diff --git a/.github/workflows/lifetime-moment-boundary.yml b/.github/workflows/lifetime-moment-boundary.yml new file mode 100644 index 000000000..db56ae55c --- /dev/null +++ b/.github/workflows/lifetime-moment-boundary.yml @@ -0,0 +1,33 @@ +name: Lifetime and radial moment exact controls +'on': + pull_request: + paths: + - 'frontiers/lifetime_moment_boundary_20260930/**' + - 'reviews/radial_moment_corollaries_20260930/**' + - '.github/workflows/lifetime-moment-boundary.yml' + push: + branches: [main] + paths: + - 'frontiers/lifetime_moment_boundary_20260930/**' + - 'reviews/radial_moment_corollaries_20260930/**' + - '.github/workflows/lifetime-moment-boundary.yml' + workflow_dispatch: +permissions: + contents: read +jobs: + exact-moments: + runs-on: ubuntu-latest + timeout-minutes: 5 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - run: python -B -S frontiers/lifetime_moment_boundary_20260930/verify.py + - name: Receipt and frozen-source regression controls + run: | + python -B -S -m unittest discover -s frontiers/lifetime_moment_boundary_20260930 -p 'test_*.py' -v + python -B -O -S -m unittest discover -s frontiers/lifetime_moment_boundary_20260930 -p 'test_*.py' -v + - run: git diff --exit-code diff --git a/frontiers/lifetime_moment_boundary_20260930/CHECK_RECEIPT.json b/frontiers/lifetime_moment_boundary_20260930/CHECK_RECEIPT.json new file mode 100644 index 000000000..2ab7e45dc --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/CHECK_RECEIPT.json @@ -0,0 +1,54 @@ +{ + "good_quartic_cases": 54, + "normalization_exponent_cases": 12, + "outside_hypothesis_axial_negative_controls": 18, + "proof_sha256": "3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8", + "runner_sha256": "2d8b4a2cc61908412574734c0266e5b568c32a653d19eb70edd1c0e46a3c8cab", + "scope": 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+ "coefficient_of_c_t_minus_2": "0", + "leading": "1", + "p": "0" + }, + { + "coefficient_of_c_t_minus_2": "-4/525", + "leading": "22/21", + "p": "1/2" + }, + { + "coefficient_of_c_t_minus_2": "-1/60", + "leading": "11/10", + "p": "1" + }, + { + "coefficient_of_c_t_minus_2": "-4/99", + "leading": "11/9", + "p": "2" + }, + { + "coefficient_of_c_t_minus_2": "-20/3", + "leading": "11", + "p": "10" + }, + { + "coefficient_of_c_t_minus_2": "-2180/21", + "leading": "110", + "p": "109/10" + }, + { + "coefficient_of_c_t_minus_2": "-21998000/2001", + "leading": "11000", + "p": "10999/1000" + } + ], + "quantile_controls": "exact rational brackets for Pareto mixtures", + "scientific_effect": "NONE", + "scope": "PR176 equation17 conditional moment consequence only", + "tests": 11 + }, + "observed_control_log": "test_actual_probability_mixture_has_claimed_moment_remainder_bound (__main__.MomentConsequenceTests) ... ok\ntest_all_order_finite_tail_matches_direct_pareto_mixture (__main__.MomentConsequenceTests) ... ok\ntest_all_order_log_powers_and_normalization (__main__.MomentConsequenceTests) ... ok\ntest_direct_log_tail_integral (__main__.MomentConsequenceTests) ... ok\ntest_hand_checked_special_cases (__main__.MomentConsequenceTests) ... ok\ntest_offered_rational_coefficient_factors (__main__.MomentConsequenceTests) ... ok\ntest_quantile_expansion_brackets_an_actual_conditional_law_exactly (__main__.MomentConsequenceTests) ... ok\ntest_tail_and_density_derivations_agree_for_negative_zero_positive_p (__main__.MomentConsequenceTests) ... ok\ntest_threshold_and_above_diverge (__main__.MomentConsequenceTests) ... ok\ntest_uniform_in_p_remainder_overclaim_has_exact_counterexample (__main__.MomentConsequenceTests) ... ok\ntest_wrong_normalizer_sign_and_missing_p_are_detected (__main__.MomentConsequenceTests) ... ok\n\n----------------------------------------------------------------------\nRan 11 tests in 0.004s\n\nOK\n" + }, + "checks": [ + "C28 source P size+SHA256+blob", + "C28 source ERR size+SHA256+blob", + "C28 source BOREL size+SHA256+blob", + "C28 source REC size+SHA256+blob", + "C28 source MARK_D size+SHA256+blob", + "C28 source MARK_2 size+SHA256+blob", + "C28 source CAP size+SHA256+blob", + "C27 artifact MOMENT_CONSEQUENCE.md size+SHA256", + "C27 artifact moment_consequence_checks.py size+SHA256", + "C27 artifact moment-checks-normal.json size+SHA256", + "C27 artifact moment-checks-normal.log size+SHA256", + "C27 artifact moment-checks-optimized.json size+SHA256", + "C27 artifact moment-checks-optimized.log size+SHA256", + "C27 artifact AUTHOR_COMMENT_5907459466.json size+SHA256", + "C27 artifact AUTHOR_COMMENT_5903080058.json size+SHA256", + "C27 artifact SOURCES.json size+SHA256", + "C27 source PR176 size+SHA256+blob", + "C27 source R size+SHA256+blob", + "C27 source SC size+SHA256+blob", + "C27 source P size+SHA256+blob" + ] +} diff --git a/frontiers/lifetime_moment_boundary_20260930/LOSS_CHECK_RECEIPT.json b/frontiers/lifetime_moment_boundary_20260930/LOSS_CHECK_RECEIPT.json new file mode 100644 index 000000000..cfa0b0070 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/LOSS_CHECK_RECEIPT.json @@ -0,0 +1,8 @@ +{ + "frozen_proof_sha256": "3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8", + "loss_corollary_sha256": "8d202bf2adbf25cb10ba7b114024e7772f2e6839746dc744e1494027a25a178c", + "rational_probability_cases": 48, + "runner_sha256": "20b5f8e4c1330e1a4e0b0ee3256cebf002ca9f6114b0686b30e9d63ba98441bd", + "scope": "Finite normalization, clipped versus signed loss, and deterministic physical scaling only; analytic premises and model-realized probabilities are not tested.", + "status": "PASS" +} diff --git a/frontiers/lifetime_moment_boundary_20260930/LOSS_CHECK_RECEIPT_OPTIMIZED.json b/frontiers/lifetime_moment_boundary_20260930/LOSS_CHECK_RECEIPT_OPTIMIZED.json new file mode 100644 index 000000000..cfa0b0070 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/LOSS_CHECK_RECEIPT_OPTIMIZED.json @@ -0,0 +1,8 @@ +{ + "frozen_proof_sha256": "3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8", + "loss_corollary_sha256": "8d202bf2adbf25cb10ba7b114024e7772f2e6839746dc744e1494027a25a178c", + "rational_probability_cases": 48, + "runner_sha256": "20b5f8e4c1330e1a4e0b0ee3256cebf002ca9f6114b0686b30e9d63ba98441bd", + "scope": "Finite normalization, clipped versus signed loss, and deterministic physical scaling only; analytic premises and model-realized probabilities are not tested.", + "status": "PASS" +} diff --git a/frontiers/lifetime_moment_boundary_20260930/LOSS_COROLLARY.md b/frontiers/lifetime_moment_boundary_20260930/LOSS_COROLLARY.md new file mode 100644 index 000000000..f12bd82d2 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/LOSS_COROLLARY.md @@ -0,0 +1,45 @@ +# Actual weighted-Palm lifetime loss + +30 September 2026. Additive author-side corollary proposed by the root OpenAI agent and checked by reader_first_independent_review. Zero organizational-independence credit; scientific effect NONE. The frozen [PROOF.md](PROOF.md), SHA256 `3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8`, remains unchanged. Its hypotheses, original weight, full normalizer, source identities, and reading rules are retained. + +The success event is exact. Source #175 §1 defines F_r as failure of S to be the actual ordinary elder partner, and identifies finite death with that partner's height on the almost-sure Morse distinct-value locus. Source #170 §§1,8.1 gives the same definition and explicitly writes p_r=Q_r^W{d_f(M)=f(S)}. Both have f(S)=b-kr^3. Thus, Q_r^W-almost surely, + + F_r^c implies finite death and X_r=(b-d_f(M))/(kr^3)=1. + +The essential event E_r is contained in F_r. These are the exact sources MARK_D and MARK_2 in [SOURCES.json](SOURCES.json), not a count-defined replacement event. + +Fix q>0 and the fixed model parameters. Using the notation of PROOF.md, define + + m_q=integral lambda(theta)^q dmu_fail(theta), + c_q=a-m_q=integral [1-lambda(theta)^q] dmu_fail(theta). + +Because a>0 and 00. QED. + +This is a pair-Palm expectation under the existing marked-law premises. It is not a new Kac–Rice intensity, an unrestricted parameter statement, or an acceptance of those premises. No existing weak-limit scope was defective. The separate rational control checks finite probability bookkeeping, the finite-conditioning denominator, and the difference between clipped loss and signed deficit; it is not a Gaussian-model simulation or a proof of the analytic premises. diff --git a/frontiers/lifetime_moment_boundary_20260930/LOSS_IDENTITY_AND_CONTROL_OBSERVATIONS.json b/frontiers/lifetime_moment_boundary_20260930/LOSS_IDENTITY_AND_CONTROL_OBSERVATIONS.json new file mode 100644 index 000000000..6ebb1c1ce --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/LOSS_IDENTITY_AND_CONTROL_OBSERVATIONS.json @@ -0,0 +1,20 @@ +{ + "LOSS_COROLLARY.md": { + "bytes": 3374, + "sha256": "8d202bf2adbf25cb10ba7b114024e7772f2e6839746dc744e1494027a25a178c", + "git_blob": "f12bd82d2a0d0804f4c262d0d080b3efcf269f59" + }, + "check_loss_exact.py": { + "bytes": 2680, + "sha256": "20b5f8e4c1330e1a4e0b0ee3256cebf002ca9f6114b0686b30e9d63ba98441bd", + "git_blob": "630938818e6230e9df39211885eb86ddcb7ea4d6" + }, + "control_observation": { + "frozen_proof_sha256": "3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8", + "loss_corollary_sha256": "8d202bf2adbf25cb10ba7b114024e7772f2e6839746dc744e1494027a25a178c", + "rational_probability_cases": 48, + "runner_sha256": "20b5f8e4c1330e1a4e0b0ee3256cebf002ca9f6114b0686b30e9d63ba98441bd", + "scope": "Finite normalization, clipped versus signed loss, and deterministic physical scaling only; analytic premises and model-realized probabilities are not tested.", + "status": "PASS" + } +} diff --git a/frontiers/lifetime_moment_boundary_20260930/LOSS_NONAUTHOR_REVIEW.md b/frontiers/lifetime_moment_boundary_20260930/LOSS_NONAUTHOR_REVIEW.md new file mode 100644 index 000000000..003f36db8 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/LOSS_NONAUTHOR_REVIEW.md @@ -0,0 +1,15 @@ +# Nonauthor review of the additive lifetime-loss consequence + +Reviewer: OpenAI / Codex subagent `/root/workflow_bottleneck_audit`, 30 September 2026. No drafting or amendment of this consequence was performed by the reviewer. Same-provider/source exposure and shared-account work grant zero organizational-independence credit. + +**VERDICT: ACCEPT (L1)–(L3) of `LOSS_COROLLARY.md`, SHA256 `8d202bf2adbf25cb10ba7b114024e7772f2e6839746dc744e1494027a25a178c`, as conditional consequences of the separately reviewed frozen C28 proof and its original actual-elder event definition. No blocking amendment found.** + +The source check is substantive: MARK_D §1 defines F_r as failure of the actual ordinary elder partner to equal S and identifies finite death with that partner's height on the full-probability Morse/distinct-value locus. MARK_2 uses the same event and expressly equates success with d_f(M)=f(S). Since f(S)=b-kr³, success implies finite death and X_r=1 exactly. No witness-count event is substituted. + +The finite numerator decomposes exactly into Q^W(F_r complement) plus the finite-death failure moment. Substituting Q^W(F_r)=a r³+o(r³) and the frozen proof's failure moment m_q r³+o(r³) gives 1-(a-m_q)r³+o(r³). The essential mass e_r is o(r³), so division by 1-e_r leaves this coefficient unchanged. Positivity and strict upper bound of c_q=a-m_q follow from positive failure mass and limiting fractions strictly between zero and one. + +For the clipped formula, success contributes one and essential deaths also have clipped value one; subtracting its expectation from one leaves the bounded nonnegative loss. For the finite signed deficit the exact identity is Q^W(finite)-E[X_r^q;finite]=(1-e_r)-E[X_r^q;finite]. The candidate correctly distinguishes it from a pointwise nonnegative loss: it is negative on finite X_r>1. The frozen axial-tail bound makes the expectation of this discrepancy negligible at r³ scale. Neither formula silently treats infinity times an indicator as an ordinary finite moment. + +Multiplication by the deterministic positive factor (kr³)^q yields the physical-lifetime expansion at fixed q,k. Both essential-inclusive positive moments remain infinite. These remain weighted pair-Palm expectations, not counts or intensities of distinct replacement bars, and there is no extra failure normalization. + +The additive file leaves frozen `PROOF.md` SHA256 `3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8` unchanged. The separate exact control was inspected and replayed once with `python3 -B -S`; its 48 rational probability-bookkeeping cases passed. The fixture retains separate success, ordinary failure, rare finite long lifetime, and essential atoms, so it checks the conditioning denominator and the distinction between signed and clipped loss. It does not prove any Gaussian-model premise. File identities and the observed receipt are in `LOSS_IDENTITY_AND_CONTROL_OBSERVATIONS.json`. diff --git a/frontiers/lifetime_moment_boundary_20260930/NONAUTHOR_REVIEW.md b/frontiers/lifetime_moment_boundary_20260930/NONAUTHOR_REVIEW.md new file mode 100644 index 000000000..24c571bfb --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/NONAUTHOR_REVIEW.md @@ -0,0 +1,44 @@ +# Nonauthor review of the actual-lifetime moment boundary + +Reviewer: OpenAI / Codex subagent `/root/workflow_bottleneck_audit`. +Date: 30 September 2026. + +**VERDICT: ACCEPT the frozen consumer's axial escape lemma, clipped and finite-death positive-moment consequences, and essential-inclusive infinite-moment obstruction, conditional on its exact stated parent and failure-mark premises. No blocking amendment found.** + +The reviewer did not draft, suggest, or edit the candidate proof. This is a separate mathematical read after the authors froze it. The reviewer has source-custody and queue-triage exposure to this research chain, shares the authors' provider and GitHub account, and earns zero organizational-independence credit. This acceptance does not revalidate the Gaussian parent or the draft marked-law sources and does not change their disposition. + +## Exact reviewed object + +- `payload/proof/PROOF.md`: 13,952 bytes; SHA256 `3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8`; Git blob `d5ee8abee120a8bec2f6c470e517ff9db4645183`. +- `payload/proof/SOURCES.json`: 5,148 bytes; SHA256 `45d36269416ba0033edf184a540c32b3007e91fb2031ab8b77ae89eb1942db2e`; Git blob `46a791e3313df1954a332db2da7d3489d29bc48c`. +- The seven retained source bodies P, ERR, BOREL, REC, MARK_D, MARK_2 and CAP match every declared byte count, SHA256 and Git blob. The identity observations are recorded separately in `IDENTITY_AND_CONTROL_OBSERVATIONS.json`. + +The mathematical read covered the complete candidate; P §§1–5 and the actual-death/Morse interface in §8; REC's source identity set and reading rules; MARK_2 §§8.7–8.8; and MARK_D's actual-death definition and §6 marked-law interface. The two source failure-mark theorems remain explicitly assumed inputs. Their full proofs are not re-reviewed here. + +## Reconstructed checks + +1. **The estimand is the actual elder lifetime.** P §8 and both marked-law sources define death by the supremum of path minima over endpoints strictly older than the pinned birth. Empty endpoint sets give death minus infinity. The candidate retains this definition, the original endpoint weight, and one full normalizer. It does not relabel a saddle count or a local witness as the actual death. Under the tilted law the maximum type is enforced. Essential maxima are failures because they cannot die at the finite pinned saddle. + +2. **The clipped consequence follows from the supplied weak law.** For fixed positive q, the clipped power is a bounded continuous function of the finite normalized death mark, with the isolated essential sentinel assigned value one. The source gives vanishing scaled sentinel mass, so its treatment does not create an extra limiting contribution. A finite death remains available even if its position is outside the chart; the candidate correctly does not replace that height by a position sentinel. The limiting fraction lies strictly between zero and one and the limiting failure mass a is positive, hence 0 < m_q < a. Dividing by the convergent failure mass is valid. + +3. **The repeated-Rolle step is valid with C4 regularity.** The Hermite cubic satisfies both values and both zero derivatives. Thus g minus the cubic has zeros of total multiplicity four on [a,c], and a third-derivative zero exists in its interior. The cubic's third derivative is 12k. The fourth-derivative bound therefore gives g''' at least 12k minus 2rH on the extended interval [a,e], which is at least 6k under rH ≤ 3k. No simultaneous vector Rolle zero is used: this is the scalar axial restriction. + +4. **Strict convexity identifies the path minimum.** Since (g')'' = g''' > 0, g' is strictly convex. Its exact zeros at a and c force it below zero between the zeros and above zero to their right. Hence the axial path has its minimum exactly at c, with value b minus kr³. The Hermite extrapolation remainder at e uses five zeros counting multiplicities after subtracting a multiple of (x-a)²(x-c)². Four differentiations are legitimate. Its factor is 4r⁴/24; the cubic value at e is b plus 4kr³. Therefore the asserted lower endpoint margin b plus (7/2)kr³ is correct. This produces one admissible path to an older endpoint and proves death at least f(S); it does not assert that S is the winning partner. + +5. **The weighted joint moments are actually present in the parent.** P(4.1) gives all finite moments of the global C4 norm uniformly at the stated compact parameter scope. The sentence following P(5.3) explicitly gives all finite moments of W_r/r². Hölder therefore bounds E_Q[(W_r/r²)K^s] for every finite s while retaining correlations. P(5.5), read with ERR and REC, supplies the positive full-normalizer floor. There is no assumption that the field norm and weight are independent, and no substitution of a good-event normalizer. + +6. **The rare large-norm estimate has the correct powers.** The axial result implies that an essential death or a finite fraction above one requires K > const/r. Weighted Markov bounds its tilted mass by every positive power of r. For a finite death on the connected torus, some older endpoint exists, every path minimum is at least the global minimum, and birth equals f(M); therefore the finite lifetime is bounded by 2K. Multiplication by r^(-3q) and the failure scaling r^(-3) produces r^(-3q-3). On K > const/r, using K^q 1_B ≤ const r^p K^(q+p) gives the displayed exponent p-3q-3. Arbitrarily high parent moments allow any prescribed positive remainder power. The q+epsilon version supplies uniform integrability at the scaled failure-measure level. + +7. **The finite-death conditioning is legitimate.** The difference between the finite qth moment and the clipped numerator consists only of the finite tail above one and the clipped essential contribution. Both vanish after r^(-3) scaling. Essential probability is smaller than every power, so Q^W(F_r, finite)/r³ tends to the same positive a. This proves both formulas in (MOM) for each fixed q>0. It neither proves inverse moments nor parameter-uniform convergence of the assumed marked law. + +8. **Full affine conditional support follows for the actual model.** P §2 has strictly positive variance for every real Fourier mode, including the constant mode, and summable coefficients in every Cq norm. Finite coefficient neighborhoods and an independent small C2 tail give positive probability near every trigonometric polynomial. Their C2 density gives full unconditioned C2 support. For fixed distinct pins, the value/gradient covariance is positive definite. The regression map A is a continuous right inverse of the pin map L, so T = Id minus AL is a continuous projection onto ker L. Its Gaussian residual is independent of the observations, and the pushforward of full support has support closure(T(B)) = ker L. Thus the canonical pinned law has full relative support on the precise affine pin space for the prescribed values, without additional conditioning or a changed normalizer. + +9. **The essential maximum is a robust, admissible typed configuration.** Disjoint pin balls can be chosen at every fixed r. The M quadratic has a strict negative Hessian. The S quadratic has one positive and d-1 negative directions while its entire support lies below birth by a strict margin. Convex cutoff interpolation with the lower constant stays strictly below b away from M. Exact pinning plus negative Hessian controls a neighborhood of M; the compact complement has a strict value gap. These conditions persist in a relative C2 neighborhood. Both determinant factors stay bounded away from zero there, so the original tilted law gives the neighborhood positive probability. Intersecting the source's full-measure Morse/distinct-value locus preserves that probability. There is no older endpoint, hence this is the actual essential class. + +10. **Positive essential mass and a superpolynomial upper bound are compatible.** The support neighborhood is constructed separately for each fixed r and no uniform probability lower bound is asserted. Its fourth derivatives may become large as the pins coalesce, exactly as the axial lemma requires. On its positive-probability essential subset the extended lifetime is infinity, so every positive essential-inclusive moment is infinite, including after conditioning on failure. This does not conflict with convergence of all finite-death moments or clipped moments. + +## Verification and boundary + +I read the finite control script and replayed it once with `python3 -B -S`. It passed all seven source identities, 54 admissible quartic controls, 18 outside-hypothesis axial negative controls and 12 normalization-exponent controls. These are arithmetic and custody observations, not the justification for the infinite-dimensional support argument, weak convergence, or uniform integrability; those were checked analytically above. + +The candidate's fixed-radius axial lemma and conditional support argument are new consumer arguments. The marked failure limit, its positive mass, the Gaussian model and the reconciled parent interfaces retain their existing assumptions. No essential-class intensity or uniquely counted replacement-bar intensity is inferred. No source file, register, graph, readiness state or external review was changed by this read. diff --git a/frontiers/lifetime_moment_boundary_20260930/PROOF.md b/frontiers/lifetime_moment_boundary_20260930/PROOF.md new file mode 100644 index 000000000..d5ee8abee --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/PROOF.md @@ -0,0 +1,179 @@ +# Actual lifetime moments: axial escape, finite-death moments, and the essential class + +30 September 2026. Additive local analytic proposal, not a parent source edit or acceptance decision. R17 claim `19650bdb-2c21-42b9-b4a9-d67cabb12d4c` / Work Event 319; exact public scope pickup on Math- #175 comment 5907574791. Root owns integration; no branch write is authorized before review. +Authors: the reader_first_independent_review OpenAI subagent and the root OpenAI agent. Root supplied the clipped-moment formulation and independently reconstructed the essential-class support argument; the subagent supplied the axial escape lemma and its finite-death uniform-integrability consequence. This is author-side technical work with zero organizational-independence credit. It does not repeat the completed exact-head reviews of the existing proofs. + +## Source boundary and current ownership + +At 08:44:53 UTC, fresh GitHub reads returned: + +- Math- #175: open draft, head `9c6a73489c5ed2465a3cf65dcc950eeeae074b05`, base `8d48d02a6a580666a879d719a9e61f460baa5f40`. Its marks/lifetime slice already has same-provider review 5363612238. Its consumed `frontiers/concave_fibre_elder_20260930/PROOF.md` is SHA256 `5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e`. +- Math- #170: open draft, head `79f18f0cb318fceccd5bdf4e6dad29815165bc7d`, base `358f2562efbccc59f9549e5a79650e2e936395c3`. Its remaining rare-measure/marks/lifetime slice already has same-provider review 5363625339. Its consumed `frontiers/local_elder_geometry_20260930/PROOF.md` is SHA256 `f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b`. +- Parent P: `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, commit `ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f`, blob `dfed3b8d318a3ab1950957f393307733a4bef3f2`, SHA256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. + +Cached bytes were rehashed before this derivation. Exact consumed source bytes are retained under `sources/` beside this note; immutable URLs, commits, Git blob IDs, SHA256 values, byte counts, consumption roles, and every retained reading rule are in [SOURCES.json](SOURCES.json). Relevant source slices are #175 §§1,5–6; #170 §§8.1,8.7–8.8; P §§2,4–5,7–8. P §12 concerns integrated finite-bar intensity moments, which are a different estimand. The full parent identity set includes P, CAP, the congruence erratum ERR, Borel replacement BOREL, and reconciliation REC. Their reading rules remain in force; no new formal alignment or scientific acceptance is claimed here. + +The bounded question is what the existing actual failure-mark weak law implies about moments of the actual lifetime fraction. Its intentional weak scope is not a defect. The conclusion below is stronger than a generic statement that weak convergence needs an extra uniform-integrability assumption: the original pin constraints and already available all-order global moments supply that assumption for the finite-death submeasure. + +## Exact estimand + +Fix the parameters and the original pinned Gaussian law of the relevant source, in dimension 2 for #170 or a fixed dimension at least 3 for #175. Retain the reconciliation embedding