructures.Trace.TraceMonoid + informal: Defines the trace monoid as lists of events quotiented by the congruence + generated by swapping adjacent concurrent events; paths in the configuration graph + are quotiented by the same trace equivalence to form asynchronous paths. + source_ref: A. Mazurkiewicz, Trace theory, Advances in Petri Nets 1986, LNCS 255, + Springer, 1987, 278-324, doi:10.1007/3-540-17906-2_30. +tags: +- concurrency +- order-theory +- reversible-computation +- event-structures +msc: +- 68Q85 +- 06A06 diff --git a/LeanPool/projects/exponential-chaos.yaml b/LeanPool/projects/exponential-chaos.yaml new file mode 100644 index 0000000000..65f07ddce6 --- /dev/null +++ b/LeanPool/projects/exponential-chaos.yaml @@ -0,0 +1,66 @@ +slug: exponential-chaos +title: Chaos and the Julia set of the complex exponential +summary: 'Formalizes the dynamics of z ↦ exp(z): escaping points, points with dense + forward orbit, and repelling periodic points are dense in the complex plane; the + map is chaotic in Devaney''s sense and its Julia set is the whole plane. The development + proves eventual covering of every compact set omitting zero, constructs contracting + inverse branches, and rules out every locally uniform spherical subsequential limit + on nonempty open sets. It includes general normal-sequence restriction lemmas, quantitative + open mapping, and a two-target obstruction for arbitrary separated uniform codomains. + The Lean proofs were primarily generated by AI under Lasse Rempe''s direction. The + initial six-lemma architecture adapts John Harrison''s HOL Light formalization, + credited with its retained BSD notice in the entry module.' +branch: complex dynamics +entry_module: LeanPool.ExpChaotic +authors: +- Lasse Rempe +source: + title: 'The Exponential Map is Chaotic: An Invitation to Transcendental Dynamics' + authors: + - Zhaiming Shen + - Lasse Rempe-Gillen + arxiv: '1408.1129' + doi: 10.4169/amer.math.monthly.122.10.919 + url: https://github.com/LR-UK/exp-chaotic + github_repo: LR-UK/exp-chaotic + commit: 303b2d3d1121ebc3a86399de8f1e71c17a8b6dd8 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- ExpChaotic.devaney_chaotic +- ExpChaotic.juliaSet_exp +main_results: +- declaration: ExpChaotic.devaney_chaotic + informal: The complex exponential is continuous, has dense periodic points, and + the monoid action generated by its iterates is topologically transitive on the + complex plane. +- declaration: ExpChaotic.juliaSet_exp + informal: The Julia set of the complex exponential is the whole plane, where the + Fatou set is defined by local normality of sphere-valued iterates with arbitrary + locally uniform sphere-valued limits. +- declaration: ExpChaotic.dense_repellingPeriodicSet + informal: Points p with a positive period n and derivative norm greater than one + for the nth exponential iterate are dense in the complex plane. +- declaration: ExpChaotic.eventually_covers_compact + informal: For every nonempty open set U in the complex plane and every compact set + K omitting zero, all sufficiently large iterates of the exponential map send U + onto a set containing K. +- declaration: ExpChaotic.dense_escapingSet + informal: Points whose forward exponential orbit eventually leaves every centered + Euclidean ball are dense in the complex plane. +- declaration: ExpChaotic.not_countable_denseOrbitSet + informal: The set of complex points with dense forward exponential orbit is uncountable. +- declaration: ExpChaotic.not_tendstoLocallyUniformly_sphericalExpIterate + informal: On every nonempty open set in the complex plane, no strictly increasing + subsequence of exponential iterates converges locally uniformly to any function + into the Riemann sphere. +tags: +- complex-dynamics +- exponential-map +- julia-sets +- chaos +- holomorphic-functions +msc: +- 37F10 +- 30D05 diff --git a/LeanPool/projects/factorizationsystems.yaml b/LeanPool/projects/factorizationsystems.yaml new file mode 100644 index 0000000000..47ba06b089 --- /dev/null +++ b/LeanPool/projects/factorizationsystems.yaml @@ -0,0 +1,38 @@ +slug: factorizationsystems +title: Factorization Systems +summary: Formalizes factorization systems, orthogonality, orthogonal complements, + and the standard construction from weak factorization plus orthogonality. +branch: category theory +entry_module: LeanPool.FactorizationSystems +authors: +- Ivan Kobe +source: + title: Category Theory in Context + authors: + - Emily Riehl + url: https://emilyriehl.github.io/files/context.pdf + github_repo: ivankobe/FactorizationSystems +license: MIT +status: verified +provenance: human +main_declarations: +- CategoryTheory.FactorizationSystemCharacterization +main_results: +- declaration: CategoryTheory.FactorizationSystem + informal: Defines factorization systems on a category with specified left and right + classes. +- declaration: CategoryTheory.FactorizationSystemCharacterization + informal: Constructs a factorization system from repleteness, weak factorizations, + and orthogonality. +- declaration: CategoryTheory.orthogonal + informal: Defines orthogonality of two morphisms by existence and uniqueness of + diagonal fillers. +- declaration: CategoryTheory.right_orthogonal_complement + informal: Defines the right orthogonal complement of a morphism property. +tags: +- category-theory +- factorization-systems +- orthogonality +msc: +- 18A32 +- 18A40 diff --git a/LeanPool/projects/fd1d-lean.yaml b/LeanPool/projects/fd1d-lean.yaml new file mode 100644 index 0000000000..e0687e5797 --- /dev/null +++ b/LeanPool/projects/fd1d-lean.yaml @@ -0,0 +1,31 @@ +title: Optimal fully dynamic matching on the line +summary: Constructs a measurable hierarchical policy for fully dynamic matching of + uniformly arriving demand and replenishment on the unit interval. Proves logarithmic-over-inventory + bounds for long-run root-mean-square distance from every initial inventory and for + initialized finite-horizon expected average cost. +branch: probability and optimization +main_declarations: +- FD1D.V5.Palomar.optimalDynamicMatchingUpperBound +main_results: +- declaration: FD1D.V5.Palomar.optimalDynamicMatchingUpperBound + informal: For every inventory size m at least two, an online policy has long-run + RMS cost at most (72000/log 2) log(m+1)/m from every initial inventory, and the + same expected-average-cost bound after initialization for horizons at least 2m². +tags: +- online-matching +- probability +- stochastic-optimization +msc: +- 60J20 +- 90B15 +provenance: AI +slug: fd1d-lean +entry_module: LeanPool.FullyDynamicMatching +authors: +- Yash Kanoria +source: + url: https://github.com/ykanoria/fd1d-lean + github_repo: ykanoria/fd1d-lean + commit: 26a5b4e2fd83fa86053239e09e92af370c62f18c +license: Apache-2.0 +status: verified diff --git a/LeanPool/projects/feige.yaml b/LeanPool/projects/feige.yaml new file mode 100644 index 0000000000..c5f26bfb2b --- /dev/null +++ b/LeanPool/projects/feige.yaml @@ -0,0 +1,53 @@ +slug: feige +title: Feige's sharp unit-slack inequality +summary: 'For small-universe probability spaces, proves the sharp fixed-dimensional + unit-slack lower bound for sums of independent nonnegative random variables: if + each coordinate has mean at most one, then the probability that the sum lies below + its mean plus one is at least (n/(n+1))^n, and this constant is optimal. The supporting + library formalizes the Vlassis--Thomas exact distribution-free calibration theorem, + Grünbaum''s centroid-halfspace inequality and sharp simplex example, and the normalized-exponential + bridge joining them.' +branch: probability theory and convex geometry +entry_module: LeanPool.Feige +authors: +- Zhengqing Zhou +source: + title: Sharp Small-Deviation Inequalities for Sums of Independent Nonnegative Random + Variables + authors: + - Weibo Fu + - Yanjun Han + - Guanyang Wang + - Jun Yan + - Peng Zhang + - Zhengqing Zhou + arxiv: '2607.23980' + github_repo: pengzhang91/Feige + commit: 98ab466e74280ae9d40622c19dc7f24f01b60864 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Feige.sharp_unit_slack_feige_complete +main_results: +- declaration: Feige.sharp_unit_slack_feige_complete + informal: For every positive dimension n and every small-universe probability space, + the sharp fixed-dimensional lower bound for the strict event that a sum of independent + nonnegative random variables is below its expectation plus one is (n/(n+1))^n, + and an explicit finite family proves optimality. +- declaration: VlassisThomas.exactCalibration + informal: On every small-universe probability space, the Dirichlet statistic used + in the proof is super-uniform for every independent family of measurable, integrable, + nonnegative random variables whose coordinate means are at most one. +- declaration: Grunbaum.grunbaum_centroid_halfspace + informal: Every closed halfspace containing the centroid of a positive-volume convex + body in finite-dimensional Euclidean space contains at least the sharp Grünbaum + fraction of the body's volume. +tags: +- probability +- concentration-inequalities +- convex-geometry +- sharp-constants +msc: +- 60E15 +- 52A20 diff --git a/LeanPool/projects/fel-conjecture.yaml b/LeanPool/projects/fel-conjecture.yaml new file mode 100644 index 0000000000..41486333e8 --- /dev/null +++ b/LeanPool/projects/fel-conjecture.yaml @@ -0,0 +1,29 @@ +slug: fel-conjecture +title: Fel's Conjecture on Syzygies of Numerical Semigroups +summary: Formalizes Fel's conjectural formula for syzygies of numerical semigroups. +branch: commutative algebra +entry_module: LeanPool.FelConjecture +authors: +- Evan Chen +- Kenny Lau +- Ken Ono +- Jujian Zhang +source: + arxiv: '2602.03716' + github_repo: AxiomMath/fel-polynomial +license: MIT +status: verified +provenance: AI +main_declarations: +- fels_conjecture +main_results: +- declaration: fels_conjecture + informal: Proves Fel's conjecture for the polynomial attached to numerical-semigroup + syzygies. +tags: +- number-theory +- combinatorics +- polynomials +msc: +- 20M14 +- 13D02 diff --git a/LeanPool/projects/fineqs.yaml b/LeanPool/projects/fineqs.yaml new file mode 100644 index 0000000000..380698bd24 --- /dev/null +++ b/LeanPool/projects/fineqs.yaml @@ -0,0 +1,37 @@ +slug: fineqs +title: FinEqs - reducing equations defining a subset of n-space over a finite field +summary: Formalizes the finite-field equation-reduction theorem and sharpness examples. +branch: number theory +entry_module: LeanPool.Fineqs +authors: +- Stefan Barańczuk +- Aristotle +source: + arxiv: '1906.11174' + doi: 10.5802/afst.1766 + github_repo: nasqret/fineqs +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.Fineqs.theorem_1 +- LeanPool.Fineqs.prop_1 +main_results: +- declaration: LeanPool.Fineqs.theorem_1 + informal: Reduces equations on a small finite set to at most n equations in the + same span. +- declaration: LeanPool.Fineqs.prop_1 + informal: Shows the cardinality bound in the reduction theorem is sharp. +- declaration: LeanPool.Fineqs.corollary_2 + informal: Applies the reduction theorem to affine space over a finite field. +- declaration: LeanPool.Fineqs.corollary_3 + informal: Applies the reduction theorem to nonempty projective zero loci. +- declaration: LeanPool.Fineqs.remark_example + informal: Shows n - 1 coordinate-function combinations have at least q common zeros. +tags: +- number-theory +- finite-fields +- algebraic-geometry +msc: +- 14G15 +- 11T06 diff --git a/LeanPool/projects/finite-graph-fundamental-group.yaml b/LeanPool/projects/finite-graph-fundamental-group.yaml new file mode 100644 index 0000000000..23be3fa8e1 --- /dev/null +++ b/LeanPool/projects/finite-graph-fundamental-group.yaml @@ -0,0 +1,56 @@ +slug: finite-graph-fundamental-group +title: Fundamental groups of finite connected graphs +summary: Constructs the standard interval-cell realization of a finite connected directed + multigraph and proves that its topological fundamental group is a free group of + rank E - V + 1. The proof uses an explicit spanning-tree basis and develops a verified + path-lifting cover, a contraction of the covering tree, root-change equivalences, + the tree/triviality criterion, and the corresponding free-abelian result. +branch: algebraic topology and graph theory +entry_module: LeanPool.FiniteGraphFundamentalGroup +authors: +- Arthur Freitas Ramos +- David Hulak +- Ruy de Queiroz +source: + title: 'Algebraic Topology, Additional Topic 1.A: Graphs and Free Groups' + authors: + - Allen Hatcher + url: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf + github_repo: Arthur742Ramos/finite-graph-fundamental-group + commit: dd57e3ab8bc5a7042fcc5498e778b008d87ac032 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- FiniteGraphFreeGroup.graphTopologicalFundamentalGroupEquiv +main_results: +- declaration: FiniteGraphFreeGroup.graphTopologicalFundamentalGroupEquiv + informal: For a finite weakly connected directed multigraph, the topological fundamental + group of its interval-cell quotient realization at any vertex is explicitly isomorphic + to the free group on E + 1 - V generators. +- declaration: FiniteGraphFreeGroup.graphCombinatorialToTopologicalEquiv + informal: The endomorphism group at a vertex in the graph's free groupoid is explicitly + isomorphic to the topological fundamental group of the interval-cell realization. +- declaration: FiniteGraphFreeGroup.graphCoverProjection_isCovering + informal: The canonical path-lifting projection from rooted free-groupoid paths + to graph vertices is a quiver covering, with bijective maps on every incoming + and outgoing edge star. +- declaration: FiniteGraphFreeGroup.graphRealization_tree_simplyConnected + informal: The interval-cell realization of a directed arborescence is simply connected, + via an explicit continuous contraction to its root. +- declaration: FiniteGraphFreeGroup.graphFundamentalGroup_subsingleton_iff_cycleRank_zero + informal: The combinatorial fundamental group is trivial exactly when the graph's + cycle rank E + 1 - V is zero. +- declaration: FiniteGraphFreeGroup.graphFundamentalGroupFreeAbelianizationEquiv + informal: The additive abelianization of the graph fundamental group is the free + abelian group on E + 1 - V generators. +tags: +- algebraic-topology +- graph-theory +- fundamental-groups +- free-groups +- covering-spaces +msc: +- 05C25 +- 55Q05 +- 20F65 diff --git a/LeanPool/projects/finite-max-flow-min-cut.yaml b/LeanPool/projects/finite-max-flow-min-cut.yaml new file mode 100644 index 0000000000..c2e68884f8 --- /dev/null +++ b/LeanPool/projects/finite-max-flow-min-cut.yaml @@ -0,0 +1,40 @@ +slug: finite-max-flow-min-cut +title: Finite max-flow / min-cut +summary: Develops finite networks with real nonnegative capacities, proves cut-flow + weak duality and maximum-flow attainment, and derives max-flow/min-cut equality + by an augmenting-path argument. +branch: combinatorial optimization and graph theory +entry_module: LeanPool.MaxFlowMinCut +authors: +- Juan Pablo Traverso Gianini +- Aristotle +source: + title: Reusable max-flow/min-cut component accompanying Linear-Error Clique Partitions + of Split Graphs via Structured Triangle Packing + authors: + - Juan Pablo Traverso Gianini + url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/blob/b3423f3e8c8db7c2d1b279673293ef3079faf903/preprints/PAPER_III/05_formalization/lean_v1.4_freeze/Contrib/MaxFlowMinCut.lean + github_repo: jtraverso/erdos-81-chordal-clique-partitions +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Contrib.MaxFlowMinCut.maxflow_eq_mincut +main_results: +- declaration: Contrib.MaxFlowMinCut.maxflow_eq_mincut + informal: Every finite network with real nonnegative capacities and distinct source + and sink admits a feasible flow and a source-sink cut whose flow value equals + the cut capacity. +- declaration: Contrib.MaxFlowMinCut.value_le_capacity + informal: The value of every feasible flow is at most the capacity of every source-sink + cut. +- declaration: Contrib.MaxFlowMinCut.max_flow_exists + informal: Every finite network with real nonnegative capacities admits a maximum-value + feasible flow. +tags: +- combinatorial-optimization +- graph-theory +- network-flows +msc: +- 05C21 +- 90C35 diff --git a/LeanPool/projects/five-distance-sharp.yaml b/LeanPool/projects/five-distance-sharp.yaml new file mode 100644 index 0000000000..4678dec621 --- /dev/null +++ b/LeanPool/projects/five-distance-sharp.yaml @@ -0,0 +1,47 @@ +slug: five-distance-sharp +title: Sharp Five-Distance and Sup-Norm Gap Theorems +summary: 'Formalizes sharp gap theorems for Kronecker sequences on tori: the Euclidean + five-distance theorem on the 2-torus and the sup-norm gap bound, together with explicit + arithmetic witnesses realizing the extremal cases - five distinct nearest-neighbour + distances for the Euclidean and sup-norm metrics in dimension two, and nine distinct + sup-norm distances in dimension three (Haynes-Marklof-Ramirez), via exact modular + arithmetic on best simultaneous Diophantine approximations.' +branch: number theory +entry_module: LeanPool.ThreeGap +authors: +- Vico Bonfioli +source: + title: Higher dimensional gap theorems for the maximum metric + authors: + - Alan Haynes + - Jens Marklof + - Ana Cristina Moreira Ramirez + arxiv: '2010.08842' + github_repo: ElVec1o/five-distance-sharp +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- ThreeGap.LinftyRecords3.nine_attained +main_results: +- declaration: ThreeGap.EuclideanRecords.sharp_attained + informal: Exhibits a base point on the 2-torus with an irrational coordinate and + an N whose Euclidean nearest-neighbour distances realize exactly five distinct + values, showing the Euclidean five-distance bound is sharp. + source_ref: A. Haynes and J. Marklof, A five distance theorem for Kronecker sequences, + International Mathematics Research Notices 2022(24):19747-19789, doi:10.1093/imrn/rnab205, + arXiv:2009.08444. +- declaration: ThreeGap.LinftyRecords.sharp_attained + informal: Exhibits a base point on the 2-torus with an irrational coordinate and + an N whose sup-norm nearest-neighbour distances realize exactly five distinct + values, showing the sup-norm gap bound is sharp in dimension two. +- declaration: ThreeGap.LinftyRecords3.nine_attained + informal: Exhibits a base point on the 3-torus with an irrational coordinate and + an N whose sup-norm nearest-neighbour distances realize at least nine distinct + values, showing the sup-norm gap bound is sharp in dimension three. +tags: +- number-theory +- kronecker-sequences +- three-gap +msc: +- 11J71 diff --git a/LeanPool/projects/five-eighths-theorem.yaml b/LeanPool/projects/five-eighths-theorem.yaml new file mode 100644 index 0000000000..353f8f5248 --- /dev/null +++ b/LeanPool/projects/five-eighths-theorem.yaml @@ -0,0 +1,36 @@ +slug: five-eighths-theorem +title: The 5/8 theorem +summary: Proves Gustafson's 5/8 theorem — the Mathlib commuting probability of a nonabelian + group is at most 5/8, with the infinite case immediate from commProb being zero + — via the finite-group index bounds for the center and centralizers of noncentral + elements. +branch: group theory +entry_module: LeanPool.FiveEighthsTheorem +authors: +- Xuanji Li +source: + title: What is the probability that two group elements commute? + authors: + - W. H. Gustafson + doi: 10.2307/2317564 + github_repo: ldct/lean-monorepo +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- FiveEighths.commProb_le_five_eighths +main_results: +- declaration: FiveEighths.commProb_le_five_eighths + informal: The commuting probability of a nonabelian group is at most 5/8. +- declaration: FiveEighths.centralFraction_le_quarter + informal: The center of a finite nonabelian group contains at most a quarter of + its elements. +- declaration: FiveEighths.centralizerFraction_le_half + informal: The centralizer of a noncentral element of a finite group contains at + most half of its elements. +tags: +- group-theory +- finite-groups +- commuting-probability +msc: +- 20P05 diff --git a/LeanPool/projects/flean.yaml b/LeanPool/projects/flean.yaml new file mode 100644 index 0000000000..692ae5b41f --- /dev/null +++ b/LeanPool/projects/flean.yaml @@ -0,0 +1,46 @@ +slug: flean +title: 'Flean: Floating-Point Numbers in Lean' +summary: Formalizes a configurable model of floating-point numbers in Lean 4, including + normal and subnormal representations, conversions between rationals and floats, + round-trip correctness, and rounding error bounds. +branch: numerical analysis +entry_module: LeanPool.Flean +authors: +- Joseph McKinsey +source: + doi: 10.1109/IEEESTD.2019.8766229 + url: https://github.com/josephmckinsey/flean + github_repo: josephmckinsey/flean +license: Apache-2.0 +status: verified +provenance: human +msc: +- 65G50 +- 65G30 +- 68V20 +main_declarations: +- Flean.Float +- toFloat +- toRat +- to_float_to_rat +- roundf_close +main_results: +- declaration: Flean.Float + informal: Defines the floating-point number type parameterized by a precision and + exponent configuration, with normal, subnormal, infinite, and NaN cases. +- declaration: toFloat + informal: Rounds a rational number to the nearest representable floating-point value + under a given rounding mode. +- declaration: toRat + informal: Interprets a floating-point value as the exact rational number it represents. +- declaration: to_float_to_rat + informal: Proves the round-trip identity that converting a finite nonzero float + to a rational and back recovers the original float. +- declaration: roundf_close + informal: Bounds the distance between a rational and its rounded floating-point + representation. +tags: +- floating-point +- numerical-analysis +- ieee-754 +- rounding diff --git a/LeanPool/projects/flt-regular.yaml b/LeanPool/projects/flt-regular.yaml new file mode 100644 index 0000000000..eb48e9626e --- /dev/null +++ b/LeanPool/projects/flt-regular.yaml @@ -0,0 +1,51 @@ +slug: flt-regular +title: Fermat's Last Theorem for regular primes +summary: Formalizes Kummer's proof of Fermat's Last Theorem for every odd regular + prime exponent. The development proves both cases of the theorem and includes the + required theory of cyclotomic integers, relative units, Hilbert 90 and 92, ramification, + class groups, and Kummer's lemma. +branch: algebraic number theory +entry_module: LeanPool.FltRegular +authors: +- FltRegular contributors +source: + title: Fermat's Last Theorem for regular primes + authors: + - Riccardo Brasca + - Ruben Van de Velde + - Andrew Yang + - Chris Birkbeck + - Alex J. Best + - Eric Rodriguez + url: https://github.com/leanprover-community/flt-regular + github_repo: leanprover-community/flt-regular + commit: 94a1d78956a348568c80975ec9005dc5762ab10a +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- flt_regular +main_results: +- declaration: flt_regular + informal: Fermat's Last Theorem holds for every odd regular prime exponent. +- declaration: FltRegular.caseI + informal: 'The first case of Fermat''s Last Theorem holds for every regular prime: + no primitive solution exists when the exponent does not divide the product of + the three terms.' +- declaration: FltRegular.caseII + informal: 'The second case of Fermat''s Last Theorem holds for every odd regular + prime: no primitive nonzero solution exists when the exponent divides the product + of the three terms.' +- declaration: eq_pow_prime_of_unit_of_congruent + informal: A unit of the cyclotomic field of an odd regular prime that is congruent + to an integer modulo the prime is itself a prime-th power. +tags: +- algebraic-number-theory +- fermats-last-theorem +- cyclotomic-fields +- class-groups +- regular-primes +msc: +- 11D41 +- 11R18 +- 11R29 diff --git a/LeanPool/projects/fo-zfc.yaml b/LeanPool/projects/fo-zfc.yaml new file mode 100644 index 0000000000..0c184f258c --- /dev/null +++ b/LeanPool/projects/fo-zfc.yaml @@ -0,0 +1,34 @@ +slug: fo-zfc +title: First Order Language of ZF Set Theory +summary: Formalizes ZF set theory inside Lean's `FirstOrder.Language` framework, enabling + meta-mathematical arguments such as elementary submodels. +branch: mathematical logic +entry_module: LeanPool.FoZfc +authors: +- Tetsuya Ishiu +source: + title: Set Theory + authors: + - Thomas Jech + url: https://doi.org/10.1007/3-540-44761-X + github_repo: ishiut/fo_zfc +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- FirstOrder.ZFC.ModelZF +- FirstOrder.ZFC.ext_induction +main_results: +- declaration: FirstOrder.ZFC.ModelZF + informal: Bundles the ZF axioms (Emptyset, Pairing, Union, Powerset, Infinity, Regularity, + Comprehension, Replacement) into a single model class. +- declaration: FirstOrder.ZFC.ext_induction + informal: States the principle of mathematical induction over ω externally for any + bounded formula in LZFC. +tags: +- model-theory +- set-theory +- zf +msc: +- 03B10 +- 03E30 diff --git a/LeanPool/projects/formal-learning-theory.yaml b/LeanPool/projects/formal-learning-theory.yaml new file mode 100644 index 0000000000..caa0db142b --- /dev/null +++ b/LeanPool/projects/formal-learning-theory.yaml @@ -0,0 +1,44 @@ +slug: formal-learning-theory +title: Formal Learning Theory Kernel +summary: Formalizes major equivalences and separations in PAC, VC, Gold-style, and + online learning theory. +branch: computational learning theory +entry_module: LeanPool.FormalLearningTheory +authors: +- Dhruv Gupta +source: + url: https://github.com/Zetetic-Dhruv/formal-learning-theory-kernel + github_repo: Zetetic-Dhruv/formal-learning-theory-kernel +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- fundamental_theorem +- littlestone_characterization +- pac_not_implies_online +main_results: +- declaration: vc_characterization + informal: Characterizes PAC learnability by finite VC dimension. +- declaration: fundamental_theorem + informal: Bundles the central PAC, VC, compression, Rademacher, and sample-complexity + equivalences. +- declaration: fundamental_vc_compression + informal: Relates finite VC dimension to compression with side information. +- declaration: gold_theorem + informal: States the Gold-style characterization for identification in the limit. +- declaration: mind_change_characterization + informal: Characterizes learnability by a bounded mind-change ordinal. +- declaration: littlestone_characterization + informal: Characterizes online learnability by finite Littlestone dimension. +- declaration: pac_not_implies_online + informal: Separates PAC learnability from online learnability. +- declaration: ex_not_implies_pac + informal: Separates explanatory learnability from PAC learnability. +tags: +- learning-theory +- probability +- combinatorics +- online-learning +msc: +- 68Q32 +- 68T05 diff --git a/LeanPool/projects/formalization-beyond-bethe.yaml b/LeanPool/projects/formalization-beyond-bethe.yaml new file mode 100644 index 0000000000..ff95bb65ae --- /dev/null +++ b/LeanPool/projects/formalization-beyond-bethe.yaml @@ -0,0 +1,36 @@ +title: Beyond the Bethe approximation of the permanent +summary: Constructs a deterministic polynomial-time approximation algorithm for the + permanent of every nonnegative rational matrix, with exponential base strictly smaller + than the square root of two. The development includes stable-polynomial inequalities, + rational convex optimization, finite-precision arithmetic, and a Turing-machine + complexity proof. +branch: theoretical computer science +provenance: AI +main_declarations: +- BeyondBethe.theoremOne +main_results: +- declaration: BeyondBethe.theoremOne + informal: There is a fixed rational algorithm approximating the permanent of every + nonnegative rational n-by-n matrix within c^n for an absolute c below sqrt(2), + with a deterministic polynomial-time Turing-machine implementation. +tags: +- permanent +- approximation-algorithms +- computational-complexity +- stable-polynomials +msc: +- 68W25 +- 15A15 +slug: formalization-beyond-bethe +entry_module: LeanPool.BeyondBethe +authors: +- Nima Anari +- Samuel Schlesinger +- Bolton Bailey +- Christian Reitwiessner +source: + url: https://github.com/nimaanari/formalization-beyond-bethe + github_repo: nimaanari/formalization-beyond-bethe + commit: 325cda6d2118870f7f121a9a986b7ea9ffdd7a26 +license: Apache-2.0 +status: verified diff --git a/LeanPool/projects/formalization-of-bounded-arithmetic.yaml b/LeanPool/projects/formalization-of-bounded-arithmetic.yaml new file mode 100644 index 0000000000..f8c0251797 --- /dev/null +++ b/LeanPool/projects/formalization-of-bounded-arithmetic.yaml @@ -0,0 +1,57 @@ +slug: formalization-of-bounded-arithmetic +title: Strengthened V0 Bounded Arithmetic Interfaces +summary: Formalizes first-order and two-sorted bounded-arithmetic model infrastructure, + including arithmetic model laws and string-addition laws for strengthened bundled + V0-style interfaces with propositional induction, order, cancellation, decidability, + and algebraic laws assumed as fields rather than derived from a separate raw V0 + class. +branch: mathematical logic +entry_module: LeanPool.FormalizationOfBoundedArithmetic +authors: +- ruplet +source: + title: Logical Foundations of Proof Complexity + authors: + - Stephen Cook + - Phuong Nguyen + doi: 10.1017/CBO9780511676277 + github_repo: ruplet/formalization-of-bounded-arithmetic +license: MIT +status: verified +provenance: human +main_declarations: +- V0Model.ind_strengthened_v0 +- str_add_assoc_strengthened_v0 +main_results: +- declaration: IOPENModel.add_assoc + informal: Proves associativity of addition in IOPEN models. +- declaration: IOPENModel.add_comm + informal: Proves commutativity of addition in IOPEN models. +- declaration: IDelta0Model.pred_exists + informal: Shows that nonzero elements in IDelta0 models have predecessors. +- declaration: IDelta0Model.mul_cancel_right + informal: Proves right cancellation for multiplication by a nonzero element in IDelta0 + models. +- declaration: V0Model.xind + informal: Exposes string induction inside the strengthened bundled V0Model structure, + whose fields include the comprehension and induction/order principles used downstream. +- declaration: V0Model.ind_strengthened_v0 + informal: Exposes propositional induction inside the strengthened bundled V0Model + structure, rather than deriving it from only the raw finite V0 axiom list. +- declaration: str_add_comm_strengthened_v0 + informal: Proves commutativity of string addition in V0 extension models extending + the strengthened V0Model bundle. +- declaration: str_add_assoc_strengthened_v0 + informal: Proves associativity of string addition in V0 extension models extending + the strengthened V0Model bundle, using the bundled propositional induction field. +- declaration: str_succ_assoc_strengthened_v0 + informal: Proves that string addition associates with string successor in V0 extension + models extending the strengthened V0Model bundle. +tags: +- logic +- bounded-arithmetic +- model-theory +- computational-complexity +msc: +- 03F30 +- 03D15 diff --git a/LeanPool/projects/formalizing-gmt.yaml b/LeanPool/projects/formalizing-gmt.yaml new file mode 100644 index 0000000000..a4205e9c79 --- /dev/null +++ b/LeanPool/projects/formalizing-gmt.yaml @@ -0,0 +1,53 @@ +slug: formalizing-gmt +title: Formalizing geometric measure theory +summary: A development of foundations and central results in geometric measure theory, + including Hausdorff content and density bounds, weak convergence and compactness + for Radon measures, tangent-measure machinery, Mattila's Lemmas 14.5--14.7, and + Marstrand's integrality theorem for positive finite densities. +branch: measure theory and geometric analysis +entry_module: LeanPool.FormalizingGMT +authors: +- Theodore Meek +- Ignacio Tejeda +- Annie Cao +- Nathan Pao +source: + url: https://github.com/uw-math-ai/FormalizingGMT + github_repo: uw-math-ai/FormalizingGMT + commit: 0acca38a91c4121b813f8e0b6ff719a4590bf126 + notes: The geometric-measure-theory results follow Mattila, Geometry of Sets and + Measures in Euclidean Spaces, Chapter 14. The weak-convergence, compactness, covering, + and density results follow Evans--Gariepy, Measure Theory and Fine Properties + of Functions, revised edition. +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- mattila_14_10 +- mattila_14_11 +main_results: +- declaration: mattila_14_10 + informal: If a Radon measure on Euclidean n-space has positive finite s-dimensional + density on a set of positive measure, then s is a natural number. + source_ref: Mattila, Theorem 14.10 (Marstrand's theorem). +- declaration: mattila_14_11 + informal: Every n-uniform measure on Euclidean n-space is a constant multiple of + Lebesgue measure. + source_ref: Mattila, Theorem 14.11. +- declaration: MeasureTheory.Measure.weaklyConverges_iff_compactOpenBounds + informal: Vague convergence of Radon measures on Euclidean space is equivalent to + the Portmanteau compact-set upper and open-set lower bounds. + source_ref: Evans--Gariepy, revised edition, Theorem 1.40. +- declaration: MeasureTheory.exists_vaguelyConvergent_subseq_of_compact_bounded + informal: A sequence of Radon measures on Euclidean space that is uniformly bounded + on every compact set has a vaguely convergent subsequence. + source_ref: Evans--Gariepy, revised edition, Theorem 1.41. +tags: +- geometric-measure-theory +- Hausdorff-measure +- tangent-measures +- weak-convergence +msc: +- 28A75 +- 28A78 +- 49Q15 diff --git a/LeanPool/projects/four-ap-free-permutation.yaml b/LeanPool/projects/four-ap-free-permutation.yaml new file mode 100644 index 0000000000..c757fdb050 --- /dev/null +++ b/LeanPool/projects/four-ap-free-permutation.yaml @@ -0,0 +1,38 @@ +slug: four-ap-free-permutation +title: A 4AP-free permutation of the positive integers +summary: Formalizes Boon Suan Ho's construction of a permutation of the positive integers + with no four-term arithmetic progression as a subsequence, proving the binary-order + lemmas, extension lemma, and computable limit. +branch: additive combinatorics +entry_module: LeanPool.FourAP +authors: +- Boon Suan Ho +source: + title: A 4AP-free permutation of the positive integers + authors: + - Boon Suan Ho + arxiv: '2609.12780' + url: https://github.com/boonsuan/4ap + github_repo: boonsuan/4ap + commit: 4630aefbbcb7f6537aceaeb1afac8dd3af455871 +license: MIT +status: verified +provenance: human +main_declarations: +- FourAP.exists_fourAPFree_positive_permutation +main_results: +- declaration: FourAP.exists_fourAPFree_positive_permutation + informal: There exists a bijection of the positive integers such that no four increasing + positions map to a nonconstant four-term arithmetic progression, including negative + common differences. +- declaration: FourAP.explicitPositivePermutation_apFree + informal: The explicit computable positive-integer permutation constructed in the + development has the stated four-term progression avoidance property. +tags: +- additive-combinatorics +- arithmetic-progressions +- Erdos-problem-196 +- permutations +msc: +- 11B25 +- 05D10 diff --git a/LeanPool/projects/frieze-patterns.yaml b/LeanPool/projects/frieze-patterns.yaml new file mode 100644 index 0000000000..4d6b2a8f43 --- /dev/null +++ b/LeanPool/projects/frieze-patterns.yaml @@ -0,0 +1,49 @@ +slug: frieze-patterns +title: Maxima of Coxeter frieze patterns are Fibonacci numbers +summary: Formalizes the claim that the sequence of maxima of values of (Coxeter) arithmetic + frieze patterns is the Fibonacci sequence, using a correspondence between arithmetic + frieze patterns and n-flutes. +branch: combinatorics +entry_module: LeanPool.FriezePatterns +authors: +- Antoine de Saint-Germain +- Akselai +- Jon Cheah +- Bockman Cheung +- Eaton Liu +source: + title: On upper bounds in frieze patterns + authors: + - Antoine de Saint-Germain + arxiv: '2407.16717' + doi: 10.1080/00150517.2024.2430958 + github_repo: Antoine-dSG/frieze_patterns +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- main3 +- FluteBounded +- glideSymm +- translationInvariance +main_results: +- declaration: main3 + informal: Proves that the maximum value of an arithmetic frieze pattern of width + n is exactly the n-th Fibonacci number, and that this bound is attained. +- declaration: FluteBounded + informal: Proves that every entry of an n-flute is bounded above by the n-th Fibonacci + number. +- declaration: glideSymm + informal: Proves the glide-reflection symmetry of Coxeter frieze patterns. +- declaration: translationInvariance + informal: Proves the (n+1)-periodicity (horizontal translation invariance) of Coxeter + frieze patterns. +tags: +- combinatorics +- frieze-patterns +- fibonacci +- coxeter +msc: +- 05E15 +- 11B39 +- 13F60 diff --git a/LeanPool/projects/frontiermath-open-hypergraphs.yaml b/LeanPool/projects/frontiermath-open-hypergraphs.yaml new file mode 100644 index 0000000000..ee9ded84e4 --- /dev/null +++ b/LeanPool/projects/frontiermath-open-hypergraphs.yaml @@ -0,0 +1,52 @@ +slug: frontiermath-open-hypergraphs +title: FrontierMath Ramsey Hypergraphs +summary: Formalizes lower bounds for the Brian-Larson hypergraph extremal function + H(n), including a uniform 26/25 improvement over the recursive benchmark k_n for + n ≥ 15 and a Lubell-frame asymptotic lower bound. +branch: combinatorics +entry_module: LeanPool.FrontierMathOpenHypergraphs +authors: +- Dean Cureton +source: + title: Choosing between incompatible ideals + authors: + - Will Brian + - Paul B. Larson + doi: 10.1016/j.ejc.2021.103349 + arxiv: '1908.10914' + github_repo: math-inc/FrontierMathOpen-Hypergraphs +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- HypergraphLowerBound.thm_main_H_lower_bound +main_results: +- declaration: HypergraphLowerBound.thm_main_H_lower_bound + informal: Proves H(n) ≥ 26 * k(n) / 25 for every n ≥ 15. + source_ref: Epoch AI FrontierMath, A Ramsey-style Problem on Hypergraphs. +- declaration: HypergraphLowerBound.uniform_26_25 + informal: Proves the uniform recursive construction bound A(n) ≥ 26 * k(n) / 25 + for every n ≥ 15. +- declaration: HypergraphLowerBound.frame_recurrence + informal: The substitution-frame recurrence that combines block witnesses and a + valid support frame into a larger witness. +- declaration: HypergraphLowerBound.lubell_is_frame + informal: Proves that the Lubell support family forms a valid frame for every arity + t ≥ 2. +- declaration: HypergraphLowerBound.fixed_t_bound + informal: Derives a fixed-t lower bound for the exact witness supremum from Lubell + frames. +- declaration: HypergraphLowerBound.asymptotic_lower_bound + informal: Proves the asymptotic lower bound liminf H(n) / k(n) ≥ 2 * log 2. +- declaration: HypergraphLowerBound.thm_asymptotic + informal: States the asymptotic consequence in terms of H(n) divided by n log n. +tags: +- combinatorics +- hypergraphs +- ramsey-theory +- extremal-combinatorics +- frontiermath +msc: +- 05C65 +- 05D10 +- 03E02 diff --git a/LeanPool/projects/fundamental-inequality.yaml b/LeanPool/projects/fundamental-inequality.yaml new file mode 100644 index 0000000000..6aa5fb0c06 --- /dev/null +++ b/LeanPool/projects/fundamental-inequality.yaml @@ -0,0 +1,41 @@ +slug: fundamental-inequality +title: The Fundamental Inequality of Valued Fields +summary: Defines the ramification index and residue degree of an extension of valued + fields and proves the fundamental inequality e * f ≤ finrank K L for finite extensions, + via the valuation independence of products of coset representatives and residue-field + basis lifts. +branch: number theory +entry_module: LeanPool.FundamentalInequality +authors: +- Alessandro Linzi +source: + title: Valued Fields + authors: + - Franz-Viktor Kuhlmann + url: https://github.com/linzialessandro/FundamentalInequality + github_repo: linzialessandro/FundamentalInequality +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Valuation.fundamentalInequality +- Valuation.ramificationIndex +main_results: +- declaration: Valuation.ramificationIndex + informal: Defines the ramification index e(w/v) as the relative index of the value + group of the restricted valuation inside the value group of the extension. +- declaration: Valuation.residueDegree + informal: Defines the residue degree f(w/v) as the rank of the residue field extension. +- declaration: Valuation.valuation_independence + informal: Elements with valuations in distinct cosets and residues that are linearly + independent give products that are linearly independent over the base field. +- declaration: Valuation.fundamentalInequality + informal: The fundamental inequality e(w/v) * f(w/v) ≤ finrank K L for finite extensions + of valued fields. + source_ref: F.-V. Kuhlmann, Valued Fields, Chapter 4, §4.4. +tags: +- valued-fields +- number-theory +- valuation-theory +msc: +- 12J20 diff --git a/LeanPool/projects/gap-cvp.yaml b/LeanPool/projects/gap-cvp.yaml new file mode 100644 index 0000000000..b1d032ee9e --- /dev/null +++ b/LeanPool/projects/gap-cvp.yaml @@ -0,0 +1,51 @@ +slug: gap-cvp +title: Polynomial-factor hardness of the closest vector problem +summary: Formalizes polynomial-factor NP-hardness of the closest vector problem through + an explicit reduction from 3-SAT. The development also derives NP-hardness of binary + nearest codeword, binary syndrome decoding, and finite p-norm variants of GapCVP. + It includes machine-level polynomial-time bounds, binary encodings, affine systems, + Reed--Solomon-style interpolation, and the lattice and coding-theory reductions + needed by the final results. +branch: computational complexity and lattice problems +entry_module: LeanPool.GapCVP +authors: +- OpenAI +- Dean Cureton +source: + title: Ten Advances in Mathematics and Theoretical Computer Science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- GapCVP.Comparator.gapCVP400IsNPHard +main_results: +- declaration: GapCVP.Comparator.gapCVP400IsNPHard + informal: 'GapCVP in the Euclidean norm with approximation factor n^(1/400), where + n is the lattice dimension, is NP-hard as a promise problem: every NP language + reduces in polynomial time to the promise problem on encoded well-formed instances + (positive dimension, nonsingular integer basis, positive rational radius) with + an integer target, whose YES instances have a lattice vector within the radius + of the target and whose NO instances have every lattice vector at distance greater + than n^(1/400) times the radius.' +- declaration: GapCVP.Comparator.binaryNearestCodewordIsNPHard + informal: Binary nearest codeword is NP-hard as a promise problem. +- declaration: GapCVP.Comparator.binarySyndromeDecodingIsNPHard + informal: Binary syndrome decoding is NP-hard as a promise problem. +- declaration: GapCVP.Comparator.finitePNormGapCVPIsNPHard + informal: For every rational p at least one, the finite p-norm version of GapCVP + is NP-hard as a promise problem. +tags: +- computational-complexity +- np-hardness +- lattice-problems +- coding-theory +msc: +- 68Q17 +- 68Q25 +- 11H06 +- 94B35 diff --git a/LeanPool/projects/gaussian-moments-counterexamples.yaml b/LeanPool/projects/gaussian-moments-counterexamples.yaml new file mode 100644 index 0000000000..f53d19ef27 --- /dev/null +++ b/LeanPool/projects/gaussian-moments-counterexamples.yaml @@ -0,0 +1,54 @@ +slug: gaussian-moments-counterexamples +title: Gaussian Moments Conjecture counterexamples +summary: Proves explicit complex polynomial counterexamples to the Gaussian Moments + Conjecture in every dimension at least three, using genuine expectations under finite + products of standard real Gaussian measures. Retains reusable polynomial integrability, + Stein identities, complex contractions, coefficient formulas, injective-coordinate + marginals, and related moment results. +branch: probability and polynomial algebra +entry_module: LeanPool.GaussianMomentsCounterexamples +authors: +- Christopher D. Long +source: + arxiv: '2607.18186' + url: https://github.com/long-mathematics/gaussian-moments-counterexamples + github_repo: long-mathematics/gaussian-moments-counterexamples + commit: c31bb63aaa7cfccf1191893f8e7d04fa71bae025 + announcement: https://github.com/long-mathematics/gaussian-moments-counterexamples/pull/2 + completion_date: '2026-09-17T15:47:31Z' + notes: Imports the completed Gaussian core and related moment infrastructure. The + July manuscript predates the September Lean proof release. Five and six term counts + refer to the natural complex coordinates; total degrees four and three also hold + in the original real coordinates. The separate Jacobian-map and route-arithmetic + files are excluded. No general inversion, Jacobian-reduction route, or resolution + of the two-dimensional conjecture is claimed. Upstream MIT notice is retained + in the project entry module. Upstream Lean authorship is inferred from the public + codex/lean-formalization branch; its exact proof-authoring model and extent of + human editing are unreported. This port and optimization are Codex-authored. +license: MIT +status: verified +provenance: AI +main_declarations: +- GaussianMomentsCounterexamples.not_GMC_of_three_le +main_results: +- declaration: GaussianMomentsCounterexamples.not_GMC_of_three_le + informal: 'For every n at least three, the Gaussian Moments Conjecture fails: some + complex polynomial has all positive pure Gaussian moments zero, while a polynomial + multiplier prevents eventual mixed-moment vanishing.' +- declaration: GaussianMomentsCounterexamples.master_three + informal: For every complex univariate polynomial A and positive m, the expectation + of A(Q3) times P3 to power m equals m! times the degree-m coefficient of A(X) + times (1+X) to power m-1. The explicit P3 has degree four and five terms in its + natural complex coordinates. +- declaration: GaussianMomentsCounterexamples.master_four + informal: The analogous arbitrary-A coefficient identity for the explicit four-variable + polynomial P4, of degree three and six terms in its natural complex coordinates, + gives an independent counterexample. +tags: +- gaussian-measure +- polynomial-moments +- counterexamples +- formal-power-series +msc: +- 60E05 +- 13F20 diff --git a/LeanPool/projects/gkp-density-one.yaml b/LeanPool/projects/gkp-density-one.yaml new file mode 100644 index 0000000000..87f76aa1f6 --- /dev/null +++ b/LeanPool/projects/gkp-density-one.yaml @@ -0,0 +1,63 @@ +slug: gkp-density-one +title: Density-One GKP Divisibility and Its Carry-Language Characterization +summary: Proves that the exponents k satisfying the Graham--Knuth--Patashnik divisibility + condition have natural density one. The proof gives an exact all-depth modular sieve, + quantitative bounds for possible failures in initial intervals, and a density-zero + limit for the failure proportion. It also develops the Kummer digit-sum bridge, + the binary-popcount reduction, and a verified ternary carry automaton. The universal + GKP conjecture remains open because the possible density-zero failure set may be + nonempty. +branch: number theory +entry_module: LeanPool.GKPCarry +authors: +- Egor Lyfar +source: + title: 'Concrete Mathematics: A Foundation for Computer Science, Second Edition' + authors: + - Ronald L. Graham + - Donald E. Knuth + - Oren Patashnik + url: https://cs.stanford.edu/~knuth/gkp.html + github_repo: lyfar/gkp-carry-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- GKPCarry.tendsto_gkpSuccessProportion_one +main_results: +- declaration: GKPCarry.tendsto_gkpSuccessProportion_one + informal: The proportion of exponents k below N for which nine divides the central + binomial coefficient indexed by 2^k tends to one as N tends to infinity. +- declaration: GKPCarry.card_gkpFailureExponentsBelow_le + informal: At each ternary depth n+3, the number of possible failures below N is + at most (N divided by 2*3^(n+2), plus one) times (n+9)*2^n. +- declaration: GKPCarry.gkp_exact_modular_sieve + informal: At depth n+3, exactly (n+9)*2^n of the 2*3^(n+2) exponent classes remain + uncertified; every other class gives divisibility by nine and avoids the deficient + carry language for all exponents in that class. +- declaration: GKPCarry.gkpConjecture_iff_badCarryLanguageExclusion + informal: The GKP conjecture is equivalent to the assertion that every nonexceptional + power of two avoids the explicitly defined deficient-carry regular language. This + is an exact reformulation and does not prove the remaining avoidance assertion. +- declaration: GKPCarry.nine_dvd_centralBinom_iff_two_le_ternaryDoubleCarryCount + informal: For every natural number n, its central binomial coefficient is divisible + by nine exactly when doubling the canonical ternary expansion of n creates at + least two carries. +- declaration: GKPCarry.badCarryLanguage_true_iff_shape + informal: 'The deficient-carry automaton accepts exactly three concrete little-endian + ternary word shapes: all digits zero or one; a unique top digit two; or a unique + two followed by a higher zero with all other digits zero or one.' +- declaration: GKPCarry.card_badCarryWords + informal: Among all ternary words of length n+2, exactly (n+7) times 2^n create + fewer than two carries when doubled. +- declaration: GKPCarry.nine_dvd_centralBinom_four_pow_in_finite_range + informal: For every m from 27 through 6560, nine divides the central binomial coefficient + indexed by 4^m; this is a bounded computational corollary, not the headline result. +tags: +- number-theory +- finite-automata +- asymptotic-density +msc: +- 11A63 +- 11B65 +- 68Q45 diff --git a/LeanPool/projects/goemans-cost-counterexample.yaml b/LeanPool/projects/goemans-cost-counterexample.yaml new file mode 100644 index 0000000000..7435201149 --- /dev/null +++ b/LeanPool/projects/goemans-cost-counterexample.yaml @@ -0,0 +1,57 @@ +slug: goemans-cost-counterexample +title: Counterexample to Goemans' unsplittable-flow cost conjecture +summary: Refutes Goemans' cost conjecture for single-source unsplittable flow, Conjecture + 1.3 of Traub, Vargas Koch, and Zenklusen's paper. A simple, loopless, acyclic graph + with seven vertices and nine arcs admits a splittable flow of cost 58, but every + unsplittable routing within the additive maximum-demand capacity bound costs at + least 60, attained by an explicit routing. The development proves completeness of + the walk enumeration and transfers the refutation from integers to any linearly + ordered commutative ring, including the rationals and reals. Jason Hickey directed + Claude's formal verification of the counterexample discovered by Dmitry Rybin with + GPT-5.6 Pro; upstream also acknowledges Katherine Schlitz. Planarity is not part + of the formalized claim. +branch: combinatorial optimization +entry_module: LeanPool.GoemansFlow +authors: +- Jason Hickey +source: + title: A verified counterexample to Goemans' cost conjecture + authors: + - Jason Hickey + url: https://github.com/jyh/dinitz-verify/tree/ffba3523f0edd14be3460d039f22a6b98c02fd9e + github_repo: jyh/dinitz-verify + commit: ffba3523f0edd14be3460d039f22a6b98c02fd9e + arxiv: '2308.02651' + announcement: https://leanprover.zulipchat.com/#narrow/channel/583339-AI-authored-projects/topic/Counterexample.20to.20Dinitz-Garg-Goemans.20conjecture/near/612748998 + announced_at: '2026-07-25T17:18:11Z' +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- GoemansFlow.goemans_cost_conjecture_false +main_results: +- declaration: GoemansFlow.goemans_cost_conjecture_false + informal: The rational form of Goemans' cost conjecture is false, even for simple, + loopless, acyclic finite digraphs with explicit capacities, positive demands, + and distinct terminals different from the source. + source_ref: Conjecture 1.3 of arXiv:2308.02651v1 +- declaration: GoemansFlow.not_GoemansCostConjectureFull + informal: The same cost conjecture is false over every linearly ordered commutative + ring, by transfer of the integer counterexample. +- declaration: GoemansFlow.optimum_is_sixty + informal: On the explicit seven-vertex, nine-arc instance, the minimum cost among + unsplittable routings satisfying the additive maximum-demand capacity bound is + exactly 60. +- declaration: GoemansFlow.fractional_cost + informal: The given splittable flow on the counterexample instance costs exactly + 58. +- declaration: GoemansFlow.path_complete + informal: For each of the three terminals, the lists of arcs giving a walk from + the source are exactly the two explicitly exhibited paths. +tags: +- network-flows +- combinatorial-optimization +- counterexamples +msc: +- 90C35 +- 05C21 diff --git a/LeanPool/projects/granville-moore.yaml b/LeanPool/projects/granville-moore.yaml new file mode 100644 index 0000000000..68cbff1b35 --- /dev/null +++ b/LeanPool/projects/granville-moore.yaml @@ -0,0 +1,57 @@ +slug: granville-moore +title: Integer Moore determinants and iterated Fermat quotients +summary: Proves Granville's integrality and congruence formulas for iterated Fermat + quotients at odd primes, including the exceptional order p - 1. For 1 ≤ d ≤ p and + integers x_j prime to p, it proves that the nonzero integer Moore determinant det(x_j^(p^i)) + is divisible by p^C(d+1,3), and that the next power does not divide it exactly when + the Fermat quotients q_p(x_j) are pairwise distinct modulo p. The development supplies + polynomial coefficient identities, denominator-clearing and valuation estimates, + a determinant reduction ladder, and weighted Vandermonde formulas over general commutative + rings. +branch: number theory +entry_module: LeanPool.GranvilleMoore +authors: +- Ken Ono +source: + title: The p-divisibility of the integer Moore determinant and iterated Fermat quotients + url: https://github.com/AxiomMath/GranvilleMoore + github_repo: AxiomMath/GranvilleMoore + commit: 16181f1f6a46d3001c1011c8947df13bcede948c +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- GranvilleMoore.not_pow_succ_dvd_det_mooreMatrix_iff +main_results: +- declaration: GranvilleMoore.exists_intCast_iteratedFermatQuot + informal: For an odd prime p, an integer x not divisible by p, any j ≤ p - 1, and + any k, the iterated Fermat quotient F^(j)_k(x) is an integer. + source_ref: Upstream README, first main result; attributed there to Granville. +- declaration: GranvilleMoore.dvd_factorial_mul_sub_mul_pow + informal: Under the same prime and unit hypotheses, for j ≤ p - 2 and any k, j! + F^(j)_k(x) is congruent to x q_p(x)^j modulo p, where q_p(x) = (x^(p-1) - 1)/p. + source_ref: Upstream README, second main result. +- declaration: GranvilleMoore.dvd_factorial_mul_sub_exceptional + informal: For an odd prime p and an integer x not divisible by p, (p-1)! F^(p-1)_0(x) + is congruent to x q_p(x)^(p-1) - x q_p(x) modulo p. + source_ref: Upstream README, exceptional congruence. +- declaration: GranvilleMoore.pow_dvd_det_mooreMatrix + informal: For an odd prime p, 1 ≤ d ≤ p, and integers x_j not divisible by p with + nonzero Moore determinant, p^C(d+1,3) divides det(x_j^(p^i)). + source_ref: Upstream README, Moore determinant lower bound, in divisibility form. +- declaration: GranvilleMoore.not_pow_succ_dvd_det_mooreMatrix_iff + informal: Under those determinant hypotheses, its p-adic valuation is exactly C(d+1,3) + if and only if the integer Fermat quotients q_p(x_j) have pairwise distinct residues + modulo p. The Lean statement expresses exactness as failure of divisibility by + p^(C(d+1,3)+1). + source_ref: Upstream README, equality criterion, including d = p. +tags: +- number-theory +- fermat-quotients +- determinants +- p-adic-valuations +- polynomial-identities +msc: +- 11A07 +- 11C20 +- 15A15 diff --git a/LeanPool/projects/grothendieck-vanishing.yaml b/LeanPool/projects/grothendieck-vanishing.yaml new file mode 100644 index 0000000000..e92f6afd34 --- /dev/null +++ b/LeanPool/projects/grothendieck-vanishing.yaml @@ -0,0 +1,28 @@ +slug: grothendieck-vanishing +title: Grothendieck's Vanishing Theorem +summary: Formalizes Grothendieck's vanishing theorem for sheaf cohomology on Noetherian + topological spaces. +branch: algebraic geometry +entry_module: LeanPool.GrothendieckVanishing +authors: +- Vasily Ilin +- Brian Nugent +source: + doi: 10.1007/978-1-4757-3849-0 + url: https://github.com/Vilin97/Clawristotle/tree/grothendieck-vanishing + github_repo: Vilin97/Clawristotle +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- GrothendieckVanishing +main_results: +- declaration: GrothendieckVanishing + informal: Proves vanishing of sheaf cohomology above the topological Krull dimension. +tags: +- algebraic-geometry +- sheaf-theory +- topology +msc: +- 14F06 +- 18F20 diff --git a/LeanPool/projects/hadwiger-nelson-bounds.yaml b/LeanPool/projects/hadwiger-nelson-bounds.yaml new file mode 100644 index 0000000000..7f160e0aa0 --- /dev/null +++ b/LeanPool/projects/hadwiger-nelson-bounds.yaml @@ -0,0 +1,43 @@ +slug: hadwiger-nelson-bounds +title: Known Bounds for the Hadwiger-Nelson Problem +summary: Formalizes the currently known bounds 5 <= chi(R^2) <= 7 for the chromatic + number of the Euclidean plane. The upper bound constructs an explicit Isbell-style + seven-coloring. The lower bound checks Parts' 481-vertex coloring certificate, develops + a finite doubled triangular-lattice forcing gadget, proves its exact Euclidean embedding, + and closes the four-color contradiction with a spindle construction. The exact chromatic + number remains open. +branch: graph theory and discrete geometry +entry_module: LeanPool.HadwigerNelsonBounds +authors: +- Egor Lyfar +source: + title: A human-verifiable proof of the chromatic number of the plane being at least + 5 + authors: + - Jaan Parts + arxiv: '2010.12661' + url: https://www.erdosproblems.com/508 + github_repo: lyfar/hadwiger-nelson-bounds-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- HadwigerNelsonBounds.hadwiger_nelson_known_bounds +main_results: +- declaration: HadwigerNelsonBounds.hadwiger_nelson_known_bounds + informal: The chromatic number of the unit-distance graph on the Euclidean plane + lies between five and seven, inclusive. +- declaration: HadwigerNelsonBounds.unitDistanceGraph_not_colorable_four + informal: The unit-distance graph on the Euclidean plane admits no proper coloring + with four colors. +- declaration: HadwigerNelsonBounds.chromaticNumber_le_seven + informal: An explicit Isbell-style coloring proves that the unit-distance graph + on the Euclidean plane is colorable with seven colors. +tags: +- graph-theory +- geometric-graph-theory +- graph-coloring +- hadwiger-nelson +msc: +- 05C15 +- 52C10 diff --git a/LeanPool/projects/hanson-wright.yaml b/LeanPool/projects/hanson-wright.yaml new file mode 100644 index 0000000000..dec1f4eac1 --- /dev/null +++ b/LeanPool/projects/hanson-wright.yaml @@ -0,0 +1,52 @@ +slug: hanson-wright +title: The Hanson-Wright inequality for sub-Gaussian quadratic forms +summary: Proves the finite-dimensional Hanson-Wright concentration inequality for + quadratic forms of independent mean-zero sub-Gaussian real random variables whose + maximum least global-MGF sub-Gaussian scale is positive, for matrices with positive + Frobenius and operator norms. The bound uses the fixed universal coefficient 1 / + (256 * exp(1)^2). This is the MGF normalization of the usual result; no formal identification + with an Orlicz psi-2 norm is claimed. The development derives the required moment-generating-function + certificate instead of assuming it, and contributes reusable scalar Bernstein, Chernoff, + cumulant, sub-Gaussian-process, and finite-maximum infrastructure. +branch: high-dimensional probability and concentration inequalities +entry_module: LeanPool.HansonWright +authors: +- Yuanhe Zhang +- Jason D. Lee +- Fanghui Liu +source: + title: High-Dimensional Probability (second edition) + authors: + - Roman Vershynin + doi: 10.1017/9781009490672 + url: https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html + github_repo: Lean-MoDS/StatsMLlib + commit: 37286c3d2c7e17642fbe26988be40f3ff982f944 +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LeanPool.HansonWright.hanson_wright_inequality_hdp +main_results: +- declaration: LeanPool.HansonWright.hanson_wright_inequality_hdp + informal: For independent mean-zero real random variables with finite least global-MGF + sub-Gaussian scales and positive maximum coordinate MGF scale, and a matrix with + positive Frobenius and operator norms, the centered quadratic form satisfies the + Hanson-Wright two-regime exponential tail bound with the fixed universal coefficient + 1 / (256 * exp(1)^2). +- declaration: LeanPool.HansonWright.hasHansonWrightMGF_of_subgaussian + informal: Given positive K and operator norm and C at least 64 * exp(1)^2, independent + coordinates with global-MGF sub-Gaussian parameter K^2 satisfy the local quadratic + cumulant-generating-function estimate needed for Hanson-Wright. +- declaration: LeanPool.bernstein_inequality + informal: A finite family of independent, centered, uniformly bounded real random + variables satisfies the classical scalar Bernstein tail inequality. +tags: +- probability-theory +- concentration-inequalities +- high-dimensional-probability +- random-matrices +msc: +- 60E15 +- 60B20 +- 62H12 diff --git a/LeanPool/projects/hasse-minkowski.yaml b/LeanPool/projects/hasse-minkowski.yaml new file mode 100644 index 0000000000..679075c744 --- /dev/null +++ b/LeanPool/projects/hasse-minkowski.yaml @@ -0,0 +1,44 @@ +slug: hasse-minkowski +title: The Hasse–Minkowski theorem over ℚ +summary: 'Proves the local–global principle for quadratic forms over the rationals: + a form is isotropic over ℚ exactly when it is isotropic over ℝ and over every p-adic + completion, together with Meyer''s theorem that an indefinite rational form in at + least five variables is isotropic. The development builds the Hilbert symbol at + every place of ℚ with Serre''s explicit formulas (including the 2-adic case), proves + Hilbert reciprocity and the existence theorem for prescribed local symbols, the + Hasse invariant and the rank-two and rank-three local representability criteria, + Legendre''s ternary descent, and the rank-four and rank-≥5 inductions.' +branch: number theory +entry_module: LeanPool.HasseMinkowski +authors: +- jayyswan +- Nirvana Coppola +- María Inés de Frutos-Fernández +source: + title: The Hasse–Minkowski theorem over ℚ + url: https://github.com/jayyswan/hasse-minkowski + github_repo: jayyswan/hasse-minkowski + commit: 47f0e9b5cba6e6354b882bc5856a11962ba61f4a +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- HasseMinkowski.hasseMinkowski +- HasseMinkowski.meyer +main_results: +- declaration: HasseMinkowski.hasseMinkowski + informal: A quadratic form over ℚ is isotropic if and only if it is isotropic over + ℝ and over every p-adic completion ℚ_[p] (the Hasse–Minkowski local–global principle, + in isotropy form). +- declaration: HasseMinkowski.meyer + informal: Every indefinite quadratic form over ℚ in at least five variables is isotropic + (Meyer's theorem). +tags: +- number-theory +- quadratic-forms +- local-global +- p-adic +msc: +- 11E12 +- 11E08 +- 11E88 diff --git a/LeanPool/projects/hopf-problem.yaml b/LeanPool/projects/hopf-problem.yaml new file mode 100644 index 0000000000..da20c2e986 --- /dev/null +++ b/LeanPool/projects/hopf-problem.yaml @@ -0,0 +1,47 @@ +slug: hopf-problem +title: A complex structure on the six-sphere +summary: Formalizes Levent Alpöge's construction of a complex-analytic atlas on the + standard six-sphere. The development builds a compact complex threefold from torus + fibrations with cusp and elliptic fibres, computes its homology and fundamental + group, proves that its underlying topological space is homeomorphic to the six-sphere, + and transports the complex atlas across that homeomorphism. It adapts Lean developments + by Yury G. Kudryashov and Sebastian Kumar, together with the statement from the + Formal Conjectures Authors. +branch: complex geometry and differential topology +entry_module: LeanPool.HopfProblem +authors: +- Boris Alexeev +- Yury G. Kudryashov +- Sebastian Kumar +- The Formal Conjectures Authors +source: + title: 'HopfProblem: formalization of Alpöge''s six-sphere construction' + authors: + - Boris Alexeev + url: https://github.com/plby/HopfProblem/tree/9ac8a456b526527837d7082ff775213ca8bc9809 + github_repo: plby/HopfProblem + commit: 9ac8a456b526527837d7082ff775213ca8bc9809 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Mathoverflow1973.mathoverflow_1973 +main_results: +- declaration: Mathoverflow1973.SixSphereComplexAtlas.exists_complex_analytic_atlas + informal: The standard unit six-sphere admits a complex three-dimensional charted-space + structure making it a complex-analytic manifold. +- declaration: Mathoverflow1973.SixSphereComplexAtlas.exists_complex_atlas + informal: The standard unit six-sphere admits a complex three-dimensional charted-space + structure making it a continuously differentiable complex manifold. +- declaration: Mathoverflow1973.mathoverflow_1973 + informal: 'The formalized statement of MathOverflow question 1973: the standard + six-sphere carries a complex three-dimensional manifold atlas.' +tags: +- complex-geometry +- differential-topology +- complex-manifolds +- six-sphere +- torus-fibrations +msc: +- 32Q55 +- 57R15 diff --git a/LeanPool/projects/hs-virial-nbc-identity.yaml b/LeanPool/projects/hs-virial-nbc-identity.yaml new file mode 100644 index 0000000000..6eb1c897d2 --- /dev/null +++ b/LeanPool/projects/hs-virial-nbc-identity.yaml @@ -0,0 +1,42 @@ +slug: hs-virial-nbc-identity +title: Hard-sphere Mayer NBC identity and fork-packing volume bound +summary: A kernel-checked Lean formalization of the exact no-broken-circuit volume + identity for hard-sphere Mayer cluster integrals and the stronger dimension-three + fork-packing upper bound for each tree-owned NBC region. The development proves + finite broken-circuit cancellation, the measurable finite-region decomposition, + the anchored Euclidean product-measure theorem, and its explicit flat-coordinate + transport. Its geometric result is the attachment-aware estimate Vol(NBC_T) <= + (17/32)^hardSphereNu T * hardSphereKappa^(k-1), where hardSphereNu T is the maximum + cardinality of a packing of pairwise vertex-disjoint, order-compatible forks in + T. A formal fork is an ordered triple (a,b,c) with a < b < c, center a, leaves + b,c, and tree edges {a,b} and {a,c}. The central contribution is the complete pointwise-to-integral + bridge together with the orientation-aware measure-preserving factorization that + realizes the per-fork volume suppression. +branch: mathematical physics +entry_module: LeanPool.HardSphereNBC +authors: +- ukiyois +- OpenCode agent sessions +source: + url: https://github.com/ukiyois/hs-virial-nbc-identity + github_repo: ukiyois/hs-virial-nbc-identity + commit: fd0a1e73c336574fa9655f9ab3c0cd9ce5c4fcce +license: Apache-2.0 +status: verified +main_declarations: +- PalomarHS.main_result +- PalomarHS.nbc_region_real_volume_le_fork_factor +main_results: +- declaration: PalomarHS.main_result + informal: The absolute hard-sphere Mayer coefficient, multiplied by the factorial, + equals the sum of nonnegative no-broken-circuit region volumes over spanning trees. +- declaration: PalomarHS.nbc_region_real_volume_le_fork_factor + informal: For a spanning tree in three dimensions, its no-broken-circuit region + has volume at most (17/32) to the maximum disjoint compatible-fork packing size + times the unit-ball volume to the number of tree edges. +tags: +- mathematical-physics +msc: +- 05C15 +- 82B05 +provenance: AI diff --git a/LeanPool/projects/hsd-interior-point-lp.yaml b/LeanPool/projects/hsd-interior-point-lp.yaml new file mode 100644 index 0000000000..9232734457 --- /dev/null +++ b/LeanPool/projects/hsd-interior-point-lp.yaml @@ -0,0 +1,50 @@ +slug: hsd-interior-point-lp +title: Homogeneous self-dual interior-point method for linear programming +summary: Formalizes a homogeneous self-dual interior-point method for linear programming, + including primal-dual data, Newton block solvability from full row rank, fixed Ye-Todd-Mizuno + predictor/corrector local estimates, contraction bounds, and logarithmic stopping + certificates for generated iterates. +branch: linear optimization +entry_module: LeanPool.HSDInteriorPointLP +authors: +- Makoto Yamashita +source: + title: An O(√nL)-iteration homogeneous and self-dual linear programming algorithm + authors: + - Yinyu Ye + - Michael J. Todd + - Shinji Mizuno + doi: 10.1287/moor.19.1.53 + github_repo: makoto-yamashita/proof-on-a-homogeneous-self-dual-interior-point-method-for-linear-programming +license: MIT +status: verified +provenance: human +main_declarations: +- HSDInteriorPointLP.YTM_fixed_local_theory_from_paper +main_results: +- declaration: HSDInteriorPointLP.YTM_fixed_local_theory_from_paper + informal: Packages the fixed Ye-Todd-Mizuno local theory, combining predictor and + corrector estimates for the tight and wide central neighborhoods. + source_ref: Y. Ye, M. J. Todd, and S. Mizuno, An O(√nL)-iteration homogeneous and + self-dual linear programming algorithm, Mathematics of Operations Research 19(1):53-67, + 1994. +- declaration: HSDInteriorPointLP.HSDGeneratedAlgorithm.gap_stop_even_at_logPairBound + informal: Proves that the generated even iterate satisfies the gap stopping test + at the logarithmic pair bound. +- declaration: HSDInteriorPointLP.HSDGeneratedAlgorithm.mu_stop_even_at_logPairBound + informal: Proves that the generated even iterate satisfies the duality-measure stopping + test at the scaled logarithmic pair bound. +- declaration: HSDInteriorPointLP.HLP_newton_block_system_solvable_from_full_row_rank + informal: Shows solvability of the Newton block system from the full-row-rank assumption + through finite-dimensional linear algebra. +- declaration: HSDInteriorPointLP.YTM_contraction_power_le_exp + informal: Bounds the fixed YTM contraction power by the corresponding exponential + decay estimate. +tags: +- linear-programming +- interior-point-methods +- optimization +- homogeneous-self-dual +msc: +- 90C05 +- 90C51 diff --git a/LeanPool/projects/incompleteness.yaml b/LeanPool/projects/incompleteness.yaml new file mode 100644 index 0000000000..73d032a415 --- /dev/null +++ b/LeanPool/projects/incompleteness.yaml @@ -0,0 +1,32 @@ +slug: incompleteness +title: Gödel's First and Second Incompleteness Theorems +summary: Formalizes Gödel's first and second incompleteness theorems for first-order + arithmetic, building on formalized arithmetization and provability-logic infrastructure. +branch: mathematical logic +entry_module: LeanPool.Incompleteness +authors: +- Palalansoukî +source: + doi: 10.1007/BF01700692 + github_repo: FormalizedFormalLogic/Incompleteness +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LeanPool.Incompleteness.goedelFirst +- LeanPool.Incompleteness.goedelSecond +main_results: +- declaration: LeanPool.Incompleteness.goedelFirst + informal: 'Proves Gödel''s first incompleteness theorem: a Sigma1-sound Delta1-definable + theory extending arithmetic is incomplete.' +- declaration: LeanPool.Incompleteness.goedelSecond + informal: 'Proves Gödel''s second incompleteness theorem: such a theory cannot prove + its own consistency.' +tags: +- incompleteness +- provability +- first-order-arithmetic +- mathematical-logic +msc: +- 03F40 +- 03F30 diff --git a/LeanPool/projects/infinitary-logic.yaml b/LeanPool/projects/infinitary-logic.yaml new file mode 100644 index 0000000000..4883d0ceaa --- /dev/null +++ b/LeanPool/projects/infinitary-logic.yaml @@ -0,0 +1,126 @@ +slug: infinitary-logic +title: Infinitary logic and countable model theory +summary: 'Formalizes the syntax and semantics of infinitary logic and a broad collection + of its classical theorems: Scott sentences and ranks, Karp''s theorem, model existence, + the sharp Morley--Hanf bound, Craig, Lyndon, and Malitz interpolation, López--Escobar + definability, undefinability of well-ordering, the Silver--Burgess and Morley counting + dichotomies, and an honest coded-fragment interface whose HF compactness theorem + is derived from first-order compactness.' +branch: mathematical logic and model theory +entry_module: LeanPool.InfinitaryLogic +authors: +- Cameron Freer +source: + title: Infinitary Logic in Lean 4 + authors: + - Cameron Freer + url: https://github.com/cameronfreer/infinitary-logic + github_repo: cameronfreer/infinitary-logic + commit: 4d0c43c5288952267e7d5478d79302cae17945a9 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- FirstOrder.Language.scottSentence_characterizes +main_results: +- declaration: FirstOrder.Language.scottSentence + informal: Every countable structure in a countable relational language has an associated + infinitary sentence. +- declaration: FirstOrder.Language.scottSentence_characterizes + informal: A countable structure satisfies the Scott sentence of another countable + structure exactly when the two structures are isomorphic. +- declaration: FirstOrder.Language.scottRank + informal: The Scott rank of a countable structure is the supremum of its element + ranks plus one. +- declaration: FirstOrder.Language.scottRank_lt_omega1 + informal: The Scott rank of every countable structure in a countable relational + language is a countable ordinal. +- declaration: FirstOrder.Language.karp_theorem_w + informal: Two relational structures admit a nonempty potential isomorphism exactly + when they agree on every infinitary sentence over every branching carrier in their + shared universe. +- declaration: FirstOrder.Language.model_existence + informal: Every countable set of infinitary sentences in an equality-respecting + consistency property over a countable language has a countable model. +- declaration: FirstOrder.Language.morley_hanf + informal: Beth omega-one is a Hanf bound for every infinitary sentence. +- declaration: FirstOrder.Language.Lomega1omegaHanfNumber_eq_beth_omega1 + informal: The Hanf number of countable-conjunction infinitary logic is exactly beth + omega-one. +- declaration: FirstOrder.Language.exists_small_model_of_hasArbLargeModels + informal: A sentence with arbitrarily large models has, at every infinite cardinality, + a model realizing only countably many complete types. +- declaration: FirstOrder.Language.exists_complete_sentence_of_lomega1omegaSmall + informal: Every infinitary-small model belongs to a complete infinitary subclass + contained in any prescribed sentence it satisfies. +- declaration: FirstOrder.Language.exists_complete_kCategorical_of_hasArbLargeModels + informal: A categorical sentence with arbitrarily large models has a complete categorical + refinement at the same infinite cardinality. +- declaration: FirstOrder.Language.craig_interpolation + informal: Every sentence entailment has an interpolant using only the function and + relation symbols shared by its two endpoints. +- declaration: FirstOrder.Language.craig_pcSeparation + informal: Disjoint projective infinitary classes can be separated by a single infinitary + sentence. +- declaration: FirstOrder.Language.lyndon_interpolation + informal: Every sentence entailment has an interpolant whose positive and negative + relation occurrences lie in the corresponding shared occurrence sets. +- declaration: FirstOrder.Language.malitz_interpolation + informal: An entailment with universal consequent in a relational language has a + universal interpolant over the shared vocabulary. +- declaration: FirstOrder.Language.exists_model_relPreserving + informal: If a sentence has chains of every countable length, some model carries + a relation-preserving map from the rationals. +- declaration: FirstOrder.Language.wellOrder_type_boundedness + informal: The order types of all well-ordered models of one infinitary sentence + are bounded below omega-one. +- declaration: FirstOrder.Language.wellOrdering_undefinable + informal: No single infinitary sentence defines exactly the well-orders. +- declaration: FirstOrder.Language.lopezEscobar_iff + informal: A class of coded countable structures is Borel and isomorphism-invariant + exactly when it is the model class of one infinitary sentence. +- declaration: FirstOrder.Language.lopezEscobar_action_iff + informal: The López--Escobar characterization equivalently uses invariance under + the natural permutation action on coded structures. +- declaration: FirstOrder.Language.wellOrderClass_not_measurableSet + informal: The class of coded well-orders is not a Borel set in the logic space. +- declaration: gandy_harrington_for_relation + informal: A Borel equivalence relation on a Polish space with uncountably many classes + has a continuous Cantor injection of pairwise inequivalent points. +- declaration: FirstOrder.Language.silverBurgessDichotomy + informal: A standard-Borel quotient has at most countably many classes or exactly + continuum many classes. +- declaration: FirstOrder.Language.morley_counting + informal: Assuming the Silver--Burgess dichotomy, the number of isomorphism classes + of coded countable models of an infinitary sentence is at most aleph-one or exactly + the continuum. +- declaration: FirstOrder.Language.iso_borel_of_bounded_scottHeight + informal: Isomorphism among models of an infinitary sentence is Borel whenever their + Scott heights have a common countable bound. +- declaration: FirstOrder.Language.FamilyPresentation + informal: A coded-family presentation packages codes with their indexed decoding + and a certificate for admissible infinitary formation. +- declaration: FirstOrder.Language.CodedFamily + informal: A coded family packages a certified family code with an explicitly indexed + decoding, made unique by the presentation's decoding law. +- declaration: FirstOrder.Language.AdmissibleFragment + informal: An admissible fragment is closed under exactly the coded infinitary families + certified by its presentation. +- declaration: FirstOrder.Language.hfFragment + informal: The hereditarily finite fragment consists of the infinitary formulas whose + syntax contains no infinitary connective. +- declaration: FirstOrder.Language.hfAdmissibleFragment + informal: The hereditarily finite fragment forms an admissible coded fragment without + assuming compactness data. +- declaration: FirstOrder.Language.finitaryFragment_compact + informal: Every finitely satisfiable theory contained in the first-order image fragment + has a model. +tags: +- mathematical-logic +- infinitary-logic +- model-theory +- descriptive-set-theory +msc: +- 03C75 +- 03E15 +- 03C30 diff --git a/LeanPool/projects/infinite-connes-rigidity.yaml b/LeanPool/projects/infinite-connes-rigidity.yaml new file mode 100644 index 0000000000..8fe62cfeef --- /dev/null +++ b/LeanPool/projects/infinite-connes-rigidity.yaml @@ -0,0 +1,44 @@ +slug: infinite-connes-rigidity +title: Infinite counterexamples to Connes's rigidity conjecture +summary: Constructs one finitely generated property-(T) ICC group and an infinite + family of finitely generated property-(T) ICC groups that are pairwise nonisomorphic + and are each nonisomorphic to the first group, while all of their tracial group + von Neumann algebras are normally star-isomorphic. This unconditional infinite-family + result is independent of the pooled conditional two-group Connes-rigidity formalization. +branch: operator algebras and geometric group theory +entry_module: LeanPool.InfiniteConnesRigidity +authors: +- OpenAI +- Dean Cureton +source: + title: Ten advances in mathematics and theoretical computer science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- ConnesRigidity.infiniteConnesRigidity +main_results: +- declaration: ConnesRigidity.infiniteConnesRigidity + informal: Registry-friendly alias of the exact canonical infinite-family Connes-rigidity + counterexample theorem below. +- declaration: ConnesRigidity.exists_infinite_pairwise_nonisomorphic_propertyT_icc_groups_with_isomorphic_factors + informal: There exist a finitely generated property-(T) ICC group and an infinite + sequence of finitely generated property-(T) ICC groups such that all their tracial + group von Neumann algebras are normally star-isomorphic, although the sequence + is pairwise nonisomorphic and none of its groups is isomorphic to the distinguished + group. +tags: +- operator-algebras +- von-neumann-algebras +- property-t +- group-theory +- connes-rigidity +msc: +- 46L10 +- 20F65 +- 22D10 diff --git a/LeanPool/projects/inflation-termination.yaml b/LeanPool/projects/inflation-termination.yaml new file mode 100644 index 0000000000..0d5d4bac68 --- /dev/null +++ b/LeanPool/projects/inflation-termination.yaml @@ -0,0 +1,36 @@ +slug: inflation-termination +title: Inflation termination for classical pair-source networks +summary: 'For finite binary pair-source networks without isolated vertices, classifies + when the inflation hierarchy characterizes compatibility with finite latent alphabets: + precisely forests of double stars, where order two suffices. Also proves triangle + nontermination for arbitrary measurable latent probability spaces, quantitative + obstructions, finite and measurable Finner inequalities, and graph witnesses.' +branch: causal inference and probability +entry_module: LeanPool.InflationTermination +authors: +- William Blair +source: + url: https://github.com/williamjblair/inflation-termination + github_repo: williamjblair/inflation-termination + commit: 2aa1f05ce932fdeef3896c83d287abd77cd8befb +license: MIT +status: verified +provenance: AI +main_declarations: +- TriangleInflation.Graph.classification_NW +main_results: +- declaration: TriangleInflation.Graph.classification_NW + informal: For finite binary pair-source graphs without isolated vertices, some finite + order of the Navascués–Wolfe hierarchy characterizes compatibility with finite + latent alphabets exactly for double-star forests; order two suffices there. +- declaration: TriangleInflation.no_finite_characterizing_orderM + informal: At every positive inflation order, a triangle law passes both the AI and + NW tests but has no compatible model over any measurable latent probability spaces. +tags: +- causal-inference +- probability +- inflation-hierarchy +- graph-theory +msc: +- 60A05 +- 05C05 diff --git a/LeanPool/projects/isograph-canonical-labelling.yaml b/LeanPool/projects/isograph-canonical-labelling.yaml new file mode 100644 index 0000000000..f040858935 --- /dev/null +++ b/LeanPool/projects/isograph-canonical-labelling.yaml @@ -0,0 +1,42 @@ +slug: isograph-canonical-labelling +title: Verified canonical labelling of finite graphs +summary: Implements an individualisation-refinement canonical-labelling algorithm + in the style of nauty and proves that its canonical adjacency form is invariant + under vertex relabelling and classifies finite graphs up to isomorphism. The imported + core includes the executable search, refinement, orbit-pruning and backjumping machinery + together with its equivariance, termination, soundness and optimality proofs. +branch: graph theory and verified algorithms +entry_module: LeanPool.IsoGraph +authors: +- Alex Meiburg +source: + title: IsoGraph + authors: + - Alex Meiburg + url: https://github.com/Timeroot/IsoGraph + github_repo: Timeroot/IsoGraph + commit: bbfcefd15087d1dcb63ebdad015a087e1d6890a1 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- IsoGraph.Canon.canonAdj_eq_iff +- IsoGraph.Canon.canonAdj_relabel +main_results: +- declaration: IsoGraph.Canon.canonAdj_eq_iff + informal: Two Boolean adjacency functions on Fin n have equal canonical forms exactly + when one is a relabelling of the other by a vertex permutation. +- declaration: IsoGraph.Canon.canonAdj_relabel + informal: Relabelling a finite graph by any permutation leaves its canonical adjacency + form unchanged. +- declaration: IsoGraph.Canon.exists_relabel_of_canonAdj_eq + informal: Equality of canonical adjacency forms yields an explicit permutation relating + the original adjacency functions. +tags: +- graph-theory +- graph-isomorphism +- canonical-labelling +- verified-algorithms +msc: +- 05C60 +- 68R10 diff --git a/LeanPool/projects/isoperimetric.yaml b/LeanPool/projects/isoperimetric.yaml new file mode 100644 index 0000000000..30636db315 --- /dev/null +++ b/LeanPool/projects/isoperimetric.yaml @@ -0,0 +1,32 @@ +slug: isoperimetric +title: Prekopa-Leindler, Brunn-Minkowski, and the isoperimetric inequality +summary: Formalizes the Prekopa-Leindler inequality, Brunn-Minkowski inequality, and + an isoperimetric inequality. +branch: geometric analysis +entry_module: LeanPool.Isoperimetric +authors: +- Jonathan Ho +source: + url: https://terrytao.wordpress.com/2011/09/16/the-brunn-minkowski-inequality-for-nilpotent-groups/ + github_repo: hojonathanho/isoperimetric +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- prekopa_leindler +- brunn_minkowski +- isoperimetric_inequality +main_results: +- declaration: prekopa_leindler + informal: Proves the Prekopa-Leindler inequality. +- declaration: brunn_minkowski + informal: Proves the Brunn-Minkowski inequality. +- declaration: isoperimetric_inequality + informal: Proves the isoperimetric inequality derived from Brunn-Minkowski. +tags: +- measure-theory +- geometric-inequalities +msc: +- 28A75 +- 52A40 +- 49Q20 diff --git a/LeanPool/projects/istranscendentalpi.yaml b/LeanPool/projects/istranscendentalpi.yaml new file mode 100644 index 0000000000..8b5ea3cc12 --- /dev/null +++ b/LeanPool/projects/istranscendentalpi.yaml @@ -0,0 +1,34 @@ +slug: istranscendentalpi +title: The transcendence of π +summary: Formalizes a self-contained proof, following Niven's 1939 argument, that + the real and complex number π is transcendental over the rationals. +branch: number theory +entry_module: LeanPool.IsTranscendentalPi +authors: +- James Huang +- Samuël Borza +source: + url: https://github.com/samuelborza/IsTranscendentalPi + github_repo: samuelborza/IsTranscendentalPi +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- IsTranscendentalPi +- IsTranscendentalPiReal +main_results: +- declaration: IsTranscendentalPi + informal: Proves that the complex number π is transcendental over ℚ. +- declaration: IsTranscendentalPiReal + informal: Proves that the real number π is transcendental over ℚ. +- declaration: isAlgebraic_I_mul_pi_of_isAlgebraic_pi + informal: Shows that if π is algebraic over ℚ then so is iπ. +- declaration: exists_monic_rat_poly_aeval_Ipi_eq_zero + informal: Produces a monic rational polynomial vanishing at iπ when iπ is algebraic. +tags: +- transcendence +- pi +- number-theory +- niven +msc: +- 11A41 diff --git a/LeanPool/projects/jacobian-diffgeo.yaml b/LeanPool/projects/jacobian-diffgeo.yaml new file mode 100644 index 0000000000..376af8f34e --- /dev/null +++ b/LeanPool/projects/jacobian-diffgeo.yaml @@ -0,0 +1,48 @@ +slug: jacobian-diffgeo +title: The Jacobian of a Compact Riemann Surface +summary: 'Constructs the Jacobian of a compact Riemann surface as the period-lattice + quotient of the dual of holomorphic 1-forms, and proves the surrounding theory it + needs: the genus as the dimension of the space of holomorphic 1-forms, discreteness + and full rank 2g of the period lattice, the Abel-Jacobi map and its injectivity + in positive genus, functoriality (pushforward, pullback, degree, and the projection + formula), Riemann-Roch, Serre duality via Laurent tails, the residue theorem, the + Dolbeault-Cech comparison, and finite dimensionality of the first cohomology by + a Schwartz/Montel argument. The headline no-hack guard is that the genus vanishes + exactly when the surface is homeomorphic to the sphere.' +branch: complex analysis and geometry +entry_module: LeanPool.JacobianDiffgeo +authors: +- Rado Kirov +source: + title: Jacobians of compact Riemann surfaces (AI challenge) + authors: + - Kevin Buzzard + url: https://gist.github.com/kbuzzard/778bc714030b3e974ab5f4038783d1a9 + github_repo: rkirov/jacobian-fable +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- genus_eq_zero_iff_homeo +- Jacobian.ofCurve_inj +main_results: +- declaration: genus_eq_zero_iff_homeo + informal: A compact connected Riemann surface has genus zero if and only if it is + homeomorphic to the 2-sphere. +- declaration: Jacobian.ofCurve_inj + informal: 'The Abel-Jacobi map of a compact Riemann surface of positive genus is + injective: two points with the same image coincide.' +- declaration: Jacobian.pushforward_pullback + informal: For a non-constant holomorphic map between compact Riemann surfaces, the + pushforward composed with the pullback on Jacobians is multiplication by the degree + of the map. +tags: +- riemann-surfaces +- complex-geometry +- abel-jacobi +- riemann-roch +- serre-duality +msc: +- 14H40 +- 30F30 +- 32G20 diff --git a/LeanPool/projects/johnson-lindenstrauss-lean.yaml b/LeanPool/projects/johnson-lindenstrauss-lean.yaml new file mode 100644 index 0000000000..ec24f25667 --- /dev/null +++ b/LeanPool/projects/johnson-lindenstrauss-lean.yaml @@ -0,0 +1,54 @@ +slug: johnson-lindenstrauss-lean +title: The Johnson–Lindenstrauss Lemma and Quantized JL +summary: 'Formalizes the Johnson–Lindenstrauss lemma end to end — a Gaussian random + projection preserves pairwise distances of a finite point set up to a (1 ± ε) factor, + so a distortion-ε embedding of any m-point set exists once k is logarithmic in m + — together with the quantized JL (QJL) one-bit sign sketch: unbiasedness of the + asymmetric 1-bit estimator and its sub-Gaussian distortion bound.' +branch: probability +entry_module: LeanPool.JohnsonLindenstraussLean +authors: +- claytomode +source: + title: Extensions of Lipschitz mappings into a Hilbert space + authors: + - William B. Johnson + - Joram Lindenstrauss + doi: 10.1090/conm/026/737400 + github_repo: claytomode/johnson-lindenstrauss-lean +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- JL.johnson_lindenstrauss +- JL.johnson_lindenstrauss_pointset +main_results: +- declaration: JL.johnson_lindenstrauss + informal: The abstract probabilistic-method union bound — if the summed per-pair + distortion-failure probability is below one, a distortion-preserving map exists. +- declaration: JL.johnson_lindenstrauss_pointset + informal: The end-to-end Johnson–Lindenstrauss lemma — a Gaussian random projection + into k dimensions embeds any m-point set with (1 ± ε) distortion once k is logarithmic + in m. + source_ref: S. Dasgupta, A. Gupta, An elementary proof of a theorem of Johnson and + Lindenstrauss, Random Structures and Algorithms 22 (2003), 60-65, doi:10.1002/rsa.10073. +- declaration: JL.qjlEstimator_unbiased + informal: The QJL asymmetric 1-bit estimator is unbiased for the normalized inner + product between key and query. + source_ref: 'A. Zandieh, M. Daliri, I. Han, QJL: 1-Bit Quantized JL Transform for + KV Cache Quantization with Zero Overhead, AAAI 2025, arXiv:2406.03482.' +- declaration: JL.qjlEstimator_concentration_exp + informal: 'Sub-Gaussian distortion bound for the QJL 1-bit estimator: deviation + at least ε has probability at most 2·exp(-m·ε²/(π·‖q‖²)).' + source_ref: 'A. Zandieh, M. Daliri, I. Han, QJL: 1-Bit Quantized JL Transform for + KV Cache Quantization with Zero Overhead, AAAI 2025, arXiv:2406.03482.' +tags: +- dimensionality-reduction +- random-projection +- johnson-lindenstrauss +- gaussian +- quantization +msc: +- 68W20 +- 60G15 +- 68P05 diff --git a/LeanPool/projects/kahn-kalai.yaml b/LeanPool/projects/kahn-kalai.yaml new file mode 100644 index 0000000000..2c9310c494 --- /dev/null +++ b/LeanPool/projects/kahn-kalai.yaml @@ -0,0 +1,46 @@ +slug: kahn-kalai +title: Kahn–Kalai expectation-threshold theorem +summary: Formalizes Tran and Vu's short proof of the Park–Pham theorem resolving the + Kahn–Kalai expectation-threshold conjecture. For an ℓ-bounded family of subsets + of a finite ground set, it proves the strengthened covering theorem with the explicit + constant L = 1000 and derives the threshold bound p_c(F) ≤ K q(F) log₂ ℓ with the + explicit witness K = 100000. The supporting library develops finite-set cover cost, + level-density, double-counting, binomial-tail, and numeric estimates. +branch: probabilistic combinatorics +entry_module: LeanPool.KahnKalai +authors: +- Dan Clemens Posch +source: + title: A Short Proof of Kahn-Kalai Conjecture + authors: + - Phuc Tran + - Van Vu + arxiv: 2303.02144 + doi: 10.37236/12266 + url: https://github.com/dcposch/kahn-kalai-lean + github_repo: dcposch/kahn-kalai-lean + commit: 641aa75f8e873d31442f2f7c317b2e7582a26d94 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- KahnKalai.covering_theorem +- KahnKalai.park_pham +main_results: +- declaration: KahnKalai.covering_theorem + informal: If ℓ ≤ N, p lies in [0, 1], and an ℓ-bounded family H has cover cost at + least 1/2 - 2^(-(ℓ+2)), then its upset has at least (2/3 + 2^(-(ℓ+2))) choose(N, + m) members of cardinality m, where m = floor(1000 p N log₂(ℓ+1)). For m ≤ N this + is the corresponding level-density bound; for N < m both cardinalities are zero. +- declaration: KahnKalai.park_pham + informal: There is an absolute positive constant K such that every ℓ-bounded finite + family with ℓ at least two satisfies p_c(F) ≤ K q(F) log₂ ℓ; the proof supplies + K = 100000. +tags: +- probabilistic-combinatorics +- random-structures +- threshold-phenomena +- set-systems +msc: +- 05C80 +- 60C05 diff --git a/LeanPool/projects/kalton-roberts.yaml b/LeanPool/projects/kalton-roberts.yaml new file mode 100644 index 0000000000..bd824bbe97 --- /dev/null +++ b/LeanPool/projects/kalton-roberts.yaml @@ -0,0 +1,41 @@ +slug: kalton-roberts +title: Halving the Kalton-Roberts upper bound +summary: Formalizes the paper-style Boolean-algebra theorem that every approximately + additive real-valued set function is uniformly close to a finitely additive signed + measure, proving the certified Kalton-Roberts constant bound below 19.838. +branch: analysis +entry_module: LeanPool.KaltonRoberts +authors: +- Ho Boon Suan +source: + title: Halving the original Kalton-Roberts upper bound for nearly additive set functions + authors: + - Ho Boon Suan + - Tomasz Kania + arxiv: '2606.06807' + url: https://github.com/boonsuan/KaltonRoberts + github_repo: boonsuan/KaltonRoberts +license: MIT +status: verified +provenance: AI +main_declarations: +- KaltonRoberts.KR_constant_lt +main_results: +- declaration: KaltonRoberts.KR_constant_lt + informal: Proves the headline certified bound that the Kalton-Roberts constant is + strictly less than 9919/500, i.e. 19.838. +- declaration: KaltonRoberts.KR_constant_le_C₂ + informal: Proves the exact rational upper bound C₂ = 694198146664396294486127753 + / 34994834677886019996000000 for the Kalton-Roberts constant. +- declaration: KaltonRoberts.set_algebra_bound_C₂ + informal: States the paper-style set-algebra formulation, where every approximately + additive function on a Boolean subalgebra of sets is within C₂ times the error + of a finitely additive signed measure. +tags: +- functional-analysis +- finitely-additive-measures +- kalton-roberts +msc: +- 46B20 +- 28A12 +- 05C35 diff --git a/LeanPool/projects/kaltonpeck.yaml b/LeanPool/projects/kaltonpeck.yaml new file mode 100644 index 0000000000..6cb49b4036 --- /dev/null +++ b/LeanPool/projects/kaltonpeck.yaml @@ -0,0 +1,32 @@ +slug: kaltonpeck +title: Rank parity and complex structures on the Kalton–Peck space +summary: Develops symplectic Fredholm theory and the real Kalton–Peck space to prove + rank parity of finite-rank perturbations and rule out a complex structure on a hyperplane. +branch: functional analysis +entry_module: LeanPool.KaltonPeck +authors: +- Avik Das +source: + url: https://github.com/adas1236/KaltonPeck + github_repo: adas1236/KaltonPeck + commit: 752dec47b173d05df3eeb579c1cd392335b87b60 +license: MIT +status: verified +provenance: human +main_declarations: +- KaltonPeck.noHyperplaneComplexStructure +main_results: +- declaration: KaltonPeck.rankParityGeneral + informal: The abstract symplectic Fredholm construction yields a rank-parity theorem. +- declaration: KaltonPeck.rankParityZ2 + informal: On the real Kalton–Peck space, T² + 1 has even rank whenever its rank + is finite. +- declaration: KaltonPeck.noHyperplaneComplexStructure + informal: A hyperplane of the real Kalton–Peck space cannot carry a complex structure. +tags: +- functional-analysis +- banach-spaces +- fredholm-theory +msc: +- 46B03 +- 47A53 diff --git a/LeanPool/projects/kasami-cyclic-additive.yaml b/LeanPool/projects/kasami-cyclic-additive.yaml new file mode 100644 index 0000000000..d2549e0863 --- /dev/null +++ b/LeanPool/projects/kasami-cyclic-additive.yaml @@ -0,0 +1,50 @@ +slug: kasami-cyclic-additive +title: Carlet's Kasami cyclic-additive conjecture +summary: 'Proves Carlet''s cyclic-additive conjecture for the Kasami power function: + over every finite field of characteristic two with cardinality 2^n and every k coprime + to n, the normalized derivative image has the exact coefficient triple count 2^(2n-3) + for all distinct nonzero slopes. The development combines the Müller--Cohen--Matthews + permutation theorem, multiplicative character sums, Fourier inversion, and a Fermat-cubic + Frobenius-kernel argument.' +branch: finite fields and coding theory +entry_module: LeanPool.KasamiCyclicAdditive +authors: +- D.S. McNeil +- Gábor P. Nagy +- Attila Vajda +source: + title: On a conjecture on the Kasami APN function + authors: + - Gábor P. Nagy + - Attila Vajda + arxiv: '2608.18584' + url: https://github.com/dsm054/kasami_cyclic_additive + github_repo: dsm054/kasami_cyclic_additive + commit: c8ab49ebd8e9e53f5aa7585f1d43acf1ff68d10f +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- KasamiCyclicAdditive.carlet_kasami_cyclic_additive_literature +main_results: +- declaration: KasamiCyclicAdditive.carlet_kasami_cyclic_additive_literature + informal: For every finite field K of characteristic two with |K| = 2^n, every k + coprime to n, and all distinct nonzero v₁ and v₂ in K, the coefficient triple + count of the normalized Kasami derivative image is exactly 2^(2n-3). +- declaration: KasamiCyclicAdditive.carlet_kasami_cyclic_additive_core + informal: In the normalized range 1 ≤ k < n, the same coefficient triple count is + 2^(2n-3), assuming characteristic two, |K| = 2^n, coprimality, and distinct nonzero + coefficients. +- declaration: KasamiCyclicAdditive.kasami_half_size + informal: Under the normalized admissibility hypotheses, the image of the Kasami + derivative has exactly half as many elements as the field. +tags: +- finite-fields +- coding-theory +- APN-functions +- difference-sets +- character-sums +msc: +- 11T06 +- 11T71 +- 94A60 diff --git a/LeanPool/projects/koethe-conjecture-disproof.yaml b/LeanPool/projects/koethe-conjecture-disproof.yaml new file mode 100644 index 0000000000..7d5cb3e801 --- /dev/null +++ b/LeanPool/projects/koethe-conjecture-disproof.yaml @@ -0,0 +1,64 @@ +slug: koethe-conjecture-disproof +title: Disproof of the Köthe conjecture (Krempa's matrix form) +summary: 'Refutes Köthe''s 1930 conjecture in Krempa''s matrix form: there is a ring + R (the unitization of a nil algebra of three weighted backward shifts over the algebraic + closure of F_2) with a nil two-sided ideal I such that the matrix ideal M_2(I) contains + a non-nilpotent matrix. The compared theorem is exactly the negation of the research-open + statement `Koethe.KotherConjecture.variants.general_matrix` from Google DeepMind''s + Formal Conjectures, in every universe. The supporting library proves a multiprojective + common-zero theorem via the Segre cone, the Nullstellensatz and Krull''s height + theorem, a mask-mortality theorem for one-row matrix pencils, a diagonal construction + of universal mortal sequences, and a linearization theorem reducing nilness of a + three-generator algebra to nilness of its root-row pencils.' +branch: noncommutative ring theory +entry_module: LeanPool.Koethe +authors: +- Tom Adamczewski +source: + title: A counterexample to Köthe's conjecture and a question of Rowen + authors: + - Tom Adamczewski + - Bernhard Böhmler + - Rene Marczinzik + arxiv: '2609.07996' + url: https://github.com/tadamcz/koethe + github_repo: tadamcz/koethe + commit: a94a72aec957cb28d76aa30fb97a01d440990d7f +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Koethe.KotherConjecture.variants.general_matrix.disproof +main_results: +- declaration: Koethe.KotherConjecture.variants.general_matrix.disproof + informal: It is not the case that for every ring R, every nil two-sided ideal I + of R and every finite index type n, the matrix ideal M_n(I) of M_n(R) is nil. + This is the negation of Krempa's matrix form of Köthe's conjecture, stated verbatim + as in Formal Conjectures. +- declaration: KoetheCounterexample.counterexample + informal: In every universe there is a ring R with a nil two-sided ideal I such + that some 2 x 2 matrix with entries in I is not nilpotent. +- declaration: KoetheMultiProjective.exists_common_zero + informal: Over an algebraically closed field, at most 2N polynomials that are multihomogeneous + of a common positive degree in each of N blocks of three variables have a common + zero with no block equal to zero. +- declaration: KoetheCounterexample.maskMortality + informal: Over an algebraically closed field, every one-row matrix pencil and every + periodic mask with more than half of its residues free admit a mask-compatible + word of nonzero letters whose forward product is the zero matrix. +- declaration: KoetheCounterexample.exists_universalMortalSequence + informal: Over a countable field satisfying mask mortality there is a sequence of + nonzero letter triples such that every pencil has all its windows of some fixed + positive length equal to zero. +- declaration: KoetheCounterexample.nil_of_all_pencils_nil + informal: If every root-row pencil lifted along three elements of an algebra is + nilpotent, then the non-unital subalgebra they generate is nil. +tags: +- ring-theory +- nil-ideals +- koethe-conjecture +- counterexample +- matrix-rings +msc: +- 16N40 +- 16S50 diff --git a/LeanPool/projects/komlos.yaml b/LeanPool/projects/komlos.yaml new file mode 100644 index 0000000000..7c9030fd2a --- /dev/null +++ b/LeanPool/projects/komlos.yaml @@ -0,0 +1,64 @@ +slug: komlos +title: The Komlós and Beck–Fiala bounds with constant 36 +summary: Proves that finitely many real vectors of Euclidean norm at most one admit + signs whose sum has every coordinate bounded by 36, and deduces the Beck–Fiala discrepancy + bound 36 times the square root of the maximum vertex degree for finite hypergraphs. + The Karingula–Lovett splitting argument is supported by reusable finite-distribution, + total-variation, transport, convex-hull, and matrix-discrepancy infrastructure. + The near-invariant distribution uses discrete squared tent weights, and real vectors + are handled by coordinatewise truncation toward zero. +branch: discrepancy theory +entry_module: LeanPool.Komlos +authors: +- Gabriel Dahia +source: + title: An elementary proof of the Komlós conjecture + authors: + - Sankeerth Rao Karingula + - Shachar Lovett + url: https://eccc.weizmann.ac.il/report/2026/188/ + github_repo: gdahia/Komlos + commit: d802857449234318d557361b1dbb0a27e0258528 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Komlos.discrepancy_le_of_forall_sum_sq_le_one +main_results: +- declaration: Komlos.discrepancy_le_of_forall_sum_sq_le_one + informal: For a real matrix with arbitrary finite row and column index types, if + each column has sum of squared entries at most one, its minimum signed-column + discrepancy is at most 36. + source_ref: Theorem 1.2, matrix form. +- declaration: Komlos.exists_isColouring_forall_abs_sum_apply_le + informal: Any finite family of vectors in a finite-dimensional real Euclidean space, + each of norm at most one, admits signs in {-1, 1} whose signed sum has absolute + value at most 36 in every coordinate. + source_ref: Theorem 1.2. +- declaration: Hypergraph.exists_isColouring_forall_abs_finsum_le + informal: If a hypergraph has finitely many vertices and each vertex belongs to + at most t edges, there is a {-1, 1} colouring of all vertices whose sum on each + edge has absolute value at most 36 times sqrt(t). + source_ref: Beck–Fiala consequence discussed in Section 1. +- declaration: Komlos.exists_isColouring_mean_add_sum_mem_convexHull + informal: For a finitely supported probability distribution on any real module and + n vectors, if translation by six times each vector has total variation distance + at most one third, some signed sum added to the mean belongs to the convex hull + of the distribution's support. + source_ref: Lemma 1.4, generalized to arbitrary real modules. +- declaration: Komlos.exists_nearInvariant + informal: Every positive-denominator integer grid in finite-dimensional real coordinate + space admits a finitely supported mean-zero probability distribution in [-6, 6]^d + whose total variation distance under any grid translation of Euclidean norm at + most one is at most one third. + source_ref: Lemma 1.5, using discrete squared tent weights. +tags: +- discrepancy +- vector-balancing +- hypergraphs +- probability +- total-variation +msc: +- 11K38 +- 05C65 +- 05D40 diff --git a/LeanPool/projects/krafftsieve.yaml b/LeanPool/projects/krafftsieve.yaml new file mode 100644 index 0000000000..ce035cef73 --- /dev/null +++ b/LeanPool/projects/krafftsieve.yaml @@ -0,0 +1,35 @@ +slug: krafftsieve +title: A Conditional Sieve Criterion for Twin Primes via Krafft Geometry +summary: Introduces an original conditional sieve criterion for the Twin Prime Conjecture. + Using a centered modular alignment (termed the Krafft alignment), proves that if + the Rayleigh-Ritz quotient over the orthogonal unit sphere satisfies muMin < 1, + then a twin prime pair must exist in the corresponding interval. +branch: analytic number theory +entry_module: LeanPool.KrafftSieve +authors: +- Fernando Portela +source: + doi: 10.5281/zenodo.19763833 + github_repo: ElNando888/KrafftSieve +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- KrafftSieve.mu_min_lt_one_implies_tpc +main_results: +- declaration: KrafftSieve.krafft_sieve_guarantee_with_mu_min + informal: Proves that muMin n < 1 is a sufficient condition for the existence of + a twin prime pair in the corresponding interval. +- declaration: KrafftSieve.mu_min_lt_one_implies_tpc + informal: If there are infinitely many intervals where the optimal multidimensional + weight achieves a ratio strictly less than 1, then there are infinitely many twin + primes. This is a conditional reduction; the hypothesis muMin n < 1 is not established + here for any n. +tags: +- analytic-number-theory +- sieve-theory +- twin-primes +- optimization +msc: +- 11N05 +- 11N35 diff --git a/LeanPool/projects/krohn-rhodes.yaml b/LeanPool/projects/krohn-rhodes.yaml new file mode 100644 