restriction r0. + +Keep the original weight and full normalizer + + W_r=F_d(H_M) F_(d-1)(H_S), Z_r=E_Q W_r, + Q_r^W=(W_r/Z_r)Q_r. + +Let F_r be failure of S to be the ordinary elder partner, and let E_r be the essential event d_f(M)=-infinity. Under Q_r^W, M is a maximum; E_r is contained in F_r. Define the extended lifetime fraction + + X_r=(b-d_f(M))/(k r^3) in [0,infinity]. + +For finite deaths it is nonnegative because every path in the death definition includes M, so d_f(M)<=b. Define its clipped version C_r=min(X_r,1), setting C_r=1 on E_r. A location outside the local chart is not a reason to replace an available finite death height by a sentinel. + +Assume the source's failure-mark conclusion: r^-3 Q_r^W(F_r)->a>0 and the finite marked measures converge weakly, with isolated sentinel mass o(r^3), to mu_fail and lifetime fraction lambda(theta)=-h_*(theta)/k in (0,1). Here a=a_fail in #175 and a=alpha_1+alpha_2 in #170. Put + + m_q=integral lambda(theta)^q dmu_fail(theta), q>0. + +Thus 00, + + r^-3 E_QW[C_r^q;F_r] -> m_q, + E_QW[C_r^q | F_r] -> m_q/a. (B1) + +Proof: on finite death marks use the bounded continuous function + + t -> min(max(-t/k,0),1)^q + +of D_r=(d_f(M)-b)/r^3, and assign value 1 at the isolated essential sentinel. Its limit is lambda^q because the limiting death lies in (-k,0). The exceptional position sentinel is irrelevant to this test. Divide by r^-3 Q_r^W(F_r)->a for the second formula. This is only a corollary of the stated weak law. + +## 2. Deterministic axial escape lemma + +Let g(x)=f(x,0) on [a,e]=[-r/2,3r/2] in the embedded chart; put c=r/2. Let H be the supremum of the absolute fourth directional derivative along that segment. If + + r H <= 3 k, (A1) + +there is a path from M to a point with field value greater than b whose minimum is exactly f(S). In particular + + d_f(M)>=f(S), E_r cannot occur, and 0<=X_r<=1. (A2) + +This assertion does not say S is the actual partner. Another path can give an earlier death at a higher level. + +Proof: the exact cubic Hermite interpolant at a,c is + + p(x)=b+k r^3 [2(x/r)^3-(3/2)(x/r)-1/2]. + +It has p(a)=b, p(c)=b-kr^3, p'(a)=p'(c)=0 and p'''=12k. Since g-p has double zeros at a and c, repeated Rolle's theorem gives xi in (a,c) with g'''(xi)=12k. For every x in [a,e], + + g'''(x)>=12k-2rH>=6k>0. + +Therefore g' is strictly convex, has its exact zeros at a,c, is negative on (a,c), and is positive on (c,e]. Consequently g has its minimum on [a,e] at c. + +For clarity the endpoint exceeds b with a strict quantitative margin. The same repeated-Rolle argument for the fourth-order Hermite remainder gives, for some zeta in (a,e), + + g(e)-p(e)=g''''(zeta)(e-a)^2(e-c)^2/24. + +Here p(e)=b+4kr^3 and (e-a)^2(e-c)^2=4r^4, so + + g(e)>=b+4kr^3-Hr^4/6>=b+(7/2)kr^3>b. + +The axial path x from a to e is therefore one admissible older-endpoint path, with minimum g(c)=f(S), proving (A2). Generalized Rolle's theorem here is elementary: subtract the displayed multiple of (x-a)^2(x-c)^2 so that the difference also vanishes at e, and differentiate four times. No probabilistic estimate or hard-fibre geometry enters this lemma. + +## 3. The current parent bounds supply finite-death uniform integrability + +Take K=1+||f||_(C4) in a fixed global coordinate norm and a dimension constant C_d with H<=C_d K for every axial orientation. Define + + B_r={K>3k/(C_d r)}. + +The lemma shows that E_r and {finite death, X_r>1} are contained in B_r. The parent supplies, uniformly for small r at fixed parameters, + + Z_r/r^2 >= z_*>0, + sup_r E_Q[(W_r/r^2) K^s] < infinity for every finite s>=0. (P1) + +The second assertion follows from P(4.1), the all-order W_r/r^2 bound following P(5.3), and Holder's inequality. It retains the weight-field correlations. Its constants can also be taken common on the parent's compact b,k,R sets with k bounded away from zero. + +For every N>0, direct weighted Markov gives + + Q_r^W(B_r) + = E_Q[(W_r/r^2)1_B]/(Z_r/r^2) + <= C_N r^N. (U1) + +For a finite death the torus is connected and an older endpoint exists. Its path minimum is at least min f, so + + 0<=b-d_f(M)<=b-min f<=2||f||_infinity<=2K. + +For q>0 and any N>0, using a Markov power p>N+3q+3 in (P1), + + r^-3 E_QW[X_r^q; finite death, X_r>1] + <= C r^(-3q-3) E_Q[(W_r/r^2) K^q 1_B] + <= C_p r^(p-3q-3) + = O(r^N). (U2) + +Changing p slightly if necessary gives the stated exponent for any prescribed N. The same estimate with q+epsilon supplies the conventional sufficient criterion + + sup_(small r) r^-3 E_QW[X_r^(q+epsilon);F_r,finite death]0. (U3) + +This is an upper bound only. It does not imply the essential event has zero probability at any fixed r. + +Combining (B1), (U2) and (U3) gives the finite-death moment corollary: + + r^-3 E_QW[X_r^q;F_r,d_f(M)>-infinity] -> m_q, + E_QW[X_r^q | F_r,d_f(M)>-infinity] -> m_q/a. (MOM) + +Indeed the difference between the first numerator and the clipped numerator consists of the finite part above 1, bounded by (U2), and the clipped contribution Q_r^W(E_r), bounded by (U3). Also Q_r^W(F_r,finite)/r^3->a. These statements hold for every fixed positive q. They do not assert inverse moments (q<0), convergence uniform in all unbounded parameters, or an intensity for uniquely counted replacement bars. + +## 4. Essential-inclusive moments are nevertheless infinite in the actual model + +For every fixed sufficiently small r>0 and fixed parameter choice, the original tilted law has + + Q_r^W(E_r)>0. (S1) + +Therefore, with the literal extended-value convention, + + E_QW[X_r^q;F_r]=infinity, + E_QW[X_r^q | F_r]=infinity, q>0. (S2) + +This is a model-realized obstruction, not an abstract rare-tail example. It is compatible with the faster-than-every-power upper bound (U3). + +Proof of conditional support: P §2 gives strictly positive variance for every real Fourier mode and convergence of the Fourier expansion in C2 with its expected tail norm tending to zero. Every smooth trigonometric polynomial is in the C2 support: finitely many coefficients can fall in any prescribed neighborhood with positive probability, independently of a tail whose C2 norm is small with positive probability. Smooth trigonometric polynomials are dense in C2 on the torus (smooth approximation followed by Fourier approximation). Thus the unconditioned Gaussian field has full support in the Banach space B=C2(T^d). + +Let L:B->R^(2(d+1)) be the original value/gradient pin map and Sigma its positive-definite covariance. Define the continuous regression right inverse + + A y = Cov(f,Lf) Sigma^-1 y, so L A=Id, + T=Id-A L. + +Then T is a continuous projection of B onto ker L, and Tf is a centered Gaussian field independent of Lf. For the prescribed pin vector y, Q_r is the law of Ay+Tf. Since f has full support in B, Tf has support the closure of T(B)=ker L. Therefore Q_r has full relative C2 support on {h:Lh=y}. This uses the original pin law, not a second conditioning or a restricted normalizer. + +Proof of a stable essential typed configuration: write Delta=kr^3. Choose disjoint small embedded balls about M,S, a constant c00 so q_S=w0>0 on U. Conditional full support gives Q_r(U)>0. Thus + + Q_r^W(E_r)>=E_Q[W_r;U]/Z_r>=w0 Q_r(U)/Z_r>0. + +The prototype itself need not be globally Morse outside the pin balls: the source's almost-sure Morse distinct-value event has Q_r-measure one and intersects U with the same positive measure. There is no older endpoint on U, so d_f(M)=-infinity and E_r is indeed a failure event. No lower bound uniform in r or quantitative probability asymptotic is asserted. + +## Assessment and retained boundaries + +There is no defect in the existing weak mark claims. There is a useful author-side positive extension: the finite-death conditional law has convergence of every fixed positive moment, because an axial path and the parent's existing all-order moments already prove the needed uniform integrability. Clipped positive moments are an immediate bounded-test corollary. With the actual essential value retained as infinity, every positive moment is infinite at each r despite the vanishing essential mass. The finite-death convention must therefore be stated explicitly. + +The source failure measure and cubic mark identification remain required hypotheses. This proposal supplies neither their acceptance nor a provider-distinct review. It changes no scientific status, evidence JSON, formal-required-check contract, or historical source. No repository source was edited, no external message or review was posted, and no Lean execution or full test suite was run. Root retains integration and external coordination ownership. diff --git a/frontiers/lifetime_moment_boundary_20260930/README.md b/frontiers/lifetime_moment_boundary_20260930/README.md new file mode 100644 index 000000000..508e39633 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/README.md @@ -0,0 +1,31 @@ +# Which lifetime moments exist? + +The answer depends on whether the maximum eventually dies. For finite deaths, the existing pinned Gaussian estimates give convergence of **every fixed positive lifetime moment**. Including the essential maximum makes every positive moment infinite, even though its probability becomes smaller than every power of the pin separation. + +Let `X_r=(b-d_f(M))/(k r³)` be the actual elder lifetime divided by the prescribed pin gap, under the original weighted law `Q_r^W`. If `F_r` is pairing failure, the existing marked law has mass `a r³+o(r³)` and limiting fraction `λ∈(0,1)`. Write `m_q=∫λ^q dμ_fail`. + +| Quantity, for fixed `q>0` | Result | +| --- | --- | +| Failure moment, restricted to finite deaths | `r^-3 E[X_r^q; F_r, finite] → m_q` | +| Moment conditional on failure and finite death | `E[X_r^q | F_r, finite] → m_q/a` | +| Finite lifetime tail above the nominal gap | `r^-3 E[X_r^q; finite, X_r>1] = O(r^N)` for each fixed `N>0` | +| Essential maximum | Positive probability at each fixed small `r`; `O(r^N)` for every fixed `N>0` | +| Moment including essential lifetime `∞` | Infinite at every fixed small `r` | + +The key geometric step is short. A cubic matching the two values and two zero axial derivatives reaches a value above the maximum after passing the saddle. A fourth-derivative bound preserves that path. Thus a lifetime above the nominal gap requires an unusually large global derivative norm; the parent's all-order weighted bounds control its contribution. + +[Read the proof](PROOF.md) · [Nonauthor analytic review](NONAUTHOR_REVIEW.md) · [Exact source identities and reading rules](SOURCES.json) · [Companion radial moment results](../../reviews/radial_moment_corollaries_20260930/README.md) + +The [weighted moment and loss corollary](LOSS_COROLLARY.md), with its [separate review](LOSS_NONAUTHOR_REVIEW.md), gives a direct consequence for the full pinned law. Put `c_q=a−m_q>0`. Then the finite-death moment and the moment conditional on finite death both have expansion `1−c_q r³+o(r³)`. The bounded nonnegative loss `E[1−min(X_r,1)^q]` has expansion `c_q r³+o(r³)`. These use actual pairing success, on which `X_r=1` exactly; their finite-death convention is essential. + +These are conditional mathematical consequences of the frozen parent estimates and the marked-law premises from PRs [#170](https://github.com/d6g8k5htny-coder/Math-/pull/170) and [#175](https://github.com/d6g8k5htny-coder/Math-/pull/175). Their review and acceptance obligations remain. The analysis uses actual elder pairing, the full weight and normalizer, and fixed parameters. It proves neither inverse moments nor an intensity for uniquely counted replacement bars. + +## Reproduce the finite checks + +From the repository root, using Python 3.11: + +```sh +python3 -B -S frontiers/lifetime_moment_boundary_20260930/verify.py +``` + +This checks exact rational examples, negative controls, both Python optimization modes, and frozen source bytes. It does not replace the analytic proof, the separate nonauthor review, or the repository's required hosted formal gate. The proof's author-stage operational notes are preserved with its reviewed bytes; publication state belongs to the PR and delivery handoff. Scientific effect: NONE. diff --git a/frontiers/lifetime_moment_boundary_20260930/SOURCES.json b/frontiers/lifetime_moment_boundary_20260930/SOURCES.json new file mode 100644 index 000000000..46a791e33 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/SOURCES.json @@ -0,0 +1,85 @@ +{ + "repository": "d6g8k5htny-coder/Math-", + "review_independence_credit": 0, + "scientific_effect": "NONE", + "scope": "Additive author-side lifetime moment boundary: axial escape, clipped and finite-death moment corollaries, positive-probability essential obstruction. 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this note does not re-prove it.", + "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", + "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/9c6a73489c5ed2465a3cf65dcc950eeeae074b05/frontiers/concave_fibre_elder_20260930/PROOF.md" + }, + { + "bytes": 39722, + "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "id": "MARK_2", + "local_path": "sources/MARK_2.md", + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "role": "Required planar stated failure-mark limit #170 S2/S3 and \u00a78.8, consumed as a premise; this note does not re-prove it.", + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/79f18f0cb318fceccd5bdf4e6dad29815165bc7d/frontiers/local_elder_geometry_20260930/PROOF.md" + }, + { + "bytes": 15160, + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "id": "CAP", + "local_path": "sources/CAP.md", + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "role": "Retained parent-selection dependency: CAP \u00a7\u00a71\u20135, consumed through the reconciled P failure O(r\u00b3) premise; the new axial escape lemma does not use it.", + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md" + } + ] +} diff --git a/frontiers/lifetime_moment_boundary_20260930/check_exact.py b/frontiers/lifetime_moment_boundary_20260930/check_exact.py new file mode 100644 index 000000000..a15e2b4df --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/check_exact.py @@ -0,0 +1,97 @@ +#!/usr/bin/env python3 +"""Finite rational controls and source-identity checks, not analytic proof acceptance.""" +from fractions import Fraction as F +from pathlib import Path +from itertools import product +import hashlib +import json + +HERE = Path(__file__).resolve().parent + +def require(ok, message): + if not ok: + raise RuntimeError(message) + +def add(a, b): + return [(a[i] if i < len(a) else F(0)) + (b[i] if i < len(b) else F(0)) for i in range(max(len(a), len(b)))] + +def mul(a, b): + result = [F(0)] * (len(a) + len(b) - 1) + for i, x in enumerate(a): + for j, y in enumerate(b): + result[i + j] += x * y + return result + +def der(p): + return [i * p[i] for i in range(1, len(p))] + +def at(p, x): + return sum(c * x ** i for i, c in enumerate(p)) + +def cubic(b, k, r): + return [b-k*r**3/2, -3*k*r**2/2, F(0), 2*k] + +source_inventory = json.loads((HERE / 'SOURCES.json').read_text()) +source_checks = [] +proof_text = (HERE / 'PROOF.md').read_text() +for source in source_inventory['sources']: + raw = (HERE / source['local_path']).read_bytes() + sha256 = hashlib.sha256(raw).hexdigest() + blob = hashlib.sha1(f'blob {len(raw)}\0'.encode() + raw).hexdigest() + require(len(raw) == source['bytes'], f"size {source['id']}") + require(sha256 == source['sha256'], f"SHA256 {source['id']}") + require(blob == source['git_blob'], f"Git blob {source['id']}") + if source['id'] in {'P', 'MARK_2', 'MARK_D'}: + require(sha256 in proof_text, f"stated source hash {source['id']}") + source_checks.append({'id': source['id'], 'bytes': len(raw), 'sha256': sha256, 'git_blob': blob}) + +quartic_cases = 0 +negative_controls = 0 +for r, k, b in product([F(1,8), F(1,100)], [F(1,3), F(1), F(7,2)], [F(-4), F(0), F(5)]): + a, c, e = -r/2, r/2, 3*r/2 + p = cubic(b, k, r) + require(at(p,a) == b and at(p,c) == b-k*r**3, 'cubic values') + require(at(der(p),a) == 0 and at(der(p),c) == 0, 'cubic gradient pins') + require(der(der(der(p))) == [12*k], 'third derivative normalization') + require(at(p,e) == b+4*k*r**3, 'cubic escape value') + require((e-a)**2*(e-c)**2 == 4*r**4, 'Hermite remainder factor') + bubble = mul([-a, F(1)], [-c, F(1)]) + bubble = mul(bubble,bubble) + for t in [F(-1), F(0), F(1)]: + coeff = t*k/(8*r) + g = add(p, [coeff*v for v in bubble]) + H = abs(der(der(der(der(g))))[0]) + require(r*H <= 3*k, 'good fourth-derivative condition') + require(at(g,a) == b and at(g,c) == b-k*r**3, 'quartic exact values') + require(at(der(g),a) == 0 and at(der(g),c) == 0, 'quartic exact gradients') + require(der(g) == mul([-r*r/4,F(0),F(1)],[6*k,4*coeff]), 'exact derivative factorization') + require(6*k - 6*abs(coeff)*r >= 21*k/4, 'positive derivative factor throughout axial segment') + require(at(g,e) >= b+F(7,2)*k*r**3, 'escape endpoint bound') + require(12*k-2*r*H >= 6*k, 'strict convexity margin') + quartic_cases += 1 + # Dropping the derivative condition is not harmless, even with the exact pins + # and the correct axial max/min types. This is only an axial negative fixture. + coeff = -2*k/r + g = add(p, [coeff*v for v in bubble]) + require(at(g,e) == b-4*k*r**3, 'negative fixture endpoint') + require(r*abs(der(der(der(der(g))))[0]) == 48*k, 'negative fixture outside hypothesis') + require(at(der(der(g)),a) < 0 < at(der(der(g)),c), 'negative fixture axial types retained') + negative_controls += 1 + +power_cases = 0 +for q, desired in product([F(1,2),F(1),F(2),F(29,4)], [F(1),F(4),F(10)]): + markov_power = 3*q + 3 + desired + 1 + require(markov_power - 3*q - 3 == desired + 1, 'weighted tail exponent') + power_cases += 1 + +print(json.dumps({ + 'status': 'PASS', + 'scope': 'Exact finite algebra controls, hypothesis-sensitive negative fixtures, normalization exponents, and source identities only; no Gaussian simulation, infinite-dimensional proof certification, or scientific acceptance.', + 'source_identity_checks': source_checks, + 'good_quartic_cases': quartic_cases, + 'outside_hypothesis_axial_negative_controls': negative_controls, + 'normalization_exponent_cases': power_cases, + 'proof_sha256': hashlib.sha256((HERE/'PROOF.md').read_bytes()).hexdigest(), + 'sources_sha256': hashlib.sha256((HERE/'SOURCES.json').read_bytes()).hexdigest(), + 'runner_sha256': hashlib.sha256(Path(__file__).read_bytes()).hexdigest(), +}, indent=2, sort_keys=True)) diff --git a/frontiers/lifetime_moment_boundary_20260930/check_loss_exact.py b/frontiers/lifetime_moment_boundary_20260930/check_loss_exact.py new file mode 100644 index 000000000..630938818 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/check_loss_exact.py @@ -0,0 +1,52 @@ +#!/usr/bin/env python3 +"""Exact finite probability bookkeeping; not a Gaussian-model simulation.""" +from fractions import Fraction as F +from itertools import product +from pathlib import Path +import hashlib +import json + +HERE = Path(__file__).resolve().parent +FROZEN = '3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8' + +def require(ok, message): + if not ok: + raise RuntimeError(message) + +require(hashlib.sha256((HERE/'PROOF.md').read_bytes()).hexdigest() == FROZEN, 'frozen proof changed') +cases = 0 +for t,a,lam,q,k in product([F(1,100),F(1,1000)],[F(1,3),F(3,4)],[F(1,4),F(2,3)],[1,2,3],[F(1,3),F(2)]): + # Four abstract atoms: success X=1, ordinary failure X=lam, + # rare finite long lifetime X=2, essential X=infinity. + # t represents r^3 for the algebra. These probabilities are finite + # normalization fixtures, not realizations or tail rates of the model. + failure = a*t + essential, long = t**3, t**4 + ordinary = failure-essential-long + success = 1-failure + require(min(ordinary,essential,long,success) > 0, 'positive atom probabilities') + require(ordinary+essential+long+success == 1, 'full probability normalization') + finite_moment = success + ordinary*lam**q + long*2**q + clipped_moment = success + ordinary*lam**q + long + essential + genuine_loss = ordinary*(1-lam**q) + signed_deficit = ordinary*(1-lam**q) + long*(1-2**q) + coefficient = a*(1-lam**q) + remainder = -essential*lam**q + long*(2**q-lam**q) + require(finite_moment == 1-coefficient*t+remainder, 'success/failure moment decomposition') + require(genuine_loss == 1-clipped_moment, 'bounded actual loss') + require(signed_deficit == 1-essential-finite_moment, 'finite signed deficit') + require(genuine_loss-signed_deficit == long*(2**q-1), 'rare long-tail sign retained') + conditional = finite_moment/(1-essential) + require(conditional-(1-coefficient*t) == (remainder+essential*(1-coefficient*t))/(1-essential), 'finite-conditioning denominator') + physical_moment = success*(k*t)**q + ordinary*(k*t*lam)**q + long*(2*k*t)**q + require(physical_moment == (k*t)**q*finite_moment, 'physical lifetime power scaling') + cases += 1 + +print(json.dumps({ + 'status':'PASS', + 'rational_probability_cases':cases, + 'scope':'Finite normalization, clipped versus signed loss, and deterministic physical scaling only; analytic premises and model-realized probabilities are not tested.', + 'frozen_proof_sha256':FROZEN, + 'loss_corollary_sha256':hashlib.sha256((HERE/'LOSS_COROLLARY.md').read_bytes()).hexdigest(), + 'runner_sha256':hashlib.sha256(Path(__file__).read_bytes()).hexdigest(), +},indent=2,sort_keys=True)) diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/BOREL.md b/frontiers/lifetime_moment_boundary_20260930/sources/BOREL.md new file mode 100644 index 000000000..fe9b9ce49 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/BOREL.md @@ -0,0 +1,147 @@ +# D1 Section 9 repair — Borel elder mark in weighted pair Kac–Rice + +**Object:** D1-SECTION9-BOREL-REPAIR-20260925-v1.1 +**Author lane:** OpenAI / ChatGPT +**Parent:** UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1, Drive file 1foDgiDi4XIKOfbrZEE8dWKV_BkZU8LIb, 40261 B, SHA256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7. +**Disposition:** AUTHOR-SIDE AMENDMENT to Section 9 only; exact-source nonauthor re-review required. +**Scientific effect:** NONE. The parent file is unchanged. Theorem A/C, the cap theorem, residual independence, normalizer, and coefficient (15.2) are not accepted by this amendment. + +## 1. Review finding consumed — and one citation correction + +The nonauthor review on main PR112 correctly identified the load-bearing write-up gap: Section 9 names a monotone-class passage from continuous cylinder weights to the global Borel elder mark but does not explicitly construct the regular conditional law on the whole C^2 field space or state the closure hypotheses of the determining class. + +The review's theorem-number statement is not correct for arXiv:2304.07424v3. Primary-source inspection of the exact v3 PDF/HTML gives: + +- Theorem 2.2: Gaussian unweighted Kac–Rice under C^1 paths and positive-definite point covariance; compact domains have finite expected level-set measure. +- Theorem 6.1: Crofton's formula for rectifiable sets. +- Theorem 7.1: Expected integral on the level set, i.e. the weighted formula for a nonnegative weight g(t,Z(.)), under lower-semicontinuity in the location and mark and continuity of the conditional mark law. Its displayed formula is (7.2). + +Therefore the parent was right to point to Theorem 7.1 for the continuous-cylinder base case. It was also right not to apply Theorem 7.1 directly to the elder indicator: the elder equality/path mark is Borel but need not be lower semicontinuous. The repair below makes the author's separate finite-measure/monotone-class extension explicit. + +Primary source inspected: Armentano–Azaïs–León, *On a general Kac–Rice formula for the measure of a level set*, arXiv:2304.07424v3 (5 Dec 2023), especially Theorems 2.2 and 7.1. No priority claim. + +## 2. Exact replacement for parent Section 9 + +### 9. A source-bound weighted Kac–Rice interface for the global elder mark + +Fix a compact ordered-pair domain D contained in the off-diagonal set of X x X, with positive distance from the diagonal. Write + + G(x,y) = (grad f(x), grad f(y)) in R^(2d). + +By the positive Fourier spectrum and distinct-site distribution argument of parent Section 2, Cov G(x,y) is positive definite at every pair, uniformly on D. The sample field is smooth. The derivative of G with respect to (x,y) is block diagonal, so its normal Jacobian is + + Delta(x,y) = |det H_x det H_y|. + +Armentano–Azaïs–León v3, Theorem 2.2, therefore gives the unweighted zero-count formula for G=0 on D, and finiteness on compact D. We use their Theorem 7.1 only for bounded nonnegative continuous cylinder weights, for which its lower-semicontinuity hypotheses hold automatically. The Borel elder mark is reached by the independent finite-measure argument below, not by claiming it is lower semicontinuous. + +#### 9.1 A regular conditional law on the whole C^2 field + +Let E = C^2(X), equipped with its usual separable Banach topology on the compact torus. For t=(x,y) in D and z in R^(2d), finite-dimensional Gaussian regression gives a smooth conditional mean m_(t,z) and a centered residual Gaussian field R_t such that, as a random element of E, + + law(f | G(t)=z) = law(m_(t,z) + R_t). + +The residual can be constructed from the full Gaussian field by subtracting + + Cov(f,G(t)) Cov(G(t))^(-1) G(t). + +Uniform nonsingularity on D and smooth covariance imply that this subtraction is continuous in (t,z) in every finite C^q mean norm needed here. In particular, + + K(t,z,A) = P{m_(t,z) + R_t in A}, A in Borel(E), + +is a regular conditional probability kernel on E, and for every bounded continuous cylinder functional F, the map + + (t,z) -> integral F(phi) K(t,z,dphi) + +is continuous. This is the whole-field version required below; it is not merely a family of conditional laws for a finite list of cylinder coordinates. + +#### 9.2 Two finite measures on D x E + +On the almost-sure zero set of G, define + + mu(A) = E sum_{t in D : G(t)=0} 1_A(t,f), + +for A in Borel(D x E). Theorem 2.2 makes mu finite. + +Define a second finite Borel measure + + nu(A) = integral_D p_(G(t))(0) + E[ Delta(t) 1_A(t,f) | G(t)=0 ] dt. + +The conditional expectation is defined by the kernel K(t,0,.). Its total mass equals the finite unweighted Kac–Rice integral, so nu is finite. + +For every bounded nonnegative continuous cylinder function h(t,phi), Theorem 7.1 (or equivalently the ordinary Gaussian regression proof of its continuous case) gives + + integral h dmu = integral h dnu. (9.1) + +#### 9.3 Cylinder functions generate the full Borel sigma-algebra + +Choose a countable dense set Q in X. The maps + + phi -> partial^alpha phi(q), q in Q, |alpha| <= 2, + +together with a countable coordinate basis on D, separate points of D x E. Because the C^2 norm on the compact torus is recovered from suprema over the dense set for derivatives through order two, these evaluations generate Borel(D x E). + +Let H be the bounded real Borel functions h for which (9.1) holds. Since mu and nu are finite, H + +1. is a vector space, +2. contains the constants, +3. is closed under bounded pointwise convergence by dominated convergence. + +It contains the algebra of bounded continuous cylinder functions. The functional monotone-class theorem therefore implies that H contains every bounded Borel function on D x E. + +This is the missing closure. It does not extend Theorem 7.1's lower-semicontinuous hypothesis by assertion; it first obtains equality of two finite measures on a generating algebra and then identifies those measures on their full Borel sigma-algebra. + +#### 9.4 Apply the extension to the elder/type mark + +Parent Section 8 represents the maximin connection event using countably many strict rational polygonal-path tests in finitely many torus charts. Hence + + (t,phi) -> 1{ d_phi(x) = phi(y) } + +is Borel on the Morse distinct-critical-value locus, with the essential-maximum convention preserved. Hessian index indicators are Borel functions of the 2-jet. Their product with the elder mark is therefore a bounded Borel weight on D x E, so (9.1) applies. + +No local cap event is substituted for the global mark. The cap theorem is used later only to estimate the probability that this global elder mark fails. + +Disintegrate the two endpoint heights using their nondegenerate joint Gaussian density conditional on G(t)=0. At prescribed heights + + f(M)=b, f(S)=b-k r^3, + +the Jacobian Delta = |det H_M det H_S| and the type indicators give exactly the parent endpoint weight W_r. Thus candidate pair intensity contains the full-pin density times E_Q W_r; the selected intensity contains the same factor times the Q^W global elder probability p_r. There is no second determinant factor. + +Finally exhaust D toward the diagonal. The parent Section 10 bound is locally r dr, hence integrable at zero; nonnegative monotone convergence gives the near-diagonal formula. This last step imports the parent Sections 3/5/10 estimates and does not re-prove or accept them here. + +## 3. What this repair does and does not close + +Repaired if a nonauthor reviewer accepts this successor: +- exact theorem attribution for the continuous-cylinder base; +- regular conditional law on C^2; +- finite-measure formulation; +- functional monotone-class hypotheses; +- Borel product-space status of the elder/type mark. + +Still separate: +- contact covariance; +- residual-field independence in parent Section 4; +- typed boundary integration; +- full normalizer; +- Theorem A; +- off-diagonal/unbounded-mark domination; +- coefficient (15.2); +- numerical constants; +- RN/24-jet certificates; +- every authoritative scientific-status gate. + +## 4. Review request + +Review this exact file, not the parent prose by memory. Return: +- C1: primary-source theorem attribution; +- C2: C^2-valued regular conditional Gaussian kernel; +- C3: finiteness and equality of mu,nu on continuous cylinders; +- C4: Borel generation / functional monotone class; +- C5: elder/type mark measurability and determinant accounting; +- C6: scope boundary. + +Disposition per interface: ACCEPT / AMEND / COUNTEREXAMPLE / BLOCKED. The reviewer should explicitly state whether the original PR112 theorem-number objection is withdrawn, retained with a primary-source citation, or replaced by a narrower hypothesis objection. + +## 5. Source-integrity note + +The immediately preceding v1 commit on this branch was created through a string-serialization path that stripped several backslashes from LaTeX notation. It is preserved in Git provenance but is not a valid review target. This v1.1 file deliberately uses plain-text equations to remove that serialization ambiguity. Review only the exact current head and digest. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/CAP.md b/frontiers/lifetime_moment_boundary_20260930/sources/CAP.md new file mode 100644 index 000000000..0633aca3c --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/CAP.md @@ -0,0 +1,230 @@ +# A weaker separating-cylinder criterion, dimension-independent geometry, and an improved SIDE24 cubic bound + +Object: **MARKED-CYLINDER-CAP-20260924-v1**. Author: OpenAI / ChatGPT, 24 September 2026. +Disposition: author-side proof candidate. Ordinary mathematical argument below; exact finite controls accompany it. No independent acceptance or formal verification is claimed. +Tasks: CAP-01 and CAP-02, GitHub campaign #61. This is an additive successor to #57/#58, not a change to their historical statements. + +## 1. Results and exact division of scope + +There are two results, not one unqualified higher-dimensional Gaussian theorem. + +**Deterministic theorem.** Let d>=2 and use coordinates (x,y) in R x R^(d-1) on an embedded product cylinder + + D=[-2r,2r] x closed_ball(0,2r), r>0. + +Suppose f is C4 on a neighborhood of D, has critical pins M=(-r/2,0), S=(r/2,0), with f(M)=b and f(S)=b-kappa*r^3, kappa>0. For j=3,4 define the partial-block norm + + M_j=max_{a+c=j} sup_D || partial_x^a D_y^c f ||_op. + +For c=0 this is absolute value. For c>=1 the operator norm is the supremum on c transverse Euclidean unit vectors, independently chosen. In dimension two this is exactly the maximum of the coordinate partial derivatives used by #57, not the larger full-tensor operator norm. Put + + lambda=lambda_min(-D_y^2 f(M)). + +The explicit sufficient conditions are + + lambda > (4/(3*kappa))*r*M_3^2, + r*M_4 <= 3*kappa/10. (1) + +They give a unique transverse-critical ridge on D and a closed cap through S separating M from every older point. On a compact manifold, for a global Morse extension with distinct critical values, the ordinary superlevel elder death partner of M is S. Exactly one ascending unstable branch of S ends at M, for a smooth Riemannian gradient. No Morse-Smale assumption and no identification of the ridge with a flow trajectory are used. + +**Probabilistic corollary, narrower scope.** Retain EXACTLY the fixed-x-axis 2D normalized SIDE24 six-pin law of #57/#58: b=6/5, kappa=1/6, 08*r*m^2, r*n<=1/20. (4) + +The exact critical pins give the Hermite identity + + f(S)-f(M)=-(1/2) integral_{-r/2}^{r/2} + (x+r/2)(r/2-x) f_xxx(x,0) dx. + +The positive kernel has integral r^3/6, so its weighted average of f_xxx is 2. In particular m>=2 and some x0 in the pin interval satisfies f_xxx(x0,0)=2. This forced lower bound on m is useful; replacing m by the larger 1+m is unnecessary. + +The partial-block norms control increments by |Delta x|+||Delta y||, with no factor depending on dimension. Every point of D is within this product distance <5r of (x0,0), and likewise <5r from M. Thus + + f_xxx >= 2-5rn >= 7/4, + D_y^2 f <= -(lambda-5rm) I = -delta I, (5) + delta>rm(8m-5), 8m-5>=11. + +The Hessian inequality is in the quadratic-form order throughout D. A scalar lower bound on a diagonal entry cannot substitute for this matrix inequality when d>2. + +## 3. A vector ridge exists globally across the cylinder + +Let w(x)=grad_y f(x,0). It vanishes at the two pins and ||w''||<=m. Applying the scalar two-node interpolation remainder to every unit-vector projection of w gives, including outside the pin interval, + + ||w(x)|| <= (m/2)|x^2-r^2/4| <= (15/8)mr^2 <2mr^2. (6) + +The bound outside the nodes follows from the interpolation remainder on the convex hull of the three points, not from an unjustified inside-only estimate. + +Also integral_{-r/2}^{r/2} w'(t)dt=0. Subtract this average from w'(x), use the Lipschitz constant m, and integrate |x-t| to obtain + + ||w'(x)|| <= (m/r) integral_{-r/2}^{r/2}|x-t|dt + <=2mr, |x|<=2r. (7) + +This vector average argument does not posit a common point at which every component of w' vanishes. + +For fixed x, f(x,.) is delta-strongly concave on the transverse ball. On its boundary ||y||=2r, + + grad_y f(x,y).y <= ||w(x)|| ||y|| -delta||y||^2 <0. + +The maximum on the compact ball is therefore interior (moving inward increases the value at a boundary point). Strict concavity makes it unique. Call it h(x). It solves grad_y f(x,h(x))=0, has h(+-r/2)=0 and is C3 by the implicit function theorem. Strong monotonicity gives + + ||h(x)|| <= ||w(x)||/delta <2r/k, k=8m-5>=11. (8) + +Its derivative satisfies + + h'=-(D_y^2 f)^(-1) partial_x grad_y f. + +Equations (7),(8) and the mixed derivative bound give + + ||h'|| < (2rm+m||h||)/delta + <2/k+2/k^2=:u, u<=24/121. (9) + +Since m=(k+5)/8, + + m*u=(1+6/k+5/k^2)/4 <=48/121. (10) + +All these inequalities use transverse operator norms; there is no hidden dimension-dependent coordinate sum. + +## 4. The correct ridge derivative and all cap boundary faces + +Set g(x)=f(x,h(x)), F=g'=f_x(x,h(x)). With v=(1,h'), differentiating the transverse critical equation gives (H_f v)_y=0. In the third derivative of g, every term involving h'' is therefore zero. The exact identity is + + F''=f_xxx +3 f_xxy[h'] +3 f_xyy[h',h'] + + f_yyy[h',h',h']. (11) + +This is valid for a vector h. It does not assert that the ridge is an integral curve of the gradient. + +Using (9),(10), the adverse part is at most + + (48/121)[3+3*(24/121)+(24/121)^2] + =2554128/1771561. + +Consequently + + F'' > 7/4-2554128/1771561 + =2184415/7086244 >1/4. (12) + +The bound is weaker than #58's F''>1, but it is already sufficient for an older connecting endpoint. This is the second source of the improved depth condition. + +F has the prescribed zeros a=-r/2,c=r/2. Strict convexity makes them its only zeros, with F negative between and positive outside. Every critical point in D lies on the ridge, hence D contains precisely M and S. The Schur complement of D_y^2 f at these points is F'(a)<0 and F'(c)>0, proving that M is a maximum and S has exactly one positive Hessian direction. + +Define + + C=[-2r,r/2] x closed_ball(0,2r). + +Strong concavity gives f(x,y)<=g(x) and g(x)<=b on C. To control the longitudinal faces, subtract (x-a)(x-c)/8 from F. The difference is convex with zero values at a,c, hence nonnegative outside [a,c]. Integrating on either exterior interval gives + + g(-2r) <= b-(9/32)r^3 = s-(11/96)r^3, + g( 2r) >= s+(9/32)r^3 = b+(11/96)r^3, (13) + +where s=b-r^3/6. These are lower/upper inequalities at the actual ridge endpoints, not Taylor fits. + +For a transverse boundary point ||y||=2r, (5),(8) imply + + f(x,y) <=g(x)-(delta/2)||y-h(x)||^2 + rm(8m-5)>=22r and ||y-h||>20r/11. This covers the entire curved side of the cylinder, including its intersections with the end faces; in dimension two it covers both horizontal edges and their corners. + +On the right cap face x=c, h(c)=0, so + + f(c,y) <= s-(delta/2)||y||^2, + +with equality at S only. On the left face use f<=g and (13). Thus every boundary exit from C has value <=s. + +Finally, the ridge segment a<=x<=2r starts at M, passes through S and ends at z=(2r,h(2r)), where f(z)>b by (13). Its minimum is exactly s. A path through S at level s is not a preemption path staying strictly above s. + +## 5. Global elder pairing, actual branches, and mark scaling + +Every point with value >b lies outside C. Every continuous path from M to such a point must first intersect the boundary of C, so its minimum is <=s regardless of later remote excursions or reentry. The preceding ridge path has minimum exactly s and endpoint >b. Hence the maximin connection level to an older point is s. + +For a compact Morse function with distinct critical values, ordinary superlevel-component persistence identifies this maximin value with M's elder death level; the older endpoint excludes the essential/global-maximum case. The unique critical value s identifies S. This is the same separating-cap criterion established in #58, now with all its boundary premises re-proved under the weaker event. + +At S the Hessian is negative definite on the transverse hyperplane. The unstable eigenvector for gradient ascent has nonzero x component: its Hessian quadratic form is positive, whereas that hyperplane is strictly negative. This argument also holds for a smooth positive-definite Riemannian metric. One half-branch enters the cap at height >s; the other enters outside. The first cannot exit because its height increases and every boundary value is <=s. Its omega-limit is a critical point in the compact cap. Gradient monotonicity and finiteness of the critical set make this limit a single point; it cannot be S, so it is M. The other branch cannot enter C or tend to its interior point M. Exactly one branch ends at M. + +For general kappa>0 apply the normalized theorem to f/(6*kappa). Its pins have gap r^3/6 and arbitrary birth b/(6*kappa). Its derivative norms and transverse curvature are divided by 6*kappa. Condition (4) becomes exactly (1). The geometric ridge is unchanged. The reduced third derivative is >3*kappa/2, the longitudinal drop/rise is at least (27/16)*kappa*r^3, and the excess over the specified gap is (11/16)*kappa*r^3. The radial drop exceeds (26400/121)*kappa*r^3. + +An arbitrary orthonormal frame can be used in this pathwise argument as long as its product chart is embedded. Scaling or rotating the function here is not a proof that SIDE24's conditional covariance, pin energy or normalizer are invariant under that operation. + +## 6. What is imported for the 2D probability calculation + +Use the fixed-axis Q,Q^W law from Section 1 and the same random-variable coupling as #57 Sections 5 and 7. The following are retained premises, not newly rerun numerical programs: + +* Z>=(17/10)r^2, six-pin regression energy<8, and ||det H_M det H_S||_2/Z<23/5. +* q=f_yy(M) has density <=1/3 and ||q||_8<5. On typed support q<0. +* There is T>=1, JOINTLY independent of q, with ||T||_8<320 and M_3<=T+8|q|; also ||M_4||_8<340. +* In the SAME coupling, at q=-ru, the full typed weight satisfies the pointwise bound + + W<=r^4[T^2*u^2+3*T^3*u], u>=0. (15) + + Internal correlations among the residual field, axial/mixed variables and transverse innovation are retained. Independence is only from q. The coefficient 3 includes the saddle's mixed-square contribution. +* The pinned-genericity candidate #49 supplies the full-probability Morse/distinct-value locus for each fixed r. No simultaneous null-set statement over all parameters is imported. + +These facts are read from exact source bytes and their explicit chain. Hash verification does not establish their truth; H3 and LPW interval programs were not rerun by this delivery. The stronger geometry does not depend on #56's endpoint spectral candidate. + +## 7. The smaller exceptional layer and its retained far branch + +Set G_new={-q>8r M_3^2, rM_4<=1/20}. Section 2-5 give global pairing on G_new. Unlike an inference from a local-event probability, this uses a direct bound on G_new's complement. + +Put lambda=-q>0 on typed support. Failure of the depth condition implies + + lambda<=8r(T+8lambda)^2 + <=16rT^2+1024r lambda^2. + +Thus the failure lies in + + {01/(2048r)}. (16) + +The large-lambda branch is essential to the quadratic inequality and has NOT been discarded. + +For the near event, disintegrate on q independent of T. Use its density cap 1/3, dq=r du, and (15): + + E[W 1_near] <= (r^5/3) E integral_0^(32T^2) + [T^2 u^2+3T^3 u]du + = (r^5/3)[(32^3/3)E T^8+(3*32^2/2)E T^7]. (17) + +Division by the full Z and the moment bounds gives C3_new*r^3. Weighted Cauchy-Schwarz, INCLUDING its square root, and eighth-moment Markov give + + Q^W{lambda>1/(2048r)} < (23/5)(10240r)^4, + Q^W{M_4>1/(20r)} < (23/5)(6800r)^4. (18) + +Together (17),(18) prove (2),(3) at the imported premises. The same single complement controls nonadjacency and the event that both unstable branches end at M; no independence or equality of these events off G_new is asserted. + +The earlier LPW lower premise, when separately consumed, is unaffected: c_LPW=13/468574870385996070912000 and r<=1/356352 still compose with this upper bound for two-sided cubic order in the one fixed-axis law. This delivery does not re-establish that lower premise or give a limiting coefficient. + +## 8. Exact comparison and remaining work + +The predecessor constants from #57/#58 are + + C3_old=(10/51)[(256^3/3)320^8+(3*256^2/2)320^7], + C4_old=(23/5)[81920^4+6800^4]. + +Exact arithmetic checks C3_old>500*C3_new, C4_old>500*C4_new and + + C3_new+C4_new/20 <2.4*10^23. + +Their absorbed-constant ratio is about 511.803 (display only; the comparison >500 is rational). The tighter good event is strictly weaker than the old sufficient event; neither claims to characterize all successful pairs. + +For the three-dimensional probability program the deterministic matrix/ridge obstacle now has a concrete candidate solution, but the probability of a small transverse eigenvalue and intersections of matrix boundary strata remains to be proved under the correct eight-pin law. Full-annulus expected counts, 24-jet modulus certificates, all-angle/all-mark estimates and lifetime integration remain distinct targets. Small failure probability alone does not bound a count on the failure event. + +Source identities and external reconnaissance are in SOURCE_OBSERVATIONS.json and RECONNAISSANCE.md. Tests check exact algebra, finite examples and deliberate mutations, not arbitrary continuous fields or the analytic imported chain. Review should focus on the vector average in (7), all partial-block norms, the h'' cancellation, entire radial side, marked scaling, and the independence/density/normalizer powers in (17). diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/ERR.md b/frontiers/lifetime_moment_boundary_20260930/sources/ERR.md new file mode 100644 index 000000000..213594d6c --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/ERR.md @@ -0,0 +1,44 @@ +# Additive erratum — endpoint Hessian congruence in UNIFORM_MATRIX_CAP_AND_LIFETIME + +**Object:** UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1-ERRATUM-1 +**Scientific effect:** correction of a displayed linear-algebra sentence only; no theorem promotion. + +Source object: `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, raw SHA256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. + +## Correction + +In Section 5, immediately after the convergence of endpoint Hessians, the source says: + +> Here the congruence uses `diag(sqrt(r),I)`, so its off-diagonal entries are `sqrt(r) beta_i`. + +Given the source's own definition + + H_i = [[r alpha_i, r beta_i^T], + [r beta_i, A_i]], + +that displayed congruence factor is incorrect. The matrix used by the argument is + + D_r = diag(r^(-1/2), I), + +because + + D_r H_i D_r + = [[alpha_i, sqrt(r) beta_i^T], + [sqrt(r) beta_i, A_i]]. + +It also satisfies + + det(D_r H_i D_r) = det(H_i)/r, + +which is exactly the determinant normalization used in Sections 5–7. + +With `alpha_M -> -6k`, `alpha_S -> 6k`, bounded `beta_i`, and `A_i -> A_0`, this corrected congruence gives + + D_r H_M D_r -> diag(-6k,A_0), + D_r H_S D_r -> diag( 6k,A_0). + +No formula (5.3), (5.4), (5.5), or the subsequent soft-factor/eigenvalue integration is changed by this correction. The original file is retained byte-for-byte for provenance; consumers of its Section 5 type-convergence argument must read this erratum with it. + +## Review requirement + +Because this correction touches the load-bearing full-normalizer/type-convergence interface A3, the pending nonauthor A1–A7 review on main #63 should review the source **together with this erratum**. A green test or obvious intended meaning is not acceptance. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/MARK_2.md b/frontiers/lifetime_moment_boundary_20260930/sources/MARK_2.md new file mode 100644 index 000000000..ef2aa5795 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/MARK_2.md @@ -0,0 +1,433 @@ +# Planar cubic elder mark: a compact-bar transfer theorem + +Object: OA-PLANAR-LOCAL-ELDER-20260930-v1. +Authors/contributors: OpenAI root, elder_bridge_audit and its local_topology child, foreground continuation, 30 September 2026. The companion scale analysis has an additional OpenAI contributor, mixed_geometry_audit. The proof is AUTHOR_SIDE / HOLD pending qualified source-bound mathematical review. All these agents have the same provider and earn zero organizational-independence credit. Existing source headers are not treated as live acceptance decisions. Publication, source hashes and engineering checks do not establish mathematical acceptance or promote any parent claim. + +This is a new full proof candidate. Only this additive directory is written; no source body, count theorem, register, graph, numerical certificate or integration disposition is changed. Exact dependency interfaces, provenance and unresolved obligations appear below and in SOURCES.json. + +## 1. Exact source cut and estimands + +The main source cut is Math- commit 98fdb54954a2d9ac0b8ba8ca03e279701f70d751. Immutable source links and blob identities are in SOURCES.json. + +* [P] imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md, blob dfed3b8d318a3ab1950957f393307733a4bef3f2, §§1,8,10–14. Read with its congruence erratum (blob 213594d6ca6a86fb938110f4d166d9ce275a02d0), §9 Borel repair (blob fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a), and [REC] §1 reading rule/W1/embedding restriction. [REC] is reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md, blob 75da2597971510f843f8d90c743950cb8c177342. +* [CUB] frontiers/planar_cubic_cluster_20260929/PROOF.md, blob bb446d08db8a944537a743ad550b88c1c2ad5758, §2 (C1)–(C14), §3 Lemma S, §4 (G6)–(G10). +* [LOW] frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md, blob aaefc8da9668e819591dd9db0b34442394f2362e, §§1–4; and frontiers/elder_lower_all_d_20260929/PROOF.md, blob 7f41c9e315be0a7985d4c770a11a2cdf717bd46a, §§0,3–5. +* [TWO] frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md at PR166 head be9b7fa95de9b7e59a67347640e157ed3027f624, blob a32fd5f7d941bbe1fe943df045b1e0fbec8d691c, §§1–4, (M1)–(M2), Theorem L and (L3). +* [Q] frontiers/c6_palm_route_20260929/PROOF.md, blob 89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5, §1.2 Theorem Q and §1.3 non-claims. +* [U] frontiers/unrestricted_selection_difference_20260929/PROOF.md, blob 5a55b179a974a90c65d257f3f93765cc6fb30bb8, §§1,3–5. + +For the actual pinned law Q_r^W, the pins are M_r=(-r/2,0), S_r=(r/2,0), heights b and b-kr^3, and both gradients zero. The weight is W_r=F_2(H_M)F_1(H_S), with exactly one FULL Z_r=E_Q W_r. + +N_r in [Q]/[TWO] counts **additional critical points of all indices in the open height window**, excluding the two pins. The ordered pair measure (L3) is a measure of two additional window witnesses. Its microscopic limit has two index-1 saddles. It is not a maximum/saddle persistence-pair measure. + +The elder selector in [P] is different: +d_f(M)=sup_{gamma(0)=M, f(gamma(1))>f(M)} min_t f(gamma(t)), +and p_r=Q_r^W{d_f(M)=f(S)} on the Morse distinct-value locus. Its candidate maximum/saddle intensity is r A_r dr db dk d sigma with A_r=12 pi_r(Z_r/r^2); its selected intensity is r A_r p_r with the same measures. The lifetime pushforward ell=kr^3 supplies r dr/dell=ell^(-1/3)/(3k^(2/3)), [P] §§10–11. There is no adjacency normalization, no factor 1/2 and no second Z_r. + +Consequently, localization of (L3) alone does not supply the elder mark. The common leading candidate/elder lifetime density is already supplied by the reconciled D1 cap/selection/Kac–Rice chain, rather than by C6 localization. [LOW] identifies the compact-window rejected density order ell^(2/3); [U] separately shows a positive far rejected density and forbids importing that compact-window decay as an unrestricted difference rate. + +The following theorem adds an actual topological mark to the planar cubic sector. It uses neither a new count argument nor any remote witness theorem. + +## 2. Deterministic theorem + +Fix k>0 and a typed cubic from [CUB]: +P(X,Z)=2kX^3-3kX/2-k/2+sZ^2/2 + +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3. +Write M=(-1/2,0), S=(1/2,0), so P(M)=0, P(S)=-k and both gradients vanish. With the determinant-one shear u=X+aZ/(12k), set +B=beta-a^2/(12k), +D=(c-a beta/(4k)+a^3/(72k^2))/2. +The typed domain is s<-|B|/2. + +Exclude the two algebraic sets +Sigma_k={12kD^2+(s-B)^2(B+2s)=0}, +Delta_k={12kD^2+B^3=0}. +Both are nonzero polynomial zero sets after clearing fixed positive k denominators, hence Lebesgue-null. The first is [CUB]'s window transition; the second is the only additional exclusion made here. + +Let C(P) be the additional critical points with -k0. Suppose their scaled functions + g_i(X,Z)=(f_i(r_iX,r_iZ)-b)/r_i^3 + converge to P in C^2 on every fixed compact disk, and retain the exact pins, gradients and values 0,-k. Then their global normalized death levels satisfy + (d_{f_i}(M_i)-b)/r_i^3 -> h_*(P). + No conditions are imposed outside