index 0000000000..0004c2d706 --- /dev/null +++ b/LeanPool/projects/krohn-rhodes.yaml @@ -0,0 +1,61 @@ +slug: krohn-rhodes +title: Aperiodic and simple-group wreath-division towers +summary: Constructs a finite recursive wreath-division tower for every finite monoid + in universe zero, with finite aperiodic or finite simple-group factors and every + simple-group factor a monoid divisor of the original monoid. Implements a variant + of the Diekert–Kufleitner–Steinberg local-divisor argument, with reusable monoid + and semigroup division, local divisors, action coverings, wreath reassociation, + and the Krasner–Kaloujnine group embedding. The endpoint permits arbitrary aperiodic + factors and recursive intermediate bases; it does not assert the three-element flip-flop + reduction, a single transformation-monoid strong division, or quantitative decomposition + bounds. +branch: algebra +entry_module: LeanPool.KrohnRhodes +authors: +- Aditya Rao +source: + title: Krohn–Rhodes formalization by Aditya Rao + url: https://github.com/adii800/krohn-rhodes-lean/tree/8152fef3507406d2b4f4ed3760404ea88298e382 + github_repo: adii800/krohn-rhodes-lean + commit: 8152fef3507406d2b4f4ed3760404ea88298e382 + arxiv: '1111.1585' +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.KrohnRhodes.krohn_rhodes_prime_decomposition +main_results: +- declaration: LeanPool.KrohnRhodes.krohn_rhodes_prime_decomposition + informal: Every finite monoid M in universe zero admits a finite list of finite + monoid factors, each witnessed to be aperiodic or a simple group, with DivTowerWreath + M factors and every simple-group factor a monoid divisor of M. An empty tower + means M divides the trivial monoid. At a nonempty step, M divides F wreath B for + a finite action of a finite monoid B that itself has the tail tower. + source_ref: Upstream PrimeDecomposition.lean, completed September 28, 2026 at 07:36:52 + UTC and publicly released at 08:48:24 UTC. Variant of Diekert–Kufleitner–Steinberg, + arXiv:1111.1585, Sections 3–4, with left actions, the larger local divisor cM + intersect Mc, arbitrary aperiodic factors, and recursive semigroup divisions. + The stronger flip-flop and transformation-monoid claims of Theorem 4.1 are not + asserted. Upstream CITATION.cff attributes the proofs primarily to Claude Code + under Aditya Rao's direction. +- declaration: LeanPool.KrohnRhodes.krasnerKaloujnine_injective + informal: For any group G and normal subgroup N, the constructed monoid homomorphism + from G to Mathlib's regular wreath product of N and G/N is injective. No finiteness + assumption is required. + source_ref: Upstream Foundations/KrasnerKaloujnine.lean; the classical Krasner–Kaloujnine + embedding theorem. +- declaration: LeanPool.KrohnRhodes.LocalDivisor.localDivides + informal: For any monoid M and c in M, the local divisor on cM intersect Mc, with + its twisted multiplication and identity c, is a quotient of a submonoid of M. + source_ref: Diekert–Kufleitner–Steinberg, arXiv:1111.1585, Section 2.5; uses the + full local divisor rather than the smaller cMc union {c}. +tags: +- semigroup-theory +- finite-monoids +- wreath-products +- automata-theory +- group-extensions +msc: +- 20M10 +- 20M35 +- 20E22 diff --git a/LeanPool/projects/kuramoto.yaml b/LeanPool/projects/kuramoto.yaml new file mode 100644 index 0000000000..2c833f2160 --- /dev/null +++ b/LeanPool/projects/kuramoto.yaml @@ -0,0 +1,41 @@ +slug: kuramoto +title: Finite-N Kuramoto Synchronization +summary: 'Formalizes the finite-N Kuramoto model of coupled phase oscillators: the + order parameter and its unit bound, the potential/force gradient identities, smoothness + and local Lipschitz continuity of the vector field with ODE existence and uniqueness, + Lyapunov descent along trajectories, pairwise and semicircle contraction estimates, + Hebbian phase/weight dynamics, and full convergence to synchrony under all-to-all + coupling from initial phases in an open semicircle.' +branch: dynamical systems +entry_module: LeanPool.Kuramoto +authors: +- Ben Cassie +source: + title: 'Finite-N Kuramoto synchronization: order-parameter bounds and Lyapunov descent' + authors: + - Ben Cassie + url: https://github.com/velvetmonkey/kuramoto-lean + github_repo: velvetmonkey/kuramoto-lean +license: MIT +status: verified +provenance: AI +main_declarations: +- kuramotoR_norm_le_one +- weighted_lyapunov_descent +main_results: +- declaration: kuramotoR_norm_le_one + informal: The Kuramoto order parameter has norm at most one for any phase configuration. +- declaration: weighted_lyapunov_descent + informal: The weighted Kuramoto potential is non-increasing along the gradient flow, + since the sum of force times potential-derivative over all oscillators is at most + zero. +- declaration: allToAll_convergence_to_synchrony + informal: Under all-to-all coupling with positive strength and initial phases in + an open semicircle, the order parameter norm tends to one, i.e. the oscillators + synchronize. +tags: +- dynamical-systems +- synchronization +- kuramoto +msc: +- 34D06 diff --git a/LeanPool/projects/kurosh.yaml b/LeanPool/projects/kurosh.yaml new file mode 100644 index 0000000000..e68e22870d --- /dev/null +++ b/LeanPool/projects/kurosh.yaml @@ -0,0 +1,63 @@ +slug: kurosh +title: Kurosh and Schreier subgroup theorems via covering graphs +summary: Proves the Kurosh subgroup theorem for arbitrary subgroups of indexed free + products, with factors given by actual intersections with conjugates of the original + factors and with the decomposition maps induced by their inclusions. The development + constructs Bass–Serre trees, quotient graphs, graph-of-groups covers, path lifting, + and normal forms. It also gives an explicit finite Schreier basis and classifies + automorphisms of a finite regular Schreier action by its quotient group. The finite-graph + basis infrastructure is reused from LeanPool.FiniteGraphFundamentalGroup. +branch: combinatorial group theory +entry_module: LeanPool.Kurosh +authors: +- Arthur Freitas Ramos +- David Barros Hulak +- Ruy J. G. B. de Queiroz +source: + title: Schreier and Kurosh subgroup theorems via covering graphs + url: https://github.com/Arthur742Ramos/KuroshSubgroupTheorem + doi: 10.1007/BF01449159 + github_repo: Arthur742Ramos/KuroshSubgroupTheorem + commit: 911707126c8b9bb0c764bf853008fe1053c0aad9 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- GraphCoveringTheory.KuroshStatement.kurosh_decomposition_with_inclusions +main_results: +- declaration: GraphCoveringTheory.KuroshStatement.kurosh_decomposition_with_inclusions + informal: Every subgroup H of an arbitrary indexed free product admits a free product + decomposition into a free group and subgroups of H that are intersections with + conjugates of the original factors. The isomorphism to H is induced by the actual + subgroup inclusions and a homomorphism from the free factor. + source_ref: Kurosh subgroup theorem; upstream inclusion-preserving statement. +- declaration: GraphCoveringTheory.Kurosh.kurosh_decomposition + informal: H is isomorphic to the free product of its nontrivial vertex stabilizers + in the quotient Bass–Serre graph and the free group of that graph. +- declaration: GraphCoveringTheory.schreier_basis_fintype + informal: A finite-index subgroup H of a free group on any finite generator type + of cardinality r has an explicit free basis of size d*r + 1 - d, where d is its + index and subtraction is natural-number subtraction. This formulation also covers + the empty generator type. +- declaration: GraphCoveringTheory.schreier_basis_fintype_proved + informal: If the finite generator type is nonempty, the basis has the classical + Schreier size 1 + d*(r - 1). + source_ref: Schreier index formula; DOI 10.1007/BF02952517. +- declaration: GraphCoveringTheory.finiteIndexQuotientDeckGroupEquiv + informal: For a finite-index normal subgroup H of a group G, the automorphism group + of the finite regular Schreier G-action is isomorphic to G/H by right translations. + This is a theorem about the action, not a classification of general topological + covers. +- declaration: GraphCoveringTheory.Kurosh.kurosh_subgroup_is_free_of_trivial_stabilizers + informal: A subgroup with trivial stabilizers at every vertex of its quotient Bass–Serre + graph is a free group. +tags: +- group-theory +- free-products +- bass-serre-theory +- schreier-index-formula +- covering-graphs +msc: +- 20E07 +- 05C25 +- 18B40 diff --git a/LeanPool/projects/language-generation-characterization.yaml b/LeanPool/projects/language-generation-characterization.yaml new file mode 100644 index 0000000000..095896d60e --- /dev/null +++ b/LeanPool/projects/language-generation-characterization.yaml @@ -0,0 +1,73 @@ +slug: language-generation-characterization +title: Finite witnesses and the width hierarchy for language generation +summary: Characterizes generation in the limit for arbitrary families of infinite + languages over any countably infinite universe by finite positive witnesses. Ordered-history + generators admit a single set-driven normalization, chosen before the target language. + The separation-width invariant takes every value 0, 1, 2, ..., omega, omega+1; generation + is equivalent to width at most omega. Reusable infrastructure includes locking witnesses, + positive separation, bounded diagonal capture over arbitrary universes, transport + under equivalences, and finite-query normalization on natural numbers. The formalized + scope is the semantic characterization and hierarchy, not the paper's additional + density-one or Turing-computability results in Appendices D and E. +branch: formal language theory and learning theory +entry_module: LeanPool.LanguageGeneration +authors: +- Xiaoyu Li +- Andi Han +- Jiaojiao Jiang +- Junbin Gao +- Shuangping Li +- Peng Zhang +source: + title: 'Characterizing Language Generation in the Limit: Finite Witnesses and a + Separation-Width Hierarchy' + authors: + - Xiaoyu Li + - Andi Han + - Jiaojiao Jiang + - Junbin Gao + arxiv: '2609.10525' + url: https://github.com/xiaoyulics/language-generation-characterization + github_repo: xiaoyulics/language-generation-characterization + commit: 4f7d3f2148017ae15135ba39ddc4d655bb175ab6 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- GenLimit.FiniteWitness.Simplified.ordinary_iff_finiteWitnesses +main_results: +- declaration: GenLimit.FiniteWitness.Simplified.ordinary_iff_finiteWitnesses + informal: For any family H of infinite languages over a countably infinite universe, + there exists one ordered-history generator that eventually outputs an unseen target + element on every exact presentation of every target in H if and only if H admits + a finite positive witness assignment whose nonempty active subfamilies always + have infinite common intersection. +- declaration: GenLimit.FiniteWitness.Simplified.universal_normalization + informal: Every ordered-history generator over a countably infinite universe has + one set-driven normalization that locks on every infinite target on which the + original generator succeeds on all exact presentations. The normalization is chosen + before the target. +- declaration: GenLimit.FiniteWitness.ordinary_iff_width + informal: A family of infinite languages over a countably infinite universe is generatable + in the limit exactly when its separation width is at most omega. +- declaration: GenLimit.FiniteWitness.full_range + informal: Over every countably infinite universe, every value in WithTop (WithTop + Nat), representing 0, 1, 2, ..., omega, omega+1, is the separation width of some + family of infinite languages. +- declaration: GenLimit.FiniteWitness.bounded_capture + informal: Over an arbitrary universe, every sequence of uniformly bounded finite + sets has infinitely many terms contained in one set D such that D contains none + of a prescribed countable list of infinite cores. +- declaration: GenLimit.FiniteWitness.executable_universal_normalization + informal: The finite-search normalization on natural numbers locks on every infinite + language on which the supplied generator eventually succeeds on every exact presentation. + This is a semantic correctness theorem for the defined procedure, without a formal + Turing-computability claim. +tags: +- language-generation +- learning-theory +- finite-witnesses +- combinatorics +msc: +- 68Q32 +- 68Q45 diff --git a/LeanPool/projects/lattice-triangle.yaml b/LeanPool/projects/lattice-triangle.yaml new file mode 100644 index 0000000000..869a5a709f --- /dev/null +++ b/LeanPool/projects/lattice-triangle.yaml @@ -0,0 +1,29 @@ +slug: lattice-triangle +title: On the paucity of lattice triangles +summary: Formalizes an analytic lower bound used in the paucity theorem for lattice + triangles. +branch: additive combinatorics +entry_module: LeanPool.LatticeTriangle +authors: +- Evan Chen +- Kenny Lau +- Ken Ono +- Jujian Zhang +source: + arxiv: '2603.23928' + github_repo: AxiomMath/lattice-triangle +license: MIT +status: verified +provenance: AI +main_declarations: +- analyticEngine_lower_bound +main_results: +- declaration: analyticEngine_lower_bound + informal: Proves the analytic-engine lower bound underlying the lattice-triangle + result. +tags: +- number-theory +- combinatorics +msc: +- 11H06 +- 11B30 diff --git a/LeanPool/projects/lean-booleanfun.yaml b/LeanPool/projects/lean-booleanfun.yaml new file mode 100644 index 0000000000..44d0c11255 --- /dev/null +++ b/LeanPool/projects/lean-booleanfun.yaml @@ -0,0 +1,28 @@ +slug: lean-booleanfun +title: Analysis of Boolean functions in Lean +summary: Walsh-Fourier analysis on Boolean functions, including Arrow's theorem and + BLR linearity testing. +branch: combinatorics +entry_module: LeanPool.LeanBooleanfun +authors: +- Joris Roos +source: + arxiv: '2105.10386' + doi: 10.1017/CBO9781139814782 + github_repo: roos-j/lean-booleanfun +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LeanPool.LeanBooleanfun.BooleanFun.BV.dictator_of_condorcet_and_unanimous +main_results: +- declaration: LeanPool.LeanBooleanfun.BooleanFun.BV.dictator_of_condorcet_and_unanimous + informal: Arrow's theorem for two-candidate Boolean voting rules. +tags: +- boolean-functions +- fourier-analysis +- social-choice +msc: +- 06E30 +- 42C10 +- 91B14 diff --git a/LeanPool/projects/lean-codes.yaml b/LeanPool/projects/lean-codes.yaml new file mode 100644 index 0000000000..7bc7f9a4ac --- /dev/null +++ b/LeanPool/projects/lean-codes.yaml @@ -0,0 +1,38 @@ +title: Carlson special functions and complex Dirichlet averages +summary: Formalizes Carlson special functions through complex Dirichlet averages, + with simplex measures, Gamma-regularized analytic continuation, Euler inversion, + Euler–Poisson equations, and quadratic transformations. Includes the complete supporting + several-complex-variable and special-function libraries. +branch: complex analysis +main_declarations: +- PalomarSnapshot.r_first_quadratic +- PalomarSnapshot.joint_average_continuation +main_results: +- declaration: PalomarSnapshot.r_first_quadratic + informal: The Gamma-regularized Carlson R function satisfies the first quadratic + transformation for complex exponent and Dirichlet parameters with positive-real-part + unsquared variables. +- declaration: PalomarSnapshot.joint_average_continuation + informal: For a holomorphic scalar function on a convex open domain D, its Gamma-regularized + Dirichlet average extends jointly holomorphically to all complex Dirichlet parameters + and all node vectors in D. The extension agrees with the regularized simplex integral + when every parameter has positive real part; the ordinary normalized average differs + by the factor Gamma of the parameter sum. +tags: +- special-functions +- complex-analysis +- dirichlet-averages +msc: +- 33C65 +- 32A10 +provenance: AI +slug: lean-codes +entry_module: LeanPool.CarlsonFunctions +authors: +- Bastiaan J Braams +source: + url: https://github.com/bjbraams/lean-codes + github_repo: bjbraams/lean-codes + commit: fcc2be9a086c1bdd572db91f005e868b80a8d644 +license: Apache-2.0 +status: verified diff --git a/LeanPool/projects/lean-malliavin.yaml b/LeanPool/projects/lean-malliavin.yaml new file mode 100644 index 0000000000..1363eff979 --- /dev/null +++ b/LeanPool/projects/lean-malliavin.yaml @@ -0,0 +1,37 @@ +slug: lean-malliavin +title: Malliavin calculus and the Clark–Ocone representation +summary: Constructs the Malliavin derivative, Gaussian integration by parts, closability, + Wiener chaos, Brownian Itô integration and martingale representation. The Clark–Ocone + endpoint uses a generating pre-Brownian process with continuous linear coordinates + on a separable real Gaussian Banach space, its natural filtration, and pairs in + the closed Malliavin graph; the conditional-expectation sections agree almost everywhere + at positive times. +branch: stochastic analysis +entry_module: LeanPool.Malliavin +authors: +- Ezzeri Esa +source: + url: https://github.com/savarin/lean-malliavin + github_repo: savarin/lean-malliavin + commit: a1a6ccc6eedbdf15fc3aebf9146d1fddfa0c8272 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- PalomarClarkOcone.generated_clark_ocone +main_results: +- declaration: PalomarClarkOcone.generated_clark_ocone + informal: For a generating pre-Brownian process on a separable real Gaussian Banach + space whose coordinates are pointwise represented by continuous linear functionals, + constructs a time realization of the Malliavin derivative and an Itô isometry + for the natural filtration, with martingale representation for every terminal + L² variable. For pairs in the closed Malliavin graph, the Clark–Ocone formula + identifies the derivative's predictable projection with its conditional-expectation + sections for almost every positive time. +tags: +- probability +- malliavin-calculus +- stochastic-analysis +msc: +- 60H07 +- 60H05 diff --git a/LeanPool/projects/lean-model-checking.yaml b/LeanPool/projects/lean-model-checking.yaml new file mode 100644 index 0000000000..2b0e0e0a9b --- /dev/null +++ b/LeanPool/projects/lean-model-checking.yaml @@ -0,0 +1,55 @@ +slug: lean-model-checking +title: Formalizations of theorems related to model checking +summary: Proves that every linear temporal logic formula has an equivalent finite-state + nondeterministic Büchi automaton (via negation normal form, alternating Büchi automata, + and the Miyano-Hayashi breakpoint construction), and that every linear-time property + decomposes as the intersection of a safety property and a liveness property. +branch: model checking +entry_module: LeanPool.LeanModelChecking +authors: +- György Kurucz +source: + title: An Automata-Theoretic Approach to Linear Temporal Logic + authors: + - Moshe Y. Vardi + doi: 10.1007/3-540-60915-6_6 + url: https://github.com/kuruczgy/lean-model-checking + github_repo: kuruczgy/lean-model-checking +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LeanModelChecking.for_any_LTL_formula_exists_an_equivalent_NBW +main_results: +- declaration: LeanModelChecking.for_any_LTL_formula_exists_an_equivalent_NBW + informal: For any linear temporal logic formula there exists an equivalent finite-state + nondeterministic Büchi automaton, i.e. one whose state type is finite, accepting + the same language. + source_ref: M. Y. Vardi and P. Wolper, An automata-theoretic approach to automatic + program verification, LICS 1986, 332-344; C. Baier and J.-P. Katoen, Principles + of Model Checking, MIT Press, 2008, ISBN 978-0-262-02649-9, https://mitpress.mit.edu/9780262026499/principles-of-model-checking/ +- declaration: LeanModelChecking.LTL + informal: The syntax of linear temporal logic formulas over a set of atomic propositions. +- declaration: LeanModelChecking.NBW + informal: A nondeterministic Büchi automaton over a given alphabet of letters. +- declaration: LeanModelChecking.ABW.toNBW.lang_eq + informal: The Miyano-Hayashi breakpoint construction turns any alternating Büchi + automaton over a finite state space into a nondeterministic Büchi automaton accepting + the same language. + source_ref: S. Miyano and T. Hayashi, Alternating finite automata on omega-words, + Theoretical Computer Science 32 (1984), 321-330, doi:10.1016/0304-3975(84)90049-5 +- declaration: SafetyLivenessDecomposition.safety_liveness_decomposition + informal: Every linear-time property is the intersection of a safety property and + a liveness property. + source_ref: B. Alpern and F. B. Schneider, Defining liveness, Information Processing + Letters 21 (1985), 181-185, doi:10.1016/0020-0190(85)90056-0 +tags: +- model-checking +- linear-temporal-logic +- buchi-automaton +- safety-liveness +- omega-automata +msc: +- 68Q60 +- 03B44 +- 68Q45 diff --git a/LeanPool/projects/lean-operator-theory.yaml b/LeanPool/projects/lean-operator-theory.yaml new file mode 100644 index 0000000000..e59b0c25cb --- /dev/null +++ b/LeanPool/projects/lean-operator-theory.yaml @@ -0,0 +1,35 @@ +slug: lean-operator-theory +title: Unitary dilation and the Crouzeix–Palencia bound +summary: Proves unitary power dilation for Hilbert-space contractions, von Neumann’s + polynomial inequality, and the Crouzeix–Palencia polynomial spectral-set estimate + for the closed numerical range. +branch: operator theory +entry_module: LeanPool.OperatorTheory +authors: +- Ezzeri Esa +source: + url: https://github.com/savarin/lean-operator-theory + github_repo: savarin/lean-operator-theory + commit: 6f90bb7e6b056b54f56005e0416d77d6d7c2a478 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- PalomarCrouzeixPalencia.crouzeix_palencia +main_results: +- declaration: PalomarCrouzeixPalencia.exists_unitary_power_dilation + informal: Every contraction on a complex Hilbert space admits a unitary power dilation. +- declaration: PalomarCrouzeixPalencia.vonNeumann_inequality + informal: Polynomial evaluation on a contraction is bounded by the polynomial supremum + on the closed unit disk. +- declaration: PalomarCrouzeixPalencia.crouzeix_palencia + informal: The closure of the numerical range is a polynomial spectral set with constant + 1 + √2. +tags: +- operator-theory +- functional-analysis +- spectral-theory +msc: +- 47A12 +- 47A20 +- 47A25 diff --git a/LeanPool/projects/lean-poly-abc.yaml b/LeanPool/projects/lean-poly-abc.yaml new file mode 100644 index 0000000000..e88dff9876 --- /dev/null +++ b/LeanPool/projects/lean-poly-abc.yaml @@ -0,0 +1,36 @@ +slug: lean-poly-abc +title: Polynomial ABC (Mason–Stothers) and its corollaries +summary: Formalizes the Mason–Stothers theorem (the polynomial ABC theorem) and derives + the polynomial Fermat's Last Theorem, the Fermat–Catalan inequality, Davenport's + theorem, and a no-parametrization corollary. +branch: number theory +entry_module: LeanPool.LeanPolyABC +authors: +- Seewoo Lee +source: + title: Formalization of Polynomial ABC and FLT + arxiv: '2408.15180' + github_repo: seewoo5/lean-poly-abc +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LeanPolyABC.Polynomial.abc +- LeanPolyABC.Polynomial.flt +main_results: +- declaration: LeanPolyABC.Polynomial.abc + informal: Mason–Stothers — for coprime non-constant polynomials with a+b=c, the + max degree is below the degree of the radical of abc. +- declaration: LeanPolyABC.Polynomial.flt + informal: The polynomial Fermat's Last Theorem — no nontrivial coprime polynomial + solutions of aⁿ+bⁿ=cⁿ for n ≥ 3. +- declaration: LeanPolyABC.Polynomial.flt_catalan + informal: The polynomial Fermat–Catalan inequality bounding solutions of aᵖ+bᵍ=cʳ. +tags: +- number-theory +- polynomials +- algebra +- mason-stothers +msc: +- 11C08 +- 12E05 diff --git a/LeanPool/projects/lean-quantumalg.yaml b/LeanPool/projects/lean-quantumalg.yaml new file mode 100644 index 0000000000..edaf26cd8d --- /dev/null +++ b/LeanPool/projects/lean-quantumalg.yaml @@ -0,0 +1,52 @@ +slug: lean-quantumalg +title: Lean-QuantumAlg +summary: Provides a Mathlib-based library of formally verified quantum-algorithm definitions, + primitives, and endpoints, including QFT, QSP, quantum kernels, amplitude amplification, + Grover search, phase estimation, order finding, teleportation, and a Lie-algebraic + barren-plateau witness. +branch: quantum computing +entry_module: LeanPool.LeanQuantumAlg +authors: +- QudeLeap Team +source: + title: Lean-QuantumAlg + authors: + - QudeLeap Team + url: https://github.com/QudeLeap/Lean-QuantumAlg + github_repo: QudeLeap/Lean-QuantumAlg +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- QuantumAlg.GroverSearch.main_success_probability +main_results: +- declaration: QuantumAlg.QuantumFourierTransform.main + informal: Proves the QFT action on computational-basis states and records exact + circuit counts. +- declaration: QuantumAlg.ReflectionBasedQuantumSignalProcessing.main + informal: Characterizes reflection-based single-qubit QSP transforms in the Chebyshev + basis. +- declaration: QuantumAlg.TrigonometricQuantumSignalProcessing.main + informal: Characterizes trigonometric single-qubit QSP transforms in the YZZYZ form. +- declaration: QuantumAlg.QuantumKernel.main + informal: Proves that fidelity quantum-kernel Gram matrices are positive semidefinite. +- declaration: QuantumAlg.QuantumKernelAdvantage.separation + informal: States the conditional quantum-kernel learning separation from discrete-log + hardness. +- declaration: QuantumAlg.fullControllability_hasBarrenPlateau + informal: Exhibits an exponentially vanishing loss-variance family for a fully controllable + dynamical Lie algebra, conditional on the assumed Ragone second-moment variance + law (the Haar/Weingarten averaging identity Var = P_g(rho)*P_g(O)/dim, taken as + a hypothesis rather than derived from a circuit twirl). +- declaration: QuantumAlg.GroverSearch.main_success_probability + informal: Derives Grover search success probability from the amplitude-amplification + closed form. +tags: +- quantum-computing +- quantum-algorithms +- quantum-signal-processing +- quantum-kernels +msc: +- 81P68 +- 81P45 +- 68Q12 diff --git a/LeanPool/projects/lean-scottish-book-155.yaml b/LeanPool/projects/lean-scottish-book-155.yaml new file mode 100644 index 0000000000..7cc7b700e8 --- /dev/null +++ b/LeanPool/projects/lean-scottish-book-155.yaml @@ -0,0 +1,29 @@ +slug: lean-scottish-book-155 +title: A counterexample to Scottish Book Problem 155 +summary: A Lean proof that there are real Banach spaces X and Y and a bijection U + from X to Y which preserves every pairwise distance on every closed ball of radius + one quarter while failing to be a global isometry. This gives a counterexample to + Scottish Book Problem 155. +branch: functional analysis +entry_module: LeanPool.ScottishBook155 +authors: +- Yoshito Ishiki +source: + url: https://github.com/yoshito-ishiki-math/lean-scottish-book-155 + github_repo: yoshito-ishiki-math/lean-scottish-book-155 + commit: 233eff4af7901eadee7927a1eb1f814ad0f3c118 +license: MIT +status: verified +main_declarations: +- ScottishBook155.claim14 +main_results: +- declaration: ScottishBook155.claim14 + informal: There exist real Banach spaces and a bijection between them that preserves + all pairwise distances on each closed ball of radius one quarter but is not a + global isometry. +tags: +- functional-analysis +msc: +- 46B20 +- 54E40 +provenance: AI diff --git a/LeanPool/projects/lean-scv.yaml b/LeanPool/projects/lean-scv.yaml new file mode 100644 index 0000000000..e9a5e1f266 --- /dev/null +++ b/LeanPool/projects/lean-scv.yaml @@ -0,0 +1,41 @@ +slug: lean-scv +title: Classical several complex variables +summary: Formalizes Cauchy and Taylor theory, Hartogs phenomena, removable singularities, + analytic sets and germs, Weierstrass theory, holomorphic convexity, Cartan–Thullen + and Bochner tube theorems, plurisubharmonic functions, Levi convexity, and elementary + Runge-domain theory. Weierstrass preparation and coordinate-origin germ Noetherianity + overlap LocalComplexGeometry; this development also supplies germ unique factorization + and global extension and convexity results. +branch: complex analysis +entry_module: LeanPool.SeveralComplexVariables +authors: +- Bastiaan J Braams +source: + url: https://github.com/bjbraams/lean-scv + github_repo: bjbraams/lean-scv + commit: caef1ae776ff79933718312357980d46628d3702 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- SeveralComplexVariables.analyticOnNhd_of_separately_analytic_locally_bounded +main_results: +- declaration: SCV.cauchy_formula_polydisc + informal: Holomorphic functions on a polydisc satisfy the iterated Cauchy integral + formula. +- declaration: SeveralComplexVariables.analyticOnNhd_of_separately_analytic_locally_bounded + informal: On an open subset of a finite complex coordinate space, a Banach-valued + separately analytic map that is locally bounded is jointly analytic. + source_ref: Boas (2013), Section 2.7, Theorem 7 (the classical two-variable scalar + case). +- declaration: SCV.identity_theorem + informal: Holomorphic functions on a connected domain that agree on a nonempty open + subset agree throughout the domain. +tags: +- complex-analysis +- several-complex-variables +- holomorphic-functions +msc: +- 32A10 +- 32D05 +- 32E10 diff --git a/LeanPool/projects/lean-snumbers.yaml b/LeanPool/projects/lean-snumbers.yaml new file mode 100644 index 0000000000..859089da29 --- /dev/null +++ b/LeanPool/projects/lean-snumbers.yaml @@ -0,0 +1,35 @@ +slug: lean-snumbers +title: Pietsch s-numbers and the maximal difference theorem +summary: Formalizes the five classical s-number sequences, their comparison inequalities, + the maximal difference theorem, compact-operator singular value decomposition, John + ellipsoids, entropy numbers, and compactness characterizations. +branch: functional analysis +entry_module: LeanPool.SNumbers +authors: +- Mario Ullrich +source: + url: https://github.com/mario-ullrich/lean-snumbers + github_repo: mario-ullrich/lean-snumbers + commit: d984f39b53eb1579920fe987de10382dcf553f4e +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- SNumbers.approximationNumber_le_e_mul_hilbertNumber +main_results: +- declaration: SNumbers.approximationNumber_le_e_mul_hilbertNumber + informal: For a bounded operator, its nth approximation number is at most e times + (n+1) times its nth Hilbert number. +- declaration: SNumbers.allSNumbers_eq_on_HilbertSpace + informal: Every s-number sequence agrees with the approximation numbers on operators + between Hilbert spaces. +- declaration: John.john_decomposition + informal: In John position, the identity decomposes into a positive weighted sum + of rank-one operators from contact points. +tags: +- functional-analysis +- operator-theory +- s-numbers +msc: +- 47B06 +- 46B20 diff --git a/LeanPool/projects/lean-spectral-theory.yaml b/LeanPool/projects/lean-spectral-theory.yaml new file mode 100644 index 0000000000..cf66cae929 --- /dev/null +++ b/LeanPool/projects/lean-spectral-theory.yaml @@ -0,0 +1,36 @@ +slug: lean-spectral-theory +title: lean-spectral-theory +summary: The spectral theorem for possibly unbounded self-adjoint operators on complex + Hilbert spaces, formalized through the Cayley transform and real projection-valued + measures. The selected spectral theorem constructs a PVM whose scalar measures give + the exact finite-second-moment domain and first-moment quadratic form, with measurable-set + uniqueness, and is proved equivalent to equality with the unbounded coordinate spectral + integral. The selected Stone theorem proves both that strongly continuous one-parameter + unitary groups have self-adjoint generators and that every self-adjoint operator + generates such a group. +branch: functional analysis +entry_module: LeanPool.SpectralTheory +authors: +- Ezzeri Esa +source: + url: https://github.com/savarin/lean-spectral-theory + github_repo: savarin/lean-spectral-theory + commit: d19e279f0ab244657f02ce678e8d67fa69e2f407 +license: Apache-2.0 +status: verified +main_declarations: +- PalomarSpectralStone.spectral_theorem_intrinsic +main_results: +- declaration: PalomarSpectralStone.spectral_theorem_intrinsic + informal: Every self-adjoint partial linear operator on a complex Hilbert space + has a real projection-valued measure representing its domain and scalar moments, + unique on Borel sets. +- declaration: PalomarSpectralStone.stone_theorem_intrinsic + informal: Every strongly continuous one-parameter unitary group has a self-adjoint + generator, and every self-adjoint operator generates such a group. +tags: +- functional-analysis +msc: +- 47A10 +- 47B15 +provenance: AI diff --git a/LeanPool/projects/lean-superorthogonality.yaml b/LeanPool/projects/lean-superorthogonality.yaml new file mode 100644 index 0000000000..7b95c5e64b --- /dev/null +++ b/LeanPool/projects/lean-superorthogonality.yaml @@ -0,0 +1,31 @@ +slug: lean-superorthogonality +title: Square-function estimates from type IV superorthogonality +summary: Proves the type IV superorthogonality square-function estimate for finite + and countable index sets, including the pointwise combinatorial estimate and unconditional + convergence in L^(2r). +branch: harmonic analysis +entry_module: LeanPool.Superorthogonality +authors: +- Joris Roos +source: + url: https://github.com/roos-j/lean-superorthogonality + github_repo: roos-j/lean-superorthogonality + commit: 6d3a3645c1e6f6a3c50bfdc12c4a51250e63bc31 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Superorthogonal.sqfct_estimate_of_type_iv_superorthogonal +main_results: +- declaration: Superorthogonal.sqfct_estimate_of_type_iv_superorthogonal + informal: For an integer r >= 1, a countable type IV superorthogonal family in L^(2r), + with almost-everywhere summable squared norms and square function in L^(2r), converges + unconditionally in L^(2r) and satisfies the square-function bound with constant + C(r). +tags: +- harmonic-analysis +- superorthogonality +- square-functions +msc: +- 42B25 +- 46E30 diff --git a/LeanPool/projects/lean4-gl-coalgebras.yaml b/LeanPool/projects/lean4-gl-coalgebras.yaml new file mode 100644 index 0000000000..016c08cf90 --- /dev/null +++ b/LeanPool/projects/lean4-gl-coalgebras.yaml @@ -0,0 +1,27 @@ +slug: lean4-gl-coalgebras +title: Craig Interpolation for Gödel-Löb logic via coalgebraic proofs +summary: Formalizes a proof-theoretic Craig interpolation theorem for Gödel-Löb provability + logic, built on a coalgebraic notion of circular proof. +branch: provability logic +entry_module: LeanPool.Lean4GlCoalgebras +authors: +- Madeleine Gignoux +source: + doi: 10.1134/S0081543811060198 + github_repo: mgignoux/lean4-gl-coalgebras +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Lean4GlCoalgebras.interpolation +main_results: +- declaration: Lean4GlCoalgebras.interpolation + informal: Proves the Craig interpolation theorem for Gödel-Löb provability logic. +tags: +- modal-logic +- provability-logic +- craig-interpolation +- proof-theory +msc: +- 03B45 +- 03F45 diff --git a/LeanPool/projects/lean4-itree.yaml b/LeanPool/projects/lean4-itree.yaml new file mode 100644 index 0000000000..c5bc0e287f --- /dev/null +++ b/LeanPool/projects/lean4-itree.yaml @@ -0,0 +1,57 @@ +slug: lean4-itree +title: Coinductive Interaction Trees using QPFs +summary: Formalizes coinductive interaction trees as the M-type of a polynomial functor + (via Mathlib QPFs), proving the lawful functor and monad structure, a bisimulation + equality coinciding with propositional equality, and iteration and effect-handler + interpretation combinators, supported by a vendored parameterized-coinduction (Paco) + library. +branch: coinduction and semantics +entry_module: LeanPool.Lean4Itree +authors: +- Paul Mure +- Joonhyup Lee +source: + arxiv: '1906.00046' + doi: 10.1145/3371119 + github_repo: mit-plv/lean4-itree +license: MIT +status: verified +provenance: human +main_declarations: +- Lean4Itree.ITree +- Lean4Itree.ITree.ieq_iff_eq +- Lean4Itree.plfp_acc +main_results: +- declaration: Lean4Itree.ITree + informal: Defines the coinductive interaction tree as the final coalgebra (M-type) + of the interaction-tree polynomial functor. +- declaration: Lean4Itree.ITree.IEq + informal: Defines bisimulation equality between interaction trees as a coinductive + fixed point. +- declaration: Lean4Itree.ITree.ieq_iff_eq + informal: Proves that bisimulation equality of interaction trees coincides with + propositional equality. +- declaration: Lean4Itree.ITree.bind_assoc + informal: Proves associativity of monadic bind for interaction trees, part of the + LawfulMonad instance. +- declaration: Lean4Itree.ITree.iter + informal: Defines the iteration combinator producing an interaction tree from a + step function. +- declaration: Lean4Itree.ITree.interp + informal: Defines interpretation of an interaction tree against an effect handler + into an arbitrary iterable monad. +- declaration: Lean4Itree.plfp_acc + informal: Proves the accumulation principle for the parameterized least fixed point + underlying the Paco coinduction library. + source_ref: C.-K. Hur, G. Neis, D. Dreyer, V. Vafeiadis, The Power of Parameterization + in Coinductive Proof, POPL 2013, doi:10.1145/2429069.2429093. +tags: +- coinduction +- interaction-trees +- monads +- qpf +- semantics +msc: +- 68Q55 +- 18C50 +- 68N18 diff --git a/LeanPool/projects/leancomplexanalysis.yaml b/LeanPool/projects/leancomplexanalysis.yaml new file mode 100644 index 0000000000..8a35dd1756 --- /dev/null +++ b/LeanPool/projects/leancomplexanalysis.yaml @@ -0,0 +1,41 @@ +slug: leancomplexanalysis +title: Formalized Complex Analysis in Lean +summary: Formalizes Poisson integral formulas on the unit disc and arbitrary centered + discs, Herglotz–Riesz representation, and Harnack's inequality on the unit disc, + plus infrastructure for univalent function classes S and Σ. +branch: complex analysis +entry_module: LeanPool.LeanComplexAnalysis +authors: +- seb488 +- Aristotle +source: + url: https://github.com/seb488/LeanComplexAnalysis + github_repo: seb488/LeanComplexAnalysis +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LeanPool.LeanComplexAnalysis.harnack_ineq +main_results: +- declaration: LeanPool.LeanComplexAnalysis.poisson_integral_of_harmonicOn_unitDisc_continuousOn_closedUnitDisc + informal: Proves the Poisson integral formula for harmonic functions on the unit + disc. +- declaration: LeanPool.LeanComplexAnalysis.poisson_integral_of_harmonicOn_disc_continuousOn_closedDisc + informal: Proves the Poisson integral formula for harmonic functions on arbitrary + centered discs. +- declaration: LeanPool.LeanComplexAnalysis.HerglotzRiesz_representation_harmonic + informal: Proves that every positive harmonic function on the unit disc with value + 1 at 0 has a unique Herglotz–Riesz representation by a probability measure on + the unit circle. +- declaration: LeanPool.LeanComplexAnalysis.harnack_ineq + informal: Proves Harnack's inequality for positive harmonic functions on the unit + disc. +tags: +- complex-analysis +- harmonic-functions +- poisson-integral +- univalent-functions +msc: +- 30A99 +- 31A05 +- 30C55 diff --git a/LeanPool/projects/leanmodularforms.yaml b/LeanPool/projects/leanmodularforms.yaml new file mode 100644 index 0000000000..0f05eea017 --- /dev/null +++ b/LeanPool/projects/leanmodularforms.yaml @@ -0,0 +1,35 @@ +slug: leanmodularforms +title: Modular forms and the generalized residue theorem +summary: Formalizes the Hungerbühler-Wasem generalized residue theorem for Cauchy + principal value integrals through poles and applies it to the valence formula for + modular forms on the modular group. +branch: complex analysis +entry_module: LeanPool.LeanModularForms +authors: +- Chris Birkbeck +source: + arxiv: '1808.00997' + doi: 10.1155/2019/6130464 + github_repo: CBirkbeck/LeanModularForms +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- generalizedResidueTheorem +- valence_formula_textbook_orbit_finsum +main_results: +- declaration: generalizedResidueTheorem + informal: Proves the Hungerbühler-Wasem generalized residue theorem for piecewise-C1 + curves through poles. +- declaration: generalizedResidueTheorem_simplePoles + informal: Specializes the generalized residue theorem to the simple-pole case. +- declaration: valence_formula_textbook_orbit_finsum + informal: Proves the valence formula for modular forms on the modular group. +tags: +- modular-forms +- complex-analysis +- residue-theorem +- valence-formula +msc: +- 11F11 +- 30E20 diff --git a/LeanPool/projects/lehmer-e10.yaml b/LeanPool/projects/lehmer-e10.yaml new file mode 100644 index 0000000000..766a4d3748 --- /dev/null +++ b/LeanPool/projects/lehmer-e10.yaml @@ -0,0 +1,40 @@ +slug: lehmer-e10 +title: Lehmer's Polynomial and the E10 Coxeter Element +summary: 'Proves that Lehmer''s degree-10 polynomial — the integer polynomial of smallest + known Mahler measure above 1 — is irreducible over the integers, and that it is + the characteristic polynomial of a Coxeter element of the E10 Weyl group (the rank-10 + instance of McMullen). Against Mathlib''s Mahler-measure API it computes M(L) exactly + as the Salem root with the unconditional bounds 1 < M(L) < 7/5, states Lehmer''s + conjecture as a defined Prop (never assumed) with the strong form implying the gap + form, and contrasts the E8 Coxeter element (order exactly 30) with the E10 element + (infinite order in GL(10,Z)): the E-series crosses from torsion to a Salem matrix + at rank 10, and the crossing value is Lehmer''s number.' +branch: number theory +entry_module: LeanPool.LehmerE10 +authors: +- Dillon Ryan +source: + github_repo: dillon-11/lehmer-E10 + url: https://github.com/dillon-11/lehmer-E10 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LehmerE10.main_theorem +main_results: +- declaration: LehmerE10.main_theorem + informal: Lehmer's polynomial L(x) = x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + + x + 1 is irreducible over the integers, and the characteristic polynomial of a + Coxeter element of the E10 Weyl group equals L. +- declaration: LehmerE10.lehmer_mahlerMeasure + informal: The Mahler measure of L equals its Salem root mu, with the unconditional + bounds 1 < M(L) < 7/5. +tags: +- number-theory +- mahler-measure +- salem-numbers +- coxeter-groups +msc: +- 11R06 +- 11C08 +- 20F55 diff --git a/LeanPool/projects/lentil.yaml b/LeanPool/projects/lentil.yaml new file mode 100644 index 0000000000..1911a01eee --- /dev/null +++ b/LeanPool/projects/lentil.yaml @@ -0,0 +1,47 @@ +slug: lentil +title: 'Lentil: Temporal Logic of Actions in Lean 4' +summary: A shallow embedding of Leslie Lamport's Temporal Logic of Actions (TLA) in + Lean 4, with the temporal modalities, standard propositional and temporal proof + rules, weak-fairness and leads-to reasoning, and an interactive proof mode for TLA + judgments. +branch: temporal logic +entry_module: LeanPool.Lentil +authors: +- Qiyuan Zhao +source: + url: https://github.com/verse-lab/Lentil + github_repo: verse-lab/Lentil +license: Apache-2.0 +status: verified +provenance: human +msc: +- 03B44 +- 68Q60 +main_declarations: +- TLA.always +- TLA.eventually +- TLA.always_induction +- TLA.wf1 +main_results: +- declaration: TLA.always + informal: Defines the TLA "always" (box) temporal modality, asserting a predicate + holds on every suffix of an execution. +- declaration: TLA.eventually + informal: Defines the TLA "eventually" (diamond) temporal modality, asserting a + predicate holds on some suffix of an execution. +- declaration: TLA.always_induction + informal: The always-induction rule, equating `□ p` with `p` together with `□ (p + → ◯ p)`. +- declaration: TLA.wf1 + informal: The WF1 weak-fairness proof rule deriving a leads-to property from a next-state + action under weak fairness. +- declaration: TLA.wf_as_leads_to + informal: Characterizes weak fairness of an action as a leads-to property between + its enabledness and its occurrence. +- declaration: TLA.leads_to_trans + informal: Transitivity of the leads-to relation. +tags: +- temporal-logic +- tla +- formal-verification +- proof-mode diff --git a/LeanPool/projects/li-criterion-rh-equivalence-lean.yaml b/LeanPool/projects/li-criterion-rh-equivalence-lean.yaml new file mode 100644 index 0000000000..68b414ef96 --- /dev/null +++ b/LeanPool/projects/li-criterion-rh-equivalence-lean.yaml @@ -0,0 +1,42 @@ +slug: li-criterion-rh-equivalence-lean +title: Li's criterion for the Riemann Hypothesis +summary: 'Li''s criterion for the Riemann zeta function, formalized in Lean 4 over + Mathlib: Mathlib''s `RiemannHypothesis` holds if and only if every Li-Keiper coefficient + of the completed xi function has nonnegative real part. The zero-indexed analytic + coefficient at n is Li''s classical lambda_(n+1), the n-th Taylor coefficient at + 0 of the logarithmic derivative of s |-> xi(1/(1-s)). A second compared theorem + identifies this analytic coefficient with Li''s arithmetic coefficient, the symmetrized + sum over the nontrivial zeros counted with multiplicity, and proves that symmetrized + family summable so the sum is the genuine one. Both results are unconditional: the + biconditional, the coefficient identification, the genus-1 Hadamard factorization + of xi (with multiplicity), genus-1 summability of 1/|rho|^2, and the growth bound + order(xi) <= 1 are all proved. This is a formalization of a classically known equivalence + (X.-J. Li, 1997); it is not a proof of the Riemann Hypothesis, nor of the positivity + of the coefficients.' +branch: analytic number theory +entry_module: LeanPool.LiCriterion +authors: +- Nicholas Bulka +source: + url: https://github.com/nicholasbulka/li-criterion-rh-equivalence-lean + github_repo: nicholasbulka/li-criterion-rh-equivalence-lean + commit: 35df682f3b709ffe5fbcfdd452dfa964bd622b87 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- li_criterion +- li_coefficients_eq_zero_sum +main_results: +- declaration: li_criterion + informal: The Riemann hypothesis is equivalent to nonnegativity of every Li–Keiper + coefficient. +- declaration: li_coefficients_eq_zero_sum + informal: The analytic Li–Keiper coefficients equal the symmetrically summed arithmetic + coefficients over the nontrivial zeros. +tags: +- number-theory +- riemann-hypothesis +- complex-analysis +msc: +- 11M26 diff --git a/LeanPool/projects/line-graph-colouring-matching.yaml b/LeanPool/projects/line-graph-colouring-matching.yaml new file mode 100644 index 0000000000..ff147a4d7e --- /dev/null +++ b/LeanPool/projects/line-graph-colouring-matching.yaml @@ -0,0 +1,57 @@ +slug: line-graph-colouring-matching +title: Line-graph colourings and matching decompositions +summary: 'Connects Mathlib line-graph colourings to palette-indexed partitions of + actual graph edges into Mathlib matching subgraphs. Both conversions are inverse, + preserve each matching exactly, and allow empty colours without a default colour + or a finiteness assumption. An edge equivalence yields exact edge accounting: natural-number + counts require only finite edges and a finite palette, while indexed cardinal counts + allow arbitrary vertices, edges, and palettes. This is classical representation + interoperability, not a new colouring bound or a claim of priority.' +branch: graph theory +entry_module: LeanPool.GraphColouringMatching +authors: +- Juan Pablo Traverso Gianini +source: + title: Original Lean Pool graph-colouring and matching adapters + url: https://github.com/jtraverso/lean-pool/tree/4c1fddb73f9891afc2186f8f11066ea2a08fe4fc/LeanPool/GraphColouringMatching + github_repo: jtraverso/lean-pool + commit: 4c1fddb73f9891afc2186f8f11066ea2a08fe4fc +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- SimpleGraph.lineGraphColoringEquivMatchingDecomposition +main_results: +- declaration: SimpleGraph.lineGraphColoringEquivMatchingDecomposition + informal: Proper colourings of the line graph on any palette are equivalent to partitions + of the original edges into matching subgraphs indexed by that palette, with empty + classes permitted. + source_ref: New adapter proved in GraphColouringMatching/Basic.lean, using Mathlib's + existing lineGraph, Coloring and Subgraph.IsMatching. +- declaration: SimpleGraph.MatchingDecomposition.edgeMatching_lineGraphColoring + informal: Converting a matching decomposition to a colouring and extracting one + colour class returns exactly its original matching subgraph. +- declaration: SimpleGraph.MatchingDecomposition.edgeEquivSigma + informal: Original graph edges are in bijection with the disjoint union of the edge + types of the palette-indexed matching subgraphs. +- declaration: SimpleGraph.MatchingDecomposition.card_edges_eq_sum + informal: For a finite edge set and finite palette, the number of graph edges equals + the sum of the numbers of edges in its matching classes, even when the vertex + set is infinite. +- declaration: SimpleGraph.MatchingDecomposition.cardinal_edges_eq_sum + informal: Without any finiteness assumption on vertices, edges, or colours, the + cardinality of the graph edge set equals the indexed cardinal sum of the matching-class + edge cardinalities, with a universe lift on the graph edge cardinality when required. + source_ref: Cardinal.lean applies Mathlib's indexed cardinal sum to the existing + edgeEquivSigma; no infinite graph-colouring theorem is assumed. +- declaration: SimpleGraph.lineGraph_colorable_iff_matchingDecomposition + informal: The line graph is k-colourable if and only if the graph edges decompose + into k matchings, allowing unused colours. +tags: +- graph-theory +- line-graphs +- edge-colouring +- matchings +msc: +- 05C15 +- 05C70 diff --git a/LeanPool/projects/list-edge-coloring-complete.yaml b/LeanPool/projects/list-edge-coloring-complete.yaml new file mode 100644 index 0000000000..8970bed40e --- /dev/null +++ b/LeanPool/projects/list-edge-coloring-complete.yaml @@ -0,0 +1,43 @@ +slug: list-edge-coloring-complete +title: Häggkvist–Janssen list edge colouring of complete graphs +summary: Proves that arbitrary lists of at least n colours on the edges of a complete + graph with n vertices admit a proper list edge colouring. The proof constructs and + proves uniqueness of the local rank certificate, identifies a nonzero extended-star + polynomial coefficient, and applies Mathlib's combinatorial Nullstellensatz. This + is an alternative proof presentation, not a literal translation of the article's + argument. The conclusion uses Mathlib's actual edge type and line-graph colouring, + with no certificate hypothesis and no restriction on the colour universe. This formalizes + a classical theorem, not a new bound or a solution of the general list-edge conjecture. +branch: graph theory +entry_module: LeanPool.ListEdgeColoringComplete +authors: +- Juan Pablo Traverso Gianini +source: + title: New Bounds on the List-Chromatic Index of the Complete Graph and Other Simple + Graphs + authors: + - Roland Häggkvist + - Jeannette Janssen + doi: 10.1017/S0963548397002927 + url: https://github.com/jtraverso/haggkvist-janssen/tree/26d7b2ee0e14813f0e42c3f8628d6b216327c8df + github_repo: jtraverso/haggkvist-janssen + commit: 26d7b2ee0e14813f0e42c3f8628d6b216327c8df +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.ListEdgeColoringComplete.exists_listEdgeColoring_complete +main_results: +- declaration: LeanPool.ListEdgeColoringComplete.exists_listEdgeColoring_complete + informal: For every finite vertex type V, arbitrary colour type and finite list + on each edge of the complete graph, if every list has cardinality at least the + number of vertices, there is a proper colouring of the line graph choosing a colour + from each list. This includes orders zero and one. + source_ref: Häggkvist–Janssen (1997), Theorems 1.1 and 3.1, complete-graph case. +tags: +- graph-theory +- list-edge-colouring +- complete-graphs +- combinatorial-nullstellensatz +msc: +- 05C15 diff --git a/LeanPool/projects/local-complex-geometry.yaml b/LeanPool/projects/local-complex-geometry.yaml new file mode 100644 index 0000000000..be21455f0c --- /dev/null +++ b/LeanPool/projects/local-complex-geometry.yaml @@ -0,0 +1,63 @@ +slug: local-complex-geometry +title: Foundational local complex-analytic geometry +summary: 'Develops a common Lean foundation for several classical local theorems in + several complex variables: complex-analytic Weierstrass preparation with full germ + uniqueness, Noetherianity of holomorphic-germ rings, finite projection for hypersurface + germs, the holomorphic constant-rank normal form, and the finite-family local analytic + Nullstellensatz. The imported development also exposes the arbitrary-ideal Nullstellensatz + and regular level-set, submersion, and immersion consequences.' +branch: several complex variables and local analytic geometry +entry_module: LeanPool.LocalComplexGeometry +authors: +- BochaoKong +source: + title: Foundational theorems in local complex-analytic geometry + authors: + - BochaoKong + url: https://github.com/BochaoKong/nullstellensatz + github_repo: BochaoKong/nullstellensatz + commit: 029697242294aea7989bf5d391199696a43490df +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LocalComplexGeometry.localAnalyticNullstellensatz +main_results: +- declaration: LocalComplexGeometry.localAnalyticNullstellensatz + informal: If a finite family of functions analytic at the origin has common zero + set locally contained in the zero set of another analytic function, then a positive + power of the latter is locally an analytic linear