the shrinking coordinate disks. +3. Continue the finitely many points of C(P) to local critical points of g_i by pinned C^2 root stability, and include the exact pinned saddle S. For every sufficiently large i the global death level is exactly the highest *actual* height in this finite continued set. On the global Morse distinct-value locus its unique highest point is the actual elder partner. If C(P) is empty, that point is exactly S_i; if C(P) is nonempty, it is a continued window saddle and S_i is not the partner. Thus the selector-failure indicator converges to 1{C(P) nonempty}. + +The theorem is planar and pointwise in the cubic and coupling realization. It makes no assertion uniform over marks, dimensions, radii or approaching exceptional sets. + +## 3. The critical-chord identity supplies the lower bound + +For any polynomial of degree at most three and any two critical points M,Y, put q(t)=P(M+t(Y-M)). Since q'(0)=q'(1)=0, interpolation gives exactly +q(t)=P(M)+(P(Y)-P(M))(3t^2-2t^3). +This also holds when the restriction has lower degree. + +Here P(M)=0 and h=P(Y)<0. On 0<=t<=2 the minimum of h(3t^2-2t^3) is h, attained at t=1; the endpoint is -4h>0. Every negative critical point therefore gives a finite path from M to a strictly higher point whose bottleneck is exactly its own critical height. + +Apply this to Y=S and to each window saddle. It follows that d_P(M)>=h_*. + +For g_i, the same fixed finite path has endpoint strictly above zero and minimum at least h_* minus an error tending to zero. Hence the global normalized death has liminf at least h_*. This path is a path in the actual surface after rescaling; no claim about its being a separatrix or an ascending branch is made. + +When C(P) is nonempty, h_*>-k, so the strict margin alone proves eventual failure of selection of S_i. + +## 4. No finite-height escape at infinity + +In the sheared coordinates the leading homogeneous cubic is +H(u,Z)=2ku^3+(B/2)uZ^2+(D/3)Z^3. +Its derivatives are +H_u=6ku^2+(B/2)Z^2, +H_Z=Z(Bu+DZ). + +A nonzero common zero requires Z nonzero. If B=0 it exists exactly when D=0; if B is nonzero it exists exactly when 12kD^2+B^3=0. Thus off Delta_k, grad H has no zero on the unit circle. Compactness and homogeneity give |grad H(z)|>=c|z|^2, and lower-order terms then give +|grad P(z)|>=c_0|z|^2 for |z|>=R_0 +after increasing R_0. The invertible shear preserves such a bound up to constants. + +There are no critical points of P with value in (h_*,0), by the definition of h_* and [CUB]'s complete classification of all window points. Near M a sufficiently high negative regular level t_0 has a small closed level oval enclosing a disk K_0. The disk contains only the strict maximum M. + +For any h in (h_*,t_0), choose a smooth height cutoff chi with 0<=chi<=1, equal to one on [h,t_0], supported in a slightly larger critical-free strip inside (h_*,0), and zero near zero. The vector field +V=-chi(P) grad P/|grad P|^2 +is smooth and globally bounded after setting it to zero outside that strip. The gradient-growth bound controls it at infinity, and compactness controls it elsewhere. Its flow is defined for the finite time t_0-h; it fixes M. The initial boundary stays where chi=1, so its P-value decreases at speed exactly one and ends at P=h. The image K_h is compact, contains M and has boundary P=h. + +Moreover P is nonpositive throughout K_h: the initial disk has P<=0 and along the flow P never increases. Thus every path from M to a point with P>0 exits K_h and has minimum at most h. Letting h decrease to h_* proves d_P(M)<=h_*. Together with §3 this proves part 1. + +This is the missing compact-bar/escape argument. Merely having finitely many window roots would not exclude a finite asymptotic critical level; the Delta_k exclusion and the gradient estimate do. + +## 5. Transfer of the upper bound to the actual global field + +Fix h with h_*0, the conic forces |u|<1/2, so this is negative. If B<0, either u>=1/2, when it is below -k, or u<-1/2; (C9) and 2s/B>1 again make it negative. For B=0, the two extra possible heights are s^3/(6D^2) and -k+s^3/(6D^2), both negative. Thus M is the only critical point at value zero or above. A second interior zero maximum of K_h would be another such critical point. Boundary h is negative. + +The exact pin M and C^2 convergence imply g_i<=0 on K_h for all sufficiently large i: in a fixed small ball around M its Hessian remains negative definite and M remains critical with value zero; outside that ball P has a strictly negative maximum on the remaining compact set, so C^0 convergence suffices. Boundary values of g_i are at most h+epsilon for all large i. + +Any actual global path to a point above birth must leave the embedded r_i K_h because there is no such point inside. It therefore crosses a boundary at value at most b+r_i^3(h+epsilon). This proves the normalized limsup is at most h+epsilon. Let epsilon->0 and h->h_* to prove part 2. + +This uses a local compact boundary to control the global maximin, so exterior connectivity cannot invalidate the bound. + +## 6. Exact selection of the highest actual continued local saddle + +Convergence of death heights alone would not identify the location or exact partner. The following finite-index strengthening supplies both. + +Let the finite set be C(P) union {S}. For each Y in it, let Y_i be the unique continued critical point, using exact S_i=S for the pinned saddle. All their values are negative for large i. They lie in the window except S, whose value is exactly -k. Define + + h_i=max{g_i(Y_i): Y in C(P) union {S}}. + +Fix t_0<0 close enough to zero to have a small maximum-level oval and to satisfy t_0>h_*. Then h_i 2k/(c_0 R_0^2)+1. + +For all large i, C^1 convergence gives |grad g_i|>=c_0|z|^2/2 on R_0<=|z|<=R+1. By [CUB] Lemma S, using exact pins and excluding Sigma_k, sufficiently small C^2 perturbations on B_(R+1) have exactly the continued roots in (-k,0), together with the exact pins at 0,-k. Small disjoint neighborhoods give one root each; the rest of the compact enlarged height band has a positive gradient floor. Thus there is no critical point in (h_i,0) inside B_(R+1). The same sufficiently large index satisfies all these statements, independently of the lower height h chosen next. + +### Exact upper bound by a common finite-disk collar + +Negative-definite Hessians and exact pinning give a small maximum-level oval for g_i at t_0 inside B_(R_0), enclosing M in a disk with g_i<=0. For each h with h_ih_i>=-k. A first passage from B_(R_0) to that boundary traverses the annulus, where speed is at most 2/(c_0 R_0^2). The chosen width makes this take more than k. Thus all boundary trajectories remain in B_R and reach height h at time t_0-h. They form a Jordan curve, and its enclosed image disk is inside B_R: the complement of B_R is connected and lies in the curve's unbounded component. The cap contains M and g_i<=0 throughout since the cutoff flow never increases height. + +Every older path therefore crosses a boundary of height h. For this SAME sufficiently large i the conclusion holds for every h in (h_i,t_0). Taking h down to h_i gives exact normalized global death <=h_i, without exchanging i and h limits and regardless of the exterior geometry. + +### Exact lower bound through each continued critical point + +For any one Y, consider the moving straight critical chord + + q_i(t)=g_i(M+t(Y_i-M)), 0<=t<=2. + +The actual exact criticality of M and Y_i gives q_i'(0)=q_i'(1)=0. C^2 convergence and Y_i->Y give C^2 convergence of these one-dimensional restrictions to + + q(t)=P(Y)(3t^2-2t^3). + +Writing h_Y=P(Y)<0, the limiting derivative is negative on (0,1), positive on (1,2), and + + q''(0)=6h_Y<0, q''(1)=-6h_Y>0. + +C^2 convergence preserves the second-derivative signs near 0 and 1. Together with the *exact* derivative zeros, this preserves the required derivative signs on those endpoint neighborhoods. C^1 convergence preserves them on the remaining compact subintervals. Hence q_i decreases strictly from t=0 to t=1, then increases strictly to t=2. Its exact minimum is g_i(Y_i), and its endpoint is positive since its limit is -4h_Y>0. + +There are finitely many roots, so the same sufficiently large index works for all these paths, including S. Selecting the root attaining h_i provides an actual older path of exact minimum h_i. Thus normalized global death >=h_i. With the collar bound it equals h_i exactly. Even when two limiting saddle heights tie and the actual winning root switches, the finite collection of paths proves the same result. + +On the actual global Morse distinct-value locus, the unique critical value h_i identifies the highest continued local saddle as the ordinary elder partner. Empty sectors select S exactly; nonempty sectors select one of the continued window saddles and preempt S. This proves Theorem E part3 and identifies the actual partner, not only its limiting height. + +### A unique limiting position for almost every jet + +For the position mark only, exclude D=0 as well. This is another nonzero polynomial zero set in the physical jet coordinates, hence Lebesgue null. When B!=0, (C9) gives + + (d/du)(P/k)=-1-(4s/B)u. + +On the B>0 window conic this derivative is strictly negative (at its largest allowed u=u_w it is 1+2s/B<0); on the B<0 window conic it is strictly positive because u<-1/2 and s/B>1/2. Equal window heights therefore force the same u. If D!=0, the line s+Bu+DZ=0 then forces the same Z, so distinct window roots have distinct heights. For B=0 there is at most one window root. Accordingly outside this extra null set there is a unique highest cubic window point Y_*(theta) when n>0, and the actual elder partner's scaled position converges to it. Use S as the default position when n=0. This position exclusion is unnecessary for the exact finite-index death and partner conclusion above. + +## 7. A compact-sector marked selection law under the actual tilt + +This corollary uses only the planar compact-sector disintegration already displayed in [CUB] §4; it does not modify its count theorem. + +For a compact Borel C in the typed domain, define the finite measure +dM_C(theta)=1_C(theta) z_0^(-1) w(theta) h_0(0,a,beta,c) dtheta, +with the full z_0 of [CUB] (G3), and define lambda(theta)=-h_*(P_theta)/k in (0,1]. + +Under the regression coupling [CUB] (G6)–(G8), Theorem E applies pathwise for almost every theta and almost every coupling realization. Define I_r=1{the global partner of M is not S} and D_r=(d_f(M)-b)/r^3. For every bounded measurable psi(theta), the marked identity +r^(-3) E_QW[psi(Theta_r) I_r; Theta_r in C] + -> integral_C psi(theta) 1{n(theta)>0} dM_C(theta) +holds. + +For every bounded continuous phi(theta,t), +r^(-3) E_QW[phi(Theta_r,D_r); Theta_r in C] + -> integral_C phi(theta,h_*(theta)) dM_C(theta). +On the exceptional essential-class event define phi arbitrarily at D_r=-infinity; each good coupling eventually has an explicit local path to an older point, so that arbitrary choice disappears from the limit. + +Proof: replace the count indicator in [CUB] (G10) by the Borel elder mark, or by the indicated bounded functional of the Borel maximin. The multiplier is +1_C(theta) (r^2/Z_r) h_r(rs,a,beta,c) + E[(W_r/r^4) times mark]. +Theorem E supplies pointwise convergence of the mark. Equations (G7)–(G9) supply the same integrable domination as before. Dominated convergence proves both statements. The first works for every bounded measurable psi because theta itself is not moved by the coupling and the failure indicator converges pointwise; this also yields total variation for the joint theta/Boolean measure on the compact sector. The real-valued death-height mark is asserted weakly, not in total variation. + +The marked corollary gives a selection law on every fixed compact soft-jet sector. Compact convergence alone does not justify a full-global coefficient: it needs tightness of the rescaled *elder-failure measure*, rather than of the nonempty witness measure. The next section proves that tightness by retaining the rare scalar layer in the parent's cap-failure integral and explicitly converting endpoint control to the actual midpoint coordinates. + +## 8. Full planar elder-failure tightness and global marked selection law + +### 8.1. Fixed model and immutable interfaces + +Fix the planar periodized Gaussian model, L>0, b in R, k>0 and one frame. Write Q_r for the exact six-pin regression at M_r=(-r/2,0), S_r=(r/2,0), heights b,b-kr^3 and zero gradients. Set +W_r=F_2(H_M)F_1(H_S), Z_r=E_Q W_r, dQ_r^W=(W_r/Z_r)dQ_r. +Let +F_r={the global ordinary superlevel elder partner of M_r is not S_r}, +Theta_r=(f_zz(0)/r,f_xxz(0),f_xzz(0),f_zzz(0))=(s,a,beta,c), +D_r=(d_f(M_r)-b)/r^3. +The essential class uses d_f=-infinity. All event definitions are Borel by parent §8 and its reconciled §9 whole-field interface. + +[P] is imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md at main commit 98fdb54954a2d9ac0b8ba8ca03e279701f70d751, blob dfed3b8d318a3ab1950957f393307733a4bef3f2. It is read WITH the congruence erratum, §9 Borel replacement, and reconciliation/W1/embedding rule in SOURCES.json. The consumed equations here are (3.5), (4.2)–(4.3), (5.5), (6.2), (7.5)–(7.7), and the good-cap implication in §8. + +[CUB] is frontiers/planar_cubic_cluster_20260929/PROOF.md at the same main cut, blob bb446d08db8a944537a743ad550b88c1c2ad5758. The consumed equations are (C1)–(C14), Lemma S, (G1)–(G3), regression (G6)–(G9), physical midpoint disintegration (G10), and coefficients (G11) with finiteness (G13). + +For theta=(s,a,beta,c), let P_theta be its pinned cubic, B=beta-a^2/(12k), D=(c-a beta/(4k)+a^3/(72k^2))/2, and D_k={s<-|B|/2}. Let n(theta) be the existing cubic classifier and +h_*(theta)=max({-k} union {P_theta(Y):Y is an additional window critical point}). +The function h_* is Borel: by Theorem E it is the cubic maximin, whose strict upper-level events have the same countable finite polygonal-path representation as [P] §8; each path stays in a compact plane region. +Extend it by a fixed value on the excluded polynomial sets and outside D_k when using an ambient test function. Those Borel sets have null or zero-weight contribution to the limiting failure measure, so the extension changes none of the identities. +Write +w_+(theta)=9k^2 [B-2s]_+ [-B-2s]_+. +This equals w(theta)=9k^2(4s^2-B^2) on D_k and is zero elsewhere, including its boundary. + +Define the finite candidate limit measure +dmu_fail(theta)=z_0^(-1) w_+(theta) h_0(0,a,beta,c) + 1{theta in D_k,n(theta)>0} dtheta. (S1) +Here z_0 is the FULL normalizer limit [CUB] (G3), and h_0 is the actual raw midpoint Gaussian jet density under the exact contact law. No GOE substitution, adjacency normalization or extra pin-density factor appears. + +**Theorem S (fixed planar selection law).** Conditional on Theorem E and the exact source interfaces above, +r^(-3) Q_r^W(Theta_r in ·,F_r) -> mu_fail +in total variation as finite measures on R^4. Consequently, +r^(-3)(1-p_r) -> alpha_1+alpha_2, (S2) +with alpha_j exactly [CUB] (G11). + +Moreover, for every bounded continuous phi on R^4 x R, +r^(-3) E_QW[phi(Theta_r,D_r);F_r] + -> integral phi(theta,h_*(theta)) dmu_fail(theta). (S3) +Give the test an arbitrary bounded value on the essential-class event; that event has o(r^3) tilted mass. Thus the height-mark assertion is weak convergence, not total variation. After division by alpha_1+alpha_2>0 it is the actual conditional law given elder failure. + +There is no claim about the unmarked full r^(-3) joint theta measure. Its typical s-coordinate escapes and its total mass diverges. + +### 8.2. Good cap excludes failure + +Let G_r be the parent's good cap. On the almost-sure Morse distinct-value locus, [P] §8 proves that the global ordinary partner is S on G_r. Hence +F_r subset G_r^c (S4) +up to a Q_r^W-null set. This includes essential maxima: the good cap supplies a finite older endpoint, so the essential class cannot occur there. + +This implication concerns the same actual global selector p_r as [P], not a new local adjacency event. It is the only place where the cap is used to contain the failure event. + +### 8.3. Rare-layer tail bound from the independent endpoint residual + +In the plane set lambda=-f_zz(M)>0 on typed support. Under [P] (4.2), the residual field g_r is independent of the scalar endpoint transverse Hessian. Write +J_r=1+||g_r||_(C4). +Its moments of every finite order are bounded along small-r sequences by (4.3). The bounded C4 regression coefficients and bounded mean in (4.2) give the pathwise inequality +||f||_(C4)<=C(J_r+lambda). (S5) +The mean and E_Q f_zz(M) terms are absorbed into C using the bounded fixed-parameter pin regression and J_r>=1. + +Let D_cap>0 be the fixed scalar depth constant from [P] (7.5). By that split, depth failure is contained in +near: 01/(4D_cap r). +On the near branch, r<=1 implies +||f||_(C4)<=C'J_r^2, M3<=C'J_r^2. (S6) + +The m=1 typed weight bound (6.2) gives +W_r<=C r^2 J_r^4 lambda(lambda+C rJ_r^2) +on this branch. The density of lambda is bounded by (3.5), and J_r is independent of lambda. Retaining a tail restriction J_r>A in the SAME integration yields +E_Q[W_r;near,J_r>A] + <= C r^2 E[ J_r^4 1{J_r>A} + integral_0^(4D_cap rJ_r^2) lambda(lambda+C rJ_r^2) d lambda ] + <= C r^5 E[J_r^10;J_r>A]. +Divide by the FULL floor Z_r>=z_*r^2 from (5.5): +r^(-3) Q_r^W(near,J_r>A) + <= C E[J_r^10;J_r>A]. (S7) +Higher moments of J_r imply the right side tends to zero as A increases, uniformly along small-r sequences. + +The far branch (7.6) and the fourth-derivative exception (7.7) each have tilted mass O(r^4), hence o(r^3). Their union with the near branch covers G_r^c on typed support. Therefore the rare-layer tail estimate applies to F_r via (S4). + +An unconditional C4 tail estimate followed by Cauchy–Schwarz would lose the rare factor. The retained scalar integral above supplies r^5 before division by the one full Z_r; this is why the tilted r^3 scale survives. + +### 8.4. Explicit endpoint-to-midpoint control + +Between M and the midpoint, +|f_zz(0)|/r <= lambda/r + M3/2 +by the mean-value derivative bound. On the near branch, +lambda/r<=4D_cap J_r^2. +The other midpoint coordinates a,beta,c are third derivatives and are bounded by ||f||_(C3). Equations (S6) therefore imply +||Theta_r||<=C J_r^2. (S8) + +No density-coordinate replacement occurs here: (S8) is a deterministic control of the actual midpoint target by the scalar endpoint residual. Together with (S7) and the o(r^3) exceptions, it proves +lim_(T->infinity) limsup_(r->0) + r^(-3) Q_r^W(F_r,||Theta_r||>T)=0. (S9) +The same proof gives +lim_(T->infinity) limsup_(r->0) + r^(-3) Q_r^W(F_r,||f||_(C4)>T)=0. (S10) + +Thus the leading failed population is ambient-midpoint-soft in the plane. Since there is only one transverse direction, bounded residual sectors force f_zz(M), f_zz(0), f_zz(S)=O(r), the last by (5.1). The axial and mixed endpoint entries are O(r) by (5.1)–(5.2), so both endpoints have the r^4 determinant-product scale. At finite r the far and fourth-derivative branches still exist; their smaller mass is retained above. + +### 8.5. Compact-box disintegration, including the typed boundary + +Fix an ambient compact box K_T in R^4. Although [CUB] states its count theorem on compact subsets of D_k, its actual regression construction (G6), moment bound (G7), Taylor convergence (G8), and determinant bound (G9) apply to ANY bounded midpoint target. They rely on the nonsingular joint covariance and bounded target, not on endpoint typing. No count convergence is invoked outside D_k. + +Under this coupling, the endpoint Hessians divided by r converge to +H_M(P)=[[-6k,-a/2],[-a/2,s-beta/2]], +H_S(P)=[[6k,a/2],[a/2,s+beta/2]]. +Their determinants are 3k(B-2s) and 3k(B+2s). Since the axial signs are fixed, the limiting typed weight is exactly +W_r/r^4 -> F_2(H_M(P))F_1(H_S(P))=w_+(theta). (S11) +Each F_j is continuous even at a degenerate matrix: its determinant factor tends to zero there. Hence (S11) holds on the typed boundary as well, with zero weight. On the whole box the integrable envelope +W_r/r^4 <= C_T(1+||F||_(C4))^4 (S12) +follows from the same endpoint Taylor bound as (G9). + +The fixed-k algebraic exceptional sets +Sigma_k={12kD^2+(s-B)^2(B+2s)=0}, +Delta_k={12kD^2+B^3=0} +are nonzero polynomial zero sets, hence null for the absolutely continuous midpoint density. No distinct-height exclusion is needed between the two window saddles. + +For almost every theta in D_k and coupling realization, Theorem E gives +1_F_r -> 1{n(theta)>0}, +D_r -> h_*(theta). +For theta outside D_k, including its boundary, no convergence of the selector is required: its bounded multiplier is killed by (S11). Thus [CUB]'s physical midpoint disintegration (G10), with the count mark replaced by the Borel global failure mark, gives the density +u_r(theta) + = (r^2/Z_r) h_r(rs,a,beta,c) + E[(W_r/r^4)1_F_r]. (S13) +The one factor r in (S13) comes from f_zz(0)=rs. It is not a spectral/endpoint change of variables. The prefactor is r^(-3)*r*r^4/Z_r=r^2/Z_r. + +By (S11)–(S12), the full normalizer limit, local boundedness of h_r, and dominated convergence in theta and F, +integral_(K_T) |u_r(theta)-u_0(theta)| dtheta -> 0, +u_0(theta)=z_0^(-1) w_+(theta) h_0(0,a,beta,c) + 1{n(theta)>0}. (S14) +Here define the final indicator arbitrarily off D_k, where w_+=0. + +This simultaneously handles thin typed-boundary neighborhoods and the exceptional polynomial sets; neither can carry an unidentified atom of the leading failure measure. + +### 8.6. Exhaustion gives global total variation and the coefficient + +The limiting density u_0 is integrable. It is exactly the sum of the two nonempty coefficient densities (G11), whose integrability follows from [CUB] (G13) and Gaussian polynomial tails. Its total mass is alpha_1+alpha_2, finite and strictly positive. + +Choose T large. On K_T, the L1 error tends to zero by (S14). Outside K_T, the actual density's mass is controlled by (S9), and the limiting density's mass tends to zero by integrability. Therefore +integral_(R^4) |u_r-u_0| dtheta -> 0. +This is total-variation convergence of the finite failure measures (under either conventional factor in the TV norm), proving Theorem S and (S2). + +The proof exhausts only the marked failure measure. It never integrates the divergent n=0/unmarked total soft-jet measure and never asserts an uncentered r^(-3) probability law. + +### 8.7. The normalized death mark on actual failure + +For a bounded continuous phi(theta,t), use (S13) with multiplier +1_F_r phi(theta,D_r). +On every ambient compact box the same domination applies. Theorem E gives pointwise convergence on typed generic theta to +1{n(theta)>0} phi(theta,h_*(theta)); +outside the typed domain the zero weight suffices. Dominated convergence gives the compact-box marked limit. + +The bounded test's omitted actual mass is at most ||phi||_infinity times the failure tail in (S9). The limiting omitted mass is controlled by the integrable u_0. Exhausting boxes proves (S3). + +The essential-class mark has zero scaled limit: on a good typed cubic coupling Theorem E's finite critical-chord path supplies a local point above the birth for all small r, and outside the typed domain the weight limit is zero. The compact-box domination and (S9) again exhaust the actual essential-failure mass. Thus it is o(r^3). + +Normalizing (S3) by (S2) identifies the weak actual conditional failure-height law. This is not a density result for persistence lifetimes, not a rate, and not a spatial or real-height TV assertion. + +### 8.8. Actual replacement location and the lifetime mark + +Let T_r be the actual elder partner's coordinates divided by r in a fixed embedded midpoint chart. Give it a sentinel value when the partner is outside the chart or the class is essential. Theorem E part3 proves that on every generic typed coupling the partner is eventually a continued local window saddle on failure. Outside the additional null set D=0, it is the continuation of the unique highest root Y_*(theta), so + + T_r -> Y_*(theta), D_r -> h_*(theta). + +This is an actual partner assertion; no witness factorial pair is relabelled. The functions Y_* and h_* are Borel on the generic typed domain: the finitely many nondegenerate roots continue continuously locally, and strict height comparison selects a unique root. Arbitrary Borel extensions on the excluded sets have zero limit mass. + +For every bounded continuous test phi(theta,y,t), the same compact-box domination as S13, followed by the actual failure-tail bound S9, gives + + r^(-3) E_QW[phi(Theta_r,T_r,D_r);F_r] + -> integral phi(theta,Y_*(theta),h_*(theta)) dmu_fail(theta). + +Give bounded tests arbitrary values at either sentinel. On compact good couplings those sentinel events eventually vanish, and S9 exhausts their omitted failure mass. In particular their tilted mass is o(r^3). The joint location/death assertion is weak convergence. Dividing by the positive total mass alpha_1+alpha_2 identifies the genuine conditional replacement-partner law under the original pair-Palm law given failure. + +The corresponding normalized actual lifetime on failure is + + (b-d_f(M))/r^3 -> -h_*(theta) in (0,k), + lambda(theta)=-h_*(theta)/k in (0,1). + +Its law is the pushforward of mu_fail/(alpha_1+alpha_2); thus the local geometry determines which saddle pairs and its lifetime under this specified law. This is not an intensity obtained by duplicating rejected candidates attached to the same maximum. + +The weak finite replacement-location measures have converging total masses and a finite limit on R^2, hence they are tight. Therefore for every deterministic cutoff delta_r with delta_r/r->infinity, + + Q_r^W(F_r, dist(partner(M),midpoint)>delta_r)=o(r^3). + +For the