combination of the family. +- declaration: LocalComplexGeometry.holomorphicGerm_isNoetherian + informal: For every finite complex dimension, including dimension zero, the local + ring of holomorphic germs at the origin is Noetherian. +- declaration: LocalComplexGeometry.hypersurface_finiteProjection + informal: A nonzero hypersurface germ vanishing at the origin admits a linear coordinate + change whose local quotient is finite free over the base germ ring, together with + an explicit prepared equation and a proper, surjective finite-fiber projection + of the local zero set. +- declaration: LocalComplexGeometry.holomorphic_constantRank_normalForm + informal: An analytic map of locally constant complex rank is locally equivalent, + through analytic source and target coordinates, to the standard rank projection. +- declaration: ClassicalComplexWPT.classicalComplexWeierstrassPreparation + informal: A complex-analytic germ of exact order in one distinguished variable factors + uniquely into a nonvanishing analytic unit and a monic Weierstrass polynomial + whose analytic coefficient germs vanish at the origin. + source_ref: BochaoKong/classical-complex-wpt, commit b4a7273fe5c9752753c52e10494097569089642d + (2026-08-28). +- declaration: LocalComplexGeometry.analyticNullstellensatz_ideal + informal: For every ideal of the local holomorphic-germ ring, the ideal of germs + vanishing on its local zero set equals its radical. +- declaration: LocalComplexGeometry.holomorphic_regularLevelSet_parametrization + informal: Near a regular point, the level set of a holomorphic map admits an analytic + parametrization with an analytic local inverse on the level set. +tags: +- complex-analysis +- analytic-geometry +- nullstellensatz +- weierstrass-preparation +msc: +- 32A05 +- 32A10 +- 32B05 +- 32C25 +- 13E05 diff --git a/LeanPool/projects/long-gaps-between-primes.yaml b/LeanPool/projects/long-gaps-between-primes.yaml new file mode 100644 index 0000000000..4ab676e7fc --- /dev/null +++ b/LeanPool/projects/long-gaps-between-primes.yaml @@ -0,0 +1,44 @@ +slug: long-gaps-between-primes +title: Improved long gaps between consecutive primes +summary: Formalizes the unconditional lower bound showing that, for all sufficiently + large X, the largest gap between consecutive primes below X is at least a positive + constant times log X (log log X)^2 log log log log X / (log log log X)^2. The development + includes the short-translates sieve proposition, divisor coefficient estimates, + truncated product weights, moment bounds, and the Erdos--Rankin covering argument + used to derive the headline result. +branch: analytic number theory +entry_module: LeanPool.LongGapsBetweenPrimes +authors: +- OpenAI +source: + title: Improved Long Gaps Between Primes + authors: + - OpenAI + url: https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/long_gaps.pdf + github_repo: openai/LongGapsBetweenPrimes + commit: 03a1190d0bc5502d9f54eeb60ad3e45e22b0df0b +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LongGapsBetweenPrimes.long_prime_gaps +main_results: +- declaration: LongGapsBetweenPrimes.long_prime_gaps + informal: There are a positive constant c and a real threshold X0 such that for + every X at least X0, some pair of consecutive indexed primes below X has gap greater + than c times log X (log log X)^2 log log log log X / (log log log X)^2. +- declaration: LongGapsBetweenPrimes.long_gap_theorem + informal: For all sufficiently large X, there are consecutive primes p < q with + q at most X and q - p at least a fixed positive multiple of the improved long-gap + scale. +- declaration: LongGapsBetweenPrimes.short_translates + informal: For all sufficiently large x, every sufficiently sparse subset S of an + interval of length between x and x(log x)^2 has a translate b + Q(x)t + S consisting + entirely of composite numbers, with 1 <= t <= exp x. +tags: +- analytic-number-theory +- prime-gaps +- sieve-theory +- erdos-rankin +msc: +- 11N05 diff --git a/LeanPool/projects/low-weight-pauli-dynamics.yaml b/LeanPool/projects/low-weight-pauli-dynamics.yaml new file mode 100644 index 0000000000..34e5feafd5 --- /dev/null +++ b/LeanPool/projects/low-weight-pauli-dynamics.yaml @@ -0,0 +1,63 @@ +slug: low-weight-pauli-dynamics +title: Low-weight Pauli dynamics and truncation error +summary: Derives a normalized Pauli 2-norm truncation-error bound for concrete circuits + of disjoint-support Hermitian Pauli rotations, with explicit locality, angle, step-count, + and cutoff hypotheses. Develops a faithful matrix representation, Pauli coefficient + reconstruction, finite Schur bounds, and reusable scalar multi-jump ladder estimates. + Also proves a combinatorial light-cone bound and exact counterexamples showing why + repeated product-formula layers must count toward the layer exponent. This is the + norm-level analysis, not a formalization of expectation-value accuracy, Hamiltonian + approximation, or the paper's full algorithm. +branch: quantum information +entry_module: LeanPool.LowWeightPauliDynamics +authors: +- Jue Xu +source: + title: Classical Simulation of Noiseless Quantum Dynamics without Randomness + arxiv: '2601.15770' + url: https://github.com/Jue-Xu/Lean4LPD/tree/ad0561661c46ca4c331d21104468e5dc65b2a028 + github_repo: Jue-Xu/Lean4LPD + commit: ad0561661c46ca4c331d21104468e5dc65b2a028 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Lean4LPD.PauliString.pauliNorm_layerStep_error_le_model_of_source_regime +main_results: +- declaration: Lean4LPD.PauliString.pauliNorm_layerStep_error_le_model_of_source_regime + informal: For an observable supported on Pauli weights at most ko, a block of Γ + positive-many disjoint-support layers of Hermitian rotations of weight at most + kh, kh ≥ 2, and |sin θ| ≤ a ≤ αt/r with α > 0 and t ≥ 0, truncation to weight + ko + m(kh - 1) after each of r blocks has Pauli 2-norm error at most (2Γ(kh - + 1)αt)^(m+1) (e(m+1))^(ko/(kh-1)) times the input norm. The conditions include + m ≥ 1, r ≥ 5, m ≤ r, rΓ ≥ 8(m+1)^2, and r ≥ 8e^2(ko+kh-1)αt. Γ counts every layer + of the actual block, including repeated product-formula layers. + source_ref: Upstream STATEMENTS.md and Pauli/LayerError.lean at the pinned commit; + norm-level counterpart of the revised supplement's truncation bound. +- declaration: Lean4LPD.l2_opNorm_le_schur + informal: A finite rectangular matrix over any RCLike field, with absolute row sums + bounded by nonnegative R and absolute column sums bounded by nonnegative C, has + Euclidean operator norm at most sqrt(RC). + source_ref: Upstream Schur.lean, finite-dimensional Schur test. +- declaration: Lean4LPD.PauliString.reachable_weight_le + informal: Every Pauli string reachable through a list of disjoint-support kh-local + layers, kh ≥ 1, has weight at most its initial weight times kh raised to the number + of layers. Reachability is combinatorial; this theorem does not assert that a + reachable coefficient is nonzero. + source_ref: Upstream Pauli/LayerWitness.lean, effective-layer light cone. +- declaration: Lean4LPD.LayerCounterexample.actual_discard_exceeds_gamma_bound + informal: For the explicit eight-qubit second-order word L1,L2,L2,L1 and angles + 0 < θ < π/2, truncating the evolved weight-one input at weight one discards a + nonzero coefficient of weight greater than four. Thus the two distinct layer types + give an invalid bound 1*2^2; repetitions must be counted. + source_ref: Upstream Pauli/LayerCounterexample.lean, exact symbolic witness. +tags: +- quantum-information +- pauli-operators +- numerical-analysis +- operator-norms +- truncation-error +msc: +- 81P68 +- 65L70 +- 47A30 diff --git a/LeanPool/projects/lowdimsolvclassification.yaml b/LeanPool/projects/lowdimsolvclassification.yaml new file mode 100644 index 0000000000..2bc839108e --- /dev/null +++ b/LeanPool/projects/lowdimsolvclassification.yaml @@ -0,0 +1,32 @@ +slug: lowdimsolvclassification +title: Classification of low-dimensional solvable Lie algebras +summary: Formalises the classification of solvable Lie algebras up to dimension three. +branch: lie theory +entry_module: LeanPool.LowDimSolvClassification +authors: +- Viviana del Barco +- Gustavo Infanti +- Exequiel Rivas +- Paul Schwahn +source: + title: Classification and Identification of Lie Algebras + authors: + - Pavel Winternitz + - Libor Šnobl + doi: 10.1090/crmm/033 + url: https://bookstore.ams.org/crmm-33 + github_repo: LieLean/LowDimSolvClassification +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LieAlgebra.Dim3.classification +main_results: +- declaration: LieAlgebra.Dim3.classification + informal: Classifies three-dimensional solvable Lie algebras over arbitrary fields. +tags: +- lie-algebras +- solvable +- classification +msc: +- 17B30 diff --git a/LeanPool/projects/main-theorem-of-polytopes.yaml b/LeanPool/projects/main-theorem-of-polytopes.yaml new file mode 100644 index 0000000000..aed009c92c --- /dev/null +++ b/LeanPool/projects/main-theorem-of-polytopes.yaml @@ -0,0 +1,34 @@ +slug: main-theorem-of-polytopes +title: The main theorem of polytopes +summary: Formalizes the Minkowski–Weyl theorem — the equivalence between V-polytopes + (convex hulls of finite point sets) and bounded H-polytopes (finite intersections + of half-spaces) in a finite-dimensional real vector space. +branch: convex geometry +entry_module: LeanPool.Polytopes +authors: +- Jun Kwon +source: + title: Main theorem of polytopes + url: https://github.com/Jun2M/Main-theorem-of-polytopes + github_repo: Jun2M/Main-theorem-of-polytopes +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- MainTheoremOfPolytopes +main_results: +- declaration: MainTheoremOfPolytopes + informal: The Minkowski–Weyl theorem — a set is a V-polytope iff it is a bounded + H-polytope. + source_ref: G. M. Ziegler, Lectures on Polytopes, GTM 152, Springer, 1995, Theorem + 1.1. +- declaration: Vpolytope_of_Hpolytope + informal: Every bounded H-polytope (finite intersection of half-spaces) is the convex + hull of a finite point set. +tags: +- convex-geometry +- discrete-geometry +- polytopes +msc: +- 52B11 +- 52A20 diff --git a/LeanPool/projects/markoff-modp.yaml b/LeanPool/projects/markoff-modp.yaml new file mode 100644 index 0000000000..5dddc96505 --- /dev/null +++ b/LeanPool/projects/markoff-modp.yaml @@ -0,0 +1,28 @@ +slug: markoff-modp +title: Markoff mod p +summary: Lean 4 formalization of strong approximation for the Markoff surface modulo + sufficiently large primes. The primary source is Bourgain-Gamburd-Sarnak. +branch: number theory +entry_module: LeanPool.MarkoffModP +authors: +- Yuma Mizuno +- Guanghao Li +source: + url: https://github.com/yuma-mizuno/markoff-modp + github_repo: yuma-mizuno/markoff-modp + commit: ac1099fc96b0f1cf6e6099c74a24b783691c21aa +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- BGS.Markoff.reduction_surjective_of_explicitBound +main_results: +- declaration: BGS.Markoff.reduction_surjective_of_explicitBound + informal: For primes exceeding the explicit bound, reduction of integral Markoff + points to the Markoff surface modulo p is surjective as specified by the theorem. +tags: +- markoff-equation +- strong-approximation +- finite-fields +msc: +- 11D25 diff --git a/LeanPool/projects/markovprocess.yaml b/LeanPool/projects/markovprocess.yaml new file mode 100644 index 0000000000..04c3b3220c --- /dev/null +++ b/LeanPool/projects/markovprocess.yaml @@ -0,0 +1,32 @@ +slug: markovprocess +title: MarkovProcess +summary: A Lean formalization of continuous-path Markov processes associated with + conservative Feller transition semigroups on nonempty locally compact Polish spaces. + A uniform Kolmogorov displacement-moment bound yields a unique measurable kernel + from each starting point to continuous-path space, with the prescribed finite-dimensional + distributions. The library also develops strong Markov properties, generators and + resolvents, stopping and killing, and Brownian motion as an example. +branch: probability +entry_module: LeanPool.MarkovProcess +authors: +- Scott Armstrong +source: + url: https://github.com/scottnarmstrong/markovprocess + github_repo: scottnarmstrong/markovprocess + commit: dbade0d12c4f84179441a9df7d7147df7a8bcaa0 +license: Apache-2.0 +status: verified +main_declarations: +- MarkovProcess.existsUnique_continuousProcess +main_results: +- declaration: MarkovProcess.existsUnique_continuousProcess + informal: A conservative Feller transition semigroup on a nonempty locally compact + Polish space satisfying the Kolmogorov moment criterion determines a unique continuous-path + Markov kernel with its prescribed finite-dimensional distributions. +tags: +- probability +msc: +- 60J25 +- 60J35 +- 60G17 +provenance: AI diff --git a/LeanPool/projects/marshallhalltheorem.yaml b/LeanPool/projects/marshallhalltheorem.yaml new file mode 100644 index 0000000000..2332b933c2 --- /dev/null +++ b/LeanPool/projects/marshallhalltheorem.yaml @@ -0,0 +1,45 @@ +slug: marshallhalltheorem +title: Marshall Hall's theorem through finite cores +summary: Formalizes Marshall Hall's finite-index free-factor theorem for finitely + generated subgroups of finite-rank free groups. The proof extracts the finitely + many suffix states of a subgroup generating set, completes the resulting partial + generator actions to permutations of a finite state set, identifies the connected + orbit with a finite core, and extends a geodesic spanning tree to obtain an inclusion-compatible + free-product decomposition. It also exposes an explicit finite permutation separator + for every finitely generated subgroup and element outside it, yielding the LERF + consequence. The repository also proves binary and finite-indexed free-group instances + of Grushko--Neumann rank additivity, and the arbitrary-factor binary theorem. The + arbitrary-factor proof is carried by a finite labelled-graph fold, source-unfold, + and monochromatic-vertex contraction reduction. +branch: group theory +entry_module: LeanPool.MarshallHall +authors: +- Arthur Freitas Ramos +- David Barros Hulak +- Ruy J. G. B. de Queiroz +source: + url: https://github.com/arthur742ramos/marshallhalltheorem + github_repo: arthur742ramos/marshallhalltheorem + commit: 99f419644cebf61a5f78eac334fcc1f19b0ccc73 +license: Apache-2.0 +status: verified +main_declarations: +- MarshallHall.marshallHall +- MarshallHall.rank_freeProduct_eq_add +main_results: +- declaration: MarshallHall.marshallHall + informal: Every finitely generated subgroup of a finite-rank free group is an inclusion-compatible + free factor of a finite-index subgroup. +- declaration: LERFChallenge.freeGroup_finite_permutation_separator + informal: Every element outside a finitely generated subgroup of a finite-rank free + group can be separated by an explicit finite permutation action. +- declaration: MarshallHall.rank_freeProduct_eq_add + informal: The rank of a free product of finitely generated groups is the sum of + their ranks. +tags: +- group-theory +msc: +- 20E07 +- 20F65 +- 05C25 +provenance: AI diff --git a/LeanPool/projects/matching-logic-completeness.yaml b/LeanPool/projects/matching-logic-completeness.yaml new file mode 100644 index 0000000000..9de9c900a5 --- /dev/null +++ b/LeanPool/projects/matching-logic-completeness.yaml @@ -0,0 +1,57 @@ +slug: matching-logic-completeness +title: Global completeness of one-sorted definedness-free matching logic +summary: Formalizes the one-sorted case of Chen and Roșu's global-completeness argument + and independently discharges its local-completeness oracle by mechanizing the canonical-model + construction from their 2019 Matching μ-Logic technical report (Theorem 73, Lemmas + 80–81, and strong local completeness Theorem 83). The resulting theorem supplies + both local completeness and soundness for countably infinite variable types. +branch: mathematical logic and formal methods +entry_module: LeanPool.MatchingLogic +authors: +- Aurélien Eveil +- Anthropic +- OpenAI +source: + title: Completeness and incompleteness of basic matching logic; Matching μ-Logic + (2019 technical report) + authors: + - Xiaohong Chen + - Grigore Roșu + arxiv: '2608.13306' + url: https://hdl.handle.net/2142/102281 + github_repo: eveil-labs/matching-logic-lean + commit: 4628a26bb2db117de1a11cbedcb620db9124cdb8 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- MatchingLogic.global_completeness_entryIII +main_results: +- declaration: MatchingLogic.locality + informal: For a backward-closed subset of a matching-logic model, two valuations + that agree inside the subset and both stay outside it otherwise give identical + denotations after restriction to that subset. +- declaration: MatchingLogic.two_copies + informal: The double-cover model over a nonempty backward-closed proper subset has + exactly the denotation predicted by each of its two projections back to the original + model. +- declaration: MatchingLogic.semantic_localization + informal: For a closed theory and closed conclusion, global semantic consequence + is equivalent to local consequence from the localized theory. +- declaration: MatchingLogic.strongLocalCompleteness + informal: One-sorted basic matching logic has strong local completeness over an + arbitrary one-sorted finitary signature when the element-variable type is countably + infinite. +- declaration: MatchingLogic.global_completeness_entryIII + informal: For a one-sorted finitary signature, closed premises and conclusion, and + a countably infinite element-variable type, global semantic consequence is equivalent + to derivability, with both soundness and local completeness discharged. +tags: +- matching-logic +- mathematical-logic +- modal-logic +- completeness +- formal-methods +msc: +- 03B45 +- 03B70 diff --git a/LeanPool/projects/metric-codes.yaml b/LeanPool/projects/metric-codes.yaml new file mode 100644 index 0000000000..73ca4a8315 --- /dev/null +++ b/LeanPool/projects/metric-codes.yaml @@ -0,0 +1,50 @@ +slug: metric-codes +title: Improved asymptotic bounds for binary and spherical codes +summary: Formalizes asymptotic coding bounds obtained from multilevel harmonic analysis, + branching rules, and spectral decompositions. The development proves a binary-code + exponent strictly below the MRRW rate, constructs a strict hierarchy of spherical-code + bounds, and derives the published 0.39661 asymptotic kissing-number exponent. This + import incorporates substantial proof and performance optimizations by Dean Cureton. +branch: coding theory and discrete geometry +entry_module: LeanPool.MetricCodes +authors: +- OpenAI +- Dean Cureton +source: + title: Ten Advances in Mathematics and Theoretical Computer Science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs/tree/94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- MetricCodes.Johnson.main_binary_theorem +main_results: +- declaration: MetricCodes.Johnson.main_binary_theorem + informal: For every relative distance strictly between zero and one half, the combined + variational upper bound on the asymptotic binary-code rate is valid and is strictly + smaller than the MRRW rate. +- declaration: MetricCodes.Spherical.HigherHierarchy.main_general + informal: Every feasible finite interlacing certificate gives an eventual exponential + upper bound for spherical codes, while the spherical-code rate is bounded by the + closed hierarchy's variational rate. +- declaration: MetricCodes.Spherical.HigherHierarchy.strict_hierarchy + informal: At every angle parameter strictly between zero and one, successive levels + of both the global and localized hierarchies improve strictly, with the localized + hierarchy also strictly improving the row and classical bounds. +- declaration: MetricCodes.Spherical.HigherHierarchy.NumericalMaximum.eventually_kissingNumber_lt_published + informal: In all sufficiently large dimensions, the kissing number is at most 2 + raised to 0.39661 times the dimension. +tags: +- coding-theory +- spherical-codes +- kissing-number +- harmonic-analysis +- asymptotic-bounds +msc: +- 94B65 +- 52C17 +- 41A60 diff --git a/LeanPool/projects/microwave-networks.yaml b/LeanPool/projects/microwave-networks.yaml new file mode 100644 index 0000000000..1dbed410d6 --- /dev/null +++ b/LeanPool/projects/microwave-networks.yaml @@ -0,0 +1,50 @@ +slug: microwave-networks +title: Microwave network synthesis and power-of-two DFT implementation +summary: Characterizes the complex transmission matrices synthesized from phase shifters + and permutations, with or without hybrid couplers, using series and parallel composition. + Phase-only networks are permuted phase-diagonal matrices; networks with hybrid couplers + are products of permutation and phase-parametrized block-diagonal layers. Proves + that the normalized DFT of every power-of-two size is implementable, with reusable + finite-type reindexing, independent row and column permutations, and block-diagonal + composition infrastructure. Uses the reciprocal matched network matrix model of + the source paper; no general scattering-physics theorem is claimed. +branch: applied mathematics and signal processing +entry_module: LeanPool.MicrowaveNetworks +authors: +- Matteo Nerini +- Claude AI +source: + title: Toward a Lean Formalization of Analog Computing with Microwaves + arxiv: '2610.04774' + url: https://github.com/matteonerini/formalizing-analog-computing/tree/a89c1e9263658c20a29046f9ed1a8eeb867dd537 + github_repo: matteonerini/formalizing-analog-computing + commit: a89c1e9263658c20a29046f9ed1a8eeb867dd537 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- MiLAC.HybridCouplersPhaseShifters.implementable_dft +main_results: +- declaration: MiLAC.PhaseShifters.implementable_iff_permutedPhaseDiagonal + informal: A finite phase-only network is implementable exactly when its matrix is + a permutation matrix times a diagonal matrix of complex phases. + source_ref: Upstream PhaseShifters.lean. +- declaration: MiLAC.HybridCouplersPhaseShifters.implementable_iff_layered + informal: A finite network is implementable with hybrid couplers and phase shifters + exactly when its matrix is an initial permutation preceded by finitely many permutation + and phase-parametrized block-diagonal layers. + source_ref: Theorem 1 and Section IV of arXiv:2610.04774. +- declaration: MiLAC.HybridCouplersPhaseShifters.implementable_dft + informal: For every natural L, the normalized DFT matrix of size 2^L can be implemented + by hybrid couplers, phase shifters, and permutation networks. + source_ref: Section V of arXiv:2610.04774. +tags: +- applied-mathematics +- signal-processing +- matrix-theory +- fourier-transform +- network-synthesis +msc: +- 94A12 +- 15A23 +- 42A38 diff --git a/LeanPool/projects/min-modulus-unique-multiset-sum.yaml b/LeanPool/projects/min-modulus-unique-multiset-sum.yaml new file mode 100644 index 0000000000..1f8d5785f1 --- /dev/null +++ b/LeanPool/projects/min-modulus-unique-multiset-sum.yaml @@ -0,0 +1,40 @@ +slug: min-modulus-unique-multiset-sum +title: Minimum modulus for the unique multiset-sum problem +summary: 'Formalizes the exact minimum modulus `N_min(n) = 2^n - 2^{floor(log_2 n)}` + for the unique multiset-sum problem on the super-increasing residue set `{2^i - + 1 : 0 <= i < n}`, and proves the sharp elementary-abelian-2-group lower bound `k + >= n - 1` for unique multiset sums in `(Z/2Z)^k`.' +branch: additive combinatorics +entry_module: LeanPool.MinModulusUniqueMultisetSum +authors: +- José A. R. Fonollosa +source: + title: Minimum modulus for the unique multiset-sum problem, with an application + to computing the permanent + authors: + - José A. R. Fonollosa + arxiv: '2607.08366' + github_repo: jarfo/min-modulus +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- MinModulus.nmin_eq +- MinModulus.elementaryAbelianTwoGroups_optimal +main_results: +- declaration: MinModulus.nmin_eq + informal: 'Proves that among moduli `N >= 2` the least one making the super-increasing + residue set `{2^i - 1 : 0 <= i < n}` valid for the unique multiset-sum problem + is `2^n - 2^{floor(log_2 n)}`, for every `n >= 2` (stated as `IsLeast`).' +- declaration: MinModulus.elementaryAbelianTwoGroups_optimal + informal: 'Proves that if `g_0, ..., g_{n-1}` in `(Z/2Z)^k` have unique multiset + sums, then `k >= n - 1`: an elementary abelian 2-group admitting such a family + has order at least `2^(n - 1)`.' +tags: +- additive-combinatorics +- multiset-sums +- finite-groups +- permanent +msc: +- 11B75 +- 05D99 diff --git a/LeanPool/projects/minimum-degree-matchings.yaml b/LeanPool/projects/minimum-degree-matchings.yaml new file mode 100644 index 0000000000..91924a96d3 --- /dev/null +++ b/LeanPool/projects/minimum-degree-matchings.yaml @@ -0,0 +1,58 @@ +slug: minimum-degree-matchings +title: BKLO simultaneous perfect matchings and spread matchings +summary: 'Formalizes the r = 2 specialization of the simultaneous factor-selection + step in BKLO Lemma 10.7: under quantitative parity, minimum-degree, codegree, and + incidence hypotheses, a linearly sized, overlapping family of neighbourhood graphs + admits perfect matchings whose edge sets are pairwise disjoint. The proof replaces + the published randomized greedy process by a deterministic pessimistic-estimator + sweep. Supporting results develop perfect, near-perfect, edge-disjoint, and quantitatively + weighted spread matchings from sharp minimum-degree hypotheses.' +branch: graph theory +entry_module: LeanPool.MinimumDegreeMatching +authors: +- Juan Pablo Traverso Gianini +- Aristotle +source: + title: Edge-decompositions of graphs with high minimum degree + authors: + - Ben Barber + - Daniela Kühn + - Allan Lo + - Deryk Osthus + doi: 10.1016/j.aim.2015.09.032 + arxiv: '1410.5750' + url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions/tree/6736c816e1f9cd105c2295a8e4716ad55609ccb0/preprints/PAPER_III/05_formalization/lean_v1.4_freeze + github_repo: jtraverso/erdos-81-chordal-clique-partitions +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- BKLOK2.lemma107K2_holds +main_results: +- declaration: BKLOK2.lemma107K2_holds + informal: 'Specializes BKLO Lemma 10.7 to r = 2 in its apex-neighbourhood configuration: + for every sufficiently large admissible configuration, all indexed neighbourhood + graphs admit perfect matchings whose edge sets are pairwise disjoint. The proof + is deterministic.' +- declaration: SimpleGraph.exists_isPerfectMatching_of_minDegree + informal: Every finite simple graph of even order whose minimum degree is at least + half its order has a perfect matching. +- declaration: SimpleGraph.exists_isMatching_compl_singleton_of_minDegree + informal: Every finite simple graph of odd order n and minimum degree at least (n + - 1) / 2 has a matching whose vertex set is the complement of a singleton. +- declaration: SimpleGraph.exists_involutions_pairwise_ne + informal: If every vertex of a finite even vertex set has at least half the set + size plus t neighbors inside the set, then the graph contains t + 1 pairwise edge-disjoint + perfect matchings of that set, represented by an indexed family of partner involutions. + These involutions are pairwise distinct when the vertex set is nonempty. +- declaration: SimpleGraph.exists_spread_involution + informal: Under the same minimum-degree condition with slack t, every nonnegative + edge-weight function admits a perfect matching whose weight is at most 1 / (t + + 1) times the total weight over ordered pairs in the vertex set. +tags: +- graph-theory +- perfect-matchings +- edge-disjoint-matchings +- pseudorandomness +msc: +- 05C70 diff --git a/LeanPool/projects/misere-games.yaml b/LeanPool/projects/misere-games.yaml new file mode 100644 index 0000000000..aaa40bbac7 --- /dev/null +++ b/LeanPool/projects/misere-games.yaml @@ -0,0 +1,41 @@ +slug: misere-games +title: Misere combinatorial games +summary: Formalizes a combinatorial game theory library for misere play, including + comparison modulo universes, p-free forms, outcome-stability, Property X, and concrete + rulesets such as Hackenbush. +branch: combinatorics +entry_module: LeanPool.MisereGames +authors: +- Tomasz Maciosowski +- Alfie Davies +source: + title: misere-games + authors: + - Tomasz Maciosowski + - Alfie Davies + - Violeta Hernandez Palacios + - Yuyang Zhao + url: https://github.com/t4ccer/misere-games + github_repo: t4ccer/misere-games +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- MisereGames.Form.Promain.of_universe +main_results: +- declaration: MisereGames.Form.Promain.of_universe + informal: Shows that comparison modulo a universe is characterized by the maintenance-proviso + test. +- declaration: MisereGames.IntegerInvertible.isPFree_of_propertyX + informal: Proves that Property X makes the sum of two short p-free games p-free + in an outcome-stable, hereditary, integer-invertible monoid. +- declaration: MisereGames.Form.Preservation.endLikeSeedToUniverse_bijective + informal: Proves that universal closure gives a bijection from bounded end-like + seeds to universes. +tags: +- combinatorial-games +- misere-play +- game-theory +msc: +- 91A46 +- 05C57 diff --git a/LeanPool/projects/monlib4.yaml b/LeanPool/projects/monlib4.yaml new file mode 100644 index 0000000000..24c5e6b1ae --- /dev/null +++ b/LeanPool/projects/monlib4.yaml @@ -0,0 +1,72 @@ +slug: monlib4 +title: Monlib4 Operator-Algebra and Quantum-Set Core +summary: Formalizes monlib4's finite matrix-algebra inner-automorphism theorem, unitary + inner automorphisms, star-preserving linear maps, the structural core of quantum + sets, the core Schur-product definition of quantum graphs, and a self-contained + digit-sum divisibility theorem. +branch: operator algebras and quantum graph theory +entry_module: LeanPool.Monlib4 +authors: +- Monica Omar +source: + doi: 10.4169/amer.math.monthly.124.10.966 + arxiv: '1810.08368' + github_repo: themathqueen/monlib4 +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Matrix.aut_mat_inner +- QuantumSet +- schurMul +- QuantumGraph +main_results: +- declaration: Matrix.automorphism_matrix_inner + informal: Shows that any automorphism of a finite nontrivial matrix algebra is implemented + by an invertible intertwining matrix. +- declaration: Matrix.aut_mat_inner + informal: Packages matrix-algebra automorphisms as inner automorphisms induced by + a linear equivalence of the underlying coordinate space. +- declaration: Matrix.aut_mat_inner' + informal: Packages matrix-algebra automorphisms as conjugation by an element of + `GL n 𝕜`. +- declaration: Matrix.aut_mat_inner_trace_preserving + informal: Proves that automorphisms of finite matrix algebras preserve trace. +- declaration: Matrix.innerAut + informal: Defines the linear map induced by conjugation with a unitary matrix. +- declaration: Matrix.innerAut_apply_trace_eq + informal: Proves that unitary inner automorphisms preserve matrix trace. +- declaration: Matrix.innerAut.spectrum_eq + informal: Proves that unitary inner automorphisms preserve the spectrum of the associated + linear operator. +- declaration: LinearMap.IsReal + informal: Defines star-preserving, or real, function-like maps. +- declaration: LinearMap.real + informal: Defines the star-conjugate involution on linear maps. +- declaration: QuantumSet + informal: Defines the monlib4 quantum-set structure with modular automorphisms and + a fixed orthonormal basis. +- declaration: Complex.quantumSet + informal: Instantiates the complex numbers as the base quantum set. +- declaration: schurMul + informal: Defines the Schur product of linear maps using multiplication, tensoring, + and coalgebra comultiplication. +- declaration: QuantumGraph + informal: Defines quantum graphs as Schur-idempotent endomorphisms. +- declaration: QuantumGraph.Real + informal: Defines real quantum graphs as Schur-idempotent, star-preserving endomorphisms. +- declaration: nat_repr_mod_eq_sum_repr_mod + informal: Proves that the value represented by digits in base `a + 1` has the same + remainder modulo `a` as the sum of those digits. +- declaration: nat_dvd_repr_iff_nat_dvd_sum_repr + informal: Proves the generalized digit-sum divisibility criterion for base `a + + 1` modulo `a`. +tags: +- linear-algebra +- operator-algebras +- quantum-sets +- quantum-graphs +- representation-theory +msc: +- 15A69 +- 16W20 diff --git a/LeanPool/projects/monsky.yaml b/LeanPool/projects/monsky.yaml new file mode 100644 index 0000000000..d6c58b38f8 --- /dev/null +++ b/LeanPool/projects/monsky.yaml @@ -0,0 +1,32 @@ +slug: monsky +title: Monsky's Theorem +summary: Formalizes Monsky's theorem that a square can be dissected into n triangles + of equal area if and only if n is nonzero and even, via a 2-adic valuation, a three-coloring + of the unit square, and a rainbow-triangle argument. +branch: combinatorial geometry +entry_module: LeanPool.Monsky +authors: +- Dhyan Aranha +- contributors +source: + title: On dividing a square into triangles + authors: + - Paul Monsky + doi: 10.2307/2317329 + github_repo: dhyan-aranha/Monsky +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LeanPool.Monsky.monsky_theorem +main_results: +- declaration: LeanPool.Monsky.monsky_theorem + informal: Proves that the unit square admits a dissection into n equal-area triangles + if and only if n is nonzero and even. +tags: +- geometry +- combinatorics +- measure-theory +msc: +- 52C20 +- 05B45 diff --git a/LeanPool/projects/moser-lattice-colorings.yaml b/LeanPool/projects/moser-lattice-colorings.yaml new file mode 100644 index 0000000000..ad1885da6a --- /dev/null +++ b/LeanPool/projects/moser-lattice-colorings.yaml @@ -0,0 +1,43 @@ +slug: moser-lattice-colorings +title: Geometric Four-Colorings of the Moser Lattice and Ring +summary: Formalizes the two explicit parity colorings from Dúcz's 2026 note on the + Moser lattice. The development derives the exact quadratic norm formula, proves + that the integer-coordinate embedding is faithful, proves both colorings proper + and geometric on the entire lattice, and descends them through localization by powers + of three to the Moser ring. As a structural corollary, every unit-distance graph + realized entirely in the Moser ring is four-colorable. The uniqueness classification + of geometric four-colorings and the chromatic number of the full Euclidean plane + are not claimed. +branch: combinatorics +entry_module: LeanPool.MoserLatticeColorings +authors: +- Egor Lyfar +source: + title: A note on geometric colorings of the Moser lattice + authors: + - Ákos Dúcz + arxiv: '2606.12325' + github_repo: lyfar/moser-lattice-colorings-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.MoserLatticeColorings.MoserRing.both_colorings_are_geometric +main_results: +- declaration: LeanPool.MoserLatticeColorings.MoserRing.both_colorings_are_geometric + informal: The two parity maps defined by Dúcz descend to proper four-colorings of + the full Moser ring, and equality of colors for each map depends only on Euclidean + distance. + source_ref: Theorem 3.1 and the Moser-ring extension following it. +- declaration: LeanPool.MoserLatticeColorings.MoserRing.graph_colorable_four_of_moserRing_realization + informal: Every abstract graph with a realization in the Moser ring that sends each + edge to a unit-distance pair is four-colorable. + source_ref: Corollary of the geometric four-coloring of the Moser ring. +tags: +- combinatorics +- graph-coloring +- discrete-geometry +- unit-distance-graphs +- moser-lattice +msc: +- 05C15 diff --git a/LeanPool/projects/moving-sofa.yaml b/LeanPool/projects/moving-sofa.yaml new file mode 100644 index 0000000000..3b4a7aefe9 --- /dev/null +++ b/LeanPool/projects/moving-sofa.yaml @@ -0,0 +1,55 @@ +slug: moving-sofa +title: Optimality of Gerver's sofa +summary: 'Formalizes Baek''s solution of the moving sofa problem: Gerver''s sofa attains + the supremum of the areas of closed connected shapes that can move continuously + around a right-angled corner in a unit-width hallway. Includes certified Gerver + parameters, admissibility, convex cap geometry, Stieltjes curve area, Jordan separation, + polygon approximation, and the concave quadratic upper bound.' +branch: convex geometry +entry_module: LeanPool.MovingSofa +authors: +- Dean Cureton +- Dawid Trela +- Rado Kirov +- Kirill Alkhimov +source: + title: Optimality of Gerver's Sofa + authors: + - Jineon Baek + arxiv: '2411.19826' + url: https://github.com/deancureton/MovingSofa/tree/4d5569131940815f47a9ccf3e90a4c5043c56127 + github_repo: deancureton/MovingSofa + commit: 4d5569131940815f47a9ccf3e90a4c5043c56127 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- MovingSofa.sofaConstant_eq_volume_gerversSofa +main_results: +- declaration: MovingSofa.sofaConstant_eq_volume_gerversSofa + informal: The supremum, in extended nonnegative real numbers, of the areas of all + admissible moving sofas equals the area of the explicitly defined Gerver sofa. + source_ref: Baek, Optimality of Gerver's Sofa, Theorem 1.1. Upstream Lean completion + a0753860f1d2b4424e059c35e64ad611be71a415 (2026-09-20 10:55:24 UTC), announced + 2026-09-20 18:42:40 UTC at https://x.com/deancureton/status/2101743865392926967. + The imported commit has identical Lean sources; its later change is documentation + only. +- declaration: MovingSofa.GerversSofa.ABφθSpec.existsUnique + informal: Gerver's four defining equations have exactly one quadruple of parameters + with A and B nonnegative and 0 ≤ φ ≤ θ ≤ π/4. + source_ref: Romik, Differential equations and exact solutions in the moving sofa + problem, equations (1)-(4), arXiv:1606.08111. The GerverSofaLean v1.1.0 release + archive (MIT, Dawid Trela) has SHA-256 7c496150f709ac709ae01f4836e9a702a18f752450efa69f15cf44264c0c2f77. +- declaration: MovingSofa.isMovingSofa_gerversSofa + informal: Gerver's sofa is closed and connected and admits a continuous rigid motion + from the horizontal arm to the vertical arm, starting at the identity and remaining + within the unit-width hallway throughout. +tags: +- convex-geometry +- moving-sofa-problem +- geometric-optimization +- interval-arithmetic +msc: +- 52A40 +- 52A10 +- 49Q10 diff --git a/LeanPool/projects/mriscx.yaml b/LeanPool/projects/mriscx.yaml new file mode 100644 index 0000000000..a4dda88501 --- /dev/null +++ b/LeanPool/projects/mriscx.yaml @@ -0,0 +1,50 @@ +slug: mriscx +title: MRiscX +summary: A Hoare logic for verifying unstructured RISC-V-like assembly in Lean, following + Lundberg et al. (SEFM 2020), with a certified interpreter, proved structural Hoare + rules, per-instruction specifications, custom elaborators and tactics, and a complete + One-Time-Pad correctness proof. +branch: program verification +entry_module: LeanPool.MRiscX +authors: +- Julius Marx +source: + doi: 10.1007/978-3-030-58768-0_11 + url: https://github.com/JulsDE/MRiscX + github_repo: JulsDE/MRiscX +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- hoareTripleUp +- S_SEQ +- proof_otp +main_results: +- declaration: hoareTripleUp + informal: Defines the MRiscX Hoare triple, asserting that from any state satisfying + the precondition the code reaches a successor state satisfying the postcondition + while respecting the line whitelist and blacklist. + source_ref: D. Lundberg, R. Guanciale, A. Lindner, M. Dam, Hoare-Style Logic for + Unstructured Programs, SEFM 2020, doi:10.1007/978-3-030-58768-0_11. +- declaration: S_SEQ + informal: Proves the sequencing Hoare rule, peeling off the last instruction of + a code block. + source_ref: D. Lundberg, R. Guanciale, A. Lindner, M. Dam, Hoare-Style Logic for + Unstructured Programs, SEFM 2020, doi:10.1007/978-3-030-58768-0_11. +- declaration: PRE_STR + informal: Proves the precondition-strengthening rule for MRiscX Hoare triples. +- declaration: POST_WEAK + informal: Proves the postcondition-weakening rule for MRiscX Hoare triples. +- declaration: specification_LoadImmediate + informal: Proves the per-instruction Hoare specification of the load-immediate instruction. +- declaration: proof_otp + informal: Proves the full correctness of a One-Time-Pad implementation written in + MRiscX assembly. +tags: +- hoare-logic +- program-verification +- risc-v +- assembly +- formal-methods +msc: +- 68Q60 diff --git a/LeanPool/projects/multicolor-triangle-ramsey.yaml b/LeanPool/projects/multicolor-triangle-ramsey.yaml new file mode 100644 index 0000000000..79d172229a --- /dev/null +++ b/LeanPool/projects/multicolor-triangle-ramsey.yaml @@ -0,0 +1,40 @@ +slug: multicolor-triangle-ramsey +title: Superexponential lower bounds for multicolor triangle Ramsey numbers +summary: Resolves Erdős problem 183 by proving that the k-color triangle Ramsey number + has a k-th root tending to infinity. The development constructs saturated palette + families and recursive edge colorings, derives an explicit superexponential lower + bound, and proves matching order-of-growth bounds for the logarithm of the Ramsey + number. +branch: extremal combinatorics and Ramsey theory +entry_module: LeanPool.MulticolorTriangleRamsey +authors: +- OpenAI +source: + title: Ten Advances in Mathematics and Theoretical Computer Science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- ErdosProblems.MulticolourTriangleRamsey.erdos_problem_183_explicit +main_results: +- declaration: ErdosProblems.MulticolourTriangleRamsey.erdos_183 + informal: The k-th root of the k-color triangle Ramsey number tends to infinity, + resolving Erdős problem 183. +- declaration: ErdosProblems.MulticolourTriangleRamsey.erdos_problem_183_explicit + informal: For every k at least two, the k-color triangle Ramsey number is at least + ((k^(1/3))/(6 exp(38) log k))^k, and its k-th root tends to infinity. +- declaration: ErdosProblems.MulticolourTriangleRamsey.triangleRamseyNumber_log_isTheta + informal: The logarithm of the k-color triangle Ramsey number has order k log k. +tags: +- extremal-combinatorics +- ramsey-theory +- graph-coloring +- erdos-problems +msc: +- 05D10 +- 05C55 diff --git a/LeanPool/projects/nagatafactoriality.yaml b/LeanPool/projects/nagatafactoriality.yaml new file mode 100644 index 0000000000..556a76e01b --- /dev/null +++ b/LeanPool/projects/nagatafactoriality.yaml @@ -0,0 +1,35 @@ +slug: nagatafactoriality +title: Nagata's prime-generated factoriality theorem +summary: Nagata’s factoriality criterion for noetherian domains localized at prime-generated + multiplicative sets, with concrete and abstract localization interfaces and applications + to polynomial rings through Laurent and fraction-field localizations. +branch: commutative algebra +entry_module: LeanPool.NagataFactoriality +authors: +- Arthur F. Ramos +- Ruy J. G. B. de Queiroz +- Anjolina G. de Oliveira +source: + url: https://github.com/arthur742ramos/nagatafactoriality + github_repo: arthur742ramos/nagatafactoriality + commit: 2efdf7f86085972bd758449ca43c76a2e7bdf4af +license: Apache-2.0 +status: verified +provenance: AI +provenance_note: 'Inferred from explicit Copilot coauthor trailers in the pinned upstream + history: 83.5% of surviving noncomment Lean code lines, including 94.8% of the core + localization lemmas and 96.6% of the theorem module, trace to those commits. Commits + 5b1698879b05fc76597723347544cb0b02c268c4 and ef73ee27e4b0a8d1260350801c6915bfb56b6658 + add substantial new proofs, including prime-generated descent and the fraction-field + polynomial application. Coauthorship records support this classification but do + not identify who wrote each individual line.' +main_declarations: +- NagataFactoriality.nagata_theorem +main_results: +- declaration: NagataFactoriality.nagata_theorem + informal: A noetherian integral domain is a unique factorization domain when localization + at a prime-generated multiplicative set is a unique factorization domain. +tags: +- commutative-algebra +msc: +- 13F15 diff --git a/LeanPool/projects/nash-williams.yaml b/LeanPool/projects/nash-williams.yaml new file mode 100644 index 0000000000..0970784894 --- /dev/null +++ b/LeanPool/projects/nash-williams.yaml @@ -0,0 +1,60 @@ +slug: nash-williams +title: Nash-Williams fronts and 2-better-quasi-orders +summary: Formalizes Nash-Williams fronts through their well-founded prefix trees and + ordinal ranks, proves the finite-color Nash-Williams theorem, and derives the infinite + Ramsey theorem for arbitrary finite arity. The order-theoretic development proves + that 2-better-quasi-orders are well-quasi-orders and establishes closure under finite + products, lexicographic sums, domination of subsets, and pointwise embedding of + sequences. Throughout, TwoBQO r is the good-triple property of an arbitrary binary + relation r (every pair-sequence has m < n < l with f(m,n) related to f(n,l)); it + assumes no reflexivity or transitivity of r. +branch: infinitary combinatorics and order theory +entry_module: LeanPool.NashWilliams +authors: +- Yann Pequignot +source: + title: 'Towards better: A motivated introduction to better-quasi-orders' + authors: + - Yann Pequignot + doi: 10.4171/EMSS/4-2-2 + url: https://github.com/yannpequignot/TwoBQO + github_repo: yannpequignot/TwoBQO + commit: 8adea92d7fa41700372ba16b9ff816dc7500579b +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- Front.IsFront.nash_williams +- TwoBQO.wellQuasiOrdered +main_results: +- declaration: Front.IsFront.nash_williams + informal: Every two-coloring of a Nash-Williams front on an infinite set has an + infinite restriction whose front is monochromatic. +- declaration: Front.IsFront.nash_williams_fin + informal: Every finite coloring of a Nash-Williams front has an infinite restriction + whose front is monochromatic. +- declaration: infinite_ramsey + informal: Every finite coloring of the r-element subsets of an infinite set of natural + numbers is constant on all r-element subsets of some infinite subset. +- declaration: TwoBQO.wellQuasiOrdered + informal: Every 2-better-quasi-order is a well-quasi-order. +- declaration: TwoBQO.lexSigmaQO + informal: 'The 2-BQO good-triple property is preserved by lexicographic sums: if + the index relation and every fibre relation have it, so does the lexicographic-sum + relation on the sigma type. This asserts nothing about reflexivity or transitivity + of the sum; those are proved separately (LexSumRelQO.refl and LexSumRelQO.trans) + under the corresponding order hypotheses on the index and fibres.' +- declaration: TwoBQO.embedForAll_wqo + informal: If a preorder is a 2-better-quasi-order, then its infinite sequences are + well-quasi-ordered by the relation admitting a strictly monotone reindexing along + which corresponding terms are related. +tags: +- infinitary-combinatorics +- ramsey-theory +- better-quasi-orders +- well-quasi-orders +- ordinal-ranks +msc: +- 03E05 +- 05D10 +- 06A07 diff --git a/LeanPool/projects/nashembedding.yaml b/LeanPool/projects/nashembedding.yaml new file mode 100644 index 0000000000..fd292c87bd --- /dev/null +++ b/LeanPool/projects/nashembedding.yaml @@ -0,0 +1,46 @@ +slug: nashembedding +title: Nash's smooth isometric embedding theorem +summary: 'A Lean 4 / Mathlib formalization of Nash''s C^∞ isometric embedding theorem + for closed manifolds: every compact smooth manifold without boundary, equipped with + a smooth Riemannian metric, admits a smooth injective map into some Euclidean space + ℝ^q whose differential pulls the Euclidean inner product back to the given metric. + The proof formalized is Günther''s (1989), which replaces the Nash-Moser iteration + by an elliptic fixed-point argument, following the presentation in A. J. Wassermann''s + Cambridge Part III lecture notes as the primary source. The development has two + layers. The first proves the theorem for the flat torus: a self-contained Sobolev + toolkit on 𝕋ⁿ (weighted ℓ² Sobolev spaces on ℤⁿ, Rellich compactness, the Sobolev + sup bound, three multiplication theorems, mollifiers, periodization, Fourier inversion + and Parseval), the approximation of smooth metrics by realizable ones, Günther''s + perturbation theorem proved by a contraction in Hˢ, and their assembly into the + statement that every smooth 2π-periodic positive-definite matrix field on ℝⁿ is + the pullback metric of a smooth periodic immersion that is injective modulo 2πℤⁿ. + The second layer reduces the closed-manifold case to the torus: a Whitney embedding + from Mathlib is extended to the ambient space, the metric is extended by a pseudo-inverse + construction, cut off, symmetrized and periodized, and the torus theorem is applied. + No bound on the embedding dimension is claimed. Concrete instances (round spheres + of every dimension, products of spheres, the flat torus 𝕊¹ × 𝕊¹) are derived, and + the flat-torus instance is shown to reproduce the torus theorem''s conclusion, tying + the two layers together. Nothing here is claimed to be new.' +branch: differential geometry +entry_module: LeanPool.NashEmbedding +authors: +- David Wiygul +source: + url: https://github.com/langlerangle/nashembedding + github_repo: langlerangle/nashembedding + commit: 515a8245c51c952db59d81ef840a146ed853e4ea +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- NashEmbeddingTheorem.nash_isometric_embedding +main_results: +- declaration: NashEmbeddingTheorem.nash_isometric_embedding + informal: Every compact smooth boundaryless manifold with a smooth Riemannian metric + admits a smooth isometric embedding in some finite-dimensional Euclidean space. +tags: +- differential-geometry +- isometric-embedding +- sobolev-spaces +msc: +- 53C24 diff --git a/LeanPool/projects/naslund-counterexample.yaml b/LeanPool/projects/naslund-counterexample.yaml new file mode 100644 index 0000000000..73615a085c --- /dev/null +++ b/LeanPool/projects/naslund-counterexample.yaml @@ -0,0 +1,52 @@ +slug: naslund-counterexample +title: Square-difference-free polynomials beyond Naslund's conjectured bound +summary: Constructs square-difference-free subsets of polynomials over the field with + three elements that refute Naslund's Conjecture 13 at every degree bound divisible + by four and at least eight. An explicit degree-eight lift multiplies cardinality + by 810 and yields a lower bound of 16/21 for the lower limiting normalized logarithmic + growth rate. +branch: additive combinatorics and number theory +entry_module: LeanPool.NaslundCounterexample +authors: +- JD Jones +source: + title: Square-difference-free sets in F_3[T] past the conjectured bound + authors: + - JD Jones + url: https://github.com/JD-Jones-ASES/ns-lean + github_repo: JD-Jones-ASES/ns-lean + commit: 035e9b0c147630e35631e4401433660f695d1fba +license: MIT +status: verified +provenance: AI +main_declarations: +- NaslundCounterexample.conjecture_fails +- NaslundCounterexample.liminf_ge +main_results: +- declaration: NaslundCounterexample.exists_card_810 + informal: There is a square-difference-free set of 810 polynomials over F_3 with + degree less than eight. +- declaration: NaslundCounterexample.conjecture_fails_at_eight + informal: The upper bound 3^6 = 729 in Naslund's Conjecture 13 fails for square + differences over F_3 at degree bound eight. +- declaration: NaslundCounterexample.exists_card_pow + informal: For every positive integer e, there is a square-difference-free set of + 810^e polynomials over F_3 with degree less than 8e. +- declaration: NaslundCounterexample.exists_card_twenty_seven_mul_pow + informal: For every natural number e, there is a square-difference-free set of 27 + times 810^e polynomials over F_3 with degree less than 8e + 4. +- declaration: NaslundCounterexample.conjecture_fails + informal: For every n divisible by four with n at least eight, there is a square-difference-free + set of polynomials over F_3 of degree less than n whose cardinality exceeds 3^(3n/4). +- declaration: NaslundCounterexample.liminf_ge + informal: If D(n) is the largest cardinality of a square-difference-free set of + polynomials over F_3 of degree less than n, then the lower limit of log D(n) divided + by n log 3 is at least 16/21. +tags: +- additive-combinatorics +- square-difference-free +- function-fields +- polynomial-constructions +msc: +- 11B75 +- 05D05 diff --git a/LeanPool/projects/navier-stokes-and-euler.yaml b/LeanPool/projects/navier-stokes-and-euler.yaml new file mode 100644 index 0000000000..fbaeef3625 --- /dev/null +++ b/LeanPool/projects/navier-stokes-and-euler.yaml @@ -0,0 +1,56 @@ +slug: navier-stokes-and-euler +title: Finite-time breakdown for Navier–Stokes and Euler +summary: Formalizes forced three-dimensional Navier–Stokes breakdown on Euclidean + space and the periodic torus for every positive viscosity, and finite-time singularity + formation for unforced Euler from smooth, compactly supported, divergence-free initial + data. The Euler result specifies maximal Sobolev lifespan, divergence of the velocity + C¹ norm, and divergence of the time integral of the vorticity norm. +branch: partial differential equations and mathematical fluid dynamics +entry_module: LeanPool.NavierStokesAndEuler +authors: +- OpenAI +source: + title: Finite time blowup for Navier–Stokes and Finite time blowup for the Euler + equation + authors: + - OpenAI + url: https://github.com/openai/NavierStokesAndEuler + github_repo: openai/NavierStokesAndEuler + commit: 8937a8f4cbc7abaab5e9e97d1cc7f5d2319d9538 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- NavierStokes.Comparator.navier_stokes_breakdown_R3 +- Euler.euler_breakdown_R3 +main_results: +- declaration: NavierStokes.Comparator.navier_stokes_breakdown_R3 + informal: For every positive viscosity, there are smooth divergence-free initial + data and smooth forcing on ℝ³, with the prescribed spatial and temporal decay, + admitting no global smooth Navier–Stokes solution with uniformly bounded kinetic + energy. +- declaration: NavierStokes.Comparator.navier_stokes_breakdown_periodic + informal: For every positive viscosity, there are smooth periodic divergence-free + initial data and smooth periodic forcing with the prescribed temporal decay, admitting + no global smooth periodic Navier–Stokes solution. +- declaration: Euler.euler_breakdown_R3 + informal: There exist smooth divergence-free initial data on ℝ³ whose spatial derivatives + decay faster than every polynomial and which admit no global smooth unforced Euler + solution with uniformly bounded kinetic energy. +- declaration: Euler.exists_compact_smooth_euler_singularity + informal: There exist nonzero smooth compactly supported divergence-free initial + data, a maximal time 0 < Tstar ≤ 1, and an unforced Euler solution on [0, Tstar) + in the all-order Sobolev class with bounded kinetic energy. Smooth Sobolev solutions + on [0, T] exist exactly when T < Tstar. The velocity C¹ norm has infinite left + limsup at Tstar, and the time integral of the spatial supremum of the vorticity + norm diverges. +tags: +- navier-stokes +- euler-equations +- finite-time-blowup +- fluid-dynamics +- sobolev-spaces +msc: +- 35Q30 +- 35Q31 +- 35B44 diff --git a/LeanPool/projects/nemytskii-lebesgue.yaml b/LeanPool/projects/nemytskii-lebesgue.yaml new file mode 100644 index 0000000000..dde77aa92d --- /dev/null +++ b/LeanPool/projects/nemytskii-lebesgue.yaml @@ -0,0 +1,83 @@ +slug: nemytskii-lebesgue +title: Nemytskii operators on Lebesgue spaces +summary: 'Develops superposition operators induced by Caratheodory functions: measurability, + Lp-to-Lq growth bounds and sequential continuity, Lipschitz estimates, monotonicity + and coercivity, and Frechet remainder estimates. For Lebesgue measure on the real + line and finite-dimensional real fibers, characterizes when every Lp input has an + Lq output by the standard integrable-envelope growth condition, for 1 <= p,q < infinity. + First-order remainder estimates on arbitrary measure spaces assume a Caratheodory + derivative with the stated Holder-compatible growth. The integral-functional result + bounds the integrated remainder and proves integrability of the linearized integrand; + it does not assert integrability of the original integrand without its separate + growth hypotheses. Caratheodory here requires strongly measurable sections and almost-everywhere + continuity in the fiber variable.' +branch: nonlinear functional analysis +entry_module: LeanPool.NemytskiiLebesgue +authors: +- Petr Girg +- Petr Nečesal +- Martin Dvořák +- Jakub Psutka +source: + title: Nemytskii operators on Lebesgue spaces + url: https://github.com/madvorak/nemytskii-lebesgue/tree/d276c27a1e3ebbd9eb09f25243b5e9808ca168d7 + github_repo: madvorak/nemytskii-lebesgue + commit: d276c27a1e3ebbd9eb09f25243b5e9808ca168d7 + completion_commit: 12e914c8b05694f2ea339a27c4af3d0bdf068922 + release: https://github.com/madvorak/nemytskii-lebesgue/releases/tag/v4.33.1 + completion_date: '2026-09-21T13:49:28Z' + dependencies: + - url: https://github.com/madvorak/utilities-zcu/tree/49f60eacb1401f82358b7d66ee447ea7f2fdebf1 + license: Apache-2.0 + scope: UtilitiesZCU/Basic.lean + provenance_note: Upstream credits the four listed authors. Its README and available + commit history contain no AI disclosure; this is evidence under the repository's + human-provenance rubric, not proof of tool non-use. The imported version includes + substantial AI-assisted Mathlib porting and proof refactoring by OpenAI Codex, + so it is recorded as mixed. +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- NemytskiiLebesgue.IsCaratheodory.memLp_nemytskii_volume_iff_growth +main_results: +- declaration: NemytskiiLebesgue.IsCaratheodory.memLp_nemytskii_volume_iff_growth + informal: For f from R times R^m to R^k with strongly measurable sections and almost-everywhere + continuous fibers, and 1 <= p,q < infinity, every Lp input gives an Lq output + if and only if there exist an Lq real envelope a and a nonnegative constant C + such that almost everywhere, simultaneously for every fiber vector xi, norm(f(x,xi)) + <= abs(a(x)) + C * norm(xi)^(p/q). + source_ref: Upstream NemytskiiLebesgue/ConverseImplication.lean. +- declaration: NemytskiiLebesgue.IsCaratheodory.nemytskii_seqContinuous_of_memLp_of_growth_real + informal: On any measure space, the same Caratheodory and growth assumptions imply + that a sequence of Lp functions converging in Lp to an Lp function has superposition + images converging in Lq, for finite exponents p,q at least one and finite-dimensional + real fibers. + source_ref: Upstream NemytskiiLebesgue/Continuity.lean. +- declaration: NemytskiiLebesgue.IsCaratheodory.eLpNorm_frechet_remainder_le_ofReal_mul_eLpNorm + informal: On a measure space, suppose f and its fiber derivative f' are Caratheodory, + f' is the Frechet derivative at every fiber point almost everywhere, p,r >= 1 + satisfy 1/q = 1/r + 1/p, and f' is bounded by b + C * norm(xi)^(p/r) with nonnegative + b in Lr and C > 0. At each Lp function u, the Lq integral quantity (a quasi-norm + when q < 1) of the first-order remainder is at most epsilon times the Lp norm + of l whenever l is in Lp and its Lp norm is sufficiently small. + source_ref: Upstream NemytskiiLebesgue/Frechet.lean; generalized in this port by + removing an unused finite-measure assumption. +- declaration: NemytskiiLebesgue.IsCaratheodory.integralFunctional_frechet_deriv + informal: On a measure space with finite-dimensional real inputs, under Caratheodory + assumptions on f and its fiber derivative f', conjugate exponents p,q, and derivative + growth b + C*norm(xi)^(p-1) with nonnegative b in Lq and C > 0, the linearized + integrand is integrable for every Lp perturbation. The norm of the integral of + the first-order remainder is at most epsilon times the Lp norm of the perturbation + when that norm is sufficiently small. + source_ref: Upstream NemytskiiLebesgue/IntegralFunctional.lean; generalized in this + port by removing an unused finite-measure assumption. +tags: +- nonlinear-analysis +- nemytskii-operators +- lebesgue-spaces +- differentiation +msc: +- 47H30 +- 46E30 +- 47J05 diff --git a/LeanPool/projects/neukirch.yaml b/LeanPool/projects/neukirch.yaml new file mode 100644 index 0000000000..fc48e1be78 --- /dev/null +++ b/LeanPool/projects/neukirch.yaml @@ -0,0 +1,50 @@ +slug: neukirch +title: 'Neukirch''s Algebraic Number Theory: Hilbert ramification theory' +summary: Formalizes Hilbert's ramification theory for Galois extensions of number + fields, including the fundamental identity and the decomposition and inertia groups, + ported from the jjdishere/neukirch project. +branch: number theory +entry_module: LeanPool.Neukirch +authors: +- Hu Yongle +source: + doi: 10.1007/978-3-662-03983-0 + github_repo: jjdishere/neukirch +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- NumberField.ramificationIdx_mul_inertiaDegOfIsGalois +main_results: +- declaration: NumberField.ramificationIdx_mul_inertiaDegOfIsGalois + informal: 'Proves the fundamental identity g·e·f = [L : K] for a Galois extension + of number fields.' +- declaration: NumberField.finrank_eq_ramificationIdx_mul_inertiaDeg + informal: 'Shows that when P is the unique prime over p, the degree [L : K] equals + the ramification index times the inertia degree.' +- declaration: NumberField.IsMaximal_conjugates + informal: Proves the Galois group acts transitively on the primes lying over a fixed + prime. +- declaration: NumberField.DecompositionGroup_card_eq_ramificationIdx_mul_inertiaDeg + informal: Computes the order of the decomposition group as the product of the ramification + index and inertia degree. +- declaration: NumberField.InertiaFieldAutEquivResidueFieldAut + informal: Identifies the Galois group of the inertia field over K with the Galois + group of the residue field extension. +- declaration: NumberField.InertiaGroup_card_eq_ramificationIdx + informal: Computes the order of the inertia group as the ramification index of p + in L. +- declaration: Ideal.ramificationIdx_algebra_tower_of_eq + informal: Proves multiplicativity of the ramification index in a tower of Dedekind + domains. +- declaration: Ideal.inertiaDeg_algebra_tower_of_eq + informal: Proves multiplicativity of the inertia degree in a tower of Dedekind domains. +tags: +- number-theory +- algebraic-number-theory +- ramification +- galois-theory +msc: +- 11R32 +- 11S15 +- 13B25 diff --git a/LeanPool/projects/nf-weak-partition.yaml b/LeanPool/projects/nf-weak-partition.yaml new file mode 100644 index 0000000000..d424a5d1e0 --- /dev/null +++ b/LeanPool/projects/nf-weak-partition.yaml @@ -0,0 +1,55 @@ +slug: nf-weak-partition +title: New Foundations refutes the Weak Partition Principle +summary: Constructs a formal first-order derivation of the negation of the Weak Partition + Principle from New Foundations, with extensionality and the full integer-stratified + comprehension schema. Proves full deductive equivalence with Metamath's Hailperin-derived + finite presentation, including its added singleton axiom. Replays the source proof + through nominal syntax, explicit freshness certificates, and a de Bruijn calculus. + Includes reusable first-order soundness and completeness, stratification, lowering, + and proof-translation infrastructure. This is a derivability theorem; it does not + assert the consistency of New Foundations. +branch: mathematical logic +entry_module: LeanPool.NFWeakPartition +authors: +- Elliot Glazer +- NFNotWPP contributors +- Jesse Michael Han +- Floris van Doorn +- Ian Klatzco +source: + title: NF proves not-WPP, source-complete Lean formalization + url: https://github.com/elliotglazer/nf-not-wpp/tree/fa58a7a895a73422042e167391c78e9149979913 + github_repo: elliotglazer/nf-not-wpp + commit: fa58a7a895a73422042e167391c78e9149979913 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- NFChoice.Foundation.NFStandard.NF_proves_not_WPP +main_results: +- declaration: NFChoice.Foundation.NFStandard.NF_proves_not_WPP + informal: Standard New Foundations formally derives the negation of WPP. WPP says + that, for all sets x and y, if a function maps y onto x and an injective function + maps y into x, then an injective function maps x into y. + source_ref: 'Upstream Proof/NFStandard/NotWPP.lean and the exact source formulation + documented in README.md at the pinned commit. Standard-NF completion: commit a9f9867ee379835f91a4f05f9c4a352d44ca1775, + 2026-08-25 00:16:27 UTC. Palomar registration PALOMAR-2026-08-27-000005 is dated + 2026-08-27, without a verified time of day. The responsible public maintainer + is Elliot Glazer. Upstream Flypitch4 credits Ian Klatzco and Claude for its Lean + 4 port; this import also includes AI-assisted porting and proof repairs, hence + mixed provenance. The deterministic Metamath replay is not by itself evidence + of AI mathematical authorship. Independent novelty is not claimed by this card.' +- declaration: NFChoice.Foundation.NFStandard.nf_hailperin_deductivelyEquivalent + informal: Extensionality together with every integer-stratified comprehension instance + and Metamath's Hailperin-derived finite first-order presentation, including its + added singleton axiom, derive exactly the same sentences. + source_ref: 'Upstream Proof/NFStandard/Equivalence.lean at the pinned commit. Metamath''s + ax-sn entry credits the singleton addition to SF on 2015-01-12 and distinguishes + it from Hailperin''s original presentation: https://us.metamath.org/nfeuni/ax-sn.html.' +tags: +- new-foundations +- weak-partition-principle +- first-order-logic +msc: +- 03E70 +- 03B35 diff --git a/LeanPool/projects/nikodym.yaml b/LeanPool/projects/nikodym.yaml new file mode 100644 index 0000000000..1b4a0c2f8f --- /dev/null +++ b/LeanPool/projects/nikodym.yaml @@ -0,0 +1,37 @@ +slug: nikodym +title: The sharp finite-field Nikodym exponent +summary: Proves a lower bound for Nikodym sets over arbitrary finite fields and a + matching exponent upper construction over sufficiently large prime fields, using + interpolation, algebraic degree bounds, and multiquadratic constructions. +branch: additive combinatorics +entry_module: LeanPool.Nikodym +authors: +- Ting-Wei Chao +- Zach Hunter +- Cosmin Pohoata +- Hung-Hsun Hans Yu +- Shengtong Zhang +source: + url: https://github.com/shengtongzhang-alt/nikodym + github_repo: shengtongzhang-alt/nikodym + commit: 38d16dcdbaef04f6f85e5dee92905b7db2eba930 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Nikodym.card_ge_pow_sub +- Nikodym.exists_isNikodym_card_le +main_results: +- declaration: Nikodym.card_ge_pow_sub + informal: Every Nikodym set in dimension d ≥ 2 over a finite field of size q has + size at least q^d − (8d²+1)q^(d−2^(1−d)). +- declaration: Nikodym.exists_isNikodym_card_le + informal: For every positive ε and dimension d ≥ 2, sufficiently large prime fields + admit a Nikodym set of size at most q^d − q^(d−2^(1−d)−ε). +tags: +- combinatorics +- finite-fields +- nikodym-sets +msc: +- 05B25 +- 11T99 diff --git a/LeanPool/projects/nivat-conjecture.yaml b/LeanPool/projects/nivat-conjecture.yaml new file mode 100644 index 0000000000..5808ce7b89 --- /dev/null +++ b/LeanPool/projects/nivat-conjecture.yaml @@ -0,0 +1,80 @@ +slug: nivat-conjecture +title: Nivat's conjecture +summary: Formalizes the proof of Nivat's conjecture found by GPT-6 Pro. A configuration + on the integer lattice over a finite alphabet whose number of distinct patterns + on some m-by-n rectangle is at most m times n has a nonzero period. Develops rational + alphabet labelings, the Laurent-polynomial shift action and its annihilator ideals, + the exact line ideal of a half-plane-supported difference of orbit-closure points, + complexity descent to a smaller rectangle, and Szabados's two-factor theorem for + finite-range configurations. +branch: symbolic dynamics +entry_module: LeanPool.Nivat +authors: +- Boon Suan Ho +source: + title: Pattern complexity and Nivat's conjecture + authors: + - GPT-6 Pro + - Boon Suan Ho + url: https://github.com/boonsuan/nivat + github_repo: boonsuan/nivat + commit: 84fe839635bdebb7d5e80c209b4f578a0c767fcf + announcement_date: '2026-09-14T04:54:56Z' + announcement_url: https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-14-000003&version=1 + palomar: PALOMAR-2026-09-14-000003 + notes: The proof and manuscript were produced by GPT-6 Pro and the Lean development + by GPT-6 Astra in Codex, both prompted and directed by Boon Suan Ho, who claims + no human mathematical authorship. The manuscript ships in the upstream repository + as paper/nivat.tex. The Lean sources are identical to the commit archived by Palomar + on 2026-09-14 (0d94547e); later upstream commits only touched the README and the + paper's abstract. No human expert review is claimed upstream; Bryna Kra's guest + post on Terence Tao's blog (2026-09-13) discusses the arrival of AI proofs of + the conjecture. The upstream MIT copyright and permission notice is retained in + every imported file. +license: MIT +status: verified +provenance: AI +main_declarations: +- Nivat.nivat +- NivatSubmission.nivat +main_results: +- declaration: Nivat.nivat + informal: For every finite alphabet A and configuration c from the integer lattice + to A, if for some positive m and n the number of distinct patterns of c on the + m-by-n rectangle, counted over all integer translations, is at most m times n, + then there is a nonzero lattice vector h with c(z + h) = c(z) for every z. +- declaration: NivatSubmission.nivat + informal: The same statement written with Mathlib vocabulary only, as registered + with Palomar. The pattern set is the range of the restriction of translates of + c to the product of the integer intervals [0, m) and [0, n), its natural cardinality + is at most m times n, and the conclusion is a lattice vector h different from + (0, 0) with c(z + h) = c(z) for all z. +- declaration: Nivat.nivat_rational + informal: A finite-range rational-valued configuration whose pattern count on some + m-by-n rectangle with m, n positive is at most m times n is periodic. This is + proved by strong induction on the area of the rectangle. +- declaration: Nivat.two_factors + informal: If a finite-range rational configuration has pattern count at most m times + n on an m-by-n rectangle with m, n positive and is annihilated by the product + of two difference operators in nonzero directions h and t, then it is periodic. +- declaration: Nivat.Descent.exists_smaller_low_complexity_rectangle + informal: If two pointwise limits x and y of translates of a finite-range rational + configuration c have nonzero difference whose Laurent annihilator ideal is generated + by a nonzero polynomial, and c has pattern count at most m times n on an m-by-n + rectangle, then applying that polynomial to c yields a configuration with pattern + count at most its area on some rectangle of strictly smaller positive area. +- declaration: Nivat.Algebra.exact_line_in_basis + informal: A nonzero rational configuration that vanishes on the open lower half-plane + of a lattice basis and is periodic with positive period q along the first basis + direction has Laurent annihilator ideal generated by a single line-supported polynomial, + obtained from a monic divisor of X^q - 1 of positive degree. +tags: +- symbolic-dynamics +- pattern-complexity +- periodicity +- Nivat-conjecture +- Laurent-polynomials +msc: +- 37B10 +- 37B51 +- 68R15 diff --git a/LeanPool/projects/non-sofic-group.yaml b/LeanPool/projects/non-sofic-group.yaml new file mode 100644 index 0000000000..32862b508c --- /dev/null +++ b/LeanPool/projects/non-sofic-group.yaml @@ -0,0 +1,35 @@ +slug: non-sofic-group +title: Existence of a finitely presented non-sofic group +summary: Constructs a finitely presented group that is not sofic, resolving the question + whether every group admits arbitrarily accurate finite permutation approximations. + The development combines an explicit elementary group with property T, a finite-table + construction of a finitely presented group, finite-model expansion, and a local + obstruction to sofic approximation. +branch: geometric and combinatorial group theory +entry_module: LeanPool.NonSoficGroup +authors: +- OpenAI +- Dean Cureton +source: + title: Ten Advances in Mathematics and Theoretical Computer Science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- SoficGroups.exists_finitelyPresented_nonsofic_group +main_results: +- declaration: SoficGroups.exists_finitelyPresented_nonsofic_group + informal: There exists a group admitting a finite presentation but no sofic approximation. +tags: +- group-theory +- sofic-groups +- property-t +- finite-approximations +msc: +- 20F65 +- 20E22 diff --git a/LeanPool/projects/nrr-aak.yaml b/LeanPool/projects/nrr-aak.yaml new file mode 100644 index 0000000000..4c1ceb81de --- /dev/null +++ b/LeanPool/projects/nrr-aak.yaml @@ -0,0 +1,31 @@ +slug: nrr-aak +title: Equal-area and equal-perimeter convex partitions +summary: Formalizes the Akopyan–Avvakumov–Karasev theorem that every planar convex + body can be partitioned into any prescribed positive number of convex pieces with + equal area and equal perimeter. Area is Lebesgue measure and perimeter is Hausdorff + measure of the boundary. +branch: convex geometry +main_declarations: +- HumanVerification.equalAreaEqualPerimeterPartition +main_results: +- declaration: HumanVerification.equalAreaEqualPerimeterPartition + informal: Every compact convex planar set with nonempty interior admits a convex + partition into any positive number of pieces whose areas and perimeters are all + equal. +tags: +- convex-geometry +- equivariant-topology +- partitions +msc: +- 52A10 +- 55M20 +provenance: mix +entry_module: LeanPool.NandakumarRamanaRao +authors: +- Arseniy Akopyan +source: + url: https://github.com/akopjan/nrr-aak + github_repo: akopjan/nrr-aak + commit: 9e57be63320f29cbb8d2da7c4a8ad6afdef681eb +license: Apache-2.0 +status: verified diff --git a/LeanPool/projects/odd-prime-valuation-distribution.yaml b/LeanPool/projects/odd-prime-valuation-distribution.yaml new file mode 100644 index 0000000000..ba27ee8951 --- /dev/null +++ b/LeanPool/projects/odd-prime-valuation-distribution.yaml @@ -0,0 +1,48 @@ +slug: odd-prime-valuation-distribution +title: Exact odd-prime distributions of central-binomial valuations +summary: Formalizes the exact distribution of the p-adic valuation of the central + binomial coefficient binom(2n,n) over every complete block 0 <= n < p^k for every + odd prime p. The development proves Kummer correctness for padded digit words, constructs + the two-state doubling carry enumerator, identifies it with the arithmetic valuation + polynomial, derives its second-order recurrence, proves the rational bivariate generating + function, and obtains closed formulas for valuations zero and one. +branch: number theory +entry_module: LeanPool.OddPrimeValuationDistribution +authors: +- Egor Lyfar +source: + title: Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen + authors: + - Ernst Eduard Kummer + doi: 10.1515/crll.1852.44.93 + github_repo: lyfar/gkp-carry-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- OddPrimeValuationDistribution.oddPrimeValuationGeneratingSeries_identity +main_results: +- declaration: OddPrimeValuationDistribution.oddPrimeValuationGeneratingSeries_identity + informal: For every odd prime p = 2h+1, proves the denominator-cleared identity + for the bivariate series whose coefficient of T^k X^r counts n < p^k with nu_p(binomial(2n,n)) + = r; equivalently the series is (1-XT)/(1-(h+1)(1+X)T+pXT^2). + source_ref: A. Straub, V. H. Moll, T. Amdeberhan, The p-adic valuation of k-central + binomial coefficients, Acta Arith. 140 (2009), 31-42. +- declaration: OddPrimeValuationDistribution.oddPrimeValuationPolynomial_recurrence + informal: Proves that the exact valuation polynomials satisfy F_{k+2}(X)+pX F_k(X)=(h+1)(1+X)F_{k+1}(X), + with F_0=1 and F_1=(h+1)+hX. +- declaration: OddPrimeValuationDistribution.padicValNat_oddPrime_centralBinom_ofDigits_eq_carryCount + informal: 'Proves Kummer correctness for every valid padded base-p word: the p-adic + valuation of its central binomial coefficient equals the number of carries produced + by doubling the word.' +- declaration: OddPrimeValuationDistribution.card_oddPrimeBlock_valuation_one + informal: Gives a closed formula for the number of n below p^(k+2) whose central + binomial coefficient has p-adic valuation exactly one. +tags: +- central-binomial-coefficients +- p-adic-valuations +- digit-carries +- generating-functions +msc: +- 11A63 +- 11B65 diff --git a/LeanPool/projects/odlyzko-bound.yaml b/LeanPool/projects/odlyzko-bound.yaml new file mode 100644 index 0000000000..2151114c11 --- /dev/null +++ b/LeanPool/projects/odlyzko-bound.yaml @@ -0,0 +1,37 @@ +slug: odlyzko-bound +title: Poitou's Explicit Odlyzko Bound for Root Discriminants +summary: Formalizes the unconditional explicit-formula argument for a lower bound + on root discriminants of totally complex number fields. Develops the completed Dedekind + zeta function and its functional equation, Poitou's explicit formula, the Tartar + test function, and a certified numerical estimate proving that degree at least 18 + forces root discriminant at least 8.25. +branch: algebraic number theory +entry_module: LeanPool.Odlyzko +authors: +- The FLT Project +- √2 +source: + title: Sur les petits discriminants + authors: + - Georges Poitou + url: https://www.numdam.org/item/SDPP_1976-1977__18_1_A6_0/ + github_repo: ImperialCollegeLondon/FLT +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- NumberField.Odlyzko.odlyzkoBound +main_results: +- declaration: NumberField.Odlyzko.odlyzkoBound + informal: Proves that every totally complex number field of degree at least 18 has + absolute discriminant at least 8.25 raised to its degree. + source_ref: Georges Poitou, Sur les petits discriminants, equation (26) and the + table on p. 17; Lean source adapted from ImperialCollegeLondon/FLT commit 01370b013e55c83f1994af814cd31cb8fb2ef653. +tags: +- number-theory +- discriminants +- number-fields +- explicit-formula +msc: +- 11R29 +- 11R42 diff --git a/LeanPool/projects/order-closures.yaml b/LeanPool/projects/order-closures.yaml new file mode 100644 index 0000000000..511d6571ae --- /dev/null +++ b/LeanPool/projects/order-closures.yaml @@ -0,0 +1,69 @@ +slug: order-closures +title: Order closure, order adherence and Fatou norms +summary: Formalizes order and unbounded-order adherence in vector lattices, the Gao-Leung + characterization of order-continuous norms on sigma-order-complete Banach lattices, + and the Solovay/Stone-space construction of order-complete C(K) spaces with arbitrarily + large density and norm-closed separable sublattices whose only order-closed vector-sublattice + extension is the whole C(K) space. Constructs solid sets with a prescribed infinite + number kappa of solid generators whose adherence towers are strict below any ordinal + at most kappa's successor cardinal. Constructs a complete weakly Fatou lattice norm + with constant 2 that is not equivalent to any Fatou lattice norm. Includes reusable + vector-lattice, net-convergence, sublattice, seminorm, Boolean-algebra and transfinite-iteration + infrastructure. +branch: functional analysis +entry_module: LeanPool.OrderClosures +authors: +- Antonio Avilés +- Mitchell A. Taylor +- Pedro Tradacete +- David Muñoz-Lahoz +source: + title: Order closure, order adherence and Fatou norms + arxiv: '2609.06689' + url: https://github.com/pedrotradacete/OrderClosures/tree/6189ba7134f6975e7d8e4c8c093a9869d03a769c + github_repo: pedrotradacete/OrderClosures + commit: 6189ba7134f6975e7d8e4c8c093a9869d03a769c +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- OrderClosures.exists_weaklyFatou_not_equivalent_fatou +main_results: +- declaration: OrderClosures.gaoLeung_orderContinuous_characterization + informal: 'For a sigma-order-complete Banach lattice X, the following are equivalent: + every vector sublattice has order-closed order adherence; every vector sublattice + has equal order and unbounded-order adherence; and the norm of X is order continuous.' + source_ref: Paper Theorem 1.1, recalling Gao-Leung Theorem 2.7. +- declaration: OrderClosures.gao_counterexample + informal: For every cardinal kappa there is a compact Hausdorff space K such that + C(K, R) is order complete and has density at least kappa, with a norm-closed separable + vector sublattice Y whose only order-closed vector-sublattice extension is all + of C(K, R). + source_ref: Paper Proposition 2.1, via the Solovay Boolean algebra. +- declaration: OrderClosures.solid_sets_require_arbitrarily_many_iterations + informal: For every infinite cardinal kappa and ordinal xi at most the initial ordinal + of kappa's successor cardinal, there is a vector lattice and a solid set with + solid generator number kappa whose order-adherence tower has a proper step at + every stage below xi. The witness and cardinal/ordinal parameters are lifted one + universe in Lean. + source_ref: Paper Theorem 2.6, with its explicit infinite-cardinal hypothesis. Lean + strengthens the stated size-kappa generating-set conclusion to minimum solid generator + number exactly kappa; this exact-minimum conclusion is proved in the formalization, + not stated in the paper. +- declaration: OrderClosures.exists_weaklyFatou_not_equivalent_fatou + informal: There is a real vector lattice with a complete lattice norm p satisfying + the weak Fatou property with constant 2, such that no lattice norm q with the + Fatou property is equivalent to p. Norms, completeness and equivalence are represented + explicitly by PaperLatticeNorm, IsCompleteFor and EquivalentNorms. + source_ref: Paper Theorem 3.1; self-contained construction of the result previously + announced by Anthony Wickstead and credited to Michael Elliott. The paper credits + Jaume de Dios Pont for an earlier Lean formalization of this result. +tags: +- banach-lattices +- order-convergence +- fatou-norms +- transfinite-constructions +msc: +- 46B42 +- 46A40 +- 46A19 diff --git a/LeanPool/projects/order-p-q.yaml b/LeanPool/projects/order-p-q.yaml new file mode 100644 index 0000000000..17395262bb --- /dev/null +++ b/LeanPool/projects/order-p-q.yaml @@ -0,0 +1,34 @@ +slug: order-p-q +title: Classification of groups of order p * q +summary: Classification of noncyclic finite groups of order p*q, including the prime-square + direct-product case and the p 1, the development + proves strict-local scale-identification impossibility at LR/epsilon = 4 and a deterministic + method family with three post-secant-initialization query-rate regimes. For p > + 2 it also proves deterministic known-parameter upper and lower query exponents p/(p+2), + with explicit horizon and dimension restrictions on the lower bound. +branch: optimization +entry_module: LeanPool.ParameterFreeGradient +authors: +- Yuning Yang +source: + url: https://github.com/yuningyang19/parameter_free_gradient_p + github_repo: yuningyang19/parameter_free_gradient_p + commit: f71215fa936fe01aff8a765c62b399e131df963b +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- V7.main +main_results: +- declaration: V7.scaleIdentificationImpossibility + informal: For finite real p > 1 and positive epsilon, measurable strict-local exact + value-gradient methods satisfy scale-identification lower bounds on convex smooth + one-dimensional objectives with unique minimizers, even at LR/epsilon = 4. The + result includes deterministic and randomized finite-horizon statements and infinite + worst-case expected queried-or-returned hitting time. +- declaration: V7.knownParameterAboveTwoOptimality + informal: For finite real p > 2 and convex differentiable objectives with attained + minima and a positive global primal-to-dual Lipschitz-gradient bound, deterministic + known-parameter exact value-gradient methods have upper query exponent p/(p+2). + The matching lower exponent concerns the first T queries, requires d >= 2 and + 1 <= T <= d, and retains the explicit min(p, log d) dependence. +- declaration: V7.main + informal: For convex differentiable objectives with attained minima and positive + global L-Lipschitz gradients from l_p to its dual, one deterministic family works + for every finite real p > 1 and positive epsilon after supplied nondegenerate + secant initialization with 0 < M0 <= L. It returns a queried point with dual gradient + norm at most epsilon. Writing R for distance to the minimizer set, the post-initialization + oracle-call bound has polynomial dependence on max(1, LR/epsilon) with exponent + 1/2 for 1 < p < 2 and p = 2, and p/(p+2) for p > 2, plus additive log(e + L/M0) + calibration cost. Constants depend only on p, with a universal constant at p = + 2; discovering the secant pair is outside the count. +tags: +- optimization +- oracle-complexity +- gradient-methods +msc: +- 90C25 diff --git a/LeanPool/projects/partial-combinatory-algebras.yaml b/LeanPool/projects/partial-combinatory-algebras.yaml new file mode 100644 index 0000000000..d16b00262e --- /dev/null +++ b/LeanPool/projects/partial-combinatory-algebras.yaml @@ -0,0 +1,39 @@ +slug: partial-combinatory-algebras +title: Partial Combinatory Algebras +summary: Formalises partial combinatory algebras with the K/S combinators, programming + combinators, the free combinatory algebra, and the graph model. +branch: logic and computability +entry_module: LeanPool.PartialCombinatoryAlgebras +authors: +- Andrej Bauer +source: + title: 'Realizability: An Introduction to its Categorical Side' + authors: + - Jaap van Oosten + url: https://doi.org/10.1016/S0049-237X(08)80003-1 + github_repo: andrejbauer/partial-combinatory-algebras +license: MIT +status: verified +provenance: human +main_declarations: +- LeanPool.PartialCombinatoryAlgebras.PCA +main_results: +- declaration: LeanPool.PartialCombinatoryAlgebras.PCA + informal: Defines the class of partial combinatory algebras (PCAs) with K and S + combinators. +- declaration: LeanPool.PartialCombinatoryAlgebras.CA + informal: Defines total combinatory algebras and exhibits every total combinatory + algebra as a PCA. +- declaration: LeanPool.PartialCombinatoryAlgebras.FreeCA + informal: Constructs the free total combinatory algebra as a quotient of formal + expressions. +- declaration: LeanPool.PartialCombinatoryAlgebras.GraphModel.isCA + informal: Constructs the graph-model combinatory algebra on the powerset of any + Listing-typed carrier. +tags: +- combinatory-algebra +- lambda-calculus +- computability +msc: +- 03B40 +- 03D75 diff --git a/LeanPool/projects/partial-regularity.yaml b/LeanPool/projects/partial-regularity.yaml new file mode 100644 index 0000000000..0945f18479 --- /dev/null +++ b/LeanPool/projects/partial-regularity.yaml @@ -0,0 +1,29 @@ +slug: partial-regularity +title: Almost all primes are partially regular +summary: Formalizes an asymptotic theorem about partial regularity of primes via Bernoulli-number + irregularity. +branch: analytic number theory +entry_module: LeanPool.PartialRegularity +authors: +- Evan Chen +- Kenny Lau +- Seewoo Lee +- Ken Ono +- Jujian Zhang +source: + arxiv: '2602.05090' + github_repo: AxiomMath/partial-regularity +license: MIT +status: verified +provenance: AI +main_declarations: +- LeanPool.PartialRegularity.Extension.irregularPrimes_isBigO +main_results: +- declaration: LeanPool.PartialRegularity.Extension.irregularPrimes_isBigO + informal: Proves the asymptotic big-O bound behind the partial-regularity theorem. +tags: +- number-theory +- asymptotics +msc: +- 11B68 +- 11N05 diff --git a/LeanPool/projects/pcf-theory.yaml b/LeanPool/projects/pcf-theory.yaml new file mode 100644 index 0000000000..b50d0393fd --- /dev/null +++ b/LeanPool/projects/pcf-theory.yaml @@ -0,0 +1,29 @@ +slug: pcf-theory +title: PCF Theory +summary: Formalizes foundational results from Shelah's Possible Cofinality theory, + including the existence of club-guessing sequences. +branch: set theory +entry_module: LeanPool.PCFTheory +authors: +- YnirPaz +source: + doi: 10.1007/978-1-4020-5764-9_15 + github_repo: YnirPaz/PCF-Theory +license: MIT +status: verified +provenance: human +main_declarations: +- Ordinal.exists_isClubGuessing_of_cof_uncountable +main_results: +- declaration: Ordinal.exists_isClubGuessing_of_cof_uncountable + informal: Proves the existence of a club-guessing sequence on a stationary set whose + elements share a common uncountable cofinality bounded by the cofinality of the + ambient ordinal. +tags: +- set-theory +- cardinal-arithmetic +- club-guessing +msc: +- 03E04 +- 03E10 +- 03E55 diff --git a/LeanPool/projects/pdl-craig-interpolation.yaml b/LeanPool/projects/pdl-craig-interpolation.yaml new file mode 100644 index 0000000000..31cf8be06b --- /dev/null +++ b/LeanPool/projects/pdl-craig-interpolation.yaml @@ -0,0 +1,73 @@ +slug: pdl-craig-interpolation +title: Craig interpolation and Beth definability for propositional dynamic logic +summary: Proves Craig interpolation for full propositional dynamic logic, including + tests and Kleene star, by constructing interpolants from uniform cyclic tableaux. + Interpolants use only atomic propositions and atomic programs common to the two + formulas. The development also proves the local Beth definability property for a + propositional letter, and supplies reusable syntax, Kripke semantics, Fischer-Ladner + closure, tableau soundness and completeness, finite reachability and game-theoretic + infrastructure. The results assert existence; no executable extraction algorithm + or complexity bound is claimed. Human contributors and Aristotle jointly developed + the formalization. The completeness modules reuse the finite game determinacy theorem + already pooled in `LeanPool.Lean4GlCoalgebras` instead of duplicating it. +branch: modal logic +entry_module: LeanPool.PDL +authors: +- PDL formalization contributors +contributors: +- Malvin Gattinger +- Madeleine Gignoux +- Noam Cohen +- Haitian Wang +- Wietse Bosman +- Xiaoshuang Yang +- Amos Nicodemus +- Andrés Goens +- DdosSantosGomes +- Eshel Yaron +- Dionysis Yusofy +source: + title: 'Propositional Dynamic Logic has Craig Interpolation: a tableau-based proof' + authors: + - Manfred Borzechowski + - Malvin Gattinger + - Helle Hvid Hansen + - Revantha Ramanayake + - Francisco Trucco Dalmas + - Yde Venema + arxiv: '2503.13276' + url: https://github.com/m4lvin/lean4-pdl + github_repo: m4lvin/lean4-pdl + commit: ec3b050e59cd829a350e92c512951a381a80cddd + announcement: https://leanprover.zulipchat.com/#narrow/channel/579630-Project-announcements/near/622297905 + announcement_date: '2026-09-07T14:27:11Z' +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- PDL.interpolation +- PDL.beth +main_results: +- declaration: PDL.interpolation + informal: If the implication from PDL formula phi to psi is valid in every Kripke + model, there exists a PDL formula theta whose atomic proposition and program vocabulary + is contained in the intersection of theirs, such that phi implies theta and theta + implies psi in every model. +- declaration: PDL.beth + informal: If a PDL formula phi implicitly determines a propositional letter p (all + two-letter substitutions agree wherever both substituted formulas hold), there + is a formula psi using only phi's vocabulary other than p such that phi entails + that p is equivalent to psi. +- declaration: PDL.consIffSat + informal: A free sequent is tableau-consistent if and only if it is satisfiable + in a Kripke model. +tags: +- modal-logic +- propositional-dynamic-logic +- craig-interpolation +- beth-definability +- tableaux +msc: +- 03B45 +- 03B70 +- 03F03 diff --git a/LeanPool/projects/pebbling-hypercube.yaml b/LeanPool/projects/pebbling-hypercube.yaml new file mode 100644 index 0000000000..8e1872f8ef --- /dev/null +++ b/LeanPool/projects/pebbling-hypercube.yaml @@ -0,0 +1,37 @@ +slug: pebbling-hypercube +title: Optimal Pebbling Number of the Hypercube +summary: Formalizes that the optimal pebbling number of the n-dimensional hypercube + Q_n is Theta((4/3)^n). Builds a weight-function lower bound of (4/3)^n and a probabilistic + recursive upper bound C * (4/3)^n with an explicit constant, combining them into + a two-sided big-Theta witness. +branch: combinatorics +entry_module: LeanPool.PebblingLean +authors: +- Lior Pachter +source: + title: Optimal pebbling number of the hypercube + authors: + - Lior Pachter + url: https://github.com/pachterlab/P_2026_2 + github_repo: pachterlab/P_2026_2 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- PebblingLean.Hypercube.Paper.optimalPebblingNumber_theta +main_results: +- declaration: PebblingLean.Hypercube.Paper.optimalPebblingNumber_theta + informal: 'The optimal pebbling number of Q_n has order (4/3)^n: there is a positive + constant C with (4/3)^n <= o(Q_n) <= C * (4/3)^n for all n.' +- declaration: PebblingLean.Hypercube.Paper.lower_bound + informal: The optimal pebbling number of Q_n is at least (4/3)^n, proved by a weight-function + argument summing one-pebble weights over the cube. +- declaration: PebblingLean.Hypercube.Paper.upper_bound + informal: The optimal pebbling number of Q_n is at most C * (4/3)^n for an explicit + constant C, via a probabilistic recursive splitting construction. +tags: +- combinatorics +- pebbling +- hypercube +msc: +- 05C57 diff --git a/LeanPool/projects/pentagonal-number-theorem-analytic.yaml b/LeanPool/projects/pentagonal-number-theorem-analytic.yaml new file mode 100644 index 0000000000..476102b441 --- /dev/null +++ b/LeanPool/projects/pentagonal-number-theorem-analytic.yaml @@ -0,0 +1,80 @@ +slug: pentagonal-number-theorem-analytic +title: Euler's pentagonal number theorem, two independent proofs, and the Jacobi triple + product +summary: 'Two independent proofs of Euler''s pentagonal number theorem that share + no lemmas, so each confirms the result on its own. The primary development is combinatorial: + Franklin''s sign-reversing involution on partitions into distinct parts, giving + p_e(n) - p_o(n) = (-1)^|k| when n is the generalized pentagonal number k(3k-1)/2 + and 0 otherwise, stated under a single integer index, together with the identification + of that signed count with the coefficient of X^n in the truncated product over ℤ⟦X⟧. + It builds on Mathlib''s Nat.Partition.genFun. The second route develops q-Pochhammer + symbols, the Gaussian binomial coefficient, Euler''s two identities and the Cauchy + identity, proves the Jacobi triple product both as a formal power series identity + in (LaurentPolynomial ℂ)⟦X⟧ and analytically over ℂ for ‖q‖ < 1, and reaches the + same theorem for the infinite product by specialization. The Lean proofs in both + routes were largely generated by Aristotle; the Franklin route additionally has + an author-written blueprint that the formalization follows lemma by lemma, and carries + substantially more manual intervention and review, whereas the q-series route is + machine-generated blueprint and all. The Franklin route and the analytic pentagonal + number theorem overlap the pooled `pentagonal-number-theorem` development, which + states its analytic form over RCLike fields rather than only over the complex numbers; + the q-Pochhammer, Gaussian binomial and Euler identity definitions overlap arXiv:2607.01544. + The project''s distinct contribution is the complex-analytic layer: an analytic + Jacobi triple product over the complex numbers for norm q < 1, its extension to + the punctured disc, the analytic Cauchy identity, and the summability and locally-uniform-convergence + machinery they require. Neither the pooled entry nor the non-archimedean development + of arXiv:2607.01544 covers that setting.' +branch: number theory and enumerative combinatorics +entry_module: LeanPool.PentagonalNumberTheoremAnalytic +authors: +- Jonathan Conrad +- Paula Muermann +- Maryna Viazovska +source: + title: Pentagonal Number Theorem + authors: + - Jonathan Conrad + - Paula Muermann + - Maryna Viazovska + url: https://viazovska.github.io/PentagonalNumberTheorem + github_repo: viazovska/PentagonalNumberTheorem + revision: 5c3413265fcb44f74e67ebb5abddcac4b82e611b +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- PentagonalNumberTheorem.Franklin.euler_pentagonal_number_theorem_packaged +main_results: +- declaration: PentagonalNumberTheorem.Franklin.euler_pentagonal_number_theorem_packaged + informal: The primary result. For every n, either n is a generalized pentagonal + number k(3k-1)/2 for some integer k, in which case the number of partitions of + n into an even number of distinct parts exceeds the number with an odd number + of distinct parts by exactly (-1)^|k|; or n is not of that form, in which case + the two counts are equal. Proved by Franklin's sign-reversing involution. +- declaration: PentagonalNumberTheorem.Franklin.coeff_prod_eq_pe_sub_po + informal: The coefficient of X^n in the truncated product ∏_{k=1}^{n}(1 - X^k) over + ℤ⟦X⟧ equals p_e(n) - p_o(n), the signed count of partitions of n into distinct + parts. Truncating at n is harmless because the omitted factors are ≡ 1 mod X^{n+1}. +- declaration: QSeries.jacobiTripleProduct + informal: For ‖q‖ < 1, ‖z‖ < 1 and z ≠ 0, the product (q;q)_∞ (-z;q)_∞ (-q/z;q)_∞ + equals the bilateral theta series ∑_{k ∈ ℤ} z^k q^{k(k-1)/2}. +- declaration: QSeries.euler_pentagonal_number + informal: For ‖q‖ < 1, the infinite product (q;q)_∞ equals ∑_{k ∈ ℤ} (-1)^k q^{ω(k)} + over the generalized pentagonal numbers ω(k) = k(3k-1)/2, written as two one-sided + sums. Obtained from the Jacobi triple product by specialization. +- declaration: QSeries.FormalPowerSeries.jacobiTripleProduct + informal: 'The Jacobi triple product as an identity of formal power series in (LaurentPolynomial + ℂ)-coefficient power series, with no convergence hypothesis: the product of q-Pochhammer + symbols equals the bilateral theta series coefficientwise.' +tags: +- number-theory +- partitions +- q-series +- pentagonal-number-theorem +- jacobi-triple-product +msc: +- 11P81 +- 11P84 +- 05A17 +- 05A19 +- 33D15 diff --git a/LeanPool/projects/pentagonal-number-theorem.yaml b/LeanPool/projects/pentagonal-number-theorem.yaml new file mode 100644 index 0000000000..5a996a0af9 --- /dev/null +++ b/LeanPool/projects/pentagonal-number-theorem.yaml @@ -0,0 +1,37 @@ +slug: pentagonal-number-theorem +title: Euler's pentagonal number theorem +summary: Formalizes Euler's pentagonal number theorem as a generating-function identity + for power series and for real/complex numbers, with a partition-recurrence corollary + and a self-contained Franklin-style combinatorial proof. +branch: number theory +entry_module: LeanPool.PentagonalNumberTheorem +authors: +- Weiyi Wang +source: + url: https://en.wikipedia.org/wiki/Pentagonal_number_theorem + github_repo: wwylele/PentagonalNumberTheorem +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- pentagonalNumberTheorem_powerSeries +- Nat.Partition.sum_partition +main_results: +- declaration: pentagonalNumberTheorem_powerSeries + informal: Proves Euler's pentagonal number theorem as a formal power series identity + ∏(1-xⁿ) = Σ(-1)ᵏ x^(k(3k-1)/2). +- declaration: pentagonalNumberTheorem_complex + informal: Proves the analytic version of the pentagonal number theorem for real + or complex numbers of norm less than one. +- declaration: Nat.Partition.sum_partition + informal: Derives Euler's pentagonal recurrence for the partition function from + the pentagonal number theorem. +tags: +- number-theory +- combinatorics +- partitions +- power-series +- pentagonal-number-theorem +msc: +- 11P81 +- 05A17 diff --git a/LeanPool/projects/permanent-formula-lower-bound.yaml b/LeanPool/projects/permanent-formula-lower-bound.yaml new file mode 100644 index 0000000000..af50aa8ff4 --- /dev/null +++ b/LeanPool/projects/permanent-formula-lower-bound.yaml @@ -0,0 +1,41 @@ +slug: permanent-formula-lower-bound +title: Quartic-over-logarithmic lower bound for rational permanent formulas +summary: Proves that every valid rational arithmetic formula over the complex numbers + computing the n-by-n permanent, for n at least 32, has at least n^4 / (192 log_2 + n) variable leaves. The development derives the bound through coefficient-field + transcendence degree, Jacobian minors, matching-polynomial identities, and a cyclic + block decomposition. +branch: algebraic complexity theory +entry_module: LeanPool.PermanentFormulaLowerBound +authors: +- OpenAI +- Dean Cureton +source: + title: Ten Advances in Mathematics and Theoretical Computer Science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- PermanentFormulaLowerBound.permanent_rational_formula_logarithmic_lower_bound +main_results: +- declaration: PermanentFormulaLowerBound.permanent_rational_formula_logarithmic_lower_bound + informal: For n at least 32, every valid rational formula over the complex numbers + that computes the n-by-n permanent has at least n^4 / (192 log_2 n) variable leaves. +- declaration: PermanentFormulaLowerBound.permanent_rational_formula_lower_bound + informal: The same quartic-over-logarithmic lower bound controls variable leaves, + total leaves, and vertices, and half that bound controls internal gates. +tags: +- algebraic-complexity +- arithmetic-formulas +- permanent +- lower-bounds +- transcendence-degree +msc: +- 68Q17 +- 68Q25 +- 15A15 diff --git a/LeanPool/projects/phaseretrieval.yaml b/LeanPool/projects/phaseretrieval.yaml new file mode 100644 index 0000000000..cd697146b7 --- /dev/null +++ b/LeanPool/projects/phaseretrieval.yaml @@ -0,0 +1,53 @@ +slug: phaseretrieval +title: Stable phase retrieval for Hermite-Fock expansions +summary: Formalizes stable phase retrieval for finite Hermite-Fock polynomials and + their Gaussian L2 closure in d complex dimensions, together with the quantitative + one-dimensional Fock-space local stability estimate that underlies it. +branch: analysis +entry_module: LeanPool.PhaseRetrieval +authors: +- Susanna Bertolini +- Jaume de Dios Pont +source: + title: Stable phase retrieval for Hermite-Fock expansions + authors: + - Susanna Bertolini + - Jaume de Dios Pont + url: https://github.com/susannabertolini/PhaseRetrieval + github_repo: susannabertolini/PhaseRetrieval +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- DimdPolyShowcaseChallenge.stable_phase_retrieval +main_results: +- declaration: DimdPolyShowcaseChallenge.stable_phase_retrieval + informal: Proves stable phase retrieval for the Gaussian L2 closure of finite d-dimensional + Hermite-Fock polynomials at a fixed level, giving a uniform stability constant + bounding the phase-aligned L2 distance by the L2 distance of magnitudes. + source_ref: R. Balan, P. Casazza, D. Edidin, On signal reconstruction without phase, + Appl. Comput. Harmon. Anal. 20 (2006), 345-356, doi:10.1016/j.acha.2005.07.001. +- declaration: FockSPR.orthogonalReduction_exists_phase + informal: Proves the phase-aligned one-dimensional Fock-space local stability estimate, + producing a unit phase for which the L2 distance to one is controlled by the magnitude + defect. +- declaration: FockSPR.orthogonalReduction_core + informal: Proves the quantitative centered core estimate bounding the Gaussian Fock + norm of a polynomial with small norm by the magnitude defect against one. +- declaration: FockSPR.LocalFockSPR + informal: Proves the local Fock-space stable phase retrieval coercivity estimate + for centered polynomials. +- declaration: DimdPolyShowcaseChallenge.HermitePoly + informal: Defines the normalized one-variable complex Hermite-Fock basis polynomial. +- declaration: DimdPolyShowcaseChallenge.TrueHermiteFunctions + informal: Defines the Gaussian L2 closure of finite Hermite-Fock polynomials at + a fixed level. +tags: +- phase-retrieval +- hermite-fock +- gaussian-measure +- complex-analysis +msc: +- 42C05 +- 46E22 +- 94A12 diff --git a/LeanPool/projects/pl-accelerated-nesterov-lean.yaml b/LeanPool/projects/pl-accelerated-nesterov-lean.yaml new file mode 100644 index 0000000000..8575f090bf --- /dev/null +++ b/LeanPool/projects/pl-accelerated-nesterov-lean.yaml @@ -0,0 +1,54 @@ +slug: pl-accelerated-nesterov-lean +title: Accelerated Nesterov convergence under local Polyak-Lojasiewicz conditions +summary: Formalizes accelerated Nesterov convergence near a smooth minimizer manifold + under a local Polyak-Lojasiewicz condition, including embedded-manifold and C3 public + theorem statements with explicit exponential rate and prefactor. +branch: optimization +entry_module: LeanPool.PLAcceleratedNesterovLean +authors: +- Max Obreiter +- Tobias Steinbrecher +- Robert Foerster +source: + title: Optimal local linear convergence of Nesterov's accelerated gradient method + for C2 functions under the Polyak-Lojasiewicz inequality + authors: + - Zixu Feng + - Hao Yuan + arxiv: '2603.21516' + github_repo: M1ngXU/PL-Accelerated-Nesterov-Lean +license: MIT +status: verified +provenance: human +main_declarations: +- PLAcceleratedNesterovLean.nesterov_pl_accelerated_rate +main_results: +- declaration: PLAcceleratedNesterovLean.nesterov_pl_accelerated_rate + informal: Proves an accelerated Nesterov convergence rate for state positions near + a C2 embedded minimizer manifold, with an existential momentum parameter, explicit + exponential decay, and prefactor 2. + source_ref: Z. Feng and H. Yuan, Optimal local linear convergence of Nesterov's + accelerated gradient method for C2 functions under the Polyak-Lojasiewicz inequality, + arXiv:2603.21516. +- declaration: PLAcceleratedNesterovLean.nesterov_pl_accelerated_rate_c3 + informal: Proves the corresponding C3 public theorem, deriving the minimizer geometry + internally from smoothness and the Polyak-Lojasiewicz condition. + source_ref: Z. Feng and H. Yuan, Optimal local linear convergence of Nesterov's + accelerated gradient method for C2 functions under the Polyak-Lojasiewicz inequality, + arXiv:2603.21516. +- declaration: PLAcceleratedNesterovLean.nesterov_convergence_at_base_point_position_theta + informal: Provides the reusable local theta-retuned convergence theorem used by + the public accelerated-rate statements. + source_ref: Z. Feng and H. Yuan, Optimal local linear convergence of Nesterov's + accelerated gradient method for C2 functions under the Polyak-Lojasiewicz inequality, + arXiv:2603.21516. +tags: +- optimization +- numerical-analysis +- gradient-methods +- polyak-lojasiewicz +- differential-geometry +msc: +- 49M37 +- 65K05 +- 58C15 diff --git a/LeanPool/projects/poincare-three-body-nonintegrability.yaml b/LeanPool/projects/poincare-three-body-nonintegrability.yaml new file mode 100644 index 0000000000..72be736c9e --- /dev/null +++ b/LeanPool/projects/poincare-three-body-nonintegrability.yaml @@ -0,0 +1,65 @@ +slug: poincare-three-body-nonintegrability +title: Poincaré's Nonintegrability Theorem for the Restricted Three-Body Problem +summary: Formalizes Poincaré's classical parameter-analytic nonintegrability theorem + for the planar circular restricted three-body problem. The development defines the + rotating-frame Hamiltonian on its collision-free