proof, first fix a large scaled radius A, use tightness outside A, and then use delta_r/r>A for all sufficiently small r; let A increase. The o(r^3) sentinel mass is included. No growing-radius uniform root enclosure is asserted or needed. This localizes the **actual failed elder pairing**; it neither needs nor promotes the contextual factorial-count localization results. + +## 9. A compact rejected-candidate lifetime coefficient + +The local-to-elder bridge is now supplied by Theorem E and the full marked failure limit S2, rather than postulated from a count theorem. One precise lifetime consequence follows by consuming the parent's *existing* compact-parameter domination; no new uniform convergence theorem is needed. + +Retain exactly [P] §§10–12's populations: candidate maximum/saddle pairs and the selected subset, per unit midpoint volume, with birth b in a compact interval B and positive gap mark k in a compact interval K, both intervals of positive lengths and K bounded away from zero, and directed orientation u in S^1 with ordinary arc measure. Write + + A_r(b,k,u)=12 pi_r(u;v_r) Z_r(b,k,u)/r^2, + A_0=12 pi_0(u;v_0) z_0(b,k,u), + a_fail(b,k,u)=alpha_1(b,k,u)+alpha_2(b,k,u)>0. + +For every fixed (b,k,u), S2 identifies + + (1-p_r(b,k,u))/r^3 -> a_fail(b,k,u). + +The source [P] (7.8), with its compact parameter hypotheses, bounds this ratio by a common finite constant for all sufficiently small r, b,k and frames in these compacts. Equations (10.2)–(10.3) bound A_r uniformly and identify its limit. These are already-displayed source bounds, not uniformity deduced from our pointwise cubic topology theorem. The conditional a_fail is Borel as the limit of Borel quantities, and is bounded by that same cap constant. + +Let nu_rej^{B,K}(ell)=nu_cand^{B,K}(ell)-nu_eld^{B,K}(ell) denote the density of the **nonselected candidate counting measure**, in the parent's canonical Kac–Rice versions. With r=(ell/k)^(1/3), [P] (11.1)–(11.2) give exactly + + ell^(-2/3) nu_rej^{B,K}(ell) + = integral_(B x K x S^1) + [A_r(b,k,u)/(3k^(5/3))] + [(1-p_r(b,k,u))/r^3] db dk d sigma(u). + +The common compact bounds dominate this integrand by an integrable constant, because k is bounded below and the integration domain has finite measure. Pointwise convergence and dominated convergence prove + + nu_rej^{B,K}(ell) ~ C_fail^{B,K} ell^(2/3), + C_fail^{B,K}=integral_(B x K x S^1) + A_0 a_fail/(3k^(5/3)) db dk d sigma(u). + +The coefficient is finite and is strictly positive when B and K have positive lengths. It is an identified Gaussian integral, not a numerical estimate. The original full z_0 cancels if the integrand is written as + + A_0 a_fail/(3k^(5/3)) + = 4 pi_0(u;v_0) k^(-5/3) + integral_(D_k) w(theta) h_0(0,a,beta,c) + 1{n(theta)>0} dtheta. + +There is still only one original pin density, one determinant weight and one full normalizer. By direct integration, + + E N_rej^{B,K}(0,t] ~ (3/5) C_fail^{B,K} t^(5/3). + +This is a sharp coefficient for the selected-versus-candidate **difference** on the exact compact-mark population. It is not a separate second-order expansion of nu_eld: the convergence of A_r itself has not been given an expansion here. It is also not a density of replacement elder bars obtained by counting every rejected candidate. One actual maximum can participate in several rejected candidate pairs; doing that pushforward would require the appropriate de-biasing/selection identity. Theorem S3 instead identifies the actual death-height law under the specified candidate pair-Palm law conditioned on failure, which is a different, well-defined estimand. + +No unrestricted version of this difference asymptotic is inferred. [U] already proves a positive far rejected density and excludes such an inference. No k down to zero limit, new parameter uniformity, higher-dimensional lift or numerical coefficient is added. + +## 10. Hypotheses, exclusions and falsifiers + +The deterministic Theorem E assumes the exact displayed planar cubic, k>0, strict endpoint typing, Sigma_k and Delta_k excluded, pinned C^2 convergence on every fixed disk, and embedded shrinking charts. Exact partner statements additionally assume the actual global Morse distinct-value locus. Only the unique limiting position mark additionally excludes D=0; exact finite-index partner identification permits tied limiting heights. It does not need Morse–Smale or a separatrix identification. + +The Gaussian statements consume the exact original six-pin regression, full Z_r, nonsingular finite contact-jet covariance, full-field conditional C^4 moment coupling, and the reviewed cap implication/integration interfaces of [P] with all three reading rules. The algebraic exclusions are Lebesgue null, not discarded positive-probability events. The tightness proof includes the scalar large-lambda and fourth-derivative branches as o(r^3), handles the typed boundary by continuous determinant weight, and keeps all random-field correlations. Theorem S concerns fixed planar marks. Section9 uses only the parent's already uniform compact-mark bound for its explicitly restricted lifetime difference. + +The following would falsify the corresponding new step: + +* A generic typed displayed cubic whose maximin level differs from h_*, or a pinned local C^2 sequence on compact surfaces whose actual death fails to approach h_*, falsifies Theorem E. A sequence with no cubic window saddle but eventual preemption of S would specifically falsify its exact empty-sector argument. +* A failure of the cap's global older-path implication, or of the endpoint-independent residual and rare scalar integration with the original W_r/Z_r, would invalidate S4–S9 and reopen full failure exhaustion. +* A leading failure mass escaping every bounded midpoint-jet box would contradict S9; a nonzero leading typed-boundary mass would contradict S11–S14. These are explicit measure claims, not inferences from tests or finite samples. +* An erroneous Jacobian/full normalizer in G10 or the parent's radial lifetime ledger would invalidate the corresponding marked/lifetime identities. Unrestricted or replacement-bar uses of Section9's compact rejected-candidate coefficient are outside its scope. + +The new scientific candidate is the planar local-to-global actual elder partner, full rescaled selector-failure law, joint conditional replacement location/death/lifetime law, and compact rejected-candidate lifetime coefficient. Mixed factorial pairs remain two additional saddle witnesses rather than max/saddle persistence pairs. Higher-dimensional replacement geometry and any unrestricted refined lifetime law remain unproved by this note. Source-bound review and publication do not change any controlling status, graph, prize, premise or parent acceptance. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/MARK_D.md b/frontiers/lifetime_moment_boundary_20260930/sources/MARK_D.md new file mode 100644 index 000000000..923d3236b --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/MARK_D.md @@ -0,0 +1,468 @@ +# Hard-negative fibres identify the actual elder law in every fixed dimension + +Author-side successor by OpenAI, 30 September 2026. Scientific effect NONE. +Disposition: AUTHOR_SIDE / HOLD pending qualified source-bound review. +The root, concave_fibre_geometry and higher_d_source_interface are contributors; +their source audits are not nonauthor acceptance. All source conditions below +are retained. No count moment or remote-count theorem is a premise. + +The complete deterministic lift, including its counterexample, is proved in +[HARD_FIBRE_LEMMA.md](HARD_FIBRE_LEMMA.md). The detailed analytic source +interface and retained matrix integral are in +[SOURCE_INTERFACE.md](SOURCE_INTERFACE.md). [SOURCES.json](SOURCES.json) +binds the complete public sources by immutable commit/path/blob/SHA256. +Together these files expose the full new argument; source hypotheses are not +accepted merely by being linked or having verified hashes. + +## 1. Model and the actual pairing estimand + +Fix an integer d>=3, torus side L>0, birth b in R, positive gap mark k, and an +orthonormal axial/transverse frame R. Use the exact variance-one periodized +Bargmann–Fock Gaussian field of [P], with strictly positive Fourier weights. +At M=(-r/2,0), S=(r/2,0), impose the original observations + + f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0. + +Q_r is this original 2(d+1)-observation Gaussian regression. Set + + W_r=F_d(H_M) F_(d-1)(H_S), Z_r=E_(Q_r) W_r, + Q_r^W=(W_r/Z_r) Q_r. + +F_j is absolute Hessian determinant restricted to j negative eigenvalues. +There is one original weight and one FULL normalizer. No adjacency restriction, +extra pin density or good-event normalizer is inserted. [P] gives +Z_r/r^2 -> z0>0, with its congruence erratum [ERR] and convergence-in-law reading +rule [REC]. The ordinary global elder death is the superlevel maximin + + d_f(M)=sup_(gamma(0)=M, f(gamma(1))>b) min_t f(gamma(t)). + +An essential maximum has d_f(M)=-infinity. On the almost-sure global Morse +distinct-value locus of [P] §8, d_f(M) is the value of the actual elder merge +saddle when finite. Define + + F_r={actual ordinary elder partner of M is not S}, + p_r=Q_r^W(F_r^c), D_r=(d_f(M)-b)/r^3. + +The event is Borel by [P] §8; whole-field marked Kac–Rice uses the exact [BOREL] +repair, not an assumption that elder selection is a continuous local mark. +These are candidate pair-Palm estimands; counting every rejected candidate +does not count each replacement elder bar once. + +## 2. Original spectral coordinates and the soft cubic + +Put m=d-1. Write the signed ordered eigenvalues of -D_y^2 f(0) as +mu1<...0, -k P in C2 + +on every fixed soft disk. Write M0=(-1/2,0), S0=(1/2,0) for the normalized +soft pins, keeping M,S for the physical original pins. The graph retains +EXACT critical pins and values G_r(M0)=0, G_r(S0)=-k, since the original hard +gradients vanish there and the hard stationary point is unique. The proof of the supplied C2 interface, +including its derivative powers, is displayed in SOURCE_INTERFACE §2. + +The missing geometry is a full-dimensional barrier, not another root count. +For a fixed soft disk containing the planar collar and doubled critical chords, +let eta=min_j h_j>0. Taylor and the fixed C4 bound give + + f_ww <= -(eta/2) I + +on that soft disk at physical r scale and on a hard ball of radius c0 r, for +all sufficiently small r. This is a pointwise fixed-target bound; constants may +depend on eta. Twice integrating along a straight hard fibre gives + + f(t,zeta_r(t)+v) <= f(t,zeta_r(t))-(eta/4)|v|^2. + +Choose fixed rho with D0=eta rho^2/4>k. The centred hard tube +|v|<=rho r^(3/2) fits inside the c0 r ball, since zeta_r=O(r^2) and +r^(3/2)=o(r). On its hard boundary the normalized drop is at least D0. +This is the quantitative hard-depth hypothesis of HARD_FIBRE_LEMMA. The r^2 +scale of the critical graph alone would give a drop of order r^4 and would +not block escape across a death window of order r^3. + +For each continued soft window root Y_r, and the exact normalized S0, put + + h_r=max({-k} union {G_r(Y_r):Y in C(P)}). + +Theorem H (actual highest partner). Under the displayed fixed-target hypotheses, +for every sufficiently small r, + + d_f(M)=b+r^3 h_r. (H1) + +On the actual global Morse distinct-value locus, the unique highest continued +graph saddle is the actual ordinary elder partner. Its full Hessian has one +positive and d-1 negative directions. If n=0 the partner is EXACTLY S; if n>0 +a continued local window saddle preempts S. Limiting ties do not obstruct H1; +the comparison is between the finitely many ACTUAL heights at that r. + +Proof. [LOCAL] §§3–6 construct, in the reduced profile, compact disks K_(r,h) +for every h_rh_* pointwise in this coupling. When D!=0, the highest soft root +Y_*(Theta) is unique and the actual partner divided by r tends, in the original +physical axial frame, to + + T_*(Theta)=R(X_*, O(Z_*,0,...,0)). (H2) + +The hard graph coordinates divided by r vanish. Sign changes of an eigenframe +change Z_*,tau and O together, leaving H2 invariant. No separatrix or +Morse–Smale premise is needed. HARD_FIBRE_LEMMA §8 gives an explicit smooth +counterexample if the contained hard-depth tube is omitted, even when graph +convergence is exact and the physical hard Hessian is uniformly negative. + +## 4. Full failure-measure exhaustion from the endpoint matrix + +H1 is pointwise in fixed jets. It does not alone identify the full rare failure +law. This section supplies the required exhaustion from the actual endpoint +matrix, independently of any count or near-occurrence event. + +On typed support put B_M=-D_y^2 f(M)>0, with ordered eigenvalues + + 0=2, + v=m(m-1)/2. + +[P] (4.2)–(4.3) regress the whole field on the ENTIRE endpoint matrix and give +an independent residual g_r. Put J=1+||g_r||C4, U=J+Lambda. All fixed J moments +are uniformly bounded, J is independent of the whole matrix, and + + K=1+||f||C4 <= C(J+||B_M||F) <= C U. (E1) + +Here Lambda is among the retained eigenvalues lambda2,...,lambdam; its being +independent of lambda1 for this conditional integration is why m>=2 matters. +No independence among matrix entries, derivative suprema or internal residual +evaluations is asserted. + +The exact global good-cap implication [CAP]/[P] §8 gives F_r subset G_r^c, + + G_r={lambda1>[4/(3k)] r M3^2, r M4<=3k/10}. + +[P] (7.7), divided by the FULL Z floor once, bounds the M4 exception by O(r^4), +which is o(r^3). On the remaining depth branch, + + lambda1<=D r U^2, + W_r<=C r^2 U^(2m) lambda1(lambda1+C r U). (E2) + +The second factor is essential: it retains the saddle mixed-square term in +[P] (6.2). The endpoint matrix density has Gaussian bound [P] (3.5); its +spectral Jacobian is at most C Lambda^v. For every retained event E depending +only on J and the hard eigenvalues, the SAME integral gives + + E_Q[W_r;depth,E] + <= C r^2 E_J integral_hard exp(-c sum_hard lambda_j^2) + Lambda^v U^(2m) 1_E + integral_0^(D r U^2) lambda1(lambda1+C rU) d lambda1 + <= C r^5 E_J integral_hard exp(-c sum_hard lambda_j^2) + Lambda^v U^(2m+6) 1_E d lambda_hard. (E3) + +The extension of the lambda1 interval beyond its actual ordered interval is +an upper bound of a nonnegative majorant. U and E are retained while integrating +lambda1; its Gaussian factor may be dropped. Equation E3 is the retained-mark +version of [P] (7.4), not a new independence or cutoff inversion. + +Taking E={U>A}, using arbitrarily high J moments and Gaussian polynomial +integrability, and dividing once by Z_r>=z_*r^2, gives, for every fixed p>0, + + limsup_r r^-3 Q_r^W(F_r,U>A) <= C_p A^-p. (E4) + +Taking E={lambda2<=epsilon} in E3 gives + + lim_epsilon->0 limsup_r + r^-3 Q_r^W(F_r,lambda2<=epsilon)=0. (E5) + +Indeed integration of the hard Gaussian polynomial over this strip tends to +zero; for epsilon<=1 a constant times epsilon suffices after integrating all +other hard variables and the J moments. All higher-corank intersections are +included, with no inverse-hard-eigenvalue or inverse-gap factor. + +Weyl's eigenvalue inequality and Hessian Lipschitz control give + + |muj-lambdaj| <= r M3/2 <= C rU. + +On the depth branch this implies + + |s|<=D U^2+C U, |h|+|tau|+K<=C U. (E6) + +Consequently E4 is tightness of the FULL failure measure in the actual midpoint +jets and derivative norm, not just a compact-near count estimate. On U<=A, +midpoint h2<=epsilon implies endpoint lambda2<=2epsilon for sufficiently small +r at fixed epsilon,A. Equations E4–E5 and the M4 exception yield + + lim_epsilon->0 limsup_r + r^-3 Q_r^W(F_r,h2<=epsilon)=0. (E7) + +Midpoint hard eigenvalues of either sign near zero can carry no lost leading +failure mass. Inverse hard eigenvalues are used only at fixed positive targets +for graph convergence; they never enter E3. The complete detailed derivation +is SOURCE_INTERFACE §§4–5. No scalar shortcut is used: for d=2 one must retain +[LOCAL] §8's separate scalar near/far split, since Lambda=lambda1 there. + +## 5. The identified full elder-failure measure + +Let the limiting hard domain be 0=2) h_j^3 + product_(2<=i0} + ds dh dO d tau. (S1) + +It is extended by zero off the stated hard/typed domain. The deterministic +classifier bound [SC] (19) proves integrability: if x=-s>0 and n>0, then +x<=2|B|+(48kD^2)^(1/3), and + + integral_s w_+ 1{n>0} <=384k^2 |B|^3+2304k^3 D^2. + +The remaining hard spectral factor is a polynomial, while h0 has uniform +Gaussian decay in the ORIGINAL orthogonally rotated raw variables. All angular +variables remain inside the integral. Positive Gaussian density on open typed +nonempty sectors gives mu_fail(total)>0. In [SC] (20)'s notation its mass is + + a_fail=a1+a2. (S2) + +This consumes the displayed deterministic integral, not [SC]'s global count +limit. Its remote singleton coefficient beta_far does not enter S1 or S2. + +Theorem F (full fixed-dimensional selector-failure law). Under the exact +source interfaces and H1, in the extended spectral law of §2, + + r^-3 Q_r^W(Theta_r in dot, F_r) -> mu_fail in total variation, + (1-p_r)/r^3 -> a1+a2>0. (S3) + +Proof. On a fixed bounded spectral box with hard eigenvalues bounded below, +[SC] §2 disintegrates the actual additional full jet. Its soft eigenvalue +change has Jacobian r c_m J_r, where + + J_r=product_(j>=2)(h_j+rs) product_(2<=i product h_j^2 w_+. Its limiting +weight is zero off the typed domain and ON the typed boundary; selector +convergence is unnecessary there. For a typed generic target H1 gives the +actual indicator I=1_F eventually equal to 1{n>0}. The excluded algebraic +sets are null, not discarded positive mass. + +The independent complete-jet coupling [SC] (5)–(6) gives +K<=C_box(1+||F||C4). Its two-short-column bound [SC] (15) bounds W_r/r^4 by an +integrable fixed polynomial in that coupling norm on the box. Thus pointwise +convergence, the exact weight limit, and dominated convergence give compact +L1 convergence of S4 to S1, including near the typed boundary. Repeated hard +spectra are null; no uniform inverse-gap estimate is required on the box. +E4–E7 exhaust the ACTUAL failure measure, and S1 is integrable. First exhaust +bounded boxes and then let the positive hard lower bound decrease to zero. +The limsup of both excluded tails tends to zero. Compact L1 plus those two +tails is global L1, which is precisely the total variation statement S3. +Taking total masses proves its coefficient. QED. + +This establishes the pairing relevance of the microscopic soft geometry. +It does not equate two extra saddle witnesses with one maximum/saddle pair. +Neither a factorial moment nor fixed-remote mixing is used to infer selection. +When the fixed-r almost-sure global Morse/distinct-value locus is used in this +conditional coupling, take a countable intersection along an arbitrary fixed +sequence r_j->0. Dominated convergence along every such sequence proves the +stated limit. No common probability-one locus for uncountably many pinned laws +is assumed. + +## 6. Actual replacement location, death and lifetime + +On F_r let T_r be the actual elder partner's displacement from the midpoint, +in the local torus chart, divided by r. Use an isolated sentinel if the partner +is essential or outside that chart. For the same reason use a sentinel in D_r +when d_f(M) is not finite. These exceptions have r^-3 mass tending to zero: +on each regular compact jet sector H1 provides a local older path and actual +partner for all sufficiently small r; E4–E7 exhaust the remaining failure mass. + +Theorem M (actual marked law). The finite measures + + r^-3 Q_r^W((Theta_r,T_r,D_r) in dot, F_r) + +converge WEAKLY to the pushforward of S1 under + + Theta -> (Theta,T_*(Theta),h_*(Theta)). (M1) + +The conditional-on-failure law is M1 divided by a_fail. The lifetime mark +(b-d_f(M))/r^3 converges in that law to -h_* in (0,k), and its fraction of the +candidate lifetime converges to -h_*/k in (0,1). + +Proof. At a generic positive hard target H1 identifies the exact actual +winning continued root. Off the additional null set D=0, [LOCAL] §6's winning +soft root is unique; continuation and the vanishing r-scaled hard graph give +H2 and D_r->h_*. Insert any bounded continuous joint test function in S4. +The same compact dominated-convergence argument proves convergence of its +integral. E4–E7 remove the tails; sentinel mass has already been excluded. +No continuity across the polynomial transition sets is assumed. QED. + +Weak, not total variation, convergence is claimed for these real-valued marks. +Tightness of M1 also gives, for EVERY deterministic delta_r->0 with +delta_r/r->infinity, + + Q_r^W(F_r, dist(actual elder partner,midpoint)>delta_r)=o(r^3). (M2) + +For a given tolerance choose a fixed large radius with arbitrarily small +limiting marked tail, use weak tightness including the sentinel, then use +delta_r/r eventually larger than that radius. This is actual failed-partner +localization, rather than localization of a count-defined witness. + +## 7. Exact compact lifetime-density consequence + +Retain [P] §§10–12's exact compact population: birth b in an interval B, gap k +in an interval K, both of positive length and K bounded away from zero, axial +orientation u in S^(d-1) with ordinary area measure, per unit midpoint volume. +Choose a Borel transverse completion R(u) as in the source. The coefficient +a_fail is independent of that completion: the original selection probability +is physical and independent of a transverse coordinate choice, and S3 is its +unique limit. This does not assert torus rotational invariance or birth-factor +cancellation in higher dimensions. + +Use the source's ACTUAL radial ledger, in every fixed d: + + A_r=12 pi_r(R(u);v_r) Z_r/r^2, + nu_cand(ell)=ell^-1/3 integral_(B x K x S^(d-1)) + A_((ell/k)^(1/3))/(3k^(2/3)). + +The selected density has the additional p_r. The dimension powers have already +cancelled in [P] §10; no new spatial Jacobian or dimension-dependent lifetime +power may be inserted. [P] (7.8) supplies a common compact-mark bound on +(1-p_r)/r^3, and [P] (10.2)–(10.3) supply common bounds and A_r->A0. These are +existing domination interfaces, not uniformity inferred from pointwise H1. + +With r=(ell/k)^(1/3), the exact nonselected candidate density therefore has + + ell^-2/3 [nu_cand^(B,K)(ell)-nu_eld^(B,K)(ell)] + = integral_(B x K x S^(d-1)) + A_r/(3k^(5/3)) [(1-p_r)/r^3]. + +Dominated convergence and S3 identify + + nu_rej^(B,K)(ell) ~ C_fail^(B,K) ell^(2/3), + C_fail^(B,K)=integral_(B x K x S^(d-1)) + A0(b,k,u) a_fail(b,k,u)/(3k^(5/3)) db dk d sigma(u). (L1) + +C_fail is finite and strictly positive. The full z0 cancels if A0=12pi0z0 is +substituted into S1; there is still one original pin density and no additional +normalization. + +L1 is the compact candidate-minus-selected counting measure. It is not a +replacement-bar intensity obtained by counting all rejected candidates, nor a +standalone second-order selected-density expansion. An unrestricted refined +difference does not follow; its far rejected population is different. No new +k->0, d->infinity, volume, or escaping-mark uniformity is claimed. + +## 8. Hypotheses, explicit falsifiers and review boundary + +H1 assumes a generic typed cubic and its pinned reduced C2 convergence, an +embedded fixed soft collar, and a CONTAINED negative hard tube with drop D0>k. +The graph and tube are proved from [SC]'s actual smooth field on fixed positive +hard targets. The exact finite-r partner additionally uses the source's actual +global Morse distinct-value locus. A unique limiting position additionally +excludes D=0, a null set. The deterministic lemma does not require probability, +Morse–Smale, a separatrix assertion or control of the exterior. + +The stochastic statements require the original positive-spectrum field, +nonsingular contact/full-jet regression, independent endpoint-matrix residual +with all C4 moments, actual cap implication, double soft-factor weight bound, +full normalizer floor, and the original spectral density/coupling interfaces. +They are source-bound conditions, not consequences of tests or publication. +All inverse-hard constants are pointwise only. The actual rare failure tail +and multiple-soft exclusion are proved by E3–E7, rather than borrowed from a +near-count event. The full typed boundary is treated by its continuous zero +weight. The Haar-gauge extension is part of the precise measure statement. + +Falsifiers: + +- A full path bypassing a contained hard-depth cap at height above h_r, or a + graph-lifted moving chord whose exact minimum is not the continued root, + invalidates H1. The thin-tube counterexample in HARD_FIBRE_LEMMA §8 explains + why graph convergence alone is not a substitute for that hypothesis. +- A leading failure mass escaping every bounded midpoint jet set, or staying + at a nonvanishing fraction on h2<=epsilon as epsilon->0, contradicts E4/E7. + A defect in the parent cap implication, original residual independence or + retained matrix soft integral would reopen that analytic step. +- A dropped saddle mixed-square factor, extra restricted normalizer, omitted + hard/sign multiplicity or GOE substitution changes S1 and invalidates S3. +- A positive leading essential/sentinel mass contradicts M1. Total variation + for real-valued location/lifetime marks is outside the claim. +- Replacement-bar, unrestricted refined-density, new parameter-uniform or + numerical uses of L1 are outside this proof's interface. + +This is a new fixed-dimensional actual elder/failure/lifetime candidate, with +full proofs and source identities exposed. d=2 remains the separate [LOCAL] +scalar proof. No controlling acceptance, status, graph, prize, Boolean, parent +proof or source history is altered. Qualified nonauthor review must bind this +successor itself; acceptance of a count packet or of [LOCAL] alone is not an +acceptance of this higher-dimensional lift. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/P.md b/frontiers/lifetime_moment_boundary_20260930/sources/P.md new file mode 100644 index 000000000..dfed3b8d3 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/P.md @@ -0,0 +1,494 @@ +# Uniform matrix-cap selection and unrestricted short-lifetime asymptotics + +**Object:** UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1. +**Author:** OpenAI / ChatGPT, 24 September 2026. +**Disposition:** author-side mathematical candidate; no nonauthor acceptance, formal verification, numerical finite-band certificate, or program-level closure is asserted. + +This is an additive continuation of MARKED-CYLINDER-CAP-20260924-v1, SHA256 `0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc`, reproduced unchanged in `sources/`. It addresses the matrix-probability, angular-uniformity, compact-mark, and lifetime-composition interfaces of campaign #61. It does **not** extend the predecessor's numerical constants to new parameters. Instead it derives new, non-numerically-evaluated constants by a uniform Gaussian argument. + +## 1. Model, convention, and the three theorems + +Fix a dimension d >= 2 and a side length L > 0. On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance + + K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2) + / sum_{n in Z^d} exp(-|Ln|^2/2). + +In particular d=3, L=24 is an actual three-dimensional SIDE24 application. No replacement by a nonperiodic covariance is made. Fix compact intervals B=[b_-,b_+] and K=[k_-,k_+] with 00, use the embedded local cylinder centered at the origin with pins + + M=-(r/2)u, S=(r/2)u, + f(M)=b, f(S)=s=b-k r^3, + grad f(M)=grad f(S)=0. + +Q_{r,b,k,R} is the continuous Gaussian regression law at these 2(d+1) observations, not a conditioning on a positive-probability event. A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue. Set + + W_r = |det H_M det H_S| 1{H_M<0, index(H_S)=d-1}, + Z_r = E_Q W_r, dQ^W = (W_r/Z_r)dQ. + +The index counts negative eigenvalues. The FULL normalizer Z_r has no extra pin-density, coordinate Jacobian, or adjacency condition. Let p_r(b,k,R) be the Q^W probability that the **global ordinary superlevel elder death partner** of M is S. + +**Theorem A (uniform selection candidate).** There exist r_*>0 and C=z_* r^2 and the full transverse-matrix boundary layer, including simultaneous small transverse eigenvalues. It does not presume isotropy under arbitrary rotations of the torus. + +**Theorem B (compact-mark density candidate).** Suppose B and K have positive lengths. Count ordered maximum/saddle pairs with 00, + 0 <= nu_cand(ell)-nu_eld(ell) <= C_{B,K} ell^(2/3), (1.2) + +as ell decreases to zero. An exact finite-dimensional Gaussian integral for c_{B,K} appears in Section 11. + +Theorem B is a COMPACT MARK WINDOW result, especially its quantitative difference estimate. Theorem C below separately removes the mark and distance restrictions for the LEADING asymptotic. There is no stated numerical value of C, r_*, z_*, or c_{B,K}, and no claim that r_*=1/20. The new asymptotic normalizer proof does not replace or re-certify the historical H3 finite-band floor. + +**Theorem C (unrestricted finite-lifetime leading density candidate).** For the same fixed torus and dimension, let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume. There is a positive finite constant c_{d,L} such that + + nu_cand^all(ell) ~ c_{d,L} ell^(-1/3), + nu_eld^all(ell) ~ c_{d,L} ell^(-1/3). (1.3) + +All birth heights, all positive scaled-gap marks and all spatial separations on the torus are included in (1.3). The essential global-maximum class is excluded from the finite-lifetime count. The constant is an explicitly identified finite Gaussian/angular integral, with the birth and gap integrals analytically eliminated in Section 15. No elementary closed form or numerical enclosure is claimed. Crucially, the compact-window O(ell^(2/3)) DIFFERENCE estimate in (1.2) is not asserted for the unrestricted densities. Theorem C has a separate domination and off-diagonal proof, not a change of label on Theorem B. + +## 2. Positive Fourier spectrum and finite-jet rank + +Poisson summation gives + + K_L(z)=sum_{n in Z^d} a_n exp(2 pi i n.z/L), + a_n=exp(-2 pi^2 |n|^2/L^2) / sum_j exp(-2 pi^2 |j|^2/L^2)>0. + +Thus every Fourier mode has positive variance, and sum_n sqrt(a_n)(1+|n|)^q is finite for every q. A real sine/cosine expansion therefore converges in every C^q norm in every finite L^p norm. In particular the field has a smooth version and all finite moments of its global derivative suprema exist. + +Any finite list of distinct derivative evaluation functionals at distinct sites is linearly independent. Indeed a vanishing-variance linear combination must annihilate every Fourier mode because every a_n is positive. As a distribution it is a finite sum of derivatives of point masses with every Fourier coefficient zero, so it is zero. Smooth test functions supported at one site and with arbitrarily prescribed finite jets show that every coefficient is zero. At one site in a rotated frame the same assertion also follows from a polynomial P(R^T n) vanishing on all n in Z^d: the invertible rotation reduces this to a polynomial vanishing on the full integer lattice, hence the polynomial is zero. + +Only lists without repeated derivative functionals are called distinct. A Hessian is parametrized by its independent symmetric entries, not by d^2 allegedly independent entries. + +These arguments justify positive covariance for the original pins at each pair of distinct points and for every residual jet list modulo the pins used below. They also explain why finitely many observed modes or a numerically positive sampled covariance would not suffice for this proof. + +## 3. An exact nonsingular contact frame for all orientations + +Write x for the axial coordinate and y_1,...,y_m for the transverse coordinates, m=d-1. On the line y=0 set a=-r/2, c=r/2. Order the original observations as + + O_r=(f(a), f_x(a), f(c), f_x(c), + f_y1(a),f_y1(c), ..., f_ym(a),f_ym(c)). + +This is a permutation and orthogonal re-expression of the specified physical pins. Define U_r=T_r O_r by four axial rows + + U0 = (f(a)+f(c))/2, + U1 = (f(c)-f(a))/r, + U2 = (f_x(c)-f_x(a))/r, + U3 = (6/r^2)[f_x(a)+f_x(c)-2(f(c)-f(a))/r], (3.1) + +and, for every transverse direction j, two rows + + Vj0=(f_yj(a)+f_yj(c))/2, + Vj1=(f_yj(c)-f_yj(a))/r. + +The transformation is invertible for r>0 and has exact absolute determinant + + |det T_r| = (12/r^4)(1/r)^m = 12 r^(-(d+3)). (3.2) + +The prescribed target becomes exactly + + v_r=(b-k r^3/2, -k r^2, 0, 12k, 0,...,0). (3.3) + +This is not an approximate transformation, and the factor 12 in (3.2) is retained in the lifetime ledger. + +Taylor formulas with integral remainders give mean-square convergence, uniformly over frames, of U_r to the contact jet + + U0*=(f, f_x, f_xx, f_xxx, + f_y1,f_xy1,...,f_ym,f_xym) at 0. (3.4) + +The potentially dangerous U3 cancellation is an averaged third derivative, not a subtraction of uncontrolled random values. Uniform convergence of its covariances, and of cross-covariances with any fixed finite derivative order, follows either from those integral formulas or from the rapidly summable Fourier series. The target v_r converges uniformly on B x K to v_0=(b,0,0,12k,0,...,0). + +The functionals in (3.4) are distinct. Hence their covariance Sigma_0(R) is positive definite by Section 2. Its entries are continuous in R. The orthogonal group is compact, so its least eigenvalue has a positive uniform minimum. Consequently Sigma_r(R) remains uniformly positive definite for all sufficiently small r; Sigma_r and Sigma_r^{-1} converge uniformly. + +This compactness argument establishes existence, not an evaluated interval enclosure. It never equates Sigma_r at different orientations. Full torus rotational invariance is neither assumed nor true in general. + +Let A_r=D_y^2 f(M), represented by its m(m+1)/2 independent entries. Append these entries to U_r. At r=0 the extra functionals are f_yi_yj, i<=j; none duplicates a functional in (3.4). The enlarged covariance is therefore uniformly positive definite at contact and for small r. Its Schur complement shows that the Q-law of A_r has covariance uniformly bounded above and uniformly bounded away from zero. Its mean is uniformly bounded. In symmetric-matrix Lebesgue coordinates it has a density satisfying + + density_Q(A) <= C_0 exp(-c_0 ||A||_F^2), (3.5) + +for fixed positive C_0,c_0. This remains valid after any change of eigenvector orientation. Orthogonal invariance of the density is not required. + +## 4. Uniform conditional derivative moments and the independent residual + +Because Sigma_r^{-1} and v_r are uniformly bounded, the regression energy v_r^T Sigma_r^{-1}v_r is uniformly bounded. For each unconditioned standardized Fourier coefficient xi, covariance Cauchy-Schwarz bounds its conditional mean by the square root of this energy, and its conditional centered variance by one. Minkowski's inequality and the summable Fourier coefficients then give, for every finite p and derivative order q, + + sup_{r,b,k,R} || ||f||_{C^q(X)} ||_{L^p(Q)} < infinity. (4.1) + +The same proof applies to a centered residual after further conditioning on A_r. Write, in one full-field Gaussian coupling, + + f(z)=mu_r(z)+B_r(z).(A_r-E_Q A_r)+g_r(z), (4.2) + +where g_r is jointly independent of the entire symmetric matrix A_r. The regression coefficient B_r and its derivatives through order four are uniformly bounded, by the uniform inverse covariance from Section 3 and bounded derivative cross-covariances. The residual has uniformly bounded C^4 moments of every finite order. + +Independence in (4.2) is not asserted separately for Hessian entries, endpoint variables, or sampled derivative maxima. All residual evaluations are orthogonal to A_r; Gaussian independence on a countable dense collection and continuity give independence of the random smooth field. The residual's internal correlations are retained. + +Let J_r=1+||g_r||_{C^4(X)}, with a fixed coordinate norm. The partial-block operator norms of the cap theorem differ from this coordinate norm by finite constants depending only on d. Hence, for M3,M4 on the local cylinder, + + M3 <= K_0(J_r+||A_r||_F), + M4 <= K_0(J_r+||A_r||_F), + J_r >= 1, J_r independent of A_r, + sup ||J_r||_p < infinity for every finite p. (4.3) + +No inverse Hessian moment is used. Gaussian pin covariance and random field Hessians are different objects. + +## 5. Endpoint identities and the actual full normalizer + +For either pin i in {M,S}, put + + alpha_i=f_xx(i)/r, beta_i=grad_y f_x(i)/r, + A_i=D_y^2 f(i), + H_i=[[r alpha_i, r beta_i^T], [r beta_i,A_i]]. + +The two gradient pins imply the vector and scalar average bounds + + |alpha_i| <= M3/2, ||beta_i|| <= M3/2, + ||A_S-A_M||_op <= r M3. (5.1) + +For example integral_a^c f_xx=0; subtract this average at an endpoint and integrate the M3-Lipschitz bound. For beta use the average of grad_y f_x and its vector derivative. This does not invoke a simultaneous vector Rolle zero. + +The exact height identity gives a weighted mean of f_xxx equal to 12k on [a,c]. Variation of f_xxx over that interval is at most rM4. The same average calculation yields + + |alpha_M+6k| <= r M4/2, + |alpha_S-6k| <= r M4/2. (5.2) + +The block determinant identity, valid even when alpha_i or A_i is zero, is + + det H_i/r = alpha_i det A_i + -r beta_i^T adj(A_i) beta_i. (5.3) + +By (4.1),(5.1), det H_i/r and W_r/r^2 have uniformly bounded moments of every finite order. + +Let A_0 be D_y^2 f(0) under regression on U0*=v_0. Its law is a full-support nondegenerate Gaussian on symmetric matrices. Finite-dimensional Gaussian regression gives uniform convergence of the means and covariances of A_M to those of A_0. Equations (5.1),(5.2) imply that the congruence-scaled endpoint Hessians converge, in probability uniformly over parameters, to + + diag(-6k,A_0) at M, diag(6k,A_0) at S. + +Here the congruence uses diag(sqrt(r),I), so its off-diagonal entries are sqrt(r) beta_i. The limiting matrices are nonsingular almost surely. This follows from the full matrix density and k>=k_->0. Thus the type indicator converges to 1{A_0<0}. Uniform integrability from (5.3) gives + + Z_r/r^2 -> z_0(b,k,R) + := (6k)^2 E[det(A_0)^2 1{A_0<0}], (5.4) + +uniformly on the compact parameter set. + +For completeness, uniformity can be checked by a subsequence argument: any allegedly bad sequence of parameters has a convergent subsequence; continuous Gaussian means/covariances, (5.2), and uniform integrability give the stated limit on that subsequence. The discontinuity set of the type indicator has Gaussian probability zero. Equation (3.5) supplies a common integrable majorant. + +The right side of (5.4) is continuous and strictly positive: the Gaussian law has positive density on an open negative-definite matrix ball. Compactness therefore yields + + 00, with eigenvalues + + 00. + +Determinant monotonicity on positive-definite matrices therefore gives + + |det H_M| <= r (h/2) det B. (6.1) + +This is a canceled-pivot bound: no estimate of 1/alpha_M is taken. + +From ||A_S+B||<=rh, the ordered singular values of A_S are at most lambda_j+rh. The adjugate norm is at most the product of the largest m-1 such bounds. With the empty product equal to one when m=1, (5.3) gives + + |det H_S| <= r(h/2) [lambda_1+(3/2)rh] + product_{j=2}^m(lambda_j+rh). + +Hence the actual typed weight satisfies + + W_r <= (r^2 h^2/4) lambda_1[lambda_1+(3/2)rh] + product_{j=2}^m lambda_j(lambda_j+rh). (6.2) + +The second soft factor includes the saddle mixed-square term. Omitting it, or replacing it by an unrelated order-one factor, changes the radius power. No Gaussian-entry independence is needed for (6.2). + +## 7. Full matrix boundary-layer integration, including corank intersections + +By the predecessor's deterministic marked-cylinder theorem, pairing and the one-branch conclusion hold on + + G_r={lambda_min(-A_M)>[4/(3k)]rM3^2, + rM4<=3k/10}. + +On typed support A_M<0 automatically. It suffices to control G_r^c directly, not infer a converse from a local path-event bound. + +First suppose m>=2. In (4.3) absorb sqrt(m) into K and set U=J_r+Lambda>=1. Then h<=KU. Failure of the depth condition is contained in + + 00 and use h<=K(J_r+lambda). Depth failure implies + + lambda<=Dr(J_r+lambda)^2 + <=2DrJ_r^2+2Dr lambda^2. + +It is contained in the union + + 01/(4Dr). (7.5) + +The large-lambda branch is retained. On the near branch, h<=K(1+4D)J_r^2 for r<=1, J_r>=1. The bounded scalar density, (6.2), and direct integration in lambda give a numerator <=C r^5 E J_r^10. On the far branch, direct fourth-moment Markov with the actual weight gives + + E_Q[(W_r/r^2)1_far] + <=(4Dr)^4 E_Q[(W_r/r^2)lambda^4] <=C r^4. (7.6) + +Thus the scalar numerator is <=C r^5+C' r^6. This separate argument is why the all-dimension statement does not silently drop the scalar far branch. + +Finally, for every dimension, the fourth-derivative exception is bounded directly by + + E_Q[(W_r/r^2)1{rM4>3k_-/10}] + <=[10r/(3k_-)]^4 E_Q[(W_r/r^2)M4^4] <=C r^4. (7.7) + +This is a pointwise Markov bound followed by a joint moment estimate. It is not Cauchy-Schwarz with a missing square root. All joint moments in (7.6),(7.7) are finite by Sections 4-5. + +Divide the numerators by the FULL lower bound (5.5). The result is + + Q^W(G_r^c)<=C_3 r^3+C_4 r^4<=C r^3. (7.8) + +## 8. Pinned genericity and global event measurability + +The following argument applies at every fixed positive r and parameter, not on an asserted common probability-one set for uncountably many laws. + +Away from the pins, consider (grad f(x),Hf(x)v), with ||v||=1. Its range has dimension 2d; its parameter manifold has dimension 2d-1. Modulo the pin observations, these functionals have full covariance rank: gradient and Hessian orders are distinct, and the map H -> Hv from symmetric matrices onto R^d is surjective. For two distinct unpinned points, the map + + (grad f(x),grad f(y),f(x)-f(y)) + +has range dimension 2d+1 and parameter dimension 2d. For a tie to a pinned height use (grad f(x),f(x)-b), or the analogous s map, with dimensions d+1 and d. + +Exhaust the relevant domains by compact sets separated from pins and from coincident points; use finitely many sphere charts when v occurs. Covariance positivity from Section 2 and compactness give bounded Gaussian densities on each such set. An elementary overdetermined-zero lemma applies: on a mesh of size epsilon, a random C^1 map from dimension q to dimension p>q, with derivative norm bounded by N, could have a zero only if one of O(epsilon^-q) mesh values has norm O(N epsilon). The density bound makes the probability O(N^p epsilon^(p-q)), tending to zero. First fix N, then let N increase; the random derivative norm is almost surely finite. This proves absence of all these zeros. + +Pinned Hessians have a full density on their symmetric entries for fixed distinct pins, so their determinants are nonzero almost surely. The pinned heights differ because k,r>0. Countable exhaustion now proves that Q is almost surely Morse with distinct critical values. The transfer to Q^W follows by absolute continuity and 0f(M)} min_t f(gamma(t)), + +with supremum of an empty set equal to -infinity. The event d_f(M)>h can be represented by a countable union of polygonal-path tests, with rational interior vertices and endpoints, all staying strictly above h and with endpoint value above f(M). Strict clearances allow approximation by such paths. For variable M, use countably many coordinate charts and an initial segment depending continuously on M. These are Borel tests in (M,f). Thus equality d_f(M)=f(S) is Borel. On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint. + +On G_r the unchanged deterministic cap theorem controls every boundary exit and supplies a connection through S to a point higher than b. Therefore the GLOBAL ordinary elder partner is S, not merely a local adjacency candidate. Equations (7.8) and the full-probability locus prove Theorem A. The cap theorem also gives exactly one actual ascending branch to M. It does not identify that branch with the transverse ridge. + +## 9. A source-bound weighted Kac-Rice interface + +For clarity, this section specifies why a global elder mark is admissible; a formula for local Hessian index alone would not suffice. + +Apply Kac-Rice first on compact spatial pair domains separated from the diagonal, to the Gaussian vector field (grad f(x),grad f(y)). It is C^1 and has positive definite point covariance for x!=y by Section 2. Its derivative is block diagonal with Jacobian |det H_x det H_y|. Conditional laws of the whole smooth field depend continuously on the imposed gradient values by finite-dimensional Gaussian regression. Derivative moments are locally bounded. + +The expected-integral extension in Armentano-Azais-Leon, *On a general Kac-Rice formula for the measure of a level set*, arXiv:2304.07424v3, Theorem 7.1, gives the identity for bounded nonnegative continuous weights in the location and an additional Gaussian field mark. It is enough first to use cylinder weights depending on finitely many fixed evaluations of f and its derivatives through order two. Their conditional distributions depend continuously on the imposed gradients by Gaussian regression, so the paper's continuity requirements hold. The field is a random element of the separable C^2 space on the torus. On each compact pair domain the two measures on location x C^2 are finite, by the unweighted formula. Cylinder evaluations on a countable dense set, together with location coordinates, generate this Borel sigma-algebra. Equality on their bounded continuous test-function algebra gives equality of the Borel measures by the monotone-class theorem. This extends the identity to bounded Borel global marks, rather than pretending the elder indicator is continuous or lower semicontinuous. No independence between the mark and gradient field is assumed. + +Use the Borel elder mark from Section 8 and the type indicators. Disintegrate the two heights using their nondegenerate joint Gaussian density conditional on the gradients. This gives the full-pin density times E_Q[W] for candidates, and the same quantity times p_r for selected pairs. The continuous Gaussian regression kernel defines the canonical density version; the underlying measure identity is unaffected by height-null-set choices. + +Exhaust the off-diagonal spatial domain toward r=0. The r dr bound proved in the next section makes the expectations locally integrable, and nonnegative monotone convergence completes the near-diagonal formula. No finite-grid approximation of the continuum field is substituted. + +## 10. Exact radial ledger: the dimension cancels + +Let pi_r(R;v_r) denote the density of U_r at (3.3). Since U_r=T_r O_r, + + density_{O_r}(specified pins) + =12 r^(-(d+3)) pi_r(R;v_r). (10.1) + +Use midpoint z, directed separation h=r u, birth b, and gap mark k. The change from the ordered pair (M,S) to (z,h) has absolute determinant one. Polar separation contributes r^(d-1) dr d sigma(u), where sigma is ordinary, NOT probability-normalized, surface area on S^(d-1). The height change (f(M),f(S))=(b,b-k r^3), at fixed r, contributes r^3 db dk. There is no factor 1/2: maximum/saddle pairs are ordered by their different roles. + +Consequently the candidate intensity per unit midpoint volume is + + r A_r(b,k,u) dr db dk d sigma(u), + A_r=12 pi_r(R;v_r) (Z_r/r^2). (10.2) + +The powers are + + spatial (d-1) + height-mark 3 + pin -(d+3) + + determinant-normalizer 2 = 1. + +By Sections 3 and 5, + + A_r -> A_0=12 pi_0(R;v_0) z_0(b,k,R)>0 (10.3) + +uniformly and with a uniform finite upper bound. Although frames were used in the proof, (10.2) is invariant under a rotation of the transverse frame: the original zero-gradient law and determinant weight are unchanged by that orthogonal coordinate change. Thus it is a function of u alone. Uniformity was proved on the compact orthogonal group and descends to the sphere; no nonexistent global continuous transverse frame on the sphere is assumed. + +The selected intensity is r A_r p_r with the same measures. This supplies the domination needed below, rather than assuming a fixed-axis statement can be integrated. + +## 11. Lifetime pushforward and the exact compact coefficient + +For ell=k_->0. This proves (1.2). It is an O(ell^(2/3)) difference, not a claim that either density separately has a second-order expansion with that remainder. Convergence of A_r to A_0 was not assigned a numerical rate. + +## 12. Short bars, moments, and the exact remaining boundary + +Integrating the density versions gives the following expected per-unit-volume results for the same spatial/mark-restricted populations: + + E N_eld(0,t] ~ (3/2)c_{B,K} t^(2/3), + 0<=E[N_cand(0,t]-N_eld(0,t]]<=C t^(5/3). (12.1) + +For any real q>-2/3, + + E sum_{selected, ell<=t} ell^q + ~ c_{B,K} t^(q+2/3)/(q+2/3). (12.2) + +The same leading expression holds for candidates. For q>-5/3 the nonselected difference alone satisfies + + E sum_{candidate but not selected, ell<=t} ell^q + <= C_q t^(q+5/3). (12.3) + +For q<=-2/3 the two individual expectations in (12.2) are infinite for every sufficiently small positive t, because their densities have a positive ell^(-1/3) leading term. Equation (12.3) concerns the nonselected counting measure directly, not subtraction of two infinities. In particular the compact-window inverse-lifetime threshold is p<2/3 for integrability of ell^(-p), with logarithmic divergence at p=2/3. This is unrelated to the inverse-Hessian/Schur thresholds in issue #56. + +The corresponding unrestricted cumulative and individual-moment corollaries follow from Theorem C with c_{d,L} in place of c_{B,K}, by the same integrable-power argument. The compact-window nonselected difference rates (12.1),(12.3) are not automatically unrestricted rates. + +## 13. Removing all birth and gap cutoffs by a direct intensity majorant + +The covariance Sigma_r of U_r depends on radius and frame, NOT on the imposed target. Section 3 therefore supplies a fixed r_0>0, independent of b and k, on which its eigenvalues are bounded above and away from zero. Reduce r_0 below the torus injectivity scale and below one. Do NOT require a globally uniform lower bound for Z_r or a globally uniform constant in (1.1). + +For every b in R, k>0 and 0= c(b^2+k^2), (13.1) + +with fixed c,C>0. For the second inequality it suffices to use its two coordinates b-k r^3/2 and 12k: if r<=1, then b^2<=2(b-k r^3/2)^2+k^2/2. Thus the invertible covariance bound yields + + pi_r(R;v_r) <= C exp[-c(b^2+k^2)]. (13.2) + +Conditional Fourier-coefficient means are bounded by C|v_r| and their centered variances by one. The same summability/Minkowski proof as Section 4 now gives the TARGET-GROWTH bound + + E_Q (1+||f||_{C^3})^p <= C_p(1+|b|+k)^p (13.3) + +for every fixed finite p, uniformly over r and frames on this r_0 band. This is not the compact-target bound (4.1) applied outside its scope; it is its explicit polynomial-growth version. + +By (5.1),(5.3), for r<=1, + + |det H_i|/r <= C (1+||f||_{C^3})^d, + 0<=W_r/r^2<=C(1+||f||_{C^3})^(2d). + +These inequalities remain true when k is arbitrarily small; no division by k or by Z_r has occurred. Combining with (13.2),(13.3) gives the full unnormalized-intensity majorant + + 0<=A_r(b,k,u)=12 pi_r(R;v_r) Z_r/r^2 + <= C(1+|b|+k)^(2d) exp[-c(b^2+k^2)] =: H(b,k). (13.4) + +For every fixed positive r and every b,k>0, Z_r is positive and finite: append BOTH endpoint Hessians to the distinct-site pins; the full residual