domain, proves logarithmic blow-up + of the resonant disturbing average at an apoapsis collision boundary, and analytically + propagates the resulting dense resonant obstruction to exclude an additional analytic + first integral. The final statement is the fixed-coordinate, global uniform-domain + special case of the planar part of Yagasaki's Theorem 1.1; his local meromorphic + resonant-orbit obstruction is Theorem 3.1. +branch: dynamical systems +entry_module: LeanPool.PoincareThreeBody +authors: +- Gershon Bialer +source: + title: A new proof of Poincaré's result on the restricted three-body problem + authors: + - Kazuyuki Yagasaki + arxiv: '2111.11031' + doi: 10.1063/5.0266087 + url: https://arxiv.org/abs/2111.11031 + github_repo: gersh/lean-pool +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.PoincareThreeBody.nonintegrability_of_collisionBand +main_results: +- declaration: LeanPool.PoincareThreeBody.nonintegrability_of_collisionBand + informal: No jointly real-analytic family near zero mass can be an additional first + integral of the planar circular restricted three-body Hamiltonian with phase differential + independent from the Hamiltonian differential anywhere on the uniform collision-free + domain. + source_ref: 'Kazuyuki Yagasaki, "A new proof of Poincaré''s result on the restricted + three-body problem," Theorem 1.1 (p. 2), planar case; compare the precise local + meromorphic obstruction near resonant unperturbed elliptic orbits in Theorem 3.1 + (p. 8). The Lean result is the fixed-coordinate global uniform-domain special + case: a global real-analytic family restricts and complexifies on the local neighborhoods + used in the source theorem.' +- declaration: LeanPool.PoincareThreeBody.delaunayHomological_obstruction + informal: If the first homological equation holds and its perturbing Fourier coefficient + is nonzero at every positive rational Kepler resonance, then the leading differential + of a putative analytic first integral is nowhere independent of the unperturbed + Hamiltonian frequency. +- declaration: LeanPool.PoincareThreeBody.delaunayDenseResonance_obstruction + informal: For the rotating Kepler frequency in Delaunay actions, orthogonality of + a continuous candidate differential to every positive rational resonance vector + forces dependence with the Hamiltonian frequency at every positive action. +- declaration: LeanPool.PoincareThreeBody.homologicalEquation_obstruction + informal: At a resonant action in two degrees of freedom, a nonzero perturbing Fourier + mode and the first homological equation force the unperturbed frequency and the + differential of a putative leading first-integral coefficient to be linearly dependent. +- declaration: LeanPool.PoincareThreeBody.denseResonance_obstruction + informal: If the resonant obstruction holds on a dense set of actions and the frequency + and differential vary continuously, then they are linearly dependent at every + action. +tags: +- dynamical-systems +- celestial-mechanics +- hamiltonian-systems +- nonintegrability +msc: +- 70F07 +- 37J30 +- 37J40 diff --git a/LeanPool/projects/pointwise-birkhoff.yaml b/LeanPool/projects/pointwise-birkhoff.yaml new file mode 100644 index 0000000000..d3379d02e6 --- /dev/null +++ b/LeanPool/projects/pointwise-birkhoff.yaml @@ -0,0 +1,35 @@ +slug: pointwise-birkhoff +title: Pointwise Birkhoff Ergodic Theorem +summary: Formalizes the pointwise Birkhoff ergodic theorem for integrable real-valued + observables. +branch: ergodic theory +entry_module: LeanPool.PointwiseBirkhoff +authors: +- Lua Viana Reis +- Oliver Butterley +- Pietro Monticone +source: + title: Ergodic Theory + authors: + - Karl Petersen + url: https://doi.org/10.1017/CBO9780511608728 + github_repo: lua-vr/pointwise-birkhoff +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LeanPool.PointwiseBirkhoff.birkhoffErgodicTheorem' +main_results: +- declaration: LeanPool.PointwiseBirkhoff.birkhoffErgodicTheorem' + informal: Proves almost-everywhere convergence of Birkhoff averages to invariant + conditional expectation for any integrable real-valued observable. +- declaration: LeanPool.PointwiseBirkhoff.birkhoffErgodicTheorem + informal: Proves the measurable-observable version of the pointwise Birkhoff ergodic + theorem. +tags: +- ergodic-theory +- measure-theory +- probability +msc: +- 37A30 +- 28D05 diff --git a/LeanPool/projects/polya-enumeration-theorem.yaml b/LeanPool/projects/polya-enumeration-theorem.yaml new file mode 100644 index 0000000000..9d1f5b01f2 --- /dev/null +++ b/LeanPool/projects/polya-enumeration-theorem.yaml @@ -0,0 +1,27 @@ +slug: polya-enumeration-theorem +title: Pólya's enumeration theorem +summary: Formalizes Pólya's enumeration theorem counting orbits of colorings under + a group action. +branch: combinatorics +entry_module: LeanPool.PolyaEnumerationTheorem +authors: +- Luka Opravš +source: + url: https://github.com/Luka-O/polya-enumeration-theorem + github_repo: Luka-O/polya-enumeration-theorem +license: MIT +status: verified +provenance: human +main_declarations: +- LeanPool.PolyaEnumerationTheorem.polya_theorem +main_results: +- declaration: LeanPool.PolyaEnumerationTheorem.polya_theorem + informal: Counts distinct colorings of a finite set under a group action by averaging + powers of the number of colors over cycle indices. +tags: +- combinatorics +- group-theory +- enumeration +msc: +- 05A15 +- 20B30 diff --git a/LeanPool/projects/polylean.yaml b/LeanPool/projects/polylean.yaml new file mode 100644 index 0000000000..330a293da1 --- /dev/null +++ b/LeanPool/projects/polylean.yaml @@ -0,0 +1,34 @@ +slug: polylean +title: Polylean Unit Conjecture Counterexample +summary: Formalizes Gardam's counterexample to Kaplansky's Unit Conjecture, including + the Promislow group, group rings, torsion-freeness of the group, and the non-trivial + unit in 𝔽₂[P]. +branch: algebra +entry_module: LeanPool.Polylean +authors: +- Siddhartha Gadgil +- Anand Rao +source: + arxiv: '2102.11818' + doi: 10.4007/annals.2021.194.3.9 + github_repo: siddhartha-gadgil/Polylean +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LeanPool.Polylean.P.torsionFree +- LeanPool.Polylean.Gardam.GardamTheorem +main_results: +- declaration: LeanPool.Polylean.P.torsionFree + informal: Proves that the Promislow group used in Gardam's construction is torsion-free. +- declaration: LeanPool.Polylean.Gardam.GardamTheorem + informal: Proves Gardam's counterexample to Kaplansky's Unit Conjecture by constructing + a non-trivial unit in the group ring 𝔽₂[P]. +tags: +- algebra +- group-theory +- ring-theory +- unit-conjecture +msc: +- 16S34 +- 20F65 diff --git a/LeanPool/projects/polylog-integrals.yaml b/LeanPool/projects/polylog-integrals.yaml new file mode 100644 index 0000000000..ea60d0e723 --- /dev/null +++ b/LeanPool/projects/polylog-integrals.yaml @@ -0,0 +1,169 @@ +slug: polylog-integrals +title: Polylogarithms and sixteen log-trigonometric integral identities +summary: Formalizes the sixteen A-series identities from rainrzk's PolyLog Integrals + tutorial, together with reusable real and complex polylogarithm series, absolute + convergence on the closed unit disk, differentiation, duplication, Euler reflection + and Landen identities, logarithmic moments, Fubini kernels and radial integration + machinery. The formulas involve Catalan's constant, zeta(3), Li_4(1/2) and Im Li_3((1+i)/2). + These are classical identities, not new informal results. A004 uses the corrected + factor pi^2/24; A014 uses a proved expression entirely inside the unit disk, replacing + the tutorial's invalid series expansion. A015 uses radial differentiation instead + of a half-disc contour integral. The formalization was materially completed on 2026-10-01 + at 00:01:57 UTC. The 2022 informal tutorial is older; the eligibility date is that + of the completed Lean proofs. +branch: analysis +entry_module: LeanPool.PolylogIntegrals +authors: +- rainrzk +source: + title: LeanPolyLog, completed formalization of PolyLog Integrals + url: https://github.com/rainrzk/LeanPolyLog/tree/056b7b42fd1a26f241e2945c67ae821ec5bdc49b + github_repo: rainrzk/LeanPolyLog + commit: 056b7b42fd1a26f241e2945c67ae821ec5bdc49b +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPolyLog.Proofs.A016 +main_results: +- declaration: LeanPolyLog.Proofs.A001 + informal: Li_2(i) equals -pi^2/48 plus i times Catalan's constant. + source_ref: Upstream Statements.lean and Proofs/A001.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A002 + informal: Li_2(1/(1+i)) equals 5 pi^2/96 - log^2(2)/8 plus i times (pi log(2)/8 + - G). + source_ref: Upstream Statements.lean and Proofs/A002.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A003 + informal: Li_2((sqrt(5)-1)/2) equals pi^2/10 - log^2((1+sqrt(5))/2). + source_ref: Upstream Statements.lean and Proofs/A003.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A004 + informal: The integral from 0 to 1 of log(1+x^2)/x equals pi^2/24, correcting the + tutorial's factor of two. + source_ref: Upstream Statements.lean and Proofs/A004.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A005 + informal: The integral from 0 to 1 of log^2(1+x^2)/x^3 equals pi^2/12 - log^2(2). + source_ref: Upstream Statements.lean and Proofs/A005.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A006_cos + informal: The integral from 0 to pi/4 of log(cos x) equals G/2 - (pi/4) log(2). + source_ref: Upstream Statements.lean and Proofs/A006.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A006_sin + informal: The integral from 0 to pi/4 of log(sin x) equals -G/2 - (pi/4) log(2). + source_ref: Upstream Statements.lean and Proofs/A006.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A007 + informal: The integral from 0 to pi/4 of x log(cos x) equals pi G/8 - pi^2 log(2)/32 + - 21 zeta(3)/128. + source_ref: Upstream Statements.lean and Proofs/A007.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A008 + informal: The integral from 0 to pi/4 of x log(1+tan x) equals pi^2 log(2)/64 - + pi G/8 + 21 zeta(3)/64. + source_ref: Upstream Statements.lean and Proofs/A008.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A009 + informal: The integral from 0 to pi/2 of x^2 cot x equals pi^2 log(2)/4 - 7 zeta(3)/8. + source_ref: Upstream Statements.lean and Proofs/A009.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A010 + informal: The integral from 0 to pi/4 of x^2 tan^2 x equals -G + pi^2/16 - pi^3/192 + + pi log(2)/4. + source_ref: Upstream Statements.lean and Proofs/A010.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A011 + informal: The integral from 0 to 1 of Li_2(x)/(x sqrt(1-x^2)) equals 3 pi^2 log(2)/8 + - 7 zeta(3)/16. + source_ref: Upstream Statements.lean and Proofs/A011.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A012 + informal: The integral from 0 to 1 of arctan(x) log(x)/(1+x) equals -pi^3/64 + G + log(2)/2. + source_ref: Upstream Statements.lean and Proofs/A012.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A013 + informal: The integral from 0 to 1 of arctan(x) log^2(x)/(1+x) equals -pi^2 G/24 + - pi^3 log(2)/32 + 21 pi zeta(3)/64. + source_ref: Upstream Statements.lean and Proofs/A013.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A014 + informal: The integral from 0 to 1 of log(x) log(1+x)/(1+x^2) equals 11 pi^3/128 + + 3 pi log^2(2)/32 - 2 G log(2) - 3 Im Li_3((1+i)/2). + source_ref: Upstream Statements.lean and Proofs/A014.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A015 + informal: The integral from 0 to pi/2 of x log^2(sin x) equals Li_4(1/2) - 19 pi^4/2880 + + log^4(2)/24 + pi^2 log^2(2)/12. + source_ref: Upstream Statements.lean and Proofs/A015.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Proofs.A016 + informal: The integral from 0 to 1 of Li_3(x)/(x sqrt(1-x^2)) equals Li_4(1/2)/2 + + 41 pi^4/5760 + log^4(2)/48 + pi^2 log^2(2)/6. + source_ref: Upstream Statements.lean and Proofs/A016.lean at the pinned commit; + README prior-appearance table and blueprint/ERRATA.md label classical sources + and corrections. Lean proofs were written with Claude assistance and reviewed + by rainrzk; the informal tutorial is rainrzk (2022). +- declaration: LeanPolyLog.Dilog.hasDerivAt_Li_succ + informal: For every natural s and nonzero complex z in the open unit disk, Li_(s+1) + has derivative Li_s(z)/z at z. + source_ref: Upstream Lemmas/DilogComplex.lean, termwise differentiation. +- declaration: LeanPolyLog.Dilog.Li_two_add_Li_two_div_sub_one + informal: For complex z of norm at most 1 and real part at most 1/2, Li_2(z) + Li_2(z/(z-1)) + = -log^2(1-z)/2, with the principal logarithm. + source_ref: Upstream Lemmas/DilogComplex.lean, complex Landen identity. +- declaration: LeanPolyLog.Arcsin.integral_Lir_div_mul_sqrt + informal: For s at least 2 and a measurable nonnegative weight w on (0,1] with integrable + moments equal to 1/(n+1)^s, the integral from 0 to 1 of Li_s(x)/(x sqrt(1-x^2)) + equals the integral from 0 to pi/2 of w(sin t) times (pi/2+t). + source_ref: Upstream Lemmas/Arcsin.lean, moment representation and Fubini. +- declaration: LeanPolyLog.Radial.integral_im_logCube + informal: If (log^3((1+r)/2)-log^3((1-r)/2))/r is interval integrable from 0 to + 1, its integral equals the integral from 0 to pi of the imaginary part of the + cube of the principal complex logarithm of (1+exp(i t))/2. + source_ref: Upstream Lemmas/Radial.lean, radial differentiation and domination. +tags: +- analysis +- polylogarithms +- special-functions +- integration +- complex-analysis +msc: +- 33B30 +- 26A36 +- 30B10 diff --git a/LeanPool/projects/polynomial-freiman-ruzsa.yaml b/LeanPool/projects/polynomial-freiman-ruzsa.yaml new file mode 100644 index 0000000000..6eac07aca9 --- /dev/null +++ b/LeanPool/projects/polynomial-freiman-ruzsa.yaml @@ -0,0 +1,94 @@ +slug: polynomial-freiman-ruzsa +title: The polynomial Freiman–Ruzsa conjecture +summary: Formalizes Marton's polynomial Freiman–Ruzsa conjecture by the entropy method, + including the characteristic-two bounds with exponents twelve and nine, the bounded-torsion + theorem with an explicit exponent, and quantitative consequences for affine dimension + and approximate homomorphisms. The supporting development builds entropic Ruzsa + distance, multidistance, conditional entropy, and additive-energy machinery. This + optimized import reuses the pooled Zhang–Yeung entropy and kernel library in place + of duplicate upstream modules while preserving all six closed headline statements. +branch: additive combinatorics and information theory +entry_module: LeanPool.PFR +authors: +- PFR contributors +contributors: +- Aaron Anderson +- Mantas Bakšys +- Jonas Bayer +- Mauricio Collares +- Rémy Degenne +- Yaël Dillies +- Bhavik Mehta +- Ella Yu +- Ben Eltschig +- Sébastien Gouëzel +- Kalle Kytölä +- Rob Lewis +- Paul Lezeau +- Lorenzo Luccioli +- Heather Macbeth +- Patrick Massot +- Arend Mellendijk +- Kyle Miller +- Pietro Monticone +- Kim Morrison +- Oliver Nash +- Utensil Song +- Terence Tao +- Floris van Doorn +- Sky Wilshaw +- Lawrence Wu +source: + title: Marton's conjecture (the polynomial Freiman–Ruzsa conjecture) + authors: + - W. T. Gowers + - Ben Green + - Freddie Manners + - Terence Tao + arxiv: 2311.05762 + url: https://github.com/teorth/pfr + github_repo: teorth/pfr + commit: 07839bddb1395c8808fa47db821ce2e4fff9362e +upstream_dependencies: +- title: AddCombi + github_repo: leanprover-community/add-combi + commit: 2d061f437c1ba00ba9b16d7cba1c7603ba4807a4 + license: Apache-2.0 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- Marton.pfr_conjecture +main_results: +- declaration: Marton.pfr_conjecture + informal: A nonempty finite set A of doubling at most K in an abelian group of exponent + two is covered by fewer than 2 K^12 cosets of a finite subgroup no larger than + A. +- declaration: Marton.pfr_conjecture_nine + informal: Strengthens the characteristic-two covering theorem to fewer than 2 K^9 + cosets of a finite subgroup no larger than A. +- declaration: Marton.torsion_pfr_conjecture + informal: In an abelian group of exponent m at least two, a nonempty finite set + of doubling at most K is covered by fewer than m K^(256 m^3 + 1) cosets of a subgroup + no larger than A. +- declaration: Marton.weak_pfr_int + informal: A nonempty finite set of doubling at most K in a finite-rank free abelian + group has a subset of relative size at least K^(-34) and affine dimension at most + (80 / log 2) log K. +- declaration: Marton.homomorphism_pfr + informal: A map between finite abelian groups of exponent two whose additive defects + lie in S differs from a homomorphism by a map taking at most |S|^10 values. +- declaration: Marton.approx_hom_pfr + informal: A map between finite abelian groups of exponent two that is additive on + at least a 1/K fraction of pairs agrees with a homomorphism on at least (|G| / + (2^144 K^122) - 1) / 2 points. +tags: +- additive-combinatorics +- polynomial-freiman-ruzsa +- entropy +- approximate-homomorphisms +msc: +- 11B30 +- 11P70 +- 20K01 +- 94A17 diff --git a/LeanPool/projects/polynomial-method-restricted-sums.yaml b/LeanPool/projects/polynomial-method-restricted-sums.yaml new file mode 100644 index 0000000000..a9ac264f85 --- /dev/null +++ b/LeanPool/projects/polynomial-method-restricted-sums.yaml @@ -0,0 +1,55 @@ +slug: polynomial-method-restricted-sums +title: The polynomial method and restricted sums of congruence classes +summary: Formalizes the polynomial method (Alon–Tarsi-style combinatorial Nullstellensatz + arguments) and applies it to restricted sums of congruence classes, including Cauchy–Davenport, + Dias da Silva–Hamidoune, and a Vandermonde coefficient formula. +branch: combinatorics +entry_module: LeanPool.PolynomialMethodRestrictedSums +authors: +- Nick Adfor +source: + url: https://github.com/NickAdfor/The-polynomial-method-and-restricted-sums-of-congruence-classes + github_repo: NickAdfor/The-polynomial-method-and-restricted-sums-of-congruence-classes +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- ANR_polynomial_method +- cauchy_davenport +- dias_da_silva_hamidoune +main_results: +- declaration: ANR_polynomial_method + informal: The Alon–Nathanson–Rusza polynomial method theorem, giving a non-vanishing-coefficient + criterion for restricted sums. + source_ref: N. Alon, M. B. Nathanson, I. Z. Ruzsa, The polynomial method and restricted + sums of congruence classes, J. Number Theory 56 (1996), 404-417, doi:10.1006/jnth.1996.0029. +- declaration: cauchy_davenport + informal: The Cauchy–Davenport theorem on sums of subsets of the cyclic group of + prime order. + source_ref: N. Alon, Combinatorial Nullstellensatz, Combinatorics, Probability and + Computing 8 (1999), 7-29, doi:10.1017/S0963548398003411. +- declaration: dias_da_silva_hamidoune + informal: The Dias da Silva–Hamidoune lower bound for restricted sums of distinct + elements in a prime cyclic group. + source_ref: J. A. Dias da Silva, Y. O. Hamidoune, Cyclic spaces for Grassmann derivatives + and additive theory, Bull. London Math. Soc. 26 (1994), 140-146, doi:10.1112/blms/26.2.140. +- declaration: restricted_sum_distinct_sizes + informal: A lower bound for restricted sums when the summand sets have distinct + sizes. + source_ref: N. Alon, M. B. Nathanson, I. Z. Ruzsa, The polynomial method and restricted + sums of congruence classes, J. Number Theory 56 (1996), 404-417, doi:10.1006/jnth.1996.0029. +- declaration: Vandermonde_coefficient_formula + informal: A closed-form expression for a specific Vandermonde-style coefficient + used by the polynomial method. + source_ref: N. Alon, M. B. Nathanson, I. Z. Ruzsa, The polynomial method and restricted + sums of congruence classes, J. Number Theory 56 (1996), 404-417, doi:10.1006/jnth.1996.0029. +tags: +- combinatorics +- polynomial-method +- alon-tarsi +- restricted-sums +- congruence-classes +msc: +- 11B30 +- 11B75 +- 11P70 diff --git a/LeanPool/projects/puiseux.yaml b/LeanPool/projects/puiseux.yaml new file mode 100644 index 0000000000..fefcd1b094 --- /dev/null +++ b/LeanPool/projects/puiseux.yaml @@ -0,0 +1,48 @@ +slug: puiseux +title: Puiseux's theorem +summary: Constructs Puiseux series as the bounded-denominator subfield of rational-exponent + Hahn series. Proves that this field is algebraic over Laurent series over any coefficient + field, and is an algebraic closure when the coefficient field is algebraically closed + of characteristic zero. The Newton-Puiseux proof includes reusable Hensel splitting + and polynomial translation lemmas. Dean Cureton directed the AI formalization and + subsequently simplified its proofs. +branch: algebra +entry_module: LeanPool.Puiseux +authors: +- Dean Cureton +source: + title: Puiseux's theorem in Lean 4 + url: https://github.com/deancureton/Puiseux/tree/73e78681d7bb78ddb202e29cb92ab3c16b723126 + github_repo: deancureton/Puiseux + commit: 73e78681d7bb78ddb202e29cb92ab3c16b723126 + announcement: https://x.com/deancureton/status/2082886228517564682 + announced_at: '2026-07-30T17:49:09Z' + provenance_evidence: https://x.com/deancureton/status/2082886230266487252 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- PuiseuxSeries.isAlgClosure +main_results: +- declaration: PuiseuxSeries.isAlgClosure + informal: For an algebraically closed field K of characteristic zero, the field + of Puiseux series over K is an algebraic closure of the Laurent series field K((t)). +- declaration: PuiseuxSeries.isAlgebraic + informal: Over any field K, the Puiseux series field is algebraic over the Laurent + series field. +- declaration: PuiseuxSeries.mem_subfield_iff_support + informal: A rational-exponent Hahn series belongs to the Puiseux subfield if and + only if its support lies in (1/n) times the integers for some positive integer + n. +- declaration: Polynomial.Monic.exists_factorization_rootMultiplicity + informal: A monic degree-m polynomial over K[[t]] whose reduction has a root of + multiplicity k, with 0 < k < m, factors into monic polynomials of degrees k and + m-k. +tags: +- field-theory +- puiseux-series +- algebraic-closure +- hensel-lemma +msc: +- 12F05 +- 13F25 diff --git a/LeanPool/projects/pumping-cfg.yaml b/LeanPool/projects/pumping-cfg.yaml new file mode 100644 index 0000000000..673a9da2c0 --- /dev/null +++ b/LeanPool/projects/pumping-cfg.yaml @@ -0,0 +1,33 @@ +slug: pumping-cfg +title: Pumping Lemma for Context-Free Grammars +summary: Formalizes the pumping lemma for context-free languages via parse trees of + Chomsky-normal-form grammars and a pigeonhole argument. +branch: formal language theory +entry_module: LeanPool.PumpingCfg +authors: +- Alexander Loitzl +- Martin Dvorak +source: + url: https://github.com/AlexLoitzl/pumping_cfg + github_repo: AlexLoitzl/pumping_cfg +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Language.IsContextFree.pumping +- ChomskyNormalFormGrammar.pumping +main_results: +- declaration: Language.IsContextFree.pumping + informal: Every context-free language admits a pumping length p such that any word + of length at least p decomposes as u v x y z with vy nonempty, vxy of length at + most p, and u vⁱ x yⁱ z in the language for every i. +- declaration: ChomskyNormalFormGrammar.pumping + informal: The pumping lemma at the level of a Chomsky-normal-form grammar, obtained + from a sufficiently tall parse tree with a repeated nonterminal. +tags: +- formal-languages +- context-free-grammars +- computability +- pumping-lemma +msc: +- 68Q45 diff --git a/LeanPool/projects/pythagorean-polynomial-parametrization.yaml b/LeanPool/projects/pythagorean-polynomial-parametrization.yaml new file mode 100644 index 0000000000..06c0f884bc --- /dev/null +++ b/LeanPool/projects/pythagorean-polynomial-parametrization.yaml @@ -0,0 +1,50 @@ +slug: pythagorean-polynomial-parametrization +title: Polynomial parametrizations of Pythagorean triples +summary: Formalizes Frisch and Vaserstein's result that Pythagorean triples are not + parametrizable by one triple of integer-coefficient polynomials in any number of + variables, but are parametrizable by a single triple of integer-valued rational + polynomials, including the positive and sixteen-parameter variants. +branch: number theory +entry_module: LeanPool.PythagoreanPolynomialParametrization +authors: +- Lazar Milikic +source: + title: Parametrization of Pythagorean triples by a single triple of polynomials + authors: + - Sophie Frisch + - Leonid Vaserstein + arxiv: '0706.0290' + doi: 10.1016/j.jpaa.2007.05.019 + github_repo: epfl-lara/AutoformalizedProjects +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.PythagoreanPolynomialParametrization.exists_int_valued_parametrization +main_results: +- declaration: LeanPool.PythagoreanPolynomialParametrization.no_int_poly_parametrization + informal: Proves that no single triple of integer-coefficient polynomials in any + number of variables parametrizes all Pythagorean triples. +- declaration: LeanPool.PythagoreanPolynomialParametrization.exists_int_valued_parametrization + informal: Gives the explicit four-variable integer-valued polynomial parametrization + of all Pythagorean triples. +- declaration: LeanPool.PythagoreanPolynomialParametrization.positive_triples_parametrization + informal: Formalizes the positive Pythagorean-triple parametrization with positive + parameters and a nonnegative auxiliary parameter. +- declaration: LeanPool.PythagoreanPolynomialParametrization.exists_16_param_parametrization + informal: Converts the positive parametrization to an unrestricted sixteen-integer-parameter + parametrization using the four-square theorem. +- declaration: LeanPool.PythagoreanPolynomialParametrization.single_intValued_parametrization_yields_finite_intPoly_parametrization + informal: Proves the cited reduction from one integer-valued polynomial tuple to + finitely many integer-coefficient polynomial tuples. +- declaration: LeanPool.PythagoreanPolynomialParametrization.integerValued_polynomial_ring_not_uniqueFactorization + informal: Records the integer-valued polynomial ring non-UFD phenomenon behind the + paper's motivating discussion. +tags: +- number-theory +- pythagorean-triples +- integer-valued-polynomials +msc: +- 11D09 +- 11D85 +- 13F20 diff --git a/LeanPool/projects/qec-certificates.yaml b/LeanPool/projects/qec-certificates.yaml new file mode 100644 index 0000000000..8f393a97dd --- /dev/null +++ b/LeanPool/projects/qec-certificates.yaml @@ -0,0 +1,90 @@ +slug: qec-certificates +title: Quantum code certificates and hypergraph-product theory +summary: Formalizes general binary row reduction, kernel bases and rank certificates; + quantum-code distance bounds from witnesses and exhaustive low-weight enumeration; + arbitrary-matrix hypergraph-product dimension and X/Z cleaning bounds; and CSS orthogonality + of lifted products over finite groups. Includes a Pauli expression translation modulo + global phase, a Boolean encoding with proved soundness and model extension, and + sound checking of RUP additions in LRAT certificates, bridged to matrix-level distance + bounds. Concrete code-family datasets, large certificate replays and measurement + protocols are outside this import. +branch: quantum information and coding theory +entry_module: LeanPool.QECCertificates +authors: +- Shuoming An +source: + title: Quantum code parameters, checkable by a certificate of provable size + authors: + - Shuoming An + - Fusheng Yang + arxiv: '2610.03214' + url: https://github.com/QCL-SUAT/QECCertificates + github_repo: QCL-SUAT/QECCertificates + commit: 1ced3ae0317725f80640d4a38b9c03eaba84a9c9 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- QECCertificates.hgp_kunneth +main_results: +- declaration: QECCertificates.hgp_kunneth + informal: For arbitrary finite binary matrices A and B, the number of columns of + their hypergraph product minus its X and Z ranks equals nullity(A) times nullity(B) + plus nullity(transpose A) times nullity(transpose B), expressed with natural-number + rank differences. + source_ref: Upstream GF2/HGPKunneth.lean; the Tillich–Zémor hypergraph-product construction. +- declaration: QECCertificates.hgp_X_distance_ge + informal: If a binary vector is killed by the hypergraph-product X check and is + outside the Z row space, its Hamming weight is at least the minimum of supplied + lower bounds for nonzero vectors in the kernels of the first input and the transpose + of the second input. +- declaration: QECCertificates.hgp_Z_distance_ge + informal: Dually, a vector killed by the Z check and outside the X row space has + weight at least the minimum of supplied lower bounds for the kernels of the transpose + of the first input and the second input. +- declaration: QECCertificates.lp_orthogonal + informal: For arbitrary matrices of binary-valued functions on a finite group, the + left/right regular expansion gives lifted-product X and Z checks whose every pair + of rows has zero binary dot product. +- declaration: QECCertificates.Matrix.rank_eq_length_rowReduce + informal: The rank of a finite binary matrix equals the length of the verified row-reduction + output for its list of rows. +- declaration: QECCertificates.eq_minWeight_of_decide + informal: For two binary matrices, if d is at most the number of columns, exhaustive + enumeration finds no vector of weight below d in the first kernel outside the + second row space, and a supplied vector has those membership properties and weight + d, then the minimum such weight is d. The empty-set distance convention is the + number of columns plus one. +- declaration: QECCertificates.LRAT.unsat_of_checkSteps + informal: Acceptance by the RUP certificate checker implies unsatisfiability of + the original CNF. No correctness assumption on an external SAT solver is needed; + the clause-database interface carries proved soundness laws. +- declaration: QECCertificates.LRAT.buildPair_complete + informal: For a positive number n of input bits and nonempty row supports within + that range, a Boolean assignment satisfying the parity constraints, odd bilinear + pairing and weight bound extends to a model of buildPair while agreeing on all + 2n input variables. The complementary soundness theorem extracts those constraints + from every model. + source_ref: 'Upstream Reflect/Encode.lean and Reflect/Complete.lean. The import + retains general theorems, not the paper''s concrete instance results. First public + paper announcement: 2026-10-02 12:31:45 UTC, arXiv v1. The upstream paper discloses + AI-generated Lean proofs reviewed and validated by the author. RowspaceKernel + is derived from QCL-SUAT/Lean-QEC at a0855f8a6374c1dda2097bbf07f33ba53760fdcd + (Apache-2.0), credited to Mattias Ehatamm, Yi Lee, Xiaodi Wu and Runzhou Tao, + and their paper End-to-End Formalization of Quantum Error Correction, arXiv:2605.16523.' +- declaration: QECCertificates.LRAT.le_minWeight_of_unsat + informal: For a positive number n of columns and k + 1 ≤ n, an unsatisfiable Boolean + encoding gives a matrix-level distance lower bound of k + 1. Support-row correspondence, + nonemptiness, index bounds and duplicate-free hypotheses are explicit. This composes + with kernel-checked RUP replay. + source_ref: General bridge from upstream Reflect/Certified.lean; the minimum-weight + distance wrapper is added during the Lean Pool port. +tags: +- quantum-error-correction +- coding-theory +- hypergraph-products +- certificates +msc: +- 81P45 +- 94B05 +- 03B35 diff --git a/LeanPool/projects/quadratic-carleson-palomar-submission.yaml b/LeanPool/projects/quadratic-carleson-palomar-submission.yaml new file mode 100644 index 0000000000..3001704129 --- /dev/null +++ b/LeanPool/projects/quadratic-carleson-palomar-submission.yaml @@ -0,0 +1,36 @@ +slug: quadratic-carleson-palomar-submission +title: Endpoint estimates for Stein's purely quadratic Carleson operator +summary: 'A Lean 4 formalization of four principal endpoint assertions from Fragkos, + Krause, and Lacey''s Endpoint Estimates for Stein''s Purely Quadratic Carleson Operator: + two failures of modular estimates below t log₂ t, a full-modulation L log L endpoint + estimate, and a lacunary L (log₂ L)² log₄ L endpoint estimate.' +branch: harmonic analysis +entry_module: LeanPool.QuadraticCarleson +authors: +- Anastasios Fragkos +source: + url: https://github.com/anastasiosfragkosmath/quadratic-carleson-palomar-submission + github_repo: anastasiosfragkosmath/quadratic-carleson-palomar-submission + commit: 3d3cf4316acac4c23140059a78fb570b1440394b +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- QuadraticCarleson.StatementSurface.PaperTheorems.full_LlogL_endpoint +main_results: +- declaration: QuadraticCarleson.StatementSurface.PaperTheorems.lacunary_sub_log2_modular_failure + informal: Lacunary purely quadratic modulation fails modular estimates below the + specified iterated-logarithmic threshold. +- declaration: QuadraticCarleson.StatementSurface.PaperTheorems.full_sub_log2_modular_failure + informal: Full purely quadratic modulation fails modular estimates below the specified + iterated-logarithmic threshold. +- declaration: QuadraticCarleson.StatementSurface.PaperTheorems.full_LlogL_endpoint + informal: The full-modulation operator satisfies the L log L endpoint estimate. +- declaration: QuadraticCarleson.StatementSurface.PaperTheorems.lacunary_log2_squared_log4_endpoint + informal: The lacunary operator satisfies the L (log₂ L)² log₄ L endpoint estimate. +tags: +- harmonic-analysis +- maximal-operators +- endpoint-estimates +msc: +- 42B20 diff --git a/LeanPool/projects/quadratic-iterates.yaml b/LeanPool/projects/quadratic-iterates.yaml new file mode 100644 index 0000000000..6bbd38832e --- /dev/null +++ b/LeanPool/projects/quadratic-iterates.yaml @@ -0,0 +1,49 @@ +slug: quadratic-iterates +title: Galois groups of quadratic polynomial iterates +summary: Formalizes Stoll's criterion for when the Galois group of the n-th iterate + of X² + a over the rationals is the full n-fold iterated wreath product of C₂. It + proves the criterion in terms of two-independence of a signed critical-orbit sequence + and its pairwise-coprime Möbius factors, and derives the paper's congruence families + for which maximality holds at every level. The supporting library develops reusable + multiquadratic-extension, iterated wreath-product, Möbius-factor, and valuation + results. +branch: arithmetic dynamics and Galois theory +entry_module: LeanPool.QuadraticIterates +authors: +- Michael Stoll +source: + title: Galois groups over ℚ of some iterated polynomials + authors: + - Michael Stoll + doi: 10.1007/BF01197321 + url: https://github.com/MichaelStollBayreuth/QuadraticIterates + github_repo: MichaelStollBayreuth/QuadraticIterates + commit: b0413c173972ddf121cb4394cb06f9d2b83e501f +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- QuadraticIterates.section3_main +main_results: +- declaration: QuadraticIterates.section1_equiv + informal: For each n, assuming -a is not a rational square, the Galois group of + the n-th iterate of X² + a is the full n-fold iterated wreath product of C₂ exactly + when either the first n terms of the signed sequence c₁ = -a, cₖ = fₖ(0) for k + ≥ 2, or the first n integer Möbius factors of that sequence are independent modulo + rational squares. +- declaration: QuadraticIterates.section1_squarefree + informal: If -a is not a rational square and none of the absolute Möbius factors + from levels 2 through n is a rational square, then the n-th iterate has the full + iterated-wreath-product Galois group. +- declaration: QuadraticIterates.section3_main + informal: The Galois group is the full iterated wreath product at every positive + level when a is positive and congruent to 1 or 2 modulo 4, or when a is negative + and divisible by 4 with -a nonsquare. +tags: +- arithmetic-dynamics +- galois-theory +- iterated-polynomials +msc: +- 11R32 +- 12F10 +- 37P05 diff --git a/LeanPool/projects/quantum-parallel-repetition.yaml b/LeanPool/projects/quantum-parallel-repetition.yaml new file mode 100644 index 0000000000..d8aad06925 --- /dev/null +++ b/LeanPool/projects/quantum-parallel-repetition.yaml @@ -0,0 +1,39 @@ +slug: quantum-parallel-repetition +title: Quantum parallel repetition +summary: 'Proves a quantitative parallel-repetition theorem for finite two-player + quantum games: whenever the entangled value is below one, the entangled values of + parallel repetitions decay exponentially, with an explicit universal exponent in + the game gap and answer-alphabet sizes.' +branch: quantum information theory +entry_module: LeanPool.QuantumParallelRepetition +authors: +- OpenAI +- Dean Cureton +source: + title: Ten Advances in Mathematics and Theoretical Computer Science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- QuantumParallelRepetition.distributionUniformExponential +main_results: +- declaration: QuantumParallelRepetition.distributionUniformExponential + informal: There is a universal positive constant giving an explicit exponential + upper bound on the repeated entangled value of every finite game with positive + gap and nonempty answer alphabets. +- declaration: QuantumParallelRepetition.standardQuantumParallelRepetition + informal: Every finite two-player game whose entangled value is below one has exponentially + decaying entangled value under parallel repetition. +tags: +- quantum-information +- nonlocal-games +- parallel-repetition +- theoretical-computer-science +msc: +- 81P68 +- 68Q12 diff --git a/LeanPool/projects/quantum-query-foundations.yaml b/LeanPool/projects/quantum-query-foundations.yaml new file mode 100644 index 0000000000..ef29110d72 --- /dev/null +++ b/LeanPool/projects/quantum-query-foundations.yaml @@ -0,0 +1,56 @@ +slug: quantum-query-foundations +title: Adversary bounds and the polynomial method for quantum queries +summary: Develops finite quantum query algorithms, real adversary primal and dual + programs, strong duality, exact Boolean block composition, explicit constant-factor + bounded-error query characterizations, and degree bounds for output-probability + polynomials. Query models include idle and blank sectors; bounds count queries, + not gate complexity or running time. Uniform algorithm extraction assumes a supplied + dual certificate. +branch: quantum query complexity +entry_module: LeanPool.QuantumQuery +authors: +- Troy Lee +source: + arxiv: 1011.3020v2 + url: https://github.com/troyjlee/quantum-query-complexity/tree/517ba85eb211232771429804d8583e3f52b9d541 + github_repo: troyjlee/quantum-query-complexity + commit: 517ba85eb211232771429804d8583e3f52b9d541 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- QuantumQueryComplexity.boundedErrorQQuery_characterized_by_advPM +main_results: +- declaration: QuantumQueryComplexity.advDual_eq_advPM + informal: For every total Boolean function on a finite Boolean input cube, the all-pairs + real adversary dual infimum equals the real symmetric primal supremum. +- declaration: QuantumQueryComplexity.advPM_composeFun_eq + informal: The negative-weight adversary value of total Boolean functions composed + on disjoint blocks equals the product of their adversary values, including constant + functions and empty index types. +- declaration: QuantumQueryComplexity.boundedErrorQQuery_characterized_by_advPM + informal: For a total Boolean function at error one third, quantum value-query complexity + lies between ADV±/36 and 2^14 ADV±. The value oracle swaps a blank answer with + the queried letter and includes an idle query index. +- declaration: QuantumQueryComplexity.xorQQuery_characterized_by_advPM + informal: For the Boolean XOR oracle with idle and blank sectors, query complexity + at error one third lies between ADV±/72 and 2^15 ADV±. +- declaration: QuantumQueryComplexity.qQueryOn_third_le_of_hasDualOn_uniform + informal: A supplied all-pairs real dual certificate of nonnegative cost c gives + a finite quantum algorithm using at most 8192(1+c) value queries at error one + third, for any finite promise domain and nonempty decidable output type. +- declaration: QuantumQueryComplexity.QAlg.exists_probability_polynomial + informal: Each output probability of a q-query value-oracle algorithm on Boolean + inputs agrees on the Boolean cube with a real polynomial of total degree at most + 2q. +- declaration: QuantumQueryComplexity.exists_xor_probability_polynomial + informal: Each output probability of a q-query Boolean XOR algorithm agrees on the + Boolean cube with a real polynomial of total degree at most 2q. +tags: +- quantum-computing +- query-complexity +- adversary-method +- polynomial-method +msc: +- 68Q12 +- 81P68 diff --git a/LeanPool/projects/quasi-borel-spaces.yaml b/LeanPool/projects/quasi-borel-spaces.yaml new file mode 100644 index 0000000000..2d684c6f81 --- /dev/null +++ b/LeanPool/projects/quasi-borel-spaces.yaml @@ -0,0 +1,42 @@ +slug: quasi-borel-spaces +title: Quasi-Borel Spaces +summary: Formalizes quasi-Borel spaces and quasi-Borel pre-domains, a category for + higher-order probability theory. +branch: probability theory +entry_module: LeanPool.QuasiBorelSpaces +authors: +- Anthony Vandikas +- Kiarash Sotoudeh +source: + arxiv: '1701.02547' + doi: 10.1109/LICS.2017.8005137 + github_repo: YellPika/quasi-borel-spaces +license: MIT +status: verified +provenance: human +main_declarations: +- QuasiBorelSpace +- OmegaQuasiBorelSpace +- QuasiBorelSpace.ProbabilityMeasure +main_results: +- declaration: QuasiBorelSpace + informal: Defines a quasi-Borel space as a type with a set of random variables closed + under constants, measurable precomposition, and gluing. +- declaration: QuasiBorelSpace.IsHom + informal: Defines morphisms of quasi-Borel spaces as maps that preserve random variables + under precomposition. +- declaration: OmegaQuasiBorelSpace + informal: Combines the quasi-Borel and ω-complete partial order structures with + the compatibility axiom of Vakar-Kammar-Staton. +- declaration: QuasiBorelSpace.ProbabilityMeasure + informal: Defines the probability-measure monad on quasi-Borel spaces by quotienting + pre-probability measures by integral equivalence. +tags: +- probability +- category-theory +- measure-theory +- denotational-semantics +msc: +- 60A05 +- 18C50 +- 68Q55 diff --git a/LeanPool/projects/rado-theorem.yaml b/LeanPool/projects/rado-theorem.yaml new file mode 100644 index 0000000000..6b677ad714 --- /dev/null +++ b/LeanPool/projects/rado-theorem.yaml @@ -0,0 +1,46 @@ +slug: rado-theorem +title: Radó's theorem for Riemann surfaces +summary: Proves that every connected Hausdorff one-dimensional complex manifold is + second countable. The development constructs a nonconstant harmonic function by + Perron's method, builds the étale space of its harmonic-conjugate germs, and applies + a formalized Poincaré-Volterra lemma to its discrete-fiber evaluation map. +branch: Riemann surfaces and general topology +entry_module: LeanPool.Rado +authors: +- Rado Kirov +source: + title: Radó's theorem for Riemann surfaces + authors: + - Rado Kirov + url: https://github.com/rkirov/jordan_pick + github_repo: rkirov/jordan_pick + commit: b3c9b7cf7358bf81a077d78ad67e6e8247869ddd +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- rado_riemannSurface +main_results: +- declaration: rado_riemannSurface + informal: Every connected Hausdorff topological space carrying the structure of + a one-dimensional complex manifold has a second-countable topology. +- declaration: Rado.poincare_volterra + informal: A connected Hausdorff, locally compact, locally connected and locally + second-countable space is second countable whenever it admits a continuous, locally + injective map into a second-countable Hausdorff space. +- declaration: Rado.exists_harmonic_extension + informal: Every continuous real-valued function on a Euclidean circle extends continuously + to its closed disk and harmonically to the open disk; unlike the existing LeanComplexAnalysis + Poisson theorems, this constructs the harmonic function rather than assuming one. +- declaration: Rado.IsPerronFamily.surfaceHarmonicOn_perronSup + informal: The pointwise upper envelope of a Perron family of surface subharmonic + functions is harmonic on its open domain. +tags: +- riemann-surfaces +- second-countability +- harmonic-functions +- perron-method +msc: +- 30F15 +- 30F20 +- 54D65 diff --git a/LeanPool/projects/ramanujan-nagell.yaml b/LeanPool/projects/ramanujan-nagell.yaml new file mode 100644 index 0000000000..50a7876fcc --- /dev/null +++ b/LeanPool/projects/ramanujan-nagell.yaml @@ -0,0 +1,32 @@ +slug: ramanujan-nagell +title: The Ramanujan-Nagell theorem +summary: Formalizes the Ramanujan-Nagell theorem, proving that the Diophantine equation + x^2 + 7 = 2^n has exactly the five positive-integer solutions (|x|, n) = (1, 3), + (3, 4), (5, 5), (11, 7), (181, 15). +branch: number theory +entry_module: LeanPool.RamanujanNagell +authors: +- Barinder S. Banwait +- Xinze Li +source: + arxiv: '2604.09808' + github_repo: BarinderBanwait/ramanujan_nagell +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- ramanujanNagellExact +main_results: +- declaration: ramanujanNagellExact + informal: Proves that the integer solutions of x^2 + 7 = 2^n are exactly (x, n) + = (+/-1, 3), (+/-3, 4), (+/-5, 5), (+/-11, 7), (+/-181, 15). + source_ref: T. Nagell, The Diophantine equation x^2 + 7 = 2^n, Ark. Mat. 4 (1961), + 185-187; L. J. Mordell, Diophantine Equations, Academic Press, 1969, Ch. 20. +tags: +- number-theory +- diophantine-equations +- quadratic-integers +msc: +- 11D61 +- 11D45 +- 11R11 diff --git a/LeanPool/projects/ramanujan-tau-misses-primes.yaml b/LeanPool/projects/ramanujan-tau-misses-primes.yaml new file mode 100644 index 0000000000..5f95dd4054 --- /dev/null +++ b/LeanPool/projects/ramanujan-tau-misses-primes.yaml @@ -0,0 +1,37 @@ +slug: ramanujan-tau-misses-primes +title: ABC implies that Ramanujan's tau function misses almost all primes +summary: Formalizes the ABC-conditional proof that Ramanujan's tau function takes + prime values only rarely. +branch: analytic number theory +entry_module: LeanPool.RamanujanTauMissesPrimes +authors: +- Evan Chen +- Kenny Lau +- Ken Ono +- Jujian Zhang +source: + arxiv: '2603.29970' + github_repo: AxiomMath/ramanujan-tau-misses-primes +license: MIT +status: verified +provenance: AI +main_declarations: +- main_theorem +- reduction_lemma +- abc_bound_E2 +- abc_bound_E4 +main_results: +- declaration: main_theorem + informal: ABC implies that Ramanujan's tau function misses almost all primes. +- declaration: reduction_lemma + informal: Reduces the main prime-value statement to the key analytic estimates. +- declaration: abc_bound_E2 + informal: ABC-bound input for the Eisenstein-series term E2. +- declaration: abc_bound_E4 + informal: ABC-bound input for the Eisenstein-series term E4. +tags: +- number-theory +- modular-forms +msc: +- 11F30 +- 11N05 diff --git a/LeanPool/projects/redhill.yaml b/LeanPool/projects/redhill.yaml new file mode 100644 index 0000000000..48457ac238 --- /dev/null +++ b/LeanPool/projects/redhill.yaml @@ -0,0 +1,47 @@ +slug: redhill +title: Improved Lower Bounds for Strong n-Conjectures +summary: Formalizes the disproof of Ramaekers's conjecture, constructing tuple families + whose abc-style quality stays bounded above 1, and proves Browkin-Brzezinski lower + bounds on the quality of the n-conjecture tuples. +branch: number theory +entry_module: LeanPool.Redhill +authors: +- Jeremy Tan +source: + title: Improved Lower Bounds for Strong n-Conjectures + authors: + - Rupert Hölzl + - Sören Kleine + - Frank Stephan + doi: 10.1017/S1446788725000084 + github_repo: Parcly-Taxel/Redhill +license: MIT +status: verified +provenance: human +main_declarations: +- not_ramaekersConjecture_ge_six +- le_quality_nConjectureTuples +main_results: +- declaration: RamaekersConjecture + informal: Defines Ramaekers's conjecture for fixed n as the statement that the quality + of the relevant tuple set equals 1. +- declaration: not_ramaekersConjecture_ge_six + informal: Proves Ramaekers's conjecture fails for every n at least 6 (the general + case, Theorem 1.14). +- declaration: not_ramaekersConjecture_odd_ge_five + informal: Proves Ramaekers's conjecture fails for every odd n at least 5 (the odd + case via Pell equations, Theorem 1.13). +- declaration: not_ramaekersConjecture_five + informal: Proves Ramaekers's conjecture fails for n equal to 5 (the Konyagin prelude + bound). +- declaration: le_quality_nConjectureTuples + informal: Proves the Browkin-Brzezinski 1994 lower bound 2n-5 on the quality of + the n-conjecture tuples (Theorem 1.3). +tags: +- number-theory +- abc-conjecture +- n-conjecture +- ramaekers-conjecture +msc: +- 11A41 +- 11D75 diff --git a/LeanPool/projects/regts-sevenster.yaml b/LeanPool/projects/regts-sevenster.yaml new file mode 100644 index 0000000000..391532559e --- /dev/null +++ b/LeanPool/projects/regts-sevenster.yaml @@ -0,0 +1,45 @@ +slug: regts-sevenster +title: A proof of the Regts–Sevenster conjecture, formalized +summary: A Lean 4 proof of the Regts–Sevenster conjecture. A complex-valued graph + parameter, normalized at the empty graph and invariant under isomorphism, has exponentially + bounded edge-connection rank if and only if it is a mixed partition function. For + a natural rank base R, a representing model can be chosen with k even and 2ℓ odd + colours satisfying k + 2ℓ ≤ R. The development also proves the even connection-rank + growth limit, minimum total colour dimension, minimal parity formulas and prescribed-dimension + criterion. Schrijver's factorial argument supplies nilpotent-trace vanishing; semisimplicity + and Deligne's fibre-functor theorem give the model; a polynomial commutant estimate + gives the sharp colour budget. Deligne's theorem and the required symmetric-group + character theory are proved within the development. The appendix's trace-zeta results + give rationality and degree bounds, with an independent proof of nilpotent-trace + vanishing. The theorems of record use only propext, Classical.choice and Quot.sound, + checked by pinned audits. +branch: algebraic graph theory +entry_module: LeanPool.RegtsSevenster +authors: +- William Whistler +source: + url: https://github.com/willwhistler/regts-sevenster + github_repo: willwhistler/regts-sevenster + commit: bb0a60543d48afac3f36938104b5c85c7c0396c7 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- RS.regts_sevenster_characterisation +main_results: +- declaration: RS.regts_sevenster_characterisation + informal: Normalized isomorphism-invariant complex graph parameters with exponentially + bounded edge-connection ranks are precisely mixed partition functions. +- declaration: RS.regts_sevenster_total + informal: A natural rank bound R admits a representing mixed model with total color + dimension at most R. +- declaration: RS.deligne_theorem + informal: The development proves its stated Deligne fiber-functor theorem for the + symmetric tensor categories used in the representation. +tags: +- graph-parameters +- tensor-categories +- partition-functions +msc: +- 05C50 +- 18M20 diff --git a/LeanPool/projects/rellich-kondrachov.yaml b/LeanPool/projects/rellich-kondrachov.yaml new file mode 100644 index 0000000000..a6238406ef --- /dev/null +++ b/LeanPool/projects/rellich-kondrachov.yaml @@ -0,0 +1,40 @@ +slug: rellich-kondrachov +title: Rellich–Kondrachov compact embedding theorem +summary: 'Formalizes the Rellich–Kondrachov compact embedding theorem: the Sobolev + embedding H¹ → L² is a compact operator on compact Riemannian manifolds. The proof + builds the Euclidean core (compactness of the H¹ → L² inclusion on a fixed compact + support via the Fréchet–Kolmogorov criterion and the H¹ translation estimate ‖τₐu + − u‖₂ ≤ ‖a‖ · ‖∇u‖₂), glues chart contributions over a finite atlas, and specializes + to the Levi-Civita Riemannian volume measure.' +branch: analysis +entry_module: LeanPool.RellichKondrachov +authors: +- Adam Benenson +source: + title: Rellich–Kondrachov compact Sobolev embedding theorem + authors: + - Adam Benenson + url: https://github.com/abenenson/rellich-kondrachov + github_repo: abenenson/rellich-kondrachov +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- RellichKondrachov.Geometry.Manifold.Sobolev.exists_riemannianFiniteChartData +main_results: +- declaration: RellichKondrachov.Geometry.Manifold.Sobolev.RiemannianFiniteChartData.isCompactOperator_h1ToL2_riemannianVolume + informal: On a compact Riemannian manifold, the Sobolev inclusion H¹ → L² for the + Levi-Civita volume measure is a compact operator. +- declaration: RellichKondrachov.Analysis.FunctionalSpaces.Sobolev.Euclidean.isCompactOperator_h1OnToL2 + informal: 'Euclidean Rellich–Kondrachov compactness: on a fixed compact support + K, the H¹ → L² inclusion restricted to functions supported in K is a compact operator.' +- declaration: RellichKondrachov.Geometry.Manifold.Sobolev.exists_riemannianFiniteChartData + informal: On a compact Riemannian manifold there exists finite chart data equipped + with explicit Lipschitz neighborhoods and a subordinate partition of unity, the + input to the manifold Rellich glue. +tags: +- analysis +- pde +- sobolev-embedding +msc: +- 46E35 diff --git a/LeanPool/projects/riemann-mapping-theorem.yaml b/LeanPool/projects/riemann-mapping-theorem.yaml new file mode 100644 index 0000000000..15f4a1106a --- /dev/null +++ b/LeanPool/projects/riemann-mapping-theorem.yaml @@ -0,0 +1,36 @@ +slug: riemann-mapping-theorem +title: Riemann Mapping Theorem +summary: Formalizes a proof of the Riemann mapping theorem for proper simply connected + plane domains with primitives. +branch: complex analysis +entry_module: LeanPool.RiemannMappingTheorem +authors: +- Vincent Beffara +source: + url: https://github.com/vbeffara/RMT4 + github_repo: vbeffara/RMT4 +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- RMT +- main +- montel +- hurwitz +main_results: +- declaration: RMT + informal: Proves the Riemann mapping theorem for the project's domain hypotheses. +- declaration: main + informal: Constructs a conformal map from a good domain onto the unit disk. +- declaration: montel + informal: Proves a Montel compactness theorem for uniformly bounded holomorphic + families. +- declaration: hurwitz + informal: Proves the Hurwitz theorem used to preserve injectivity under limits. +tags: +- complex-analysis +- conformal-maps +- schwarz-lemma +msc: +- 30C35 +- 30C20 diff --git a/LeanPool/projects/riemann-roch-function-fields.yaml b/LeanPool/projects/riemann-roch-function-fields.yaml new file mode 100644 index 0000000000..d63c2b6cec --- /dev/null +++ b/LeanPool/projects/riemann-roch-function-fields.yaml @@ -0,0 +1,62 @@ +slug: riemann-roch-function-fields +title: Riemann–Roch for algebraic function fields +summary: Formalizes the Riemann–Roch theorem for algebraic function fields of one + variable through the adelic route. Divisors are indexed by intrinsic places (discrete + rank-one valuation subrings containing the constants), with an equivalence to a + two-chart coordinate model via Ostrowski's theorem. The development builds Riemann–Roch + spaces, the genus, an intrinsic adele space, Weil differentials and the canonical + class, and proves Riemann's inequality, the duality theorem, Riemann–Roch, Clifford's + inequality, and the standard corollaries. As an application it shows that a Weierstrass + elliptic curve has genus one and identifies its degree-one places with the class + group of the coordinate ring compatibly with the group law. +branch: algebraic curves and function fields +entry_module: LeanPool.RiemannRochFunctionFields +authors: +- Guanghao Li +source: + title: Riemann–Roch for function fields, in Lean 4 + authors: + - Guanghao Li + url: https://github.com/vaca22/riemann-roch-function-fields + github_repo: vaca22/riemann-roch-function-fields + commit: dbca5beed1da77e2ecd1eec207d0451fa57e8aa6 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- FunctionField.riemann_roch +main_results: +- declaration: FunctionField.riemann_roch + informal: For a function field K over its full constant field k, a canonical divisor + W, and any divisor D indexed by intrinsic places, ℓ(D) = deg D + 1 − g + ℓ(W − + D). +- declaration: FunctionField.duality + informal: For a canonical divisor W and any divisor D, ℓ(W − D) equals the index + of specialty i(D), the codimension of L(D) plus K inside the adele space. +- declaration: FunctionField.riemann_ineq + informal: Riemann's inequality ℓ(D) ≥ deg D + 1 − g holds for every divisor D. +- declaration: FunctionField.clifford + informal: Clifford's inequality 2(ℓ(D) − 1) ≤ deg D holds for every divisor D with + 0 ≤ deg D ≤ 2g − 2, ℓ(D) > 0, and ℓ(W − D) > 0 for a canonical divisor W. The + degree bounds are not needed for the inequality and are dropped in FunctionField.clifford_of_ell_pos. +- declaration: FunctionField.exists_isCanonical + informal: Every function field admits a canonical divisor, namely the divisor of + a nonzero Weil differential. +- declaration: WeierstrassCurve.Affine.genus_eq_one + informal: The function field of a Weierstrass elliptic curve over a field k that + is its full constant field (for example any algebraically closed k, in any characteristic) + has genus one. +- declaration: WeierstrassCurve.Affine.picTorsor_compat_groupLaw + informal: The bijection between degree-one places of an elliptic function field + and the class group of the coordinate ring is compatible with the elliptic group + law. +tags: +- riemann-roch +- function-fields +- algebraic-curves +- weil-differentials +- elliptic-curves +msc: +- 14H05 +- 11R58 +- 14H52 diff --git a/LeanPool/projects/rl-theory-in-lean.yaml b/LeanPool/projects/rl-theory-in-lean.yaml new file mode 100644 index 0000000000..edb525a3fa --- /dev/null +++ b/LeanPool/projects/rl-theory-in-lean.yaml @@ -0,0 +1,26 @@ +slug: rl-theory-in-lean +title: RL Theory in Lean +summary: Develops probability, Markov-chain, and stochastic-matrix infrastructure + for reinforcement-learning theory. +branch: probability theory +entry_module: LeanPool.RlTheoryInLean +authors: +- Shangtong Zhang +source: + arxiv: '2511.03618' + github_repo: ShangtongZhang/rl-theory-in-lean +license: MIT +status: verified +provenance: mix +main_declarations: +- StochasticMatrix.stationary_distribution_exists +main_results: +- declaration: StochasticMatrix.stationary_distribution_exists + informal: Proves existence of a stationary distribution for a stochastic matrix. +tags: +- probability +- reinforcement-learning +- stochastic-matrices +msc: +- 62L20 +- 60J10 diff --git a/LeanPool/projects/root-system.yaml b/LeanPool/projects/root-system.yaml new file mode 100644 index 0000000000..373f9c3227 --- /dev/null +++ b/LeanPool/projects/root-system.yaml @@ -0,0 +1,49 @@ +slug: root-system +title: Explicit type Aₙ and BCₙ root pairings +summary: Constructs the type Aₙ and BCₙ root pairings explicitly over ℤ — from signed + intervals on Fin n and from the reflective vectors of the standard dot product — + proves the pairing and reflection identities exhibiting them as Mathlib `RootPairing`s, + and identifies the BCₙ roots with the classical set {±eᵢ, ±2eᵢ, ±eᵢ ± eⱼ}. +branch: representation theory +entry_module: LeanPool.RootSystem +authors: +- Antoine de Saint Germain +- Ambrose Tang +source: + title: Introduction to Lie Algebras and Representation Theory + authors: + - James E. Humphreys + doi: 10.1007/978-1-4612-6398-2 + url: https://antoine-dsg.github.io/root_system/ + github_repo: Antoine-dSG/root_system +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- An.rootPairing +- BCn.rootPairing +- BCn.isReflective_iff_isClassicalRoot +main_results: +- declaration: An.rootPairing + informal: Constructs the type Aₙ root pairing over the integers from signed intervals + on Fin n, with finiteness, reducedness, and crystallographic properties. + source_ref: N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Springer, 2002; + J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, GTM 9, + Springer, 1972, Ch. III. +- declaration: BCn.rootPairing + informal: Constructs the type BCₙ root pairing from the reflective vectors of the + standard dot product on ℤⁿ via a symmetric nondegenerate form. +- declaration: BCn.isReflective_iff_isClassicalRoot + informal: Characterizes the reflective vectors of the standard dot product on ℤⁿ + as exactly the classical type BCₙ roots ±eᵢ, ±2eᵢ, and ±eᵢ ± eⱼ for i ≠ j. +- declaration: BCn.range_rootPairing_root + informal: Identifies the root set of the type BCₙ root pairing with the classical + type BCₙ root set. +tags: +- representation-theory +- root-systems +- lie-theory +- combinatorics +msc: +- 17B22 +- 20F55 diff --git a/LeanPool/projects/rr-equals-nonmeager-paper.yaml b/LeanPool/projects/rr-equals-nonmeager-paper.yaml new file mode 100644 index 0000000000..e6cc73e812 --- /dev/null +++ b/LeanPool/projects/rr-equals-nonmeager-paper.yaml @@ -0,0 +1,29 @@ +slug: rr-equals-nonmeager-paper +title: The nonmeagre lower bound for the rearrangement number +summary: Formalizes the inequality non(M) ≤ rr in ZFC via relational systems, category + and slalom bounds, finite-vector constructions, and rearrangements of conditionally + convergent real series. The reverse inequality from the accompanying manuscript + is outside the upstream Lean formalization. +branch: set theory +entry_module: LeanPool.RearrangementNumber +authors: +- Vinicius de Oliveira Rodrigues +source: + url: https://github.com/vo-rodrigues/rr-equals-nonmeager-paper + github_repo: vo-rodrigues/rr-equals-nonmeager-paper + commit: a8e74a91fe0b9f3534194910172349c9926e3b3b +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- NonMRR.nonM_le_rr +main_results: +- declaration: NonMRR.nonM_le_rr + informal: The uniformity of the meagre ideal is at most the rearrangement number. +tags: +- set-theory +- cardinal-characteristics +- series +msc: +- 03E17 +- 40A05 diff --git a/LeanPool/projects/runge-kutta-order-conditions.yaml b/LeanPool/projects/runge-kutta-order-conditions.yaml new file mode 100644 index 0000000000..98edf91aec --- /dev/null +++ b/LeanPool/projects/runge-kutta-order-conditions.yaml @@ -0,0 +1,50 @@ +slug: runge-kutta-order-conditions +title: Certified Runge-Kutta Order Conditions +summary: Certifies Runge-Kutta order conditions over planar rooted trees, including + RK4 order four, Dormand-Prince order five, and Gauss-Legendre order six. +branch: numerical analysis +entry_module: LeanPool.RungeKuttaOrderConditions +authors: +- Carles Marín +source: + title: What Order a Method Knows + authors: + - Carles Marín + doi: 10.5281/zenodo.20787666 + github_repo: karlesmarin/runge-kutta-order-conditions-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- RungeKuttaOrderConditions.Butcher.gauss_order6_rat +main_results: +- declaration: RungeKuttaOrderConditions.Butcher.satisfiesOrderConditions_iff + informal: Reduces order-p Runge-Kutta conditions to a finite catalogue check over + rooted trees. +- declaration: RungeKuttaOrderConditions.Butcher.rk4_order4 + informal: Certifies that classical RK4 satisfies all planar rooted-tree order conditions + through order four. +- declaration: RungeKuttaOrderConditions.Butcher.rk4_not_ord5 + informal: Shows that RK4 fails a fifth-order condition, pinning its order exactly. +- declaration: RungeKuttaOrderConditions.Butcher.orderCond_node_perm + informal: Proves order conditions are invariant under permutations of a node's children. +- declaration: RungeKuttaOrderConditions.Butcher.dp_order5 + informal: Certifies the Dormand-Prince RK45 fifth-order tableau through order five. +- declaration: RungeKuttaOrderConditions.Butcher.gauss_order4 + informal: Certifies the Gauss-Legendre three-stage tableau through order four over + Q(sqrt(15)). +- declaration: RungeKuttaOrderConditions.Butcher.gauss_order6 + informal: Certifies the denominator-cleared (360-scaled) integer Z(sqrt(15)) Gauss-Legendre + tableau through order six, the scaled certificate feeding the homogeneity bridge. +- declaration: RungeKuttaOrderConditions.Butcher.gauss_order6_rat + informal: Certifies the Gauss-Legendre s=3 tableau through order six over Q(sqrt(15)), + via the formalized homogeneity bridge from the 360-scaled integer certificate. +tags: +- numerical-analysis +- runge-kutta +- rooted-trees +- order-conditions +msc: +- 65L06 +- 65L05 +- 05C05 diff --git a/LeanPool/projects/rupert.yaml b/LeanPool/projects/rupert.yaml new file mode 100644 index 0000000000..527885a335 --- /dev/null +++ b/LeanPool/projects/rupert.yaml @@ -0,0 +1,33 @@ +slug: rupert +title: The Rupert Problem for convex polyhedra +summary: Formalizes the Rupert property for the cube, the regular tetrahedron, and + the triakis tetrahedron, exhibiting for each an explicit straight-through hole admitting + a congruent copy. +branch: convex geometry +entry_module: LeanPool.Rupert +authors: +- David Renshaw +source: + url: https://github.com/dwrensha/Rupert.lean + github_repo: dwrensha/Rupert.lean +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Cube.rupert +- Tetrahedron.rupert +- TriakisTetrahedron.rupert +main_results: +- declaration: Cube.rupert + informal: Proves that the cube has the Rupert property. +- declaration: Tetrahedron.rupert + informal: Proves that the regular tetrahedron has the Rupert property. +- declaration: TriakisTetrahedron.rupert + informal: Proves that the triakis tetrahedron has the Rupert property. +tags: +- convex-geometry +- polyhedra +- rupert-problem +msc: +- 52B10 +- 52A15 diff --git a/LeanPool/projects/sabidussi.yaml b/LeanPool/projects/sabidussi.yaml new file mode 100644 index 0000000000..68a725950a --- /dev/null +++ b/LeanPool/projects/sabidussi.yaml @@ -0,0 +1,42 @@ +slug: sabidussi +title: Sabidussi's compatibility conjecture for Eulerian multigraphs +summary: Formalizes Sabidussi's compatibility conjecture for finite labelled endpoint + multigraphs with loops and parallel edges, where a loop contributes two half-edge + incidences to the degree of its vertex. Given an Euler tour and minimum degree at + least four, the main theorem constructs a partition of all edges into nonempty, + edge-connected, 2-regular ordinary circuits such that consecutive edges of the Euler + tour never lie in the same circuit. The proof runs a four-colour (F_2^2) algebra + over the cyclic transition word, uses an odd balancing parity lemma to choose compatible + local frames, and transports the resulting colouring back to a compatible ordinary + circuit decomposition. +branch: graph theory +entry_module: LeanPool.Sabidussi +authors: +- Nikolay Ulyanov +source: + title: 'Graph Puzzles III.1: A proof of Sabidussi''s compatibility conjecture' + authors: + - Nikolay Ulyanov + url: https://github.com/gexahedron/sabidussi-lean/blob/032e640df0cc41d4d3d217f7c68628ab19cb3767/sabidussi_proof.pdf + github_repo: gexahedron/sabidussi-lean +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Sabidussi.LoopMultigraph.loop_sabidussi_compatibility_ordinary +main_results: +- declaration: Sabidussi.LoopMultigraph.loop_sabidussi_compatibility_ordinary + informal: 'For any finite endpoint multigraph with an Euler tour and minimum degree + at least four, the edges partition into ordinary, nonempty, connected, 2-regular + circuits that are compatible with the tour: no circuit contains two consecutive + tour edges.' +- declaration: Sabidussi.LoopMultigraph.loop_sabidussi_compatibility + informal: The same compatibility statement phrased with the intrinsic even-edge-set + circuit decomposition, from which the ordinary-circuit form is derived. +tags: +- graph-theory +- eulerian-graphs +- circuit-decomposition +- sabidussi +msc: +- 05C45 diff --git a/LeanPool/projects/salem-theorem.yaml b/LeanPool/projects/salem-theorem.yaml new file mode 100644 index 0000000000..d54381eeca --- /dev/null +++ b/LeanPool/projects/salem-theorem.yaml @@ -0,0 +1,35 @@ +slug: salem-theorem +title: Every Pisot number is a two-sided limit of Salem numbers +summary: Proves Salem’s two-sided approximation theorem for Pisot numbers, preserving + both the polynomial reciprocal-family construction and the exceptional quadratic-unit + construction. +branch: algebraic number theory +entry_module: LeanPool.SalemTheorem +authors: +- Stephanie Alexander +source: + url: https://github.com/stalex444/salem-theorem + github_repo: stalex444/salem-theorem + commit: 887be765bf24c6a645600dfc7bd0a1055a1761dd +license: MIT +status: verified +provenance: AI +main_declarations: +- SalemTheorem.salem_theorem +main_results: +- declaration: SalemTheorem.salem_theorem + informal: Every Pisot number is approached by Salem numbers from both below and + above. +- declaration: SalemTheorem.salem_construction_two_sided + informal: For the nondegenerate Pisot polynomial pattern, reciprocal-family polynomials + have Salem roots arbitrarily near the Pisot root from either side. +- declaration: SalemTheorem.salem_quadratic_unit + informal: The quadratic-unit family supplies Salem roots approaching each reciprocal + quadratic Pisot unit from either side. +tags: +- number-theory +- pisot-numbers +- salem-numbers +msc: +- 11R06 +- 11R09 diff --git a/LeanPool/projects/sardmoreira.yaml b/LeanPool/projects/sardmoreira.yaml new file mode 100644 index 0000000000..a482e5bba5 --- /dev/null +++ b/LeanPool/projects/sardmoreira.yaml @@ -0,0 +1,34 @@ +slug: sardmoreira +title: Moreira's version of Sard's theorem +summary: Formalizes Moreira's strengthening of Sard's theorem, bounding the Hausdorff + dimension of critical-value images of C^{k+alpha} maps. +branch: geometric measure theory +entry_module: LeanPool.SardMoreira +authors: +- Yury G. Kudryashov +source: + doi: 10.5565/PUBLMAT_45101_06 + github_repo: urkud/SardMoreira + paper: http://dmle.icmat.es/pdf/PUBLICACIONSMATEMATIQUES_2001_45_01_06.pdf +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- dimH_image_le_sardMoreiraBound_of_finrank_le +main_results: +- declaration: hausdorffMeasure_sardMoreiraBound_image_null_of_finrank_le + informal: Proves Hausdorff-measure nullity for images of points where the derivative + rank is bounded by p, at Moreira's Sard bound. + source_ref: Moreira 2001, http://dmle.icmat.es/pdf/PUBLICACIONSMATEMATIQUES_2001_45_01_06.pdf +- declaration: dimH_image_le_sardMoreiraBound_of_finrank_le + informal: Converts the measure-null theorem into the headline Hausdorff-dimension + upper bound for critical-value images. + source_ref: Moreira 2001, http://dmle.icmat.es/pdf/PUBLICACIONSMATEMATIQUES_2001_45_01_06.pdf +tags: +- analysis +- measure-theory +- sard-theorem +- hausdorff-measure +msc: +- 28A78 +- 58C25 diff --git a/LeanPool/projects/schoenflies.yaml b/LeanPool/projects/schoenflies.yaml new file mode 100644 index 0000000000..faf5f72cf0 --- /dev/null +++ b/LeanPool/projects/schoenflies.yaml @@ -0,0 +1,35 @@ +slug: schoenflies +title: Jordan–Schönflies theorem +summary: Formalizes the Jordan curve theorem and the extension of every homeomorphism + between Jordan curves to a plane homeomorphism. +branch: geometric topology +entry_module: LeanPool.Schoenflies +authors: +- Álvaro Begué +source: + title: The Jordan–Schönflies theorem in Lean + url: https://github.com/alonamaloh/schoenflies-lean + github_repo: alonamaloh/schoenflies-lean + commit: 05a43d29cde026618777db3d4e4316204ccca237 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Schoenflies.jordan_schoenflies_of_homeomorph +main_results: +- declaration: Schoenflies.jordan_schoenflies_of_homeomorph + informal: Every homeomorphism between two Jordan curves in the plane extends to + a self-homeomorphism of the plane. +- declaration: Schoenflies.jordan_schoenflies + informal: A homeomorphism between Jordan curves extends to a global homeomorphism + agreeing with it on the source curve. +- declaration: Schoenflies.forall_isSeparating_of_isJordanCurve + informal: Every Jordan curve separates the plane into an inside and an outside region. +tags: +- jordan-curve +- schoenflies-theorem +- geometric-topology +- homeomorphism +msc: +- 57K10 +- 54C25 diff --git a/LeanPool/projects/selberg-sieve4.yaml b/LeanPool/projects/selberg-sieve4.yaml new file mode 100644 index 0000000000..1a0c9797d0 --- /dev/null +++ b/LeanPool/projects/selberg-sieve4.yaml @@ -0,0 +1,41 @@ +slug: selberg-sieve4 +title: Selberg Sieve +summary: Formalizes the Selberg sieve with applications to prime-counting bounds and + a Brun-Titchmarsh-style estimate. +branch: analytic number theory +entry_module: LeanPool.SelbergSieve4 +authors: +- Arend Mellendijk +source: + title: Sieve Methods + authors: + - Heini Halberstam + - Hans-Egon Richert + url: https://archive.org/details/sievemethods0000halb + github_repo: amellendijk/selberg-sieve4 +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- fundamental_theorem_simple +- primeCounting_isBigO_atTop +- primesBetween_le +main_results: +- declaration: fundamental_theorem_simple + informal: Proves the simple Selberg sieve upper-bound theorem. +- declaration: primeCounting_isBigO_atTop + informal: Derives the asymptotic prime-counting upper bound. +- declaration: primeCounting_le_mul + informal: Gives an eventual explicit constant upper bound for the prime-counting + function. +- declaration: primesBetween_le + informal: Bounds the number of primes in an interval using the sieve estimate. +tags: +- number-theory +- analytic-number-theory +- sieve-theory +- prime-counting +msc: +- 11N35 +- 11N05 +- 11N13 diff --git a/LeanPool/projects/semicircle-catalan.yaml b/LeanPool/projects/semicircle-catalan.yaml new file mode 100644 index 0000000000..1ca41baa3d --- /dev/null +++ b/LeanPool/projects/semicircle-catalan.yaml @@ -0,0 +1,42 @@ +slug: semicircle-catalan +title: Genus-Zero Pairings and Catalan Numbers +summary: 'Formalizes the finite combinatorics behind the genus-zero term of the Wigner + semicircle law: a pairing of Fin (2n), viewed as a fixed-point-free involution, + has genus zero (total cycle count of the rotation product maximal) exactly when + it is noncrossing, and these genus-zero pairings are counted by the Catalan number + Cₙ. The Catalan recurrence is obtained from an explicit equivalence decomposing + a noncrossing pairing by the partner of vertex 0 into independent inside and outside + pairings.' +branch: combinatorics +entry_module: LeanPool.SemicircleCheck +authors: +- Wondermonger-daydreaming +source: + title: 'semicircle-catalan: genus-zero pairings and the Catalan moments of the Wigner + semicircle law' + authors: + - Wondermonger-daydreaming + url: https://github.com/Wondermonger-daydreaming/semicircle-catalan + github_repo: Wondermonger-daydreaming/semicircle-catalan +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- card_noncrossingPairing_eq_catalan +- Pairing.genus_zero_count +main_results: +- declaration: card_noncrossingPairing_eq_catalan + informal: The number of noncrossing pairings on 2n points equals the Catalan number + catalan n. +- declaration: Pairing.genus_zero_count + informal: The number of genus-zero pairings of Fin (2n) equals the Catalan number + catalan n. +- declaration: Pairing.genus_zero_iff_noncrossing + informal: A pairing has genus zero if and only if it is noncrossing. +tags: +- combinatorics +- catalan-numbers +- noncrossing-partitions +msc: +- 05A15 +- 05A18 diff --git a/LeanPool/projects/semicirclelaw.yaml b/LeanPool/projects/semicirclelaw.yaml new file mode 100644 index 0000000000..7fc1b90bde --- /dev/null +++ b/LeanPool/projects/semicirclelaw.yaml @@ -0,0 +1,42 @@ +slug: semicirclelaw +title: Wigner Semicircle Distribution +summary: Defines the real Wigner semicircle distribution, its normalized density, + affine transformations, and finite-moment integrability. +branch: probability theory +entry_module: LeanPool.SemicircleLaw +authors: +- FredRaj3 +source: + url: https://www.math.ucsd.edu/~tkemp/247A.Notes.pdf + github_repo: FredRaj3/SemicircleLaw +license: MIT +status: verified +provenance: human +main_declarations: +- LeanPool.SemicircleLaw.integral_semicirclePDFReal_eq_one +main_results: +- declaration: LeanPool.SemicircleLaw.semicirclePDFReal + informal: Defines the real-valued density of the semicircle distribution with mean + μ and variance v. +- declaration: LeanPool.SemicircleLaw.semicirclePDF + informal: Defines the ℝ≥0∞-valued density of the semicircle distribution. +- declaration: LeanPool.SemicircleLaw.integral_semicirclePDFReal_eq_one + informal: Proves that the real-valued semicircle density integrates to one for nonzero + variance. +- declaration: LeanPool.SemicircleLaw.semicircleReal + informal: Defines the semicircle measure on ℝ, equal to a Dirac mass when the variance + is zero. +- declaration: LeanPool.SemicircleLaw.semicircleReal_map_add_const + informal: Shows that translating a semicircle measure translates its mean parameter. +- declaration: LeanPool.SemicircleLaw.semicircleReal_map_const_mul + informal: Shows that scaling by a nonzero constant scales the mean and variance + parameters. +- declaration: LeanPool.SemicircleLaw.memLp_id_semicircleReal + informal: Shows that the identity has all finite Lp moments under a semicircle distribution. +tags: +- probability +- random-matrix-theory +- distributions +msc: +- 60B20 +- 60E05 diff --git a/LeanPool/projects/sendov-conjecture.yaml b/LeanPool/projects/sendov-conjecture.yaml new file mode 100644 index 0000000000..9fb869a585 --- /dev/null +++ b/LeanPool/projects/sendov-conjecture.yaml @@ -0,0 +1,111 @@ +slug: sendov-conjecture +title: Sendov's conjecture and the Phelps–Rodriguez conjecture +summary: 'Proves Sendov''s conjecture in full generality: if every zero of a complex + polynomial of degree at least two lies in the closed unit disk, then every zero + has a critical point within distance one. Also proves the Phelps–Rodriguez strengthening, + with strict inequality unless the zero lies on the unit circle and the polynomial + is a scalar multiple of z^n - a^n, and recovers Rubinstein''s boundary theorem. + Formalizes Tao''s digestion of Lech Mazur''s proof: the problem is reduced to a + numerical inequality in the degree and one real parameter, settled by exact rational + Bernstein certificates for degrees up to one hundred and by an analytic large-degree + reduction beyond. Includes the top case of Maclaurin''s inequality and multiset + AM–GM, a defect inequality on the closed unit disk, a sharp bound on log(sinh h + / h), kernel-checked polynomial certificate infrastructure, and a packed-exponentiation + bridge for exact moment integrals.' +branch: complex analysis +entry_module: LeanPool.Sendov +authors: +- Terence Tao +source: + title: Sendov's conjecture in Lean + url: https://github.com/teorth/sendov/tree/1ddea92d89f951a0a7cbbffa6c267cf7e6640b1d + github_repo: teorth/sendov + commit: 1ddea92d89f951a0a7cbbffa6c267cf7e6640b1d + announcement_url: https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/ + announced_at: '2026-08-12T23:10:54Z' + mathematical_source: T. Tao, A digestion of the proof of Sendov's conjecture, blog + post, 12 August 2026, digesting Lech Mazur's AI-assisted proof; the boundary case + follows Q. Tang and T. Zhang, arXiv:2508.10341, equation (5.1) and Remark 5.1. + prior_formalization: Lech Mazur's earlier Lean formalization of Sendov's conjecture + at ProofAtlas (https://www.proofatlas.ai/formalizations/sendov-conjecture/), which + the upstream README credits with priority for both the proof and the first formalization. + The present development shares no code with it and additionally states the Phelps–Rodriguez + equality classification. +license: Apache-2.0 +status: verified +provenance: AI +provenance_note: The upstream README states that essentially all Lean source was written + by Claude Opus 5 in Claude Code under the direction and review of Terence Tao, who + set the targets, supplied the informal proof and staged plans, and made the mathematical + decisions. +main_declarations: +- Sendov.sendov +- Sendov.phelps_rodriguez +main_results: +- declaration: Sendov.sendov + informal: For every natural number n at least two and every complex polynomial p + of degree n all of whose roots have norm at most one, every root a of p admits + a root of the derivative of p within distance at most one of a. + source_ref: Sendov's conjecture (1962); Tao's blog post of 12 August 2026, Conjecture + 1, following Mazur's proof; upstream Sendov/Conjecture.lean. First publicly announced + 2026-08-12 23:10:54 UTC in that post, with the Lean repository completed the same + day. +- declaration: Sendov.phelps_rodriguez + informal: Under the same hypotheses, either some root of the derivative lies at + distance strictly less than one from a, or a has norm one and p equals a nonzero + scalar multiple of X^n - a^n. + source_ref: Phelps–Rodriguez conjecture; Tao's blog post, Conjecture 2; upstream + Sendov/Conjecture.lean. The equality case is Rubinstein's 1968 theorem. +- declaration: SendovConjecture.sendov + informal: 'The same statement in the Mathlib-only form of the upstream statement + of record: for every complex polynomial p of degree at least two such that every + complex w with p(w) = 0 has norm at most one, every zero a of p admits a root + of the derivative within distance at most one of a.' + source_ref: Upstream Solution.lean, the declaration checked by the upstream comparator + setup. +- declaration: Sendov.rubinstein_one + informal: If a polynomial of degree n at least two with all roots in the closed + unit disk vanishes at one, then either its derivative has a root within distance + strictly less than one of the point one, or the polynomial equals its leading + coefficient times X^n - 1. + source_ref: Rubinstein's theorem; Tang–Zhang arXiv:2508.10341, equation (5.1) and + Remark 5.1, run to equality; upstream Sendov/Boundary.lean. +- declaration: Sendov.stat_lt_one + informal: For every degree n at least five and every real parameter alpha between + zero and seventeen satisfying the feasibility constraint c^2 ≤ A, the quantity + R n alpha of the blog post's equation stat is strictly less than one. + source_ref: Equation (stat) of Tao's blog post; degrees five to one hundred by Bernstein + certificates (upstream Sendov/FiniteRange), degrees at least 101 by the analytic + reduction (upstream Sendov/LargeDegree). +- declaration: Multiset.esymm_card_pred_le + informal: For a nonempty multiset of nonnegative reals of cardinality m, the (m-1)-st + elementary symmetric function is at most m times the (m-1)-st power of the mean. + source_ref: Maclaurin's inequality, top case; upstream Sendov/Analytic/Maclaurin.lean. +- declaration: Multiset.prod_le_mean_pow + informal: The product of a multiset of nonnegative reals is at most the mean raised + to the cardinality. + source_ref: Multiset AM–GM inequality; upstream Sendov/Analytic/Maclaurin.lean. +- declaration: Sendov.defect + informal: For a multiset of complex numbers in the closed unit disk, the product + of the norms times the sum of the norms of 1/z minus the conjugate of z is at + most one minus the square of that product. + source_ref: The defect lemma; upstream Sendov/Analytic/Defect.lean. +- declaration: Sendov.log_sinh_div_le + informal: For every positive real h, the logarithm of sinh h divided by h is at + most the square root of h^2 + 9 minus three. + source_ref: Upstream Sendov/Common/Sinh.lean; the crux of the beta-bound step. +- declaration: Sendov.pev_pos_of_bern + informal: If the Bernstein expansion of an integer coefficient list on an interval + from zero to p/q equals p^d times a polynomial P and every Bernstein coefficient + is positive, then P is positive on that interval. + source_ref: Lean Pool port infrastructure replacing the upstream per-certificate + ring identities; LeanPool/Sendov/FiniteRange/Certificate.lean. +tags: +- sendov-conjecture +- polynomial-zeros +- critical-points +- bernstein-certificates +msc: +- 30C15 +- 30C10 +- 12D10 diff --git a/LeanPool/projects/sensitivity.yaml b/LeanPool/projects/sensitivity.yaml new file mode 100644 index 0000000000..7b0e333e3f --- /dev/null +++ b/LeanPool/projects/sensitivity.yaml @@ -0,0 +1,30 @@ +slug: sensitivity +title: 'Sensitivity Conjecture: sqrt(deg) <= sensitivity' +summary: Formalizes the quantitative Nisan-Szegedy/Huang sensitivity theorem for Boolean + functions. +branch: extremal combinatorics +entry_module: LeanPool.Sensitivity +authors: +- Samuel Schlesinger +source: + arxiv: '1907.00847' + doi: 10.4007/annals.2019.190.3.6 + github_repo: SamuelSchlesinger/sensitivity-conjecture +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LeanPoolSensitivity.sensitivity_ge_sqrt_degree +main_results: +- declaration: LeanPoolSensitivity.sensitivity_ge_sqrt_degree + informal: Proves that the square root of multilinear degree is bounded above by + sensitivity. +- declaration: LeanPoolSensitivity.degree_le_sensitivity_sq + informal: Derives the degree bound by the square of sensitivity. +tags: +- combinatorics +- boolean-functions +- computational-complexity +msc: +- 06E30 +- 68Q17 diff --git a/LeanPool/projects/serre-mass-formula.yaml b/LeanPool/projects/serre-mass-formula.yaml new file mode 100644 index 0000000000..49bbe75b4c --- /dev/null +++ b/LeanPool/projects/serre-mass-formula.yaml @@ -0,0 +1,61 @@ +slug: serre-mass-formula +title: Serre's mass formula for totally ramified local-field extensions +summary: Proves both of Serre's mass formulas for totally ramified separable extensions + of a nonarchimedean local field of any positive degree, together with the discriminant + bound, tame-ramification criterion, finiteness dichotomy, convergence, and automorphism-weighted + orbit count. The proof develops Eisenstein monogenicity, quantitative Newton-Hensel + lifting, local constancy of root counts, and Haar measures of lattices over discrete + valuation rings. +branch: local fields and ramification theory +entry_module: LeanPool.MassFormula +authors: +- Hyeon Seung-Hyeon +source: + title: Une «formule de masse» pour les extensions totalement ramifiées de degré + donné d'un corps local + authors: + - Jean-Pierre Serre + url: https://gallica.bnf.fr/ark:/12148/bpt6k6234149b/f323.item + github_repo: 0stellensatz/MassFormula + commit: 7fa41a621f724f110183904d3fd19c6730a932ad +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- MassFormula.tsum_one_div_q_pow_c +main_results: +- declaration: MassFormula.tsum_one_div_q_pow_c + informal: For every nonarchimedean local field K with residue-field cardinality + q and every positive integer n, the sum of q^(-c(L)) over the totally ramified + degree-n subextensions L of a separable closure of K is n, where c(L) is the discriminant + exponent minus n plus one. +- declaration: MassFormula.tsum_one_div_w_mul_q_pow_c + informal: For every nonarchimedean local field K, positive n, and set of representatives + of K-isomorphism classes of totally ramified separable degree-n extensions, the + sum of 1/(|Aut_K(L)| q^c(L)) is one. +- declaration: MassFormula.sigma_infinite_iff + informal: For positive n, the set of totally ramified separable degree-n subextensions + is infinite exactly when K and its residue field have the same characteristic + p and p divides n. +- declaration: MassFormula.sub_one_le_d + informal: For positive n, the discriminant exponent of a totally ramified degree-n + extension is at least n-1. +- declaration: MassFormula.c_eq_zero_iff + informal: For positive n and a totally ramified degree-n extension, the wild discriminant + exponent vanishes exactly when the residue characteristic does not divide n. +- declaration: MassFormula.summable_one_div_q_pow_c + informal: For positive n, the real-valued series of the unweighted mass terms is + summable. +- declaration: MassFormula.ncard_isomorphic_mul_w + informal: For a nonarchimedean local field K and a totally ramified separable extension + L of positive degree n, the number of K-isomorphic copies of L inside a separable + closure, multiplied by |Aut_K(L)|, is n. +tags: +- number-theory +- local-fields +- ramification +- discriminants +- haar-measure +msc: +- 11S15 +- 11S05 diff --git a/LeanPool/projects/settheory.yaml b/LeanPool/projects/settheory.yaml new file mode 100644 index 0000000000..494d1ee8da --- /dev/null +++ b/LeanPool/projects/settheory.yaml @@ -0,0 +1,43 @@ +slug: settheory +title: The Kunen inconsistency theorem +summary: Formalizes elementary embeddings of models of ZF set theory and proves the + Kunen inconsistency theorem, that there is no nontrivial elementary embedding of + the set-theoretic universe into itself. +branch: set theory +entry_module: LeanPool.SetTheory +authors: +- Shuhao Song +source: + title: Elementary embeddings and infinitary combinatorics + authors: + - Kenneth Kunen + doi: 10.2307/2269948 + url: https://github.com/znssong/SetTheory + github_repo: znssong/SetTheory +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- SetTheory.kunen_inconsistency_V +- NontrivialElementaryEmbedding +main_results: +- declaration: SetTheory.kunen_inconsistency_V + informal: Proves the Kunen inconsistency theorem, that every elementary embedding + of the von Neumann universe V into itself is the identity. +- declaration: SetTheory.kunen_inconsistency_vonNeumann + informal: Proves the Kunen inconsistency theorem for von Neumann levels V_μ at a + successor-limit ordinal, computing the supremum of the critical-point iterates. +- declaration: SetTheory.crit + informal: Defines the critical point of a nontrivial elementary embedding as the + least ordinal it moves. +- declaration: NontrivialElementaryEmbedding + informal: Defines a nontrivial elementary embedding of a model of ZF into itself. +tags: +- set-theory +- large-cardinals +- elementary-embedding +- kunen-inconsistency +- model-theory +msc: +- 03E55 +- 03C90 diff --git a/LeanPool/projects/shannon-1948-formalization.yaml b/LeanPool/projects/shannon-1948-formalization.yaml new file mode 100644 index 0000000000..07d536553c --- /dev/null +++ b/LeanPool/projects/shannon-1948-formalization.yaml @@ -0,0 +1,36 @@ +slug: shannon-1948-formalization +title: Shannon Entropy Characterization +summary: Formalizes Shannon's finite-alphabet characterization theorem for entropy + from the 1948 paper. +branch: information theory +entry_module: LeanPool.Shannon1948Formalization +authors: +- Samuel Schlesinger +source: + doi: 10.1002/j.1538-7305.1948.tb01338.x + github_repo: SamuelSchlesinger/shannon-1948-formalization +license: MIT +status: verified +provenance: mix +main_declarations: +- LeanPool.Shannon1948Formalization.entropyNat_unique +main_results: +- declaration: LeanPool.Shannon1948Formalization.entropyNat_unique + informal: Proves that any finite-alphabet uncertainty functional satisfying Shannon's + conditions equals natural-log entropy up to a positive scale. +- declaration: LeanPool.Shannon1948Formalization.entropyBase_unique + informal: Restates the characterization for any logarithm base greater than one. +- declaration: LeanPool.Shannon1948Formalization.entropyNat_shannonAxioms + informal: Proves the converse, showing that Shannon entropy satisfies the formal + condition bundle. +- declaration: LeanPool.Shannon1948Formalization.chain_rule + informal: Proves the finite joint-entropy chain rule. +- declaration: LeanPool.Shannon1948Formalization.entropyNat_joint_le_add + informal: Proves subadditivity of finite joint entropy. +tags: +- information-theory +- entropy +- probability +msc: +- 94A17 +- 60C05 diff --git a/LeanPool/projects/singular-moduli.yaml b/LeanPool/projects/singular-moduli.yaml new file mode 100644 index 0000000000..84f205b3e7 --- /dev/null +++ b/LeanPool/projects/singular-moduli.yaml @@ -0,0 +1,60 @@ +slug: singular-moduli +title: Quadratic-order prime classification +summary: Formalizes foundational algebra for quadratic orders associated to discriminants, + including their power basis, norm and discriminant identities, odd-prime splitting/ramification/inertness + criteria via Legendre symbols, and root-counting results for quadratic congruences + modulo odd prime powers. +branch: algebraic number theory +entry_module: LeanPool.SingularModuli +authors: +- Caleb L. Geiger +source: + title: Singular Moduli and the Ideal Class Group + authors: + - C. Geiger + github_repo: ElodinLaarz/lean-thesis + url: https://github.com/ElodinLaarz/lean-thesis +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- QuadraticOrder.prime_inert_iff +- QuadraticOrder.prime_split_iff +main_results: +- declaration: QuadraticOrder.prime_inert_iff + informal: Proves that for an odd rational prime p and a valid discriminant d, the + ideal (p) is maximal in the quadratic order O_d exactly when the Legendre symbol + (d/p) is -1. + source_ref: C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.2, Proposition + 3.2.1. +- declaration: QuadraticOrder.prime_ramified_iff + informal: Proves that for an odd rational prime p and a valid discriminant d, the + ideal (p) is not radical in O_d exactly when p divides d. + source_ref: C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.2, Proposition + 3.2.1. +- declaration: QuadraticOrder.prime_split_iff + informal: Proves that for an odd rational prime p and a valid discriminant d, the + ideal (p) is radical but not maximal exactly when the Legendre symbol (d/p) is + 1. + source_ref: C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.2, Proposition + 3.2.1. +- declaration: QuadraticOrder.cardSqrts_prime_pow_coprime + informal: Counts roots of x^2 = c modulo p^n for odd prime p and p not dividing + c, giving two roots when the Legendre symbol is 1 and none otherwise. + source_ref: C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.1. +- declaration: QuadraticOrder.cardSqrts_prime_pow_even_val + informal: Counts roots of x^2 = p^(2r) u modulo p^n for odd prime p, p not dividing + u, and 2r < n, reducing to the coprime case and multiplying by p^r. + source_ref: C. Geiger, Singular Moduli and the Ideal Class Group, Section 3.1. +- declaration: QuadraticOrder.tau_sub_tauConj_sq_of_valid_disc + informal: Proves the discriminant identity (tau - tauConj)^2 = d for valid quadratic-order + discriminants d congruent to 0 or 1 modulo 4. + source_ref: C. Geiger, Singular Moduli and the Ideal Class Group, Section 2. +tags: +- algebraic-number-theory +- quadratic-orders +- legendre-symbol +msc: +- 11R11 +- 11R29 +- 11A15 diff --git a/LeanPool/projects/small-undecidable-groups.yaml b/LeanPool/projects/small-undecidable-groups.yaml new file mode 100644 index 0000000000..31fde30fd8 --- /dev/null +++ b/LeanPool/projects/small-undecidable-groups.yaml @@ -0,0 +1,57 @@ +slug: small-undecidable-groups +title: Small undecidable groups and effective Adian–Rabin families +summary: Proves the existence of a finitely presented group with three generator slots + and nine relator slots whose word problem is undecidable, and effective families + with undecidable triviality having respectively four generators and eleven relators, + and two generators and ten relators. The dependency closure includes computable + machine-to-Thue reductions, synchronized rewriting, Borisov’s group simulator, injective + HNN and Tietze compression, and general Miller–Tancer and Gordon constructions. + Formalizes the algebraic results of the cited paper; its four-manifold consequences + are outside this project. +branch: combinatorial group theory and computability +entry_module: LeanPool.SmallUndecidableGroups +authors: +- Marc Kegel +- Shana Yunsheng Li +- Qiuyu Ren +source: + title: Small undecidable groups and unrecognizable 4-manifolds + arxiv: '2609.10461' + url: https://github.com/32805433/Adian-Rabin + github_repo: 32805433/Adian-Rabin + commit: fb4a401e8fbdba36e7b35c11fde2baa15592e185 +license: MIT +status: verified +provenance: AI +main_declarations: +- Undecidability.exists_small_undecidable_presentations +main_results: +- declaration: Undecidability.exists_three_generator_nine_relator_group_with_unsolvable_word_problem + informal: There is a finite presentation with three generator slots and nine relator + slots whose word problem is not a computable predicate. + source_ref: arXiv:2609.10461v1, Theorem 1.1. +- declaration: Undecidability.exists_four_generator_eleven_relator_adian_rabin_family + informal: There is a uniformly computable family of finite presentations with four + generator slots and eleven relator slots for which the set of indices presenting + the trivial group is not computable. + source_ref: arXiv:2609.10461v1, Theorem 1.3. +- declaration: Undecidability.exists_two_generator_ten_relator_adian_rabin_family + informal: There is a uniformly computable family of finite presentations with two + generator slots and ten relator slots for which the set of indices presenting + the trivial group is not computable. + source_ref: arXiv:2609.10461v1, Theorem 1.4. +- declaration: Undecidability.exists_small_undecidable_presentations + informal: 'The three small-presentation bounds hold together: an undecidable word + problem with three generators and nine relators, and effective families with undecidable + triviality with four generators and eleven relators and with two generators and + ten relators.' + source_ref: arXiv:2609.10461v1, Theorems 1.1, 1.3, and 1.4. +tags: +- group-theory +- undecidability +- finite-presentations +- hnn-extensions +- rewriting-systems +msc: +- 20F10 +- 03D35 diff --git a/LeanPool/projects/spectral-positivity.yaml b/LeanPool/projects/spectral-positivity.yaml new file mode 100644 index 0000000000..615a6f6a6a --- /dev/null +++ b/LeanPool/projects/spectral-positivity.yaml @@ -0,0 +1,49 @@ +slug: spectral-positivity +title: Spectral positivity +summary: Perron-Frobenius theory, Jentzsch's theorem, and matrix/operator positivity + in Lean 4. At the matrix level it proves that nonnegative matrices have nonnegative + powers, the Metzler decomposition and positivity of the matrix exponential of a + Metzler matrix, M-matrix inverse positivity, and the Perron-Frobenius theorem for + irreducible nonnegative matrices (a positive dominant eigenvalue with a strictly + positive eigenvector). At the operator level it proves Jentzsch's theorem for compact + self-adjoint positivity-improving operators on L2 — a simple, strictly positive + ground state with a spectral gap — together with a kernel positivity criterion. +branch: linear algebra +entry_module: LeanPool.SpectralPositivity +authors: +- Michael R. Douglas +source: + title: 'spectral-positivity: Perron-Frobenius, Jentzsch, and matrix positivity in + Lean 4' + authors: + - Michael R. Douglas + url: https://github.com/mrdouglasny/spectral-positivity + github_repo: mrdouglasny/spectral-positivity +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- allpos_has_pos_eigenvec +- perron_frobenius +- ground_state_strictly_positive +main_results: +- declaration: allpos_has_pos_eigenvec + informal: A matrix with all strictly positive entries has a positive eigenvalue + with a strictly positive eigenvector (Perron-Frobenius for positive matrices, + via the Collatz-Wielandt characterization). +- declaration: perron_frobenius + informal: An irreducible (primitive) nonnegative matrix has a positive eigenvalue + with a strictly positive eigenvector. +- declaration: ground_state_strictly_positive + informal: The ground eigenfunction of a compact self-adjoint positivity-improving + operator on L2 is strictly positive almost everywhere (Jentzsch). +- declaration: SpectralPositivity.Matrix.MMatrix.inverse_pos + informal: The inverse of a nonsingular irreducible M-matrix is entrywise strictly + positive. +tags: +- linear-algebra +- perron-frobenius +- positivity +msc: +- 15B48 +- 15A18 diff --git a/LeanPool/projects/sphere-packing.yaml b/LeanPool/projects/sphere-packing.yaml new file mode 100644 index 0000000000..1f2330990e --- /dev/null +++ b/LeanPool/projects/sphere-packing.yaml @@ -0,0 +1,50 @@ +slug: sphere-packing +title: Sharp asymptotic upper bounds for sphere packing +summary: Formalizes the sharp high-dimensional asymptotics of the unrestricted Cohn--Elkies + linear program and bridges them to sphere-packing density. The development constructs + radial test functions through Mellin, saddle-point, and harmonic analysis, proves + radialization preserves the unrestricted optimum, and certifies the exact natural- + and base-two exponential rates. +branch: discrete geometry and harmonic analysis +entry_module: LeanPool.SpherePacking +authors: +- OpenAI +- Sidharth Hariharan +- Gareth Ma +- Dean Cureton +source: + title: Ten Advances in Mathematics and Theoretical Computer Science + authors: + - OpenAI + url: https://github.com/openai/ten-proofs/tree/94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 + github_repo: openai/ten-proofs + commit: 94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- PackingBounds.sharpFullCohnElkiesManuscriptConclusions +main_results: +- declaration: PackingBounds.FullMain.exact_limit + informal: The d-th root of the unrestricted Cohn--Elkies linear-program bound converges + to the sharp constant sqrt(e / (2 pi)). +- declaration: PackingBounds.FullMain.exact_binary_exponent + informal: The base-two logarithm of the unrestricted linear-program bound, divided + by dimension, converges to -1/2 log_2(2 pi / e). +- declaration: PackingBounds.PackingBridge.sphere_packing_sharp_asymptotic_upper + informal: There is a vanishing error sequence such that, in every positive dimension, + sphere-packing density is at most (sqrt(e / (2 pi)) plus that error) to the dimension. +- declaration: PackingBounds.sharpFullCohnElkiesManuscriptConclusions + informal: Packages the complete sharp asymptotic conclusions for the unrestricted + program, including quotient-root, natural-logarithmic, and base-two limits, vanishing-error + formulas, a universal bound for all admissible functions, and a certified decimal + exponent interval. +tags: +- sphere-packing +- discrete-geometry +- harmonic-analysis +- linear-programming-bounds +msc: +- 52C17 +- 41A60 +- 42A38 diff --git a/LeanPool/projects/stafford38-formal.yaml b/LeanPool/projects/stafford38-formal.yaml new file mode 100644 index 0000000000..db7fe8ee4d --- /dev/null +++ b/LeanPool/projects/stafford38-formal.yaml @@ -0,0 +1,32 @@ +slug: stafford38-formal +title: Stafford 3.8 +summary: Lean formalization of Stafford's Conjecture 3.8 for every characteristic-zero + field and every Weyl-algebra rank, proving that each nonzero d admits F, R, and + S with 1 = d*R + F*d*S. The development also proves an exact Bernstein-degree fixed-source + strengthening and derived localization, adjoint, and evolutionary results. +branch: noncommutative algebra +entry_module: LeanPool.Stafford38 +authors: +- Christopher Albert +source: + url: https://github.com/itpplasma/stafford38-formal + github_repo: itpplasma/stafford38-formal + commit: 1e234855eee36d2541dd69580501408288760e21 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Stafford38.universalStatement +main_results: +- declaration: Stafford38.universalStatement + informal: For every characteristic-zero field, every Weyl algebra rank, and every + nonzero d, there are F, R, S with 1 = d*R + F*d*S. +- declaration: Stafford38.universalFixedSourceStatement + informal: The stated fixed-source strengthening holds with its exact Bernstein-degree + bound. +tags: +- weyl-algebras +- noncommutative-algebra +- bernstein-degree +msc: +- 16S32 diff --git a/LeanPool/projects/stallings-folding.yaml b/LeanPool/projects/stallings-folding.yaml new file mode 100644 index 0000000000..e629591955 --- /dev/null +++ b/LeanPool/projects/stallings-folding.yaml @@ -0,0 +1,72 @@ +slug: stallings-folding +title: Verified Stallings folding for finitely generated subgroups of F2 +summary: Constructs a finite inverse automaton recognizing the subgroup generated + by any finite list of signed words in the rank-two free group. Proves exact subgroup + preservation and correctness of canonical reduced-word membership testing. Reusable + infrastructure includes inverse-automaton walks, reduction-compatible traversal, + finite fold congruences, coset potentials, flower graphs, and basis roses. Folding + uses exhaustive finite search over Boolean relations; no complexity bound, core + pruning, subgroup basis, intersection, or index algorithm is claimed. +branch: computational group theory +entry_module: LeanPool.StallingsFolding +authors: +- Arthur Freitas Ramos +source: + title: Verified finite automata for finitely generated subgroups of F2 + url: https://github.com/Arthur742Ramos/stallings-folding/tree/117dde0a8415d3da1787e15ad22a6223a744432d + github_repo: Arthur742Ramos/stallings-folding + commit: 117dde0a8415d3da1787e15ad22a6223a744432d + completion_commit: 0291588d2737f10d4fb17be947277680cd3c6462 + completion_date: '2026-09-23T04:05:29Z' + announcement: https://palomar-registry.org/entry?id=PALOMAR-2026-09-23-000004&version=1 + announcement_date: '2026-09-23T05:20:13Z' + mathematical_source: John R. Stallings, Topology of Finite Graphs, Inventiones Mathematicae + 71 (1983), 551-565, doi:10.1007/BF02095993. This is the rank-two finite-recognizer + special case, implemented through least congruences rather than a sequence of + local folds. + background: Ilya Kapovich and Alexei Myasnikov, Stallings Foldings and Subgroups + of Free Groups, Journal of Algebra 248 (2002), 608-668, doi:10.1006/jabr.2001.9033. + scope: All four Stallings library modules and the proved finite-recognizer endpoint + from Solution.lean. The standalone challenge placeholder, statement-audit tooling, + and CI files are omitted. Mathlib is the only dependency. + port_extensions: Flower vertices now contain exactly the internal positions of each + word, instead of padding all words to a common maximum length. The new Restriction + module proves preservation when folding an edge-closed finite vertex set containing + the basepoint; its potential is required only on the selected set. These AI-assisted + port extensions are not attributed to the upstream release. + provenance_note: Upstream formalization.yaml records human-directed GPT-6 Codex + proof development. Lean Pool porting, cleanup, and optimization were also performed + by OpenAI Codex. Author attribution follows upstream. +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Stallings.folded_recognizer_exists +main_results: +- declaration: Stallings.folded_recognizer_exists + informal: For every finite list S of signed words on two generators, there exist + a finite type of states, a base state, and a partial transition table with inverse-labelled + reverse edges, such that traversing the canonical reduced word of any g in FreeGroup + (Fin 2) returns to the base exactly when g lies in the subgroup generated by the + word values in S. The witness is the computed flower-graph fold. + source_ref: Upstream Solution.lean, completeStatement and