Gaussian density is positive on a product of an open maximum cone and an open index-(d-1) cone. Thus p_r is defined, even though no globally uniform lower floor has been used. For every fixed b,k>0, Theorem A on a compact neighborhood implies p_r->1 as r->0. + +At a fixed lifetime ell, the near-pair restriction r<=r_0 is exactly k>=ell/r_0^3. After the exact change (11.1), + + ell^(1/3) nu_cand^near(ell) + = integral_{R x (0,infinity) x S^(d-1)} + 1{k>=ell/r_0^3} A_{(ell/k)^(1/3)}(b,k,u) + /(3 k^(2/3)) db dk d sigma, (13.5) + +with the integrand defined to be zero where the indicator is zero. The selected near density has the additional factor p_r. Both integrands are dominated by H(b,k)/(3 k^(2/3)). This majorant is integrable: the k exponent -2/3 is strictly greater than -1 at zero, and Gaussian tails absorb the polynomial at infinity and in b. This explicitly handles the shrinking-k boundary where a compact-window selection constant can blow up. + +For each fixed b,k>0,u, the indicator tends to one, A_r->A_0 and p_r->1. Dominated convergence therefore gives the same limit for the two near densities: + + c_{d,L}=4 integral_{R x (0,infinity) x S^(d-1)} + k^(-2/3) pi_0(R;v_0) z_0(b,k,R) + db dk d sigma(u), (13.6) + 00, + D_u=E[(det A_u)^2 1{A_u<0} | V_u=0]. (15.1) + +Here A_u conditional on V_u=0 is an actual centered nondegenerate Gaussian symmetric matrix, not automatically GOE. Finite-jet rank proves tau_u>0, positive densities p_G(0),p_Vu(0), and D_u>0. All are continuous in the frame and invariant under transverse-frame changes. + +The contact density and cone moment factor as + + pi_0(R;v_0)=p_(f,V_u)(b,0) p_G(0) phi_tau_u(12k), + z_0=(6k)^2 E[(det A_u)^2 1{A_u<0} | f=b,V_u=0], + +where phi_tau is the centered scalar Gaussian density of standard deviation tau. Integrating b by ordinary Gaussian disintegration gives p_Vu(0) D_u. Thus (13.6) becomes + + c_{d,L}=144 integral_{S^(d-1)} p_G(0) p_Vu(0) D_u + [integral_0^infinity k^(4/3) + phi_tau_u(12k) dk] d sigma. + +The scalar integral can be evaluated exactly. With t=12k and the elementary gamma integral, + + 144 integral_0^infinity k^(4/3) phi_tau(12k) dk + = Gamma(7/6) tau^(4/3) / [24^(1/3) sqrt(pi)]. + +Consequently the full leading coefficient is + + c_{d,L}= Gamma(7/6) / [24^(1/3) sqrt(pi)] + * integral_{S^(d-1)} p_G(0) p_Vu(0) tau_u^(4/3) + D_u d sigma(u). (15.2) + +This eliminates both unbounded mark integrals. The remaining angular and negative-definite-cone expectation are finite, strictly positive and specified by derivatives of the EXACT periodized covariance. Formula (15.2) is an explicit finite-dimensional Gaussian representation, not a reported elementary closed form or numerical enclosure. The factor24 is a numerical Jacobian/gamma factor, NOT the side length L; it is the same factor for every L. Changing the field amplitude by a>0 scales the coefficient by a^(-2/3), consistent with nu_(af)(ell)=a^(-1)nu_f(ell/a); this is an additional normalization check. + +## 16. Supplied interfaces, retained limits, and review + +**Interfaces supplied by this candidate:** the actual transverse matrix boundary argument (including d=3); a uniform compact-mark asymptotic full normalizer; all orientations without false isotropy; the global Borel marked Kac-Rice interface; exact pin/radial/lifetime factors; compact-window leading density and quantitative selection difference; a separate integrable unnormalized Gaussian majorant removing ALL birth/gap cutoffs; an off-diagonal bound completing ALL spatial separations; the unrestricted finite-bar leading density and its cumulative/moment consequences; and analytic elimination of birth and gap in its coefficient. + +**Not supplied:** a numerical r_* or C on a prescribed band; a globally uniform O(r^3) probability constant over unbounded b,k; the compact O(ell^(2/3)) difference as an unrestricted remainder; an elementary closed form or numerical enclosure of (15.2); a rigorously evaluated lower coefficient for pairing failure; the original numerical H3/LPW replay; a shrinking-annulus RN certificate; a24-jet displacement enclosure; formal proof verification; nonauthor mathematical acceptance; or unrestricted P15 prize closure. No historical scientific status is changed by this author-side proof. + +The exact-arithmetic program checks pin transforms/targets, block determinants, finite typed-matrix envelopes, spectral Jacobians, radius/lifetime exponents, explicit small-eigenvalue integral models, global target coercivity and coefficient scaling. Semantic mutations challenge the load-bearing factors. These are finite algebra/implementation controls, NOT Gaussian simulation or proof of continuum estimates. + +Priority nonauthor review: (i) desingularized full residual covariance; (ii) uniform all-field moments and independent-matrix regression; (iii) both soft determinant factors; (iv) eigenvalue integration retaining the random largest-eigenvalue width and all corank strata; (v) scalar far branch; (vi) full-normalizer type convergence; (vii) continuous-to-Borel marked Kac-Rice extension; (viii) every factor in the pin/lifetime ledger; (ix) target-growth bound (13.3) and integrability at k=0 in (13.5); (x) distinct-site compact covariance in the far domain; and (xi) parity, b-disintegration and the gamma factor in (15.2). + +### Primary-source attribution + +Armentano, Azais and Leon, arXiv:2304.07424v3, Sections 2 and 7, supply established Gaussian Kac-Rice/expected-integral background. The precise full-field/Borel and pair application is explained in Section 9, not attributed wholesale to that paper. Curry, *The Fiber of the Persistence Map for Functions on the Interval*, arXiv:1706.06059v2, Section 3, supplies elder-rule background; the all-dimensional separating cap is the explicitly identified in-project predecessor. Beliaev, Cammarota and Wigman, *No repulsion between critical points for planar Gaussian random fields*, arXiv:1911.03455, is neighboring two-point/index-asymptotic work, not a source of our elder-selection conclusion. No priority or novelty claim follows from this bounded reconnaissance. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/REC.md b/frontiers/lifetime_moment_boundary_20260930/sources/REC.md new file mode 100644 index 000000000..75da25979 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/REC.md @@ -0,0 +1,309 @@ +# D1 Theorem A selection chain — source-bound reconciliation + +**Object:** D1-CHAIN-RECONCILIATION-20260928-v1. +**Reconciler:** Anthropic Claude, Claude Code session `session_017Mi3hxjaxV45x6zo6o1ee3`. +**Effect:** register reconciliation. This record introduces **no new mathematical claim** about the parent. It binds +existing nonauthor reviews to exact source bytes, discharges every residual item they left open, reviews the two +parent sections that no earlier review covered (§§11–12), and states the resulting scoped status. The register edits +it justifies are listed in §9. `lemma_closed`, prizes and premises are untouched. + +## 1. The reconciled object and its reading rule + +The D1 parent is OpenAI/ChatGPT's UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1, with the imported deterministic +MARKED-CYLINDER-CAP-20260924-v1. The parent bytes are immutable. The object reconciled here is the parent **read +together with** the three amendments in the table below. Every consumer must cite this identity set, not the parent +alone. + +| Role | Path (Math-) | Git blob | Bytes | SHA256 | +|---|---|---|---|---| +| Parent (P) | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | `dfed3b8d318a3ab1950957f393307733a4bef3f2` | 40261 | `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7` | +| Cap import (CAP) | `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` | `0633aca3c2a2882b0de4399da0a75d64c2e6b2e1` | 15160 | `0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc` | +| Amendment E1: congruence erratum | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` | `213594d6ca6a86fb938110f4d166d9ce275a02d0` | 1782 | `bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028` | +| Amendment E2: §9 replacement v1.1 | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` | `fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a` | 9062 | `845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f` | +| Amendment E3: wording W1 and embedding radius | this record, §5 | — | — | — | + +**Reading rule.** + +- Read P §5 with E1. The congruence factor is `D_r = diag(r^{-1/2}, I)`, not the displayed `diag(√r, I)`. +- Read P §9 as replaced by E2. +- Read P l. 165 with W1 (§5 below). +- Read P §1's "sufficiently small r" as including the explicit embedding bound `r < L/(4√2)` (§5 below). + +No formula of P changes. + +**Machine binding.** Each component above is also its own source node in the hard-gate graph +(`frontiers/downstream_gate_20260925/GRAPH.json`), required by the D1 node: E1, E2, CAP (whole file; only §§1–5 +are consumed) and this record's directory, which carries E3 and `LEDGER.json`. A byte change or deletion of any +component therefore proposes D1, D2 and D3 for revalidation even when P's bytes are unchanged (§9, §10). + +**Name collision.** "Theorem A" here always means P (1.1). The fixed-remote mean-measure theorem in +`frontiers/remote_window_20260924/PROOF.md` is also called "Theorem A". That is a different theorem (D4), and +nothing in this record concerns it. + +## 2. Reconciled statements + +The model is P §1: + +- the exact L-periodized Gaussian field on `R^d/(LZ^d)`, for fixed `d >= 2` and `L > 0`; +- the six pins, generally `2(d+1)` observations; +- the typed weight `W_r` and the FULL normalizer `Z_r = E_Q W_r`; +- `p_r(b,k,R)`, the `Q^W`-probability that the global ordinary superlevel elder partner of `M` is `S`. + +The three statements below are accepted at existential scope. There is no numerical `C`, `r_*`, `z_*`, `c_{B,K}` +or `c_{d,L}`. + +- **Theorem A (P 1.1).** For compact `B` and `K ⊂ (0,∞)`, there are `C` and `r_* > 0` such that + `0 <= 1 - p_r <= C r^3`. This holds uniformly over `b ∈ B`, `k ∈ K`, all orthonormal frames and `0 < r <= r_*`. +- **Theorem B (P 1.2).** Let `B` and `K ⊂ (0,∞)` be compact intervals **of positive length** (P l. 37; a singleton + interval gives zero mark-window intensity). The compact-window densities satisfy + `ν_cand ~ ν_eld ~ c_{B,K} ℓ^{-1/3}` with `c_{B,K} > 0`. Their difference satisfies `0 <= ν_cand - ν_eld <= C ℓ^{2/3}`. +- **Theorem C (P 1.3).** All births, all positive gaps and all separations are included, and the essential class is + excluded. Then `ν_cand^all ~ ν_eld^all ~ c_{d,L} ℓ^{-1/3}` with `0 < c_{d,L} < ∞` given by (15.2). This is the + leading term only: no unrestricted remainder or difference rate is included. +- **§12 corollaries.** These are the cumulative and moment consequences (12.1)–(12.3) and their unrestricted leading + forms. + +**Sharpness in d = 2 (composition, separately sourced).** In the plane, the direct lower bound +`1 - p_r >= c r^3` combines with Theorem A to give `1 - p_r = Θ(r^3)` and `ν_cand - ν_eld = Θ(ℓ^{2/3})` on compact +marks. The lower bound is (E2) of `frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md`, +blob `aaefc8da…`, reviewed in `reviews/d1_elder_lower_claude_20260928/`. **No lower bound is asserted for `d >= 3`.** + +## 3. The four components of the chain + +| Component | Sections | Output | Interfaces | +|---|---|---|---| +| (I) Matrix-cap normalizer | P §§2–5 with E1 and W1 | `Z_r/r^2 -> z_0 > 0` uniformly; floor `z_* r^2 <= Z_r <= z^* r^2` (5.4)–(5.5) | A1, A2, A4, A3 | +| (II) Cubic pairing-defect bound | P §§6–7 | `E_Q[W_r 1_{G_r^c}] <= C r^5`, hence `Q^W(G_r^c) <= C r^3` (7.8) | A5, A6, A7 | +| (III) Global elder selection | CAP §§2–5 and P §8 | on `G_r` ∩ {Morse, distinct values}, the global elder partner of `M` is `S`; hence `1-p_r <= Q^W(G_r^c)` | CAP, D1-A | +| (IV) Marked Kac–Rice and lifetime | E2 (§9), P §§10–15 | intensity identities, radial ledger, pushforward, majorant, off-diagonal bound, coefficient (15.2) | D1-B/E2, D1-C, §11, §12, D1-D, D1-E | + +Theorem A = (I) + (II) + (III). Theorem B = Theorem A + (IV) through §11. Theorem C = (IV) + Theorem A, which gives +`p_r -> 1` pointwise in §13. + +## 4. Interface ledger + +**Providers.** OpenAI is the author of P, CAP, E1 and E2. The nonauthor providers are xAI/Grok and Anthropic/Claude. +All lanes share one GitHub account, so no organizational independence is claimed anywhere. The full machine-readable +ledger, with every review locator, is `LEDGER.json`. + +**Review records.** + +| ID | Provider and session | Record | Verdicts | +|---|---|---|---| +| G1 | xAI Grok 4.7 via Cursor, `bc-ce4bf0bc` | [main#63 comment 5841570965](https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5841570965) | D1-A–E ACCEPT | +| C1 | Anthropic Claude, this session | `reviews/d1_theorem_a_nonauthor_20260928/REVIEW.md` (blob `bc11369c…`, Math-#106 v2) and [main#63 comment 5873873903](https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5873873903) | A1–A7, CAP, Theorem A and E2 C1–C6 ACCEPT; W1 | +| G2 | xAI Grok, Lucas lane | `reviews/d1_section9_borel_grok_lucas_20260928/REVIEW.md` (blob `48ea9ad5…`) | E2 C1–C6 ACCEPT. Narrative record; its runner is absent. | +| G3 | xAI Grok via Cursor, replay rounds 3–9 | main `incoming/grok-session-20260926-replay/` | Partial slices; see the table below | +| G4 | xAI Grok 4.7 via Cursor, `bc-81faa729` and `bc-dfbce246` | [main#67 5841270276](https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841270276), [5841782206](https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841782206) | While reviewing D2, re-derived P (6.2), the depth-failure integral, the m=1 split, and the far and M4 Markov pieces. Also checked the (11.3) = (13.6) coefficient identity. | +| C2 | Anthropic Claude, this record | §6 below | §11 and §12 ACCEPT; spot re-derivations of (13.1), (13.4) and (15.2) | +| O1 | OpenAI technical review of Math-#106, [5341593068](https://github.com/d6g8k5htny-coder/Math-/pull/106#pullrequestreview-5341593068) | — | Confirms W1 and the A5/A6 fixtures. **Author provider, so non-discharge.** | + +**Interfaces.** "Full" means a nonauthor provider re-derived the whole interface. "Partial" means a nonauthor provider +checked part of it; partial checks never discharge on their own. `reconciliation_check.py` computes the full-depth +counts from `LEDGER.json`. + +| Interface | P lines | Claim | Full-depth nonauthor ACCEPT | Partial second-provider check | +|---|---|---|---|---| +| S1 §1 | 9–53 | model, conventions, statements of A–C | G1, C1 (xAI, Anthropic) | — | +| A1 §2 | 54–68 | positive spectrum, jet rank, smooth version | C1 | G3 "qualitative positivity OK" | +| A2 §3 | 69–114 | contact frame, `12 r^{-(d+3)}`, uniform gap, (3.5) | C1 (including the gap proof) | G3 certified `det T_r` | +| A4 §4 | 115–137 | (4.1)–(4.3), independent residual | C1 | — | +| A3 §5 | 138–183 | (5.1)–(5.5), type convergence, UI, floor | C1 (O1 agrees but is author-provider) | G3 UI-majorant structure and `A_M -> A_0` | +| A3-E1 | erratum | `D_r = diag(r^{-1/2}, I)` | C1, G3 | — | +| A5 (6.1) | 184–197 | canceled-pivot bound, all m | C1 | G3 (m=1) | +| A6 (6.2) | 198–211 | two soft factors, including the 3/2 saddle term | C1, G4 | — | +| A7 §7 | 212–275 | `r^5` numerators, corank strata, m=1 far branch, M4, (7.8) | C1 | G4 numerator structure | +| CAP §§1–5 | cap 1–168 | deterministic separating cap, global maximin | C1, G3 | — | +| D1-A §8 | 276–297 | pinned genericity, Borel elder event, `G_r =>` partner S | G1, C1 | — | +| D1-B §9 | 298–309 | Borel marked Kac–Rice as written | G1 | — | +| D1-B-E2 | E2 | §9 replacement v1.1, C1–C6 | C1, G2 | — | +| D1-C §10 | 310–336 | radial ledger `r A_r dr db dk dσ` | G1 | C1 (#123 E9 check) | +| §11 | 337–364 | pushforward, (11.3), difference (1.2) | C2 | G4 ((11.1) and the (11.3) identity) | +| §12 | 365–385 | cumulative and moment corollaries | C2 | — | +| D1-D §§13–14 | 386–446 | target-growth majorant, `k -> 0`, off-diagonal (14.1) | G1 | C2 spot checks | +| D1-E §15 | 447–481 | parity, disintegration, (15.2) | G1 | C2 gamma factor | + +Every interface has a full-depth ACCEPT from at least one nonauthor provider. Six have full-depth ACCEPTs from both +nonauthor providers: S1, A3-E1, A6, CAP, D1-A and D1-B-E2. Nine more have a partial check by the other provider. Three +have a single provider at any depth: A4, §12, and D1-B. D1-B is the original §9 text, which the reading rule replaces by +D1-B-E2 (two providers), so it needs no second check. The project's established reconciliations (D2, D3, D4, D6 in STATUS) +each rest on one provider-distinct nonauthor review, so this meets the existing bar. §8 invites second-provider checks, +prioritized by that gap, but they are not a precondition. + +## 5. Residual items and their discharge + +Every item that an earlier record left open, AMEND or "not closed" is listed below with where it is discharged. + +| Open item | Raised in | Discharge | +|---|---|---| +| A3: wrong congruence factor `diag(√r,I)` | main#63 5841947606; G3 `A3_TYPE_CONVERGENCE` C1 | E1, landed by Math-#64 at `d8f5505`. Algebra confirmed by G3 and C1. | +| A3: UI of `W_r/r^2` | G3 C4 | C1 A3 item 4: `sup E(W_r/r^2)^2 < ∞` from (5.3), (5.1), `‖A_S‖ <= ‖A_M‖ + rM3` and A4. G3 `A3_UI_MAJORANT` gives the same majorant. | +| A3: type-indicator convergence under `D_r` | G3 | C1 W1: Slutsky joint convergence in law. The discontinuity set lies in `{det A_0 = 0}`, which is null (C1 item 5). | +| A3: `z_* > 0` | G3 C3 | C1: `A_0` is a nondegenerate Gaussian on `Sym_m`, so it charges the open cone `{A < 0}`. Continuity plus compactness gives (5.5). | +| A3: `A_M -> A_0` uniformly | G3 `A_M_TO_A0` (conditional on A1–A2) | C1 A3 item 3 and A1–A2 below. | +| A2: "single missing estimate", the uniform Gramian gap | G3 `A2_RESIDUAL_GAP` | C1 A2. Each coordinate of `U_r ⊕ vec A_r` is `∫ ∂^α f(tu) dμ_r(t)` with `μ_r -> δ_0`, so the covariance is jointly continuous on `[0,r_0] × O(d)`. It is positive definite at `r=0` by A1. Compactness gives `λ_min >= c_* > 0`. This is existential, which is all Theorem A claims. | +| A5 for `m >= 2` | G3 table | C1 A5: `det H_M = rα det(B' )` with `B' = A - (r/α)ββ^T`, and `0 < -B' <= B` on typed support. Symbolic for m=1,2,3; probed on 1,962 exact instances. | +| A7 `r^5` integrals; far branch | G3 table; C1 v1 | C1 A7, and G4. The v2 correction is that `λ >= 1/(Dr)` lies in the **majorant** event, not necessarily in depth failure. The far branch is kept in the upper bound. | +| Cap pairing: embedded chart and Morse/§8 | G3 `EMBEDDED_CHART_AND_MORSE`, `CAP_PAIRING_IDENTITIES` | §8 by G1 and C1. The embedding radius is made explicit below. | +| Embedding radius not displayed | G3 `EMBEDDED_CHART_AND_MORSE` ("AMEND the slogan") | Below: `r < L/(4√2)`. | +| §9 write-up gap (monotone-class passage) | Math-PR112 review, recorded in E2 | E2 v1.1, reviewed by C1 and G2. C1 withdrew the theorem-number objection on primary-source grounds (arXiv:2304.07424v3, Theorems 2.1, 2.2, 7.1, Remarks 7–8). | +| §§11–12 never reviewed | This reconciliation (line audit) | §6 below. | +| "Second non-Claude review of W1 and A2 before integration" | C1 request | W1 is wording only and is confirmed by O1 (non-discharge) and G3's in-law route. For A2 see the provider note in §4. The request stands as §8's open invitation. It is not a blocker under the existing bar. | + +**W1 (wording; proof unchanged).** P l. 165 says the scaled endpoint Hessians converge "in probability" to +`diag(∓6k, A_0)`. Correctly: + +- `D_r H_M D_r - diag(-6k, A_M)` and `D_r H_S D_r - diag(6k, A_M)` tend to 0 in probability, by (5.1), (5.2) and + `‖A_S - A_M‖ <= rM3`; +- `A_M -> A_0` in law. + +Slutsky then gives joint convergence in law, and continuous mapping applies off the null set `{det A_0 = 0}`. P's +subsequence paragraph (l. 176) already argues in law. + +**Embedding radius.** Let `D = [-2r, 2r] × B̄(0, 2r)`, the cap cylinder, which contains all of `G_r`'s geometry. Then +`D ⊂ B̄(0, 2√2 r)`. On `R^d/(LZ^d)` the exponential chart at any point is injective on the open ball of radius +`L/2`. So `D` is embedded whenever `2√2 r < L/2`, that is `r < L/(4√2)`. Theorem A's existential `r_*` may be, and is, +taken below this. This makes explicit what P §1 says with "for sufficiently small r". + +## 6. Review of the previously unreviewed §§11–12 (C2) + +The line audit found that no earlier record reviewed P lines 337–384. G1 lists 215–217, 276–296, 298–308, +88–94 and 310–335, 386–445, and 447–480. C1 assumed that D1-A–E covered "§§10–15", which it did not. Both sections +were reviewed here from the bytes. + +**§11 — ACCEPT.** + +- (11.1): `r = (ℓ/k)^{1/3}` gives `r dr/dℓ = (1/3) k^{-2/3} ℓ^{-1/3}`. +- Restriction: for `ℓ < k_- r_*^3`, every `k ∈ K` has `r < r_*`, so the whole pushforward stays inside Theorem A's + range. +- (11.2): at fixed `(b,k,u)`, pushing `r A_r dr` forward to `ℓ` gives `ℓ^{-1/3} A_{(ℓ/k)^{1/3}}/(3k^{2/3}) dℓ`. + Integrating over `B × K × S^{d-1}` gives the density version; `ν_eld` carries the extra factor `p_r`. +- (11.3): as `ℓ -> 0`, `A_r -> A_0` uniformly with a uniform bound (10.3), and `p_r -> 1` uniformly by Theorem A. + Dominated convergence on the compact domain gives `ℓ^{1/3} ν -> ∫ A_0/(3k^{2/3}) = 4∫ k^{-2/3} π_0 z_0 = c_{B,K}`, + using `A_0 = 12 π_0 z_0`. Positivity follows from `π_0, z_0 > 0` on a set of positive measure. G4 verified the same + identity in (13.6)'s form. +- Difference: `ν_cand - ν_eld = ℓ^{-1/3}∫ A_r(1-p_r)/(3k^{2/3})`. Theorem A gives `<= ℓ^{-1/3} ∫ A^* C r^3/(3k^{2/3})` + with `r^3 = ℓ/k`, so the bound is `C ℓ^{2/3} ∫ A^*/(3k^{5/3})`. This is finite because `k >= k_- > 0`, which proves + (1.2). The exponent `ℓ^{2/3} k^{-5/3}` is checked exactly. + +**§12 — ACCEPT.** + +- (12.1): `∫_0^t c ℓ^{-1/3}(1+o(1)) dℓ = (3/2)c t^{2/3}(1+o(1))`, and `∫_0^t C ℓ^{2/3} = (3/5)C t^{5/3}`. +- (12.2): for `q > -2/3` the integrand `ℓ^{q-1/3}` is integrable at 0, giving `c t^{q+2/3}/(q+2/3)`. +- (12.3): for `q > -5/3`, `∫_0^t ℓ^q C ℓ^{2/3} = C t^{q+5/3}/(q+5/3)`. +- For `q <= -2/3`, each individual expectation diverges because `c > 0`. (12.3) is stated for the nonselected measure + directly, not as a difference of infinities. +- The unrestricted leading forms follow from Theorem C in the same way. The compact difference rates are correctly + **not** extended. + +**Spot re-derivations for §§13–15.** These do not replace G1; they are a second provider's check. + +- (13.1): `(b - kr^3/2)^2 + 144k^2 >= (b^2+k^2)/2` for `r <= 1`. Checked exactly at sample rationals and by the + inequality `b^2 <= 2(b - kr^3/2)^2 + k^2/2`. +- (13.4): `W_r/r^2 <= C(1+‖f‖_{C^3})^{2d}` follows from (5.1) and (5.3) with no division by `k`. +- (15.2): with `t = 12k` and `w = s^2/2`, `144∫_0^∞ k^{4/3}φ_τ(12k)dk = Γ(7/6)τ^{4/3}/(24^{1/3}√π)`. Checked + numerically to relative `1e-9` in `reconciliation_check.py`, and exactly for the power bookkeeping. + +## 7. Consumption contract + +| Consumer | Consumes from D1 | After this reconciliation | +|---|---|---| +| D2 Theorem R, `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf…`) | The §8 comparison `1-p_r <= Q^W(G_r^c)`, §9 (read as E2), §10, §14 (14.1), and (15.2) as `c`. **Not** (1.1). | Every consumed interface is reconciled. `claims` moves FAIL_CLOSED -> REVIEWED_SCOPED at Theorem R's stated `O(1)` scope. | +| D3 SIDE24 coefficient, `coefficients/side24_v1/PROOF.md` (blob `44b66f04…`) | (15.2) as the leading coefficient of Theorem C | The persistence interpretation is reconciled. `claims` moves HOLD_WITH_DOMAIN -> REVIEWED_SCOPED. The arithmetic acceptance (#65) is unchanged. | +| Direct lower bound, `ELDER_LOWER_AND_DENSITY_GAP.md` (E4)/(E11) | Theorem A (1.1) upper half, §§9–11 | The two-sided planar composition is now fully sourced. (E2) itself stays as reviewed in #123. | +| D4 fixed-remote (RM), RC (#110), IW (#107), #120 | P's model and observation **conventions** only, plus RM's own "Theorem A" | Unaffected. No D1 theorem is consumed. | +| Any new consumer | — | Cite the §1 identity set, this record, and a scope no larger than §2. | + +**Still not consumable.** + +- Numerical `C`, `r_*`, `z_*`, `c_{B,K}` or `c_{d,L}`. +- (1.2)'s difference rate outside compact marks. +- Any lower bound for `d >= 3`. +- A globally uniform `O(r^3)` constant over unbounded marks. +- RN, 24-jet, JETMOD or H3/LPW numerical certificates. +- `lemma_closed`, any prize, or "unconditional acceptance of the lifetime parent", which is a forbidden global + inference in `claims/LANDING_CLAIMS.json`. This acceptance is scoped. + +## 8. Independence, lineage and requests + +**Dual role.** The reconciler is also reviewer C1 of A1–A7, CAP and E2, and reviewer C2 of §§11–12. This is disclosed +rather than hidden. + +**What the reconciliation relies on, and what it does not.** Nothing here is accepted on the strength of a green test, +a hash, a same-provider review or O1. Every interface rests on a written nonauthor derivation. + +**Merge.** The register PR carrying this record must be merged by a non-Claude lane after reading it. Claude does not +self-merge. + +**Invitation.