folded_recognizer_exists. +- declaration: Stallings.foldWords_loopSubgroup_eq_generatorSubgroup + informal: For every finite input list of signed words in the rank-two free group, + the based-loop subgroup of the computed folded flower graph equals the subgroup + generated by the input words. + source_ref: Upstream Stallings/Flower.lean. +- declaration: Stallings.foldedMembershipTest_eq_true_iff + informal: For any finite generating list S and any rank-two free-group element g, + the executable folded-automaton test returns true if and only if g belongs to + the subgroup generated by S. There is no efficiency guarantee for constructing + the automaton. + source_ref: Upstream Stallings/Flower.lean. +tags: +- free-groups +- stallings-folding +- finite-automata +- verified-algorithms +msc: +- 20E05 +- 68Q45 +- 03B35 diff --git a/LeanPool/projects/stationary-harmonic-maps.yaml b/LeanPool/projects/stationary-harmonic-maps.yaml new file mode 100644 index 0000000000..f8bba664db --- /dev/null +++ b/LeanPool/projects/stationary-harmonic-maps.yaml @@ -0,0 +1,57 @@ +slug: stationary-harmonic-maps +title: Monotonicity Formula for Stationary Harmonic Maps +summary: 'Formalizes the monotonicity formula for stationary `W^{1,2}_{loc}` maps + with Euclidean target. The proof is target independent: it uses only the displayed + weak gradient and a vanishing domain first variation, so no target-manifold structure + enters the argument. From a bundled stationary Sobolev witness `StationaryW12LocMap + u Omega` it proves, for `0 < s <= r < R0`, that the normalized energy `Theta(Du, + a, rho) = rho^(2-n) * integral over B(a, rho) of |Du|^2` is monotone: `weakTheta + Du a s <= weakTheta Du a r`, together with the exact increment formula `weakTheta + Du a r - weakTheta Du a s = weakMonotonicityRhs Du a s r`. The Sobolev layer is + project-defined from Mathlib''s `MemLp`, weak-derivative identities, Bochner integration, + and Euclidean measure theory, following the De Giorgi-Nash-Moser engineering choice; + the internal proof routes through a radial vector-field first variation, a coarea/Radon-Nikodym + radius representation, smooth radial cutoffs, and an absolute-continuity boundary + step.' +branch: analysis +entry_module: LeanPool.LeanStationaryHarmonicMaps +authors: +- Wei Wang +source: + title: Theorems on Regularity and Singularity of Energy Minimizing Maps + authors: + - Leon Simon + doi: 10.1007/978-3-0348-9193-6 + github_repo: BrookWW/LeanStationaryHarmonicMaps +license: MIT +status: verified +provenance: mix +main_declarations: +- LeanStationaryHarmonicMaps.StationaryHarmonicMap.StationaryW12LocMap +main_results: +- declaration: LeanStationaryHarmonicMaps.StationaryHarmonicMap.StationaryW12LocMap + informal: 'The public input package: a domain-variation stationary `W^{1,2}_{loc}` + Sobolev map bundling a chosen weak gradient (`W12LocMapWitness`) with the vanishing + domain first-variation identity. This is the object users construct to invoke + the monotonicity theorem.' +- declaration: LeanStationaryHarmonicMaps.StationaryHarmonicMap.stationaryW12LocMonotonicity_euclidean + informal: 'Witness-style monotonicity: from a stationary `W^{1,2}_{loc}` map package, + the normalized weak energy `weakTheta` is monotone in the radius, `weakTheta Du + a s <= weakTheta Du a r` for `0 < s <= r < R0`.' +- declaration: LeanStationaryHarmonicMaps.StationaryHarmonicMap.stationaryW12LocMonotonicityFormula_euclidean + informal: 'Witness-style monotonicity formula: the increment of `weakTheta` equals + the nonnegative annular radial-energy term, `weakTheta Du a r - weakTheta Du a + s = weakMonotonicityRhs Du a s r` for `0 < s < r < R0`.' +- declaration: LeanStationaryHarmonicMaps.StationaryHarmonicMap.stationarySobolevMonotonicity_euclidean + informal: 'Componentwise convenience wrapper: the same monotonicity inequality stated + directly from the local `L^2`, measurability, weak-gradient, and first-variation + hypotheses, without first constructing the witness package.' +tags: +- analysis +- pde +- harmonic-maps +- monotonicity-formula +- sobolev +msc: +- 58E20 +- 35J50 diff --git a/LeanPool/projects/structural-graph-classes.yaml b/LeanPool/projects/structural-graph-classes.yaml new file mode 100644 index 0000000000..2add837100 --- /dev/null +++ b/LeanPool/projects/structural-graph-classes.yaml @@ -0,0 +1,73 @@ +slug: structural-graph-classes +title: Split, cograph and threshold graph structures +summary: Connects split partitions, cotrees and threshold creation sequences on Mathlib + SimpleGraph. Proves the finite forbidden-induced-subgraph characterizations, split + iff chordal and co-chordal, and completeness of cotrees and creation sequences. + Supplies certificate-preserving operations, twins and nonempty union/join decompositions + for cographs, maximum-clique split partitions, and induced/isomorphic transport + of existing chordal elimination orders and clique trees. Classical graph theory + is formalized; no new mathematical theorem, canonical certificate or recognition-time + bound is claimed. +branch: graph theory +entry_module: LeanPool.StructuralGraphClasses +authors: +- Juan Pablo Traverso Gianini +source: + title: Structural graph classes, original Lean development checkpoint + url: https://github.com/jtraverso/lean-pool/tree/c431fcecd3c963bcca643c379d807260585711b8/LeanPool/StructuralGraphClasses + github_repo: jtraverso/lean-pool + commit: c431fcecd3c963bcca643c379d807260585711b8 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- SimpleGraph.isThreshold_iff_nonempty_creationRepresentation +main_results: +- declaration: SimpleGraph.isSplit_iff_forbidden_induced_subgraphs + informal: A finite graph is split iff it contains no induced 2K2, C4 or C5. + source_ref: Földes and Hammer, Split graphs (1977). +- declaration: SimpleGraph.isSplit_iff_isChordal_and_compl + informal: A finite graph is split iff both it and its complement are chordal. + source_ref: Földes and Hammer, Split graphs (1977). +- declaration: SimpleGraph.isCograph_iff_nonempty_cotreeRepresentation + informal: A finite graph is P4-free iff it is isomorphic to the graph of a cotree, + whose constructors are empty, singleton, disjoint union and join. + source_ref: Seinsche (1974), doi:10.1016/0095-8956(74)90063-X; Corneil, Lerchs and + Stewart Burlingham (1981), doi:10.1016/0166-218X(81)90013-5. +- declaration: SimpleGraph.isThreshold_iff_isSplit_and_isCograph + informal: A graph has a split partition with nested independent-side neighbourhoods + iff it is both split and P4-free. No finiteness is required. + source_ref: Classical threshold characterization; local Threshold.lean and Cotree.lean. +- declaration: SimpleGraph.isThreshold_iff_forbidden_induced_subgraphs + informal: A finite graph is threshold iff it contains no induced 2K2, C4 or P4. + source_ref: Chvátal and Hammer (1977), doi:10.1016/S0167-5060(08)70731-3. +- declaration: SimpleGraph.isThreshold_iff_nonempty_creationRepresentation + informal: A finite graph is threshold iff it is isomorphic to a graph constructed + from the empty graph by successively adding an isolated or universal vertex. + source_ref: Classical threshold creation characterization; local ThresholdCreation.lean. +- declaration: SimpleGraph.IsCograph.exists_twins + informal: A finite cograph with at least two vertices has two distinct vertices + with identical adjacency outside their pair, whether adjacent or not. + source_ref: Cotree induction in CographTwins.lean; classical cograph structure. +- declaration: SimpleGraph.IsCograph.exists_union_root + informal: A finite cograph that is not preconnected admits a cotree representation + with a union root and two nonempty children. This is an existence claim, not a + claim about every unreduced cotree. + source_ref: Seinsche decomposition and CotreeRepresentation constructors. +- declaration: SimpleGraph.IsCograph.exists_join_root + informal: A preconnected finite cograph with at least two vertices admits a cotree + representation with a join root and two nonempty children. + source_ref: Seinsche decomposition and CotreeRepresentation constructors. +- declaration: SimpleGraph.IsSplit.exists_partition_with_maximum_clique + informal: Every finite split graph admits a split partition whose clique part has + maximum cardinality. Such a partition need not be unique. + source_ref: Direct finite maximization argument in SplitDecomposition.lean. +tags: +- graph-theory +- split-graphs +- cographs +- threshold-graphs +- structural-certificates +msc: +- 05C75 +- 05C05 diff --git a/LeanPool/projects/sum-difference-exponent.yaml b/LeanPool/projects/sum-difference-exponent.yaml new file mode 100644 index 0000000000..8ddea147ac --- /dev/null +++ b/LeanPool/projects/sum-difference-exponent.yaml @@ -0,0 +1,58 @@ +slug: sum-difference-exponent +title: The optimal exponent relating sumsets and difference sets +summary: For a finite integer set A with at least two elements, proves that C(A) = + log(|A+A|/|A|) / log(|A-A|/|A|) is strictly below 2, while explicit finite sets + have exponents tending to 2. Thus 2 is the supremum and is never attained. Also + gives an explicit witness within 10^(-999) of 2. This imports the upstream base-39 + row-and-column variant, not the paper's base-12 automaton construction or its quantitative + cardinality estimates. Supporting infrastructure includes injective digit expansions + and the strict Pluennecke-Petridis equality argument for finite integer sets. +branch: additive combinatorics +entry_module: LeanPool.SumDifferenceExponent +authors: +- Haowei Lin +- Shanda Li +source: + title: Settling the Optimal Exponent Relating Sumsets and Difference Sets + authors: + - Haowei Lin + - Shanda Li + arxiv: '2607.27199' + url: https://github.com/linhaowei1/sum-diff-proof + github_repo: linhaowei1/sum-diff-proof + commit: f53f84223b48457ac922836b661227309d9340c3 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- SumDifferenceExponent.sSup_admissibleExponents +main_results: +- declaration: SumDifferenceExponent.growthExponent_lt_two + informal: For every finite set A of integers with at least two elements, log(|A+A|/|A|) + / log(|A-A|/|A|) is strictly less than 2. +- declaration: SumDifferenceExponent.tendsto_growthExponent_approximatingSet + informal: The growth exponents of the explicit finite integer sets approximatingSet + l converge to 2 as l tends to infinity; these sets have at least two elements + whenever l is positive. +- declaration: SumDifferenceExponent.isLUB_admissibleExponents + informal: The number 2 is the least upper bound of the growth exponents of all finite + integer sets with at least two elements. +- declaration: SumDifferenceExponent.sSup_admissibleExponents + informal: The supremum of the growth exponents of all finite integer sets with at + least two elements equals 2. +- declaration: SumDifferenceExponent.supremum_not_attained + informal: No finite integer set with at least two elements has growth exponent equal + to 2. +- declaration: SumDifferenceExponent.quantitative_example + informal: The explicitly defined finite integer set quantitativeSet has growth exponent + strictly between 2 - 10^(-999) and 2. +tags: +- additive-combinatorics +- sumsets +- difference-sets +- Pluennecke-Ruzsa +- optimal-exponents +msc: +- 11B13 +- 11B30 +- 05D10 diff --git a/LeanPool/projects/sumsthreesquares.yaml b/LeanPool/projects/sumsthreesquares.yaml new file mode 100644 index 0000000000..001b98fa80 --- /dev/null +++ b/LeanPool/projects/sumsthreesquares.yaml @@ -0,0 +1,30 @@ +slug: sumsthreesquares +title: Sums of Three Squares +summary: Formalizes a blueprint step in the three-squares theorem using quadratic + forms and geometry of numbers. +branch: number theory +entry_module: LeanPool.SumsThreeSquares +authors: +- Bhavik Mehta +- Pietro Monticone +- Abel Donate Munoz +source: + doi: 10.1090/S0002-9939-1957-0085275-8 + github_repo: pitmonticone/SumsThreeSquares +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- LeanPool.SumsThreeSquares.blueprint_case_mod8_eq3 +main_results: +- declaration: LeanPool.SumsThreeSquares.blueprint_case_mod8_eq3 + informal: Proves the mod-8-equals-3 blueprint case for the three-squares theorem. + source_ref: H. Davenport, The geometry of numbers, The Mathematical Gazette 31 (1947), + 206-210, doi:10.2307/3608159. +tags: +- number-theory +- quadratic-forms +- geometry-of-numbers +msc: +- 11E25 +- 11H06 diff --git a/LeanPool/projects/sundogcert.yaml b/LeanPool/projects/sundogcert.yaml new file mode 100644 index 0000000000..b9c91e6507 --- /dev/null +++ b/LeanPool/projects/sundogcert.yaml @@ -0,0 +1,48 @@ +slug: sundogcert +title: Sundog certificates +summary: 'A referee-free, axiom-clean Lean development of coding-theory certificates: + syndrome decoding soundness and lossiness bounds for GF(2) parity-check schemes + (with a Reed–Solomon evaluation-dual variant), and a fully machine-checked correctness + proof of the Karp reduction chain 3SAT ≤ 3DM ≤ X3C ≤ bounded-weight syndrome decoding. + Only the logical iff of each reduction is formalized; polynomial-time bounds, NP + classes, and Cook–Levin are unformalized external inputs, so NP-hardness of decoding + is conditional on those standard facts.' +branch: complexity theory +entry_module: LeanPool.Sundogcert +authors: +- Humiliati +source: + title: Sundog certificates + authors: + - Humiliati + url: https://github.com/humiliati/sundogcert + github_repo: humiliati/sundogcert +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Sundog.SATReductionMain.sat_iff_decodes +- Sundog.Certificate.accept_sound +main_results: +- declaration: Sundog.SATReductionMain.sat_iff_decodes + informal: 'The end-to-end reduction correctness: a formula is satisfiable iff its + image under the composed 3SAT-to-decoding reduction is a yes-instance of bounded-weight + syndrome decoding.' +- declaration: Sundog.MatchingNPHard.threeDM_iff_X3C + informal: The 3-dimensional matching instance has a perfect matching iff the reduced + exact-cover-by-3-sets instance has an exact cover, both directions machine-checked. +- declaration: Sundog.Certificate.accept_sound + informal: 'Soundness of the cheap syndrome-decoding verifier when it accepts: an + exhibited light same-syndrome witness certifies the body is safe within the decoding + radius.' +- declaration: Sundog.Certificate.reject_sound + informal: 'Soundness of the verifier when it rejects: a sound column-weight lower + bound on the minimum same-syndrome weight rules out every light witness, so the + body is genuinely unsafe.' +tags: +- complexity +- coding-theory +- np-hardness +msc: +- 68Q17 +- 94B35 diff --git a/LeanPool/projects/synthetic-euclid.yaml b/LeanPool/projects/synthetic-euclid.yaml new file mode 100644 index 0000000000..9fbdce91fa --- /dev/null +++ b/LeanPool/projects/synthetic-euclid.yaml @@ -0,0 +1,47 @@ +slug: synthetic-euclid +title: 'Synthetic Euclidean geometry: Euclid''s Elements Book I' +summary: Formalizes Book I of Euclid's Elements via an axiomatic synthetic-geometry + system (incidence, betweenness, congruence, and area), culminating in the Pythagorean + theorem and its converse. +branch: synthetic euclidean geometry +entry_module: LeanPool.SyntheticEuclid4 +authors: +- André Hernandez-Espiet +- Vladimir Sedlacek +source: + title: A formal system for Euclid's Elements + authors: + - Jeremy Avigad + - Edward Dean + - John Mumma + doi: 10.1017/S1755020309990098 + url: https://github.com/ah1112/synthetic_euclid_4 + github_repo: ah1112/synthetic_euclid_4 +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- SyntheticEuclid4.pythagoras +- SyntheticEuclid4.pythagoras_converse +main_results: +- declaration: SyntheticEuclid4.IncidenceGeometry + informal: The axiomatic class of synthetic Euclidean geometry, bundling points, + lines, circles, and the incidence, betweenness, congruence, and area primitives + with Euclid's axioms. +- declaration: SyntheticEuclid4.square_of_len + informal: Constructs a square on a given segment off a given line. +- declaration: SyntheticEuclid4.pythagoras_construct + informal: Constructs the three squares and parallelograms on the sides of a triangle + used in the proof of the Pythagorean theorem. +- declaration: SyntheticEuclid4.pythagoras + informal: Euclid I.47, the Pythagorean theorem, that the square on the hypotenuse + of a right triangle equals the sum of the squares on the other two sides. +- declaration: SyntheticEuclid4.pythagoras_converse + informal: Euclid I.48, the converse of the Pythagorean theorem, that a triangle + whose square areas satisfy the Pythagorean relation has a right angle. +tags: +- euclidean-geometry +- synthetic-geometry +- pythagorean-theorem +msc: +- 51M04 diff --git a/LeanPool/projects/three-gap-theorem.yaml b/LeanPool/projects/three-gap-theorem.yaml new file mode 100644 index 0000000000..692e54bae6 --- /dev/null +++ b/LeanPool/projects/three-gap-theorem.yaml @@ -0,0 +1,37 @@ +slug: three-gap-theorem +title: The Three-Gap (Steinhaus) Theorem +summary: 'Formalizes the three-gap (Steinhaus, three-distance) theorem: for every + real `a` and every `N`, the `N`-point orbit `{i·a mod 1 : i < N}` partitions `[0,1)` + into gaps taking at most three distinct lengths, and when exactly three occur the + largest is the sum of the other two. The proof works in `[0,1)` via `Int.fract` + (not `AddCircle`) through the two first-return times `η⁺` and `η⁻`, and is uniform + in the rotation number, covering rational and irrational `a` with no case split.' +branch: number theory +entry_module: LeanPool.SteinhausThreeGap +authors: +- Dirk Kunert +source: + title: The Three Gap Theorem (Steinhaus Conjecture) + authors: + - Tony van Ravenstein + doi: 10.1017/S1446788700031062 + github_repo: dkunert/three-gap-theorem-lean +license: MIT +status: verified +provenance: AI +main_declarations: +- SteinhausThreeGap.three_gap_card_le_three +main_results: +- declaration: SteinhausThreeGap.three_gap_card_le_three + informal: 'For every real `a` and natural `N`, the gaps of the orbit `{i·a mod 1 + : i < N}` take at most three distinct lengths.' +- declaration: SteinhausThreeGap.three_gap_lengths_eq + informal: When the gaps take exactly three distinct lengths, those lengths are precisely + the two first returns `η⁺`, `η⁻`, and their sum `η⁺ + η⁻`; in particular the largest + gap is the sum of the other two. +tags: +- number-theory +- three-gap-theorem +- equidistribution +msc: +- 11K06 diff --git a/LeanPool/projects/turan3.yaml b/LeanPool/projects/turan3.yaml new file mode 100644 index 0000000000..28ab664b93 --- /dev/null +++ b/LeanPool/projects/turan3.yaml @@ -0,0 +1,40 @@ +slug: turan3 +title: Turán's theorem (the "Book" weighting proof) +summary: Formalizes the weighting/optimization proof of Turán's theorem from Proofs + from THE BOOK, bounding the number of edges of a p-clique-free simple graph by (1/2)(1 + - 1/(p-1)) n². +branch: extremal graph theory +entry_module: LeanPool.Turan3 +authors: +- Rodrigo Gutierrez +- Yves Jäckle +source: + url: https://github.com/ro-gut/turan3 + github_repo: ro-gut/turan3 +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- Turan3.turans +- Turan3.finale_bound +- Turan3.Better_forms_clique +main_results: +- declaration: Turan3.turans + informal: Turán's theorem — a p-clique-free simple graph on n vertices has at most + (1/2)(1 - 1/(p-1)) n² edges. +- declaration: Turan3.finale_bound + informal: For any weight distribution on a p-clique-free graph, the total edge weight + is at most ((p-1)(p-2)/2)·(1/(p-1))², the bound used to derive the edge count. +- declaration: Turan3.Better_forms_clique + informal: Starting from any weight function, the minimal-support weight-non-decreasing + improvement has support that forms a clique. +- declaration: Turan3.Enhance_total_weight_stricinc + informal: Equalizing weights between a maximum- and minimum-weight vertex in the + clique support strictly increases the total edge weight. +tags: +- graph-theory +- combinatorics +- turan-theorem +- extremal-graph-theory +msc: +- 05C35 diff --git a/LeanPool/projects/tutte-path.yaml b/LeanPool/projects/tutte-path.yaml new file mode 100644 index 0000000000..bde7e12397 --- /dev/null +++ b/LeanPool/projects/tutte-path.yaml @@ -0,0 +1,45 @@ +slug: tutte-path +title: Tutte’s path theorem for matroids +summary: Proves the modular-cut path theorem for finite matroids, with reusable rank-partition, + hyperplane-separation, relative-complement, indecomposable-cover, and diamond constructions. +branch: matroid theory +entry_module: LeanPool.TuttePath +authors: +- Tutte formalization contributors +source: + arxiv: 2601.02582 + url: https://github.com/mbaker386/tutte-homotopy-lean + github_repo: mbaker386/tutte-homotopy-lean + commit: ab33c23369865b5fbd30b7a19710a527aa7ebab6 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- TutteFormalization.path_theorem +main_results: +- declaration: TutteFormalization.path_theorem + informal: In a finite connected matroid, any two hyperplanes containing a proper + indecomposable flat and lying outside a modular cut are joined by a finite Tutte + path all of whose vertices contain that flat and avoid the cut. Consecutive vertices + are distinct hyperplanes with indecomposable corank-two intersection; repeated + nonconsecutive vertices are permitted. +- declaration: TutteFormalization.path_theorem_of_indecomposable + informal: In any finite matroid, two hyperplanes containing an indecomposable flat + F and lying outside a modular cut are joined by a Tutte path whose vertices contain + F and avoid the cut. This strengthens the cited path theorem by requiring only + connectedness of the contraction by F; properness of F follows from containment + in an endpoint hyperplane. +- declaration: TutteFormalization.exists_indecomposable_complement + informal: If an indecomposable flat S and a flat U span the ground, every indecomposable + flat T contained in both extends inside S to an indecomposable flat R spanning + the ground with U, with corank R equal to rank U minus rank T. +- declaration: TutteFormalization.exists_relative_complement + informal: For nested flats T contained in S, there is a flat U containing T whose + intersection with S is T, whose join with S is the ground, and whose corank equals + rank S minus rank T. +tags: +- matroids +- combinatorics +- modular-cuts +msc: +- 05B35 diff --git a/LeanPool/projects/ulms-theorem.yaml b/LeanPool/projects/ulms-theorem.yaml new file mode 100644 index 0000000000..c88b579048 --- /dev/null +++ b/LeanPool/projects/ulms-theorem.yaml @@ -0,0 +1,47 @@ +slug: ulms-theorem +title: Ulm's theorem for countable reduced abelian p-groups +summary: Formalizes Ulm's classification theorem for countable reduced abelian p-groups + in arbitrary Lean universes. The development constructs the transfinite Ulm filtration + and invariants, proves their invariance under additive equivalence, establishes + Kaplansky's one-generator extension lemma, and completes the countable back-and-forth + classification. A finite-group regression distinguishes Z/4 x Z/4 from Z/2 x Z/8 + at the zeroth Ulm invariant. +branch: abelian group theory +entry_module: LeanPool.UlmsTheorem +authors: +- Elan Roth +source: + title: Ulm's theorem for countable reduced abelian p-groups + authors: + - Elan Roth + url: https://github.com/elanroth/UlmsTheorem + github_repo: elanroth/UlmsTheorem + commit: 3bb8a10ae12eb957a0f05db248d3becaf6a7e222 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- UlmsTheorem.ulm_theorem +main_results: +- declaration: UlmsTheorem.ulm_theorem + informal: For any prime p and any two countable reduced abelian p-groups in the + same arbitrary Lean universe, the groups are additively isomorphic if and only + if all of their classical Ulm invariants, indexed by universe-zero ordinals, agree. +- declaration: UlmsTheorem.iso_of_ulm_invariants + informal: Equal Ulm invariants at every universe-zero ordinal imply an additive + equivalence between countable reduced abelian p-groups, via finite height-preserving + partial isomorphisms and a back-and-forth limit. +- declaration: UlmsTheorem.ulmInvariant_iso_invariant + informal: An additive equivalence preserves every classical Ulm invariant. +- declaration: UlmsTheorem.UlmDifferentialRegression.no_iso_44_28 + informal: The groups Z/4 x Z/4 and Z/2 x Z/8 are not additively isomorphic; the + proof exercises the classification theorem and independently detects their different + zeroth Ulm invariants. +tags: +- abelian-groups +- p-groups +- classification-theorems +- ordinal-filtrations +- ulm-invariants +msc: +- 20K10 diff --git a/LeanPool/projects/unconditional-schauder-basis.yaml b/LeanPool/projects/unconditional-schauder-basis.yaml new file mode 100644 index 0000000000..92b9b7ef62 --- /dev/null +++ b/LeanPool/projects/unconditional-schauder-basis.yaml @@ -0,0 +1,43 @@ +slug: unconditional-schauder-basis +title: Unconditional Schauder Bases +summary: 'Develops an API for Schauder bases and unconditional Schauder bases of complete + normed spaces, including ordered partial-sum convergence, rearrangement criteria + for unconditionality, and an arbitrary-index variant built on unconditional summability, + then proves a finite sign criterion: a family with dense closed span, nonzero vectors, + and a uniform finite sign bound is the basis of an unconditional Schauder basis.' +branch: functional analysis +entry_module: LeanPool.UnconditionalSchauderBasis +authors: +- Daniel Smania +source: + title: Classical Banach Spaces I and II + authors: + - Joram Lindenstrauss + - Lior Tzafriri + doi: 10.1007/978-3-662-53294-2 + github_repo: SmaniaD/UnconditionalSchauderBasis +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- UnconditionalCriterion.exists_unconditionalSchauderBasis_of_finiteSignBound +main_results: +- declaration: UnconditionalCriterion.exists_unconditionalSchauderBasis_of_finiteSignBound + informal: 'Finite sign criterion for sequences: if `x : ℕ → E` has dense closed + linear span, every `x n` is nonzero, and finite sign changes are uniformly bounded + by `C`, then `x` is the basis sequence of an unconditional Schauder basis.' + source_ref: J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II, + Chapter 1, Section c, unconditional bases, doi:10.1007/978-3-662-53294-2. +- declaration: UnconditionalCriterion.exists_unconditionalSchauderBasisAbstractIndex_of_finiteSignBound + informal: 'Abstract-index form of the finite sign criterion: under the same dense + span, nonzero, and uniform finite sign bound hypotheses for an arbitrarily indexed + family, there is an abstractly indexed unconditional Schauder basis whose basis + family is exactly `x`.' + source_ref: J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II, + Chapter 1, Section c, unconditional bases, doi:10.1007/978-3-662-53294-2. +tags: +- functional-analysis +- banach-spaces +- schauder-bases +msc: +- 46B15 diff --git a/LeanPool/projects/virasoroproject.yaml b/LeanPool/projects/virasoroproject.yaml new file mode 100644 index 0000000000..ec1ab2d597 --- /dev/null +++ b/LeanPool/projects/virasoroproject.yaml @@ -0,0 +1,36 @@ +slug: virasoroproject +title: Virasoro Project +summary: Formalizes the Virasoro algebra, its Witt-algebra central extension, Verma + modules, and the basic bosonic Sugawara construction. +branch: representation theory +entry_module: LeanPool.VirasoroProject +authors: +- Kalle Kytölä +source: + title: Infinite-Dimensional Lie Algebras + authors: + - Victor Kac + url: https://doi.org/10.1017/CBO9780511626234 + github_repo: kkytola/VirasoroProject +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- VirasoroProject.VirasoroAlgebra +- VirasoroProject.sugawaraRepresentation +main_results: +- declaration: VirasoroProject.VirasoroAlgebra + informal: Defines the Virasoro algebra as a one-dimensional central extension of + the Witt algebra. +- declaration: VirasoroProject.sugawaraRepresentation + informal: Constructs the basic bosonic Sugawara representation of the Virasoro algebra + from locally truncated Heisenberg actions. +tags: +- representation-theory +- lie-algebras +- mathematical-physics +- virasoro-algebra +msc: +- 17B68 +- 17B56 +- 81R10 diff --git a/LeanPool/projects/viscosity-solution-theory.yaml b/LeanPool/projects/viscosity-solution-theory.yaml new file mode 100644 index 0000000000..7997295088 --- /dev/null +++ b/LeanPool/projects/viscosity-solution-theory.yaml @@ -0,0 +1,62 @@ +slug: viscosity-solution-theory +title: Viscosity solutions, Aleksandrov differentiability and harmonic Dirichlet problems +summary: Develops second-order semijets, smooth test-function equivalence, Jensen + contact sets, convex Aleksandrov differentiability, the quadratic-penalty Ishii + lemma and compact-domain comparison for degenerate elliptic equations. Perron assembly + retains explicit comparison, barrier and bumping assumptions; these are discharged + for classical harmonic Dirichlet existence on bounded open domains satisfying a + uniform exterior sphere condition. +branch: partial differential equations +entry_module: LeanPool.ViscositySolutionTheory +authors: +- William M. Feldman +- AleksandrovDifferentiability contributors +- The Tau Ceti contributors +source: + arxiv: math/9207212 + url: https://github.com/willmfeldman/viscosity-solution-theory/tree/067e2540ad1594408c0b6145c5bde01b4d5791a9 + github_repo: willmfeldman/viscosity-solution-theory + commit: 067e2540ad1594408c0b6145c5bde01b4d5791a9 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- ViscositySolns.dirichlet_harmonic_modulus_of_uniformExteriorSphere +main_results: +- declaration: AleksandrovDifferentiability.convexAleksandrovAE + informal: A convex real-valued function on an open subset of any finite-dimensional + real inner product space is second-order differentiable at Lebesgue-almost every + point. +- declaration: ViscositySolns.JensenContactSetPositiveMeasureOnClosedBallTheorem.proof + informal: Under the localized semiconvexity and strict-contact hypotheses, the contact + set arising from sufficiently small linear perturbations has positive Lebesgue + measure. +- declaration: ViscositySolns.QuadraticPenaltyIshiiLemmaOn.proof + informal: For an upper semicontinuous function and a lower semicontinuous function + on locally compact subsets of real coordinate space, doubled-variable maxima with + the quadratic penalty yield closed semijets satisfying the Ishii block-matrix + bounds. +- declaration: ViscositySolns.comparison_of_constantShift_boundary_proof + informal: A viscosity subsolution lies below a supersolution on a nonempty locally + compact domain with compact closure, assuming boundary comparison, semicontinuity + on the closure, properness, operator continuity, the Ishii structural condition + and uniform scalar decrease. +- declaration: ViscositySolns.viscositySubsolution_iff_smoothTestFunctionSubsolution + informal: On an open domain, semijet and smooth C² test-function formulations of + viscosity subsolutions agree for Hessian-continuous, symmetric-part-invariant + operators. +- declaration: ViscositySolns.dirichlet_harmonic_modulus_of_uniformExteriorSphere + informal: On a bounded open subset of finite-dimensional Euclidean space with a + uniform exterior sphere condition, Lipschitz data on its closure admit a continuous + extension that agrees on the boundary and is C² and harmonic in the domain, with + a boundary modulus depending only on the domain and Lipschitz constant. +tags: +- pde +- viscosity-solutions +- convex-analysis +- harmonic-functions +- perron-method +msc: +- 35D40 +- 35J60 +- 26B25 diff --git a/LeanPool/projects/vizing-equitable-edge-colouring.yaml b/LeanPool/projects/vizing-equitable-edge-colouring.yaml new file mode 100644 index 0000000000..3655625b6d --- /dev/null +++ b/LeanPool/projects/vizing-equitable-edge-colouring.yaml @@ -0,0 +1,49 @@ +slug: vizing-equitable-edge-colouring +title: Vizing's theorem and equitable edge colourings +summary: Proves the maximum-degree-plus-one edge-colouring bound for finite simple + graphs by fans and Kempe exchanges. A separate component-swapping argument equitably + rebalances any finite proper edge colouring on its original palette. Together they + give equitable proper edge colourings on any palette of at least maximum degree + plus one colours, with every class bounded by the ceiling of the average class size. + Extracted from the published Paper IV formalization; these are classical colouring + results, not a claim to new mathematics or to the paper's clique-partition bound. +branch: graph theory +entry_module: LeanPool.Vizing +authors: +- Juan Pablo Traverso Gianini +source: + title: Vizing and equitable edge-colouring modules in the published Paper IV freeze + authors: + - Juan Pablo Traverso Gianini + url: https://github.com/jtraverso/erdos-81-chordal-clique-partitions + github_repo: jtraverso/erdos-81-chordal-clique-partitions + commit: ff2019f56a50cf300a6531c9f8d4a8fb1a541e3d +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- LeanPool.Vizing.exists_equitable_edge_colouring +main_results: +- declaration: LeanPool.Vizing.lineGraph_colorable + informal: The line graph of a finite simple graph G is colourable with maximum degree + of G plus one colours. + source_ref: Vizing's theorem for finite simple graphs; Vizing (1964). +- declaration: SimpleGraph.lineGraph_chromaticNumber_le_maxDegree_succ + informal: The chromatic number of the line graph of a finite simple graph G is at + most maximum degree of G plus one. +- declaration: LeanPool.Vizing.Equitable.exists_equitable_colouring + informal: Every proper edge colouring of a finite simple graph with a finite palette + admits a proper recolouring on the same palette in which any two colour classes + differ in size by at most one. +- declaration: LeanPool.Vizing.exists_equitable_edge_colouring + informal: For every finite simple graph G and natural c at least maximum degree + of G plus one, there is an equitable proper edge colouring with c colours; each + colour class has at most ceiling(|E(G)|/c) edges. +tags: +- graph-theory +- edge-colouring +- vizing-theorem +- equitable-colouring +- kempe-chains +msc: +- 05C15 diff --git a/LeanPool/projects/vlasov-mean-field.yaml b/LeanPool/projects/vlasov-mean-field.yaml new file mode 100644 index 0000000000..23ae4838a3 --- /dev/null +++ b/LeanPool/projects/vlasov-mean-field.yaml @@ -0,0 +1,52 @@ +slug: vlasov-mean-field +title: Mean-field derivation and well-posedness of the Vlasov equation +summary: Formalizes the mean-field limit, Dobrushin stability, and forward-in-time + well-posedness for the Vlasov equation via Wasserstein-1 optimal transport. +branch: kinetic theory +entry_module: LeanPool.Vlasov +authors: +- Joseph K. Miller +source: + title: A Formalization of the Mean-Field Derivation of the Vlasov Equation + authors: + - Joseph K. Miller + arxiv: '2607.08986' + github_repo: Hydrodynamical/Vlasov_Meanfield_Formalization +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Vlasov.vlasovWellPosedness +- Vlasov.dobrushin +- Vlasov.meanFieldLimit +main_results: +- declaration: Vlasov.vlasovWellPosedness + informal: Forward-in-time existence of a Lagrangian solution to the Vlasov equation + on any finite horizon, for interaction forces that are Lipschitz gradients of + an even pair potential (class AssW) — with no smallness restriction on the Lipschitz + constant. +- declaration: Vlasov.dobrushin + informal: A Wasserstein-1 form of Dobrushin's 1979 stability estimate (the original + uses a truncated metric) - two Lagrangian Vlasov solutions with finite first moments + at every time separate at most at an exponential Gronwall rate. +- declaration: Vlasov.meanFieldLimit + informal: Conditional mean-field limit - given Wasserstein-1 convergence of the + initial empirical data and a uniform Dobrushin stability estimate for the empirical + curves, the N-particle empirical measures converge to the Vlasov solution uniformly + on any finite horizon. +- declaration: Vlasov.weak_isLagrangianVlasovSolutionOn + informal: On a time window with L * T^2 < 1, a weak solution with the standing regularity + (even C^2 pair potential AssW2, finite first moments, narrow continuity, time-continuity + of the convolution force at each point, continuity of the convolution-derivative + field, and a uniform spatial first-moment bound on the window) is Lagrangian - + the pushforward of its initial datum along the characteristic flow. +tags: +- kinetic-theory +- mean-field-limit +- optimal-transport +- wasserstein-distance +- pde +msc: +- 35Q83 +- 82C22 +- 49Q22 diff --git a/LeanPool/projects/wallace-problem.yaml b/LeanPool/projects/wallace-problem.yaml new file mode 100644 index 0000000000..f8cf7d6b4a --- /dev/null +++ b/LeanPool/projects/wallace-problem.yaml @@ -0,0 +1,47 @@ +slug: wallace-problem +title: The Wallace problem in ZFC +summary: Formalizes in ZFC a Hausdorff countably compact group topology without nontrivial + convergent sequences on every torsion-free Abelian group of cardinality continuum. + It specializes the construction to the rational direct sum and the additive real + group, and derives a commutative Tychonoff countably compact cancellative semigroup + that is not a group. +branch: set-theoretic topology and topological algebra +entry_module: LeanPool.Wallace +authors: +- Juliane Trianon Fraga +- Vinicius de Oliveira Rodrigues +source: + title: The Wallace problem and countably compact torsion-free Abelian groups in + ZFC + authors: + - Juliane Trianon Fraga + - Vinicius de Oliveira Rodrigues + arxiv: '2608.17317' + doi: 10.48550/arXiv.2608.17317 + url: https://github.com/vo-rodrigues/wallace-problem-zfc-paper + github_repo: vo-rodrigues/wallace-problem-zfc-paper + commit: 23756864de1f14e272dafb070b97fcdc7cbc75c5 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Wallace.commutativeTychonoffWallaceCounterexampleExists +main_results: +- declaration: Wallace.torsionFreeAbelianGroup_mainTheorem_exact + informal: Every torsion-free Abelian group of cardinality continuum admits a Hausdorff + countably compact group topology with no nontrivial convergent sequences. +- declaration: Wallace.continuumRationalGroup_and_real_mainTheorem + informal: Both the continuum-indexed rational direct sum and the additive group + of real numbers admit such Hausdorff countably compact group topologies without + nontrivial convergent sequences. +- declaration: Wallace.commutativeTychonoffWallaceCounterexampleExists + informal: There exists a commutative Tychonoff countably compact topological semigroup + with two-sided cancellation which is not a group. +tags: +- wallace-problem +- topological-groups +- set-theoretic-topology +msc: +- 22A05 +- 54D30 +- 20K20 diff --git a/LeanPool/projects/whitehead-theorem.yaml b/LeanPool/projects/whitehead-theorem.yaml new file mode 100644 index 0000000000..cd185fa2f3 --- /dev/null +++ b/LeanPool/projects/whitehead-theorem.yaml @@ -0,0 +1,31 @@ +slug: whitehead-theorem +title: Whitehead's theorem for CW-complexes +summary: Formalizes Whitehead's theorem — a continuous map between CW-complexes that + induces isomorphisms on all homotopy groups is a homotopy equivalence — including + the supporting theory of CW-complexes, relative homotopy groups, the homotopy extension + property, and compression of maps. +branch: algebraic topology +entry_module: LeanPool.WhiteheadTheorem +authors: +- Jiazhen Xia +source: + url: https://github.com/jzxia/WhiteheadTheorem + github_repo: jzxia/WhiteheadTheorem +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- WhiteheadTheorem +main_results: +- declaration: WhiteheadTheorem + informal: Proves that a continuous map between CW-complexes inducing isomorphisms + on all homotopy groups is a homotopy equivalence. +tags: +- algebraic-topology +- cw-complex +- homotopy-groups +- weak-homotopy-equivalence +msc: +- 55P10 +- 55Q05 +- 55U10 diff --git a/LeanPool/projects/zeta-five-irrational.yaml b/LeanPool/projects/zeta-five-irrational.yaml new file mode 100644 index 0000000000..becd0752ac --- /dev/null +++ b/LeanPool/projects/zeta-five-irrational.yaml @@ -0,0 +1,53 @@ +slug: zeta-five-irrational +title: Irrationality of the Riemann zeta function at five +summary: Proves that Mathlib's Riemann zeta function at five is an irrational real + number, following Aabir Fauzan's September 2026 preprint. Constructs integer polynomials + of degree 37n with positive values at zeta(5) that decay exponentially in n squared. + Includes a general integer-polynomial irrationality criterion, Hankel determinant + and moment-integral identities, logarithmic-energy estimates, prime-adic integrality + bounds, and kernel-checked rational certificates. The formalization uses a different + normalizer and a positive decay constant in place of the paper's specific constant + 139/5. Reuses the prime number theorem infrastructure already in the MooreBound + project. +branch: number theory +entry_module: LeanPool.Zeta5Irrational +authors: +- Moritz Firsching +source: + title: Zeta5 formalization of Aabir Fauzan's zeta(5) is irrational + url: https://github.com/mo271/Zeta5/tree/f19a1960609f7d38e7b63fd2acb05e6f60a7b741 + github_repo: mo271/Zeta5 + commit: f19a1960609f7d38e7b63fd2acb05e6f60a7b741 +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- Zeta5Irrational.irrational_five +main_results: +- declaration: Zeta5Irrational.irrational_five + informal: There exists an irrational real number x whose complex coercion equals + Mathlib's Riemann zeta function evaluated at five. + source_ref: Fauzan, zeta(5) is irrational, Theorem 1.1, September 17, 2026, https://doi.org/10.5281/zenodo.22826419; + upstream Apery/Main.lean. The completed formalization was released on September + 23, 2026. +- declaration: Zeta5Irrational.main_estimate + informal: There is a positive real c such that, for all sufficiently large n, an + integer polynomial Q of degree 37n is a positive rational multiple of the constructed + determinant polynomial and satisfies 0 < Q(zeta(5)) < exp(-c*n^2). + source_ref: Upstream Apery/MainEstimate.lean; variant of the paper's Theorem 2.1 + with c = -800*(Aeff + U) instead of 139/5. +- declaration: Zeta5Irrational.irrational_of_eventually_exists_int_poly + informal: A real xi is irrational if, eventually, integer polynomials of degree + at most d(n) have positive values at xi bounded by epsilon(n), and b^d(n)*epsilon(n) + tends to zero for every positive natural b. + source_ref: Upstream Apery/Criterion.lean; the paper's Section 1.1. +tags: +- irrationality +- zeta-function +- hankel-determinants +- potential-theory +- certified-computation +msc: +- 11J72 +- 11M06 +- 31A15 diff --git a/LeanPool/projects/zeta-h123.yaml b/LeanPool/projects/zeta-h123.yaml new file mode 100644 index 0000000000..35ada1c902 --- /dev/null +++ b/LeanPool/projects/zeta-h123.yaml @@ -0,0 +1,47 @@ +slug: zeta-h123 +title: Thakur's hypotheses on power sums of F_q[t] +summary: Formalizes four components of Thakur's hypotheses on power sums over finite + polynomial rings, including uniqueness and monotonicity results for the carry-free + digit optimization problems and the Laurent-series expansion of finite-field power + sums. +branch: number theory +entry_module: LeanPool.ZetaH123 +authors: +- Evan Chen +- Kenny Lau +- Ken Ono +- Jujian Zhang +source: + title: Thakur's hypotheses on power sums of F_q[t] + authors: + - Evan Chen + - Ken Ono + arxiv: '2606.16239' + github_repo: AxiomMath/zeta-h123 +license: MIT +status: verified +provenance: AI +main_declarations: +- ZetaH123.H1.main_theorem +- ZetaH123.Lem41.main +main_results: +- declaration: ZetaH123.H1.main_theorem + informal: Proves uniqueness of the admissible base-q decomposition maximizing the + H1 power-sum weight functional. +- declaration: ZetaH123.H2.main + informal: Proves the H2 unique-minimizer statement for the recursive digit/carry-free + optimization problem defining the values s_d(k). +- declaration: ZetaH123.H3.main_theorem + informal: Proves the H3 monotonicity statement that s_d(k) is strictly less than + s_d(k+1) when p does not divide k. +- declaration: ZetaH123.Lem41.main + informal: Proves the Lemma 4.1 Laurent-series expansion for S_d(k), with nonzero + coefficients indexed by the carry-free tuples. +tags: +- number-theory +- function-fields +- zeta-functions +msc: +- 11M38 +- 11T55 +- 11G09 diff --git a/LeanPool/projects/zeta-zeros.yaml b/LeanPool/projects/zeta-zeros.yaml new file mode 100644 index 0000000000..57061c9ec5 --- /dev/null +++ b/LeanPool/projects/zeta-zeros.yaml @@ -0,0 +1,52 @@ +slug: zeta-zeros +title: Simple zeros of the Riemann zeta function +summary: Formalizes Lamzouri's Hilbert-space proof of lower bounds for simple and + distinct zeros of the Riemann zeta function. From explicit Riemann--von Mangoldt + and unconditional pair-correlation hypotheses, the development proves that eventually + more than 67.25% of nontrivial zeros are simple and on the critical line and more + than 83.625% are distinct. It includes the finite-multiset Hilbert-space inequality, + zeta-zero multiplicity symmetries, an admissible cutoff construction, and certified + trigonometric estimates. +branch: analytic number theory +entry_module: LeanPool.ZetaZeros +authors: +- Kenny Lau +source: + title: A new proof that more than 2/3 of the zeros of the Riemann zeta function + are simple and on the critical line + authors: + - Youness Lamzouri + arxiv: 2609.02882 + doi: 10.48550/arXiv.2609.02882 + url: https://github.com/AxiomMath/ZetaZeros + github_repo: AxiomMath/ZetaZeros + commit: 4bcaf70e544506c311d83a5a5b143a134b9fc5f7 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- ZetaZeros.simple_proportion_d4 +main_results: +- declaration: ZetaZeros.simple_proportion_d4 + informal: Assuming the stated Riemann--von Mangoldt and unconditional pair-correlation + formulas, beyond some height more than 67.25% of nontrivial zeta zeros are simple + and lie on the critical line. +- declaration: ZetaZeros.distinct_proportion_d5 + informal: Under the same two explicit analytic hypotheses, beyond some height more + than 83.625% of nontrivial zeta zeros are distinct. +- declaration: ZetaZeros.card_simpleRealPart_lower + informal: Bounds from below the number of multiplicity-one real points in a finite + nonempty conjugation-invariant multiset using the squared test-kernel sum of any + admissible function. +- declaration: ZetaZeros.card_lower + informal: Bounds from below the number of distinct points in a finite nonempty conjugation-invariant + multiset using the same Hilbert-space kernel method. +tags: +- analytic-number-theory +- riemann-zeta-function +- critical-line +- simple-zeros +- pair-correlation +msc: +- 11M06 +- 11M26 diff --git a/LeanPool/projects/zeta2-zeta3-linear-independence.yaml b/LeanPool/projects/zeta2-zeta3-linear-independence.yaml new file mode 100644 index 0000000000..5d0639bbb1 --- /dev/null +++ b/LeanPool/projects/zeta2-zeta3-linear-independence.yaml @@ -0,0 +1,48 @@ +slug: zeta2-zeta3-linear-independence +title: Linear independence of 1, zeta(2), and zeta(3) +summary: Proves that 1, the Riemann zeta function at 2, and its value at 3 are linearly + independent over the rationals, using integer polynomial approximation, p-adic coefficient + bounds, prime-edge nonvanishing, and logarithmic potential energy estimates for + Hankel determinants. Includes reusable irrationality criteria, contour integration, + Andreief identities, Newton-basis valuation bounds, and energy inequalities for + finite measures. The cited source is the completed Lean proof; independent novelty + of the informal argument is not claimed. +branch: analytic number theory +entry_module: LeanPool.Zeta32 +authors: +- Qian Tang +source: + title: Linear independence of 1, zeta(2), and zeta(3), Lean formalization + url: https://github.com/dtq1997/zeta2-zeta3-linear-independence/tree/669c92bf7a3728e0de9ad80c373bc29fb5ca3d58 + github_repo: dtq1997/zeta2-zeta3-linear-independence + commit: 669c92bf7a3728e0de9ad80c373bc29fb5ca3d58 +license: Apache-2.0 +status: verified +provenance: AI +main_declarations: +- Zeta32.one_zeta_two_zeta_three_linearIndependent +main_results: +- declaration: Zeta32.one_zeta_two_zeta_three_linearIndependent + informal: The three complex numbers 1, riemannZeta 2, and riemannZeta 3 are linearly + independent over the rational numbers. + source_ref: 'Upstream Zeta32/Final.lean and Solution.lean at the pinned commit. + First public proof release: 2026-09-30 03:47:38 UTC, commit ec4fcb5617126b0a861f9e0c16f95425b32d4638. + The upstream README and NOTICE disclose substantial AI assistance. The polynomial + argument is credited upstream to Fauzan; this card cites the Lean artifact, not + a separately verified paper.' +- declaration: Zeta32Acceptance.one_zeta_two_zeta_three_linearIndependent_series + informal: For rational a, b, and c, if a plus b times the real sum of 1/(k+1)^2 + plus c times the real sum of 1/(k+1)^3 is zero, then all three coefficients are + zero; both sums range over natural numbers k. +- declaration: Zeta32.zeta3_sub_rat_mul_zeta2_irrational + informal: For every rational r, the real series for zeta(3) minus r times the real + series for zeta(2) is irrational. +tags: +- zeta-values +- irrationality +- linear-independence +- p-adic-valuations +- potential-theory +msc: +- 11J72 +- 11M06 diff --git a/LeanPool/projects/zeta_3_irrational.yaml b/LeanPool/projects/zeta_3_irrational.yaml new file mode 100644 index 0000000000..c41d03d509 --- /dev/null +++ b/LeanPool/projects/zeta_3_irrational.yaml @@ -0,0 +1,42 @@ +slug: zeta_3_irrational +title: Irrationality of ζ(3) +summary: Formalizes Beukers' proof of Apéry's theorem for ζ(3), including the integral + identities, denominator estimates, positivity, upper bounds, and final irrationality + contradiction. +branch: analytic number theory +entry_module: LeanPool.Zeta3Irrational +authors: +- Junqi Liu +- Jujian Zhang +source: + title: A note on the irrationality of ζ(2) and ζ(3) + authors: + - Frits Beukers + doi: 10.1112/blms/11.3.268 + arxiv: '2503.07625' + github_repo: ahhwuhu/zeta_3_irrational +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- LeanPool.Zeta3Irrational.zeta3_irrational +main_results: +- declaration: LeanPool.Zeta3Irrational.zeta3_irrational + informal: Proves that ζ(3) is irrational. +- declaration: LeanPool.Zeta3Irrational.linear_int + informal: Clears denominators in the Apéry integral to produce a linear form in + 1 and ζ(3). +- declaration: LeanPool.Zeta3Irrational.JJ_pos + informal: Proves positivity of the transformed Beukers-Apéry integral. +- declaration: LeanPool.Zeta3Irrational.JJ_upper + informal: Bounds the Beukers-Apéry integral by a geometric factor times the ζ(3) + series. +- declaration: LeanPool.Zeta3Irrational.fun1_tendsto_zero + informal: Shows that the denominator-cleared Apéry linear forms tend to zero. +tags: +- number-theory +- analysis +- zeta-functions +msc: +- 11M06 +- 11J72 diff --git a/LeanPool/projects/zflean.yaml b/LeanPool/projects/zflean.yaml new file mode 100644 index 0000000000..6bd3fb8b72 --- /dev/null +++ b/LeanPool/projects/zflean.yaml @@ -0,0 +1,35 @@ +slug: zflean +title: ZFLean +summary: Develops a practical ZFC set-theory framework with von Neumann naturals, + integers, rationals, functions, embeddings, isomorphisms, booleans, and sums. +branch: set theory +entry_module: LeanPool.ZFLean +authors: +- Vincent Trélat +source: + title: Set Theory + authors: + - Thomas Jech + url: https://doi.org/10.1007/3-540-44761-X + github_repo: VTrelat/ZFLean +license: Apache-2.0 +status: verified +provenance: human +main_declarations: +- ZFSet.isIso_of_biembedding +- ZFSet.ZFNat.induction +- ZFSet.ZFInt.induction +main_results: +- declaration: ZFSet.isIso_of_biembedding + informal: Proves a Cantor-Bernstein theorem for ZF sets. +- declaration: ZFSet.ZFNat.induction + informal: Provides induction for the von Neumann natural numbers. +- declaration: ZFSet.ZFInt.induction + informal: Provides induction for the quotient construction of ZF integers. +tags: +- set-theory +- zfc +- foundations +msc: +- 03E30 +- 03B35 diff --git a/LeanPool/projects/zhang-yeung-inequality.yaml b/LeanPool/projects/zhang-yeung-inequality.yaml new file mode 100644 index 0000000000..be3f7e0b1f --- /dev/null +++ b/LeanPool/projects/zhang-yeung-inequality.yaml @@ -0,0 +1,31 @@ +slug: zhang-yeung-inequality +title: The Zhang-Yeung non-Shannon information inequality +summary: Formalizes the Zhang-Yeung inequality, the first known non-Shannon-type entropy + inequality, together with its dual and averaged forms derived through the copy lemma. +branch: information theory +entry_module: LeanPool.ZhangYeungInequality +authors: +- Christopher Boone +source: + doi: 10.1109/18.681320 + github_repo: cboone/zhang-yeung-inequality +license: Apache-2.0 +status: verified +provenance: mix +main_declarations: +- ZhangYeung.zhangYeung +main_results: +- declaration: ZhangYeung.zhangYeung + informal: Proves the Zhang-Yeung non-Shannon-type information inequality for four + discrete random variables. +- declaration: ZhangYeung.zhangYeung_dual + informal: Proves the dual form of the Zhang-Yeung inequality. +- declaration: ZhangYeung.zhangYeung_averaged + informal: Proves the averaged symmetric form of the Zhang-Yeung inequality. +tags: +- information-theory +- entropy +- non-shannon-inequality +msc: +- 94A17 +- 94A15