** A second-provider check is invited in this priority order: + +1. A4 and §12, which have one provider at any depth; +2. the full-depth gaps behind partial checks: A2's uniform gap, A5 for `m >= 2`, A7's division by the floor, and + §11's difference bound; +3. the reverse direction for D1-C, D1-D and D1-E, where xAI is full-depth and Claude is partial. + +Any lane may return COUNTEREXAMPLE with an exact equation. That would reopen only the affected interface and its +dependents, and the graph's reverse-impact rule would mark them REVALIDATION_REQUIRED. + +## 9. Register transitions justified by this record + +Math-, in the same PR: + +- `PROOF_INDEX.md`: D1 moves to "Reviewed scoped results" with the §2 scope. The §9 repair is recorded as reviewed by + C1 and G2. The D2 entry is updated to name its reconciled imports. +- `claims/LANDING_CLAIMS.json`: + - `lifetime-remainder` FAIL_CLOSED -> REVIEWED_SCOPED; + - `side24-coefficient` HOLD_WITH_DOMAIN -> REVIEWED_SCOPED; + - dependency reviews are bound to local review files. +- `frontiers/downstream_gate_20260925/GRAPH.json`: + - `math.uniform-matrix-cap-lifetime` AUTHOR_SIDE_CANDIDATE -> PROVED_REVIEWED, with this record as evidence and + the §2 scope. It is the only classification that satisfies a still-required premise, and D2 and D3 require it. + - D2 and D3 gain `review_disposition: ACCEPT` metadata under the existing convention (commit `cc3a991`). + - Four reading-rule component nodes (E1, E2, CAP and this record's directory) are added as `PROVED_REVIEWED` + source nodes with required edges from D1. Each carries the ledger reviews its role needs: C1 and G3 for E1 and + CAP, C1 and G2 for E2. This record is an Anthropic, nonauthor record; its W1 and radius items were raised in C1 + and G3 and its §6 is C2. OpenAI's technical check of this record (review 5345687857) is listed as author-provider + non-discharge. This closes the gap that review found: before it, E1, E2 and W1 were named only in metadata, so + editing them left D1's source snapshot unchanged. + - No node becomes `controlling`. Satisfying D2's and D3's required edge to D1 is not promotion: their own + classifications stay author-side, so `promotion_allowed` still refuses both. + - The fail-closed tests are kept on a pre-reconciliation fixture, so they still prove that an unreviewed parent + blocks both dependents. New tests make real two-commit edits and deletions of each component with P unchanged, + and require D1, D2 and D3 to be proposed `REVALIDATION_REQUIRED`. An unrelated edit must propose nothing, and + the pre-binding graph is shown to miss the component edit. + +main, in a companion PR pinned to this record's merged commit: + +- `STATUS.md`: D1 moves from AMEND/open to ACCEPT — scoped. +- The glossary's D1 row is updated. +- The museum and shop projections are regenerated. + +## 10. Checks + +`reconciliation_check.py` (Python standard library): + +1. Binds every §1 source and every local review record by Git blob, bytes and SHA256. +2. Cross-checks the hard-gate graph against `LEDGER.json`: + - D1 is bound to P's exact bytes; + - E1, E2, CAP and this directory are D1 source nodes behind required edges; + - each component node carries exactly the ledger reviews of its interfaces; + - the record's author provider differs from P's; + - two negative controls (a dropped edge; an author-only review basis) are rejected. +3. Validates `LEDGER.json`: + - every interface has at least one nonauthor ACCEPT; + - author-provider reviews are non-discharge; + - P's §§1–16 line ranges are fully covered; + - every residual item has a discharge pointer. +4. Re-verifies the reading-rule mathematics exactly: + - E1 congruence versus the displayed factor; + - (5.3), (6.1) and (6.2) on exact rational instances; + - (7.3) and the (7.5) split; + - the cap constants and κ-scaling; + - the embedding radius, the radial and lifetime ledgers, (11.1)–(11.3), the difference exponent and the §12 + integrals; + - the (13.1) coercivity bound; + - the (15.2) gamma factor, numerically. +5. Rejects nine semantic mutants. + +A green run is evidence that the bookkeeping and finite identities hold. It is not the analytic review, which lives in +C1, G1, G2, G4 and §6. diff --git a/frontiers/lifetime_moment_boundary_20260930/test_verify.py b/frontiers/lifetime_moment_boundary_20260930/test_verify.py new file mode 100644 index 000000000..4024f8783 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/test_verify.py @@ -0,0 +1,111 @@ +"""Real CLI controls for receipt selection and immutable local source copies.""" +from pathlib import Path +import hashlib +import json +import shutil +import subprocess +import sys +import tempfile +import unittest + +HERE = Path(__file__).resolve().parent +REPO = HERE.parents[1] +LIFETIME = Path('frontiers/lifetime_moment_boundary_20260930') +RADIAL = Path('reviews/radial_moment_corollaries_20260930') + + +class PublicationVerifierTests(unittest.TestCase): + def setUp(self): + temporary = tempfile.TemporaryDirectory() + self.addCleanup(temporary.cleanup) + self.root = Path(temporary.name) / 'repo' + for packet in (LIFETIME, RADIAL): + shutil.copytree(REPO / packet, self.root / packet, + ignore=shutil.ignore_patterns('__pycache__')) + + def replay(self): + flags = ['-O'] if sys.flags.optimize else [] + return subprocess.run( + [sys.executable, '-B', *flags, '-S', + str(self.root / LIFETIME / 'verify.py')], + cwd=self.root, capture_output=True, text=True, timeout=30) + + def test_clean_packet_replays_both_modes(self): + result = self.replay() + self.assertEqual(result.returncode, 0, result.stderr) + report = json.loads(result.stdout) + self.assertTrue(report['passed']) + self.assertEqual([row['mode'] for row in report['modes']], + ['normal', 'optimized']) + + def test_changed_optimized_lifetime_receipt_is_rejected(self): + path = self.root / LIFETIME / 'CHECK_RECEIPT_OPTIMIZED.json' + data = json.loads(path.read_text()) + data['good_quartic_cases'] += 1 + path.write_text(json.dumps(data)) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('lifetime replay differs', result.stderr) + + def test_changed_optimized_loss_receipt_is_rejected(self): + path = self.root / LIFETIME / 'LOSS_CHECK_RECEIPT_OPTIMIZED.json' + data = json.loads(path.read_text()) + data['rational_probability_cases'] += 1 + path.write_text(json.dumps(data)) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('loss replay differs', result.stderr) + + def test_changed_normal_receipt_remains_rejected(self): + path = self.root / LIFETIME / 'CHECK_RECEIPT.json' + data = json.loads(path.read_text()) + data['good_quartic_cases'] += 1 + path.write_text(json.dumps(data)) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('lifetime replay differs', result.stderr) + + def test_changed_optimized_radial_receipt_is_rejected_after_custody_check(self): + path = self.root / RADIAL / 'moment-checks-optimized.json' + data = json.loads(path.read_text()) + data['tests'] += 1 + path.write_text(json.dumps(data)) + # A self-consistent custody record must not replace actual replay. + manifest = self.root / RADIAL / 'COROLLARY_CUSTODY.json' + custody = json.loads(manifest.read_text()) + for row in custody['artifacts']: + if row['path'] == path.name: + row['bytes'] = path.stat().st_size + row['sha256'] = hashlib.sha256(path.read_bytes()).hexdigest() + manifest.write_text(json.dumps(custody)) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('radial replay differs', result.stderr) + + def check_alias_rejected(self, relative): + path = self.root / relative + target = self.root / 'byte-identical-target' + path.rename(target) + path.symlink_to(target, target_is_directory=target.is_dir()) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('symlink', result.stderr.lower()) + + def test_byte_identical_lifetime_source_symlink_is_rejected(self): + self.check_alias_rejected(LIFETIME / 'sources/P.md') + + def test_byte_identical_radial_source_symlink_is_rejected(self): + self.check_alias_rejected(RADIAL / 'P.md') + + def test_lifetime_source_directory_symlink_is_rejected(self): + self.check_alias_rejected(LIFETIME / 'sources') + + def test_lifetime_parent_directory_symlink_is_rejected(self): + self.check_alias_rejected(Path('frontiers')) + + def test_radial_parent_directory_symlink_is_rejected(self): + self.check_alias_rejected(Path('reviews')) + + +if __name__ == '__main__': + unittest.main(verbosity=2) diff --git a/frontiers/lifetime_moment_boundary_20260930/verify.py b/frontiers/lifetime_moment_boundary_20260930/verify.py new file mode 100644 index 000000000..af57cbc79 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/verify.py @@ -0,0 +1,90 @@ +"""Replay finite controls and exact packet custody; not theorem acceptance.""" +from pathlib import Path +import hashlib +import json +import stat +import subprocess +import sys + +HERE = Path(__file__).absolute().parent +REPO = HERE.parents[1] +RADIAL = REPO / 'reviews/radial_moment_corollaries_20260930' + + +def require(ok, message): + if not ok: + raise RuntimeError(message) + + +def read_regular(root, name): + part = Path(name) + require(part.parts and not part.is_absolute() and '..' not in part.parts, + 'unsafe source path') + path = root / part + cursor = REPO + require(not cursor.is_symlink() and cursor.is_dir(), 'invalid repository root') + for component in path.relative_to(REPO).parts: + cursor = cursor / component + mode = cursor.lstat().st_mode + require(not stat.S_ISLNK(mode), 'symlink source path: ' + str(cursor)) + require(stat.S_ISREG(mode) if cursor == path else stat.S_ISDIR(mode), + 'non-regular source path: ' + str(cursor)) + return path.read_bytes() + + +def verify_lifetime_paths(): + sources = json.loads(read_regular(HERE, 'SOURCES.json')) + for row in sources['sources']: + read_regular(HERE, row['local_path']) + for name in ('PROOF.md', 'check_exact.py', 'LOSS_COROLLARY.md', 'check_loss_exact.py'): + read_regular(HERE, name) + + +def verify_radial(root): + custody = json.loads(read_regular(root, 'COROLLARY_CUSTODY.json')) + for row in custody['artifacts'] + custody['sources']: + name = row.get('path', row.get('local_path')) + raw = read_regular(root, name) + require(len(raw) == row['bytes'], 'size drift: ' + name) + require(hashlib.sha256(raw).hexdigest() == row['sha256'], 'SHA256 drift: ' + name) + if 'git_blob' in row: + blob = hashlib.sha1(f'blob {len(raw)}\0'.encode() + raw).hexdigest() + require(blob == row['git_blob'], 'Git blob drift: ' + name) + return len(custody['artifacts']) + len(custody['sources']) + + +def main(): + require(sys.version_info[:2] == (3, 11), 'Python 3.11 required') + verify_lifetime_paths() + count = verify_radial(RADIAL) + rows = [] + for flags in ([], ['-O']): + mode = 'optimized' if flags else 'normal' + suffix = '_OPTIMIZED' if flags else '' + lifetime = subprocess.run([sys.executable, '-B', *flags, '-S', str(HERE/'check_exact.py')], + check=True, capture_output=True, text=True) + current = json.loads(lifetime.stdout) + expected = json.loads(read_regular(HERE, f'CHECK_RECEIPT{suffix}.json')) + require(current == expected, 'lifetime replay differs from exact receipt') + radial = subprocess.run([sys.executable, '-B', *flags, '-S', str(RADIAL/'moment_consequence_checks.py')], + check=True, capture_output=True, text=True) + current_radial = json.loads(radial.stdout) + require(current_radial == json.loads(read_regular(RADIAL, f'moment-checks-{mode}.json')), + 'radial replay differs from exact receipt') + loss = subprocess.run([sys.executable, '-B', *flags, '-S', str(HERE/'check_loss_exact.py')], + check=True, capture_output=True, text=True) + current_loss = json.loads(loss.stdout) + require(current_loss == json.loads(read_regular(HERE, f'LOSS_CHECK_RECEIPT{suffix}.json')), + 'loss replay differs from exact receipt') + rows.append({'mode': mode, + 'lifetime_quartics': current['good_quartic_cases'], + 'lifetime_negative_controls': current['outside_hypothesis_axial_negative_controls'], + 'normalization_exponents': current['normalization_exponent_cases'], + 'lifetime_sources': len(current['source_identity_checks']), + 'radial_tests': current_radial['tests'], 'loss_controls': current_loss}) + print(json.dumps({'passed': True, 'modes': rows, 'radial_custody_records': count, + 'scientific_effect': 'NONE'}, sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/reviews/radial_moment_corollaries_20260930/AUTHOR_COMMENT_5903080058.json b/reviews/radial_moment_corollaries_20260930/AUTHOR_COMMENT_5903080058.json new file mode 100644 index 000000000..0270312de --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/AUTHOR_COMMENT_5903080058.json @@ -0,0 +1,7 @@ +{ + "id": 5903080058, + "url": "https://github.com/d6g8k5htny-coder/Math-/pull/176#issuecomment-5903080058", + "created_at": "2026-09-30T02:47:55Z", + "updated_at": "2026-09-30T02:47:55Z", + "body": "Author-side corollary of unchanged §7, offered to the Slice-B reader (not a new independent verdict): at the exact aligned-planar scope, (1+12A²)/(1+A²)>=1 and D_aG0<=0. Since B<0, equation(13) therefore gives the model-independent leading-coefficient lower bound\n\n`C2/C0 > 11 U2/(2I) = 4587/856`.\n\nStrictness follows from positive Gaussian mass at a!=0. Consequently the leading coefficient of the radial TV/CDF error in(16) exceeds\n\n`(4587/856)*(2/13)*(11/13)^(11/2)`\n\n(approximately0.328937168357..., decimal diagnostic only). This is a bound on the asymptotic coefficient for each fixed model, NOT a uniform-in-L,k remainder or finite-threshold inequality.\n\nLikewise (19) rewrites as `Bsign=(11 Uabs/(2I)) E_rho[|A|/sqrt(1+A²)]` with rho proportional to H gamma^11G0, so `0t]\n = 11/(11-p)\n - [2p c / ((13-p)(11-p))] t^-2\n + O_p(t^-4).\n\nThe coefficient follows exactly from\n13/(13-p)-11/(11-p) = -2p/[(13-p)(11-p)].\nThe restriction p<11 is sharp for the Pareto limiting moment. This is AFTER the original r->0 limit and does not imply finite-r moment convergence.\n\nDifferentiating the moment identity at p=0 (justified by the same q^-12 envelope on a fixed neighborhood of p=0) gives the log-radius correction\n\nE[log(R/t) | R>t]\n = 1/11 - (2c/143) t^-2 + O(t^-4).\n\nThere is also an explicit fixed-upper-tail quantile correction. If Q_alpha(t) is defined by\nP(R/t > Q_alpha(t) | R>t)=alpha, 04587/856>0, so every fixed upper-tail quantile approaches its Pareto limit from below at second order, and E log(R/t) approaches 1/11 from below. This is an asymptotic coefficient/sign statement for each fixed model; it is NOT a model-uniform finite-threshold inequality because the O(t^-4) remainder is not uniform in L,k,b.\n\nThese are direct consequences of the frozen proof65f24b1f and can be checked inside Slice B without enlarging the source interface." +} diff --git a/reviews/radial_moment_corollaries_20260930/COROLLARY_CUSTODY.json b/reviews/radial_moment_corollaries_20260930/COROLLARY_CUSTODY.json new file mode 100644 index 000000000..5b405d890 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/COROLLARY_CUSTODY.json @@ -0,0 +1,91 @@ +{ + "artifacts": [ + { + "bytes": 11110, + "path": "MOMENT_CONSEQUENCE.md", + "sha256": "29bba56b56fd3223a2ea71d012f9abe04eb30240bc5bbfe8f5b131e0bba6f4dd" + }, + { + "bytes": 8259, + "path": "moment_consequence_checks.py", + "sha256": "0c35140179e0345eb03764bf29c3c81453ea278ce1a6319ab33edf6a548e03e6" + }, + { + "bytes": 1219, + "path": "moment-checks-normal.json", + "sha256": 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"observed_control_replay": { + "all_order_controls": "normalized finite Pareto mixtures", + "exact_rational_powers": [ + "-100", + "-7", + "-1", + "-1/2", + "0", + "1/2", + "1", + "2", + "10", + "109/10", + "10999/1000" + ], + "log_correction": "-2/143", + "log_leading": "1/11", + "moment_rows": [ + { + "coefficient_of_c_t_minus_2": "200/12543", + "leading": "11/111", + "p": "-100" + }, + { + "coefficient_of_c_t_minus_2": "7/180", + "leading": "11/18", + "p": "-7" + }, + { + "coefficient_of_c_t_minus_2": "1/84", + "leading": "11/12", + "p": "-1" + }, + { + "coefficient_of_c_t_minus_2": "4/621", + "leading": "22/23", + "p": "-1/2" + }, + { + "coefficient_of_c_t_minus_2": "0", + "leading": "1", + "p": "0" + }, + { + "coefficient_of_c_t_minus_2": "-4/525", + "leading": "22/21", + "p": "1/2" + }, + { + "coefficient_of_c_t_minus_2": "-1/60", + "leading": "11/10", + "p": "1" + }, + { + "coefficient_of_c_t_minus_2": "-4/99", + "leading": "11/9", + "p": "2" + }, + { + "coefficient_of_c_t_minus_2": "-20/3", + "leading": "11", + "p": "10" + }, + { + "coefficient_of_c_t_minus_2": "-2180/21", + "leading": "110", + "p": "109/10" + }, + { + "coefficient_of_c_t_minus_2": "-21998000/2001", + "leading": "11000", + "p": "10999/1000" + } + ], + "quantile_controls": "exact rational brackets for Pareto mixtures", + "scientific_effect": "NONE", + "scope": "PR176 equation17 conditional moment consequence only", + "tests": 11 + }, + "observed_control_log": "test_actual_probability_mixture_has_claimed_moment_remainder_bound (__main__.MomentConsequenceTests) ... ok\ntest_all_order_finite_tail_matches_direct_pareto_mixture (__main__.MomentConsequenceTests) ... ok\ntest_all_order_log_powers_and_normalization (__main__.MomentConsequenceTests) ... ok\ntest_direct_log_tail_integral (__main__.MomentConsequenceTests) ... ok\ntest_hand_checked_special_cases (__main__.MomentConsequenceTests) ... ok\ntest_offered_rational_coefficient_factors (__main__.MomentConsequenceTests) ... ok\ntest_quantile_expansion_brackets_an_actual_conditional_law_exactly (__main__.MomentConsequenceTests) ... ok\ntest_tail_and_density_derivations_agree_for_negative_zero_positive_p (__main__.MomentConsequenceTests) ... ok\ntest_threshold_and_above_diverge (__main__.MomentConsequenceTests) ... ok\ntest_uniform_in_p_remainder_overclaim_has_exact_counterexample (__main__.MomentConsequenceTests) ... ok\ntest_wrong_normalizer_sign_and_missing_p_are_detected (__main__.MomentConsequenceTests) ... ok\n\n----------------------------------------------------------------------\nRan 11 tests in 0.004s\n\nOK\n" + }, + "checks": [ + "C28 source P size+SHA256+blob", + "C28 source ERR size+SHA256+blob", + "C28 source BOREL size+SHA256+blob", + "C28 source REC size+SHA256+blob", + "C28 source MARK_D size+SHA256+blob", + "C28 source MARK_2 size+SHA256+blob", + "C28 source CAP size+SHA256+blob", + "C27 artifact MOMENT_CONSEQUENCE.md size+SHA256", + "C27 artifact moment_consequence_checks.py size+SHA256", + "C27 artifact moment-checks-normal.json size+SHA256", + "C27 artifact moment-checks-normal.log size+SHA256", + "C27 artifact moment-checks-optimized.json size+SHA256", + "C27 artifact moment-checks-optimized.log size+SHA256", + "C27 artifact AUTHOR_COMMENT_5907459466.json size+SHA256", + "C27 artifact AUTHOR_COMMENT_5903080058.json size+SHA256", + "C27 artifact SOURCES.json size+SHA256", + "C27 source PR176 size+SHA256+blob", + "C27 source R size+SHA256+blob", + "C27 source SC size+SHA256+blob", + "C27 source P size+SHA256+blob" + ] +} diff --git a/reviews/radial_moment_corollaries_20260930/MOMENT_CONSEQUENCE.md b/reviews/radial_moment_corollaries_20260930/MOMENT_CONSEQUENCE.md new file mode 100644 index 000000000..fb3bef3a9 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/MOMENT_CONSEQUENCE.md @@ -0,0 +1,228 @@ +# Conditional moments, quantiles, and coefficient bounds for the radial ratio + +Read-only nonauthor consequence review of Math- PR #176 equation (17), exact head +`71772a8906cf6663d16be2cd5f63e5d4ebdcee2b`, proof blob +`65f24b1f9a7b80522a2ee059f5f6877c9f970f0d`, 15,755 bytes, SHA-256 +`a3693664900e23e1e763d88a0b5609936a6f31d844cbe6f18c7d0379b94ba710`. + +This is an elementary conditional implication, not a second Slice B verdict. +Claude's Slice A and the competing actual Slice B reviewer retain their scopes. +The reviewer here authored none of this packet or its mathematical sources; +all are OpenAI-source-exposed, and organizational-independence credit is zero. +R17 claim: `13dc84b1-bc12-42df-8c7f-96f23cb1dd7d`, WE317; root coordinates +the actual public disposition. No proof branch or scientific status was changed. + +Exposure chronology: the second-order power/log derivations and their first +seven exact arithmetic controls were completed before this reviewer read +author comment [5907459466](https://github.com/d6g8k5htny-coder/Math-/pull/176#issuecomment-5907459466), +posted at 2026-09-30T08:38:26Z. That comment already states these power/log and +fixed-alpha quantile formulas. They are credited to the author's offered +corollaries here; no novelty or source-unexposed credit is claimed. The quantile +analysis below was made after that exposure. The all-order extension was +suggested by root, then reconstructed here using tail integration. Author +comment [5903080058](https://github.com/d6g8k5htny-coder/Math-/pull/176#issuecomment-5903080058) +was read before analysis of its coefficient inequalities. + +## Consumed hypothesis and scope + +Write Q_t=R/t under the source's point-intensity sampling conditional on R>t, +after the original r limit has already formed its microscopic point measure. +Retain the source's full normalizer; no cluster-maximum selection or extra +factor for multiplicity is introduced. Let c=C_2/C_0, with C_0>0. Consume (17) +in its quantified form: for each fixed source model there are T,K=T and q>=1, + + S_t(q)=P(Q_t>q) + =q^-11[1+c t^-2(q^-2-1)]+E_t(q), + |E_t(q)| <= K t^-4 q^-11. (M1) + +This pointwise remainder, uniform over the unbounded q range, is essential. +Unweighted L1 or total-variation convergence alone would not justify integrating +the unbounded test function Q_t^p or log Q_t. The original source's positive +tail asymptotic also supplies F(x)~C_0 x^-11 as x tends to infinity. + +## Every fixed real power below eleven + +For every fixed real p<11, + + E Q_t^p = 11/(11-p) + - [2p c/((13-p)(11-p))] t^-2 + O_p(t^-4). (M2) + +For p=0, this means the exact identity E Q_t^0=1 and a zero correction. +For p!=0, the deterministic identity for x>=1 is + + x^p = 1 + p integral_1^x q^(p-1)dq. + +For 00, the leading +correction is negative for positive p and positive for negative p. These are +asymptotic statements at each fixed model, not uniform finite-threshold bounds. + +## The divergence boundary + +For any finite t>0 for which this conditional law is defined, and every p>=11, +`E Q_t^p=infinity`. Indeed, + + S_t(q)=F(tq)/F(t) ~ [C_0 t^-11/F(t)] q^-11, + +and its positive leading coefficient yields a lower bound by a positive +multiple of q^-11 for all sufficiently large q. The positive-power tail +integral diverges at p=11 logarithmically and at every p>11 as a power. +No limit interchange at p=11 and no claim about the original finite-r field +is involved. The finite capped eleventh moment in #176 is a different object. + +## Logarithmic moment + +Since log x=integral_1^x q^-1dq for x>=1, direct Tonelli integration gives + + E log Q_t = 1/11 - (2c/143)t^-2 + O(t^-4). (M4) + +The first integral is `1/11`; the correction is +`c(1/13-1/11)=-2c/143`. The absolute remainder is at most `K t^-4/11`. +This derivation does NOT differentiate (M2) with respect to p or assume any +unstated differentiable uniformity in the p-indexed remainder. + +## Conditional all-order extension + +Consume in addition the finite-order remainder in equation (5), obtained from +the smooth even scalar C in Theorem 1. Let alpha_j=11+2j and D(t)=C(1/t)>0. +For every fixed nonnegative integer N and every t>=T, q>=1, + + t^11 F(tq) = sum_(j=0)^N C_(2j) t^(-2j) q^(-alpha_j) + + R_(N,t)(q), + |R_(N,t)(q)| <= K_N t^(-2N-2) q^(-13-2N). (M5) + +The scalar bound at s=tq provides this uniform remainder directly; no derivative +of a rough O term is taken. For every fixed real p<11, tail integration gives + + E Q_t^p = [sum_(j=0)^N alpha_j/(alpha_j-p) + C_(2j)t^(-2j) + + O_(p,N)(t^(-2N-2))] / C(1/t). (M6) + +To see the normalization explicitly, multiply the exact tail identity for +E Q_t^p by D(t). Its right side is D(t) plus p times the integral of (M5) +against q^(p-1). Expand D(t) at the same finite order and use +`1+p/(alpha_j-p)=alpha_j/(alpha_j-p)`. The remainder in this numerator is +bounded by `K_N[1+|p|/(13+2N-p)]t^(-2N-2)`. The p=0 moment is exactly one. +Keeping the denominator exact avoids a hidden normalization change. Replacing +it by `sum_(j=0)^N C_(2j)t^(-2j)+O_N(t^(-2N-2))` is equivalent. + +For every fixed integer m>=1, direct nonnegative tail integration similarly gives + + E (log Q_t)^m = [m! sum_(j=0)^N C_(2j)t^(-2j)/(11+2j)^m + + O_(m,N)(t^(-2N-2))] / C(1/t). (M7) + +Indeed, use `m integral_1^infinity (log q)^(m-1)S_t(q)dq/q` and substitute +y=log q. Each integral is m!/alpha_j^m. The numerator remainder is at most +`m! K_N/(13+2N)^m t^(-2N-2)`. These are finite-order expansions for fixed +indices, with no claim of convergence of an infinite Taylor series. + +## Fixed upper-tail quantiles + +Let 01, +and its q derivative at (0,q0) is -11 q0^-12, which is nonzero. The implicit +function theorem therefore gives a smooth even local solution q_alpha(delta). +For sufficiently large t, the separately supplied differentiated expansion +`-F'(s)=11C_0 s^-12+O(s^-14)` makes F strictly decreasing for every s>=t. +Together with S_t(1)=1 and S_t(q)->0, this identifies the local solution as +the unique upper-tail quantile of the conditional law. + +Put q=q0(1+a delta^2). Substitution in (17) gives + + S_delta(q)=alpha[1+(-11a+c(q0^-2-1))delta^2]+O_alpha(delta^4). + +Consequently + + Q_alpha(t)=alpha^(-1/11)[1+(c/11)(alpha^(2/11)-1)t^-2 + + O_alpha(t^-4)]. (M8) + +The author's quantile formula is correct under these source inputs. Strict +density at sufficiently large radii supplies uniqueness; an unquantified +pointwise distributional limit alone would not supply the error term. At +the aligned planar scope c>0 makes the displayed correction strictly negative. +The eventual approach from below is for each fixed alpha and fixed model; +no uniform claim as alpha tends to an endpoint, or over L,k,b, is needed. + +## Scope of the signed-law coefficient bound + +Let rho have probability density proportional to H gamma^11 G0 in lambda. +The positive finite normalizer is supplied by C_0 and the source Gaussian +majorant. Equations (13) and (19), with the constants in (12), yield exactly + + B_sign = (11 U_abs/(2I)) E_rho[|A|/sqrt(1+A^2)]. + +At every finite A, the integrand lies in [0,1). Positivity of G0 on the +nondegenerate Gaussian support gives positive rho mass at a!=0, so its mean +is strictly positive. Its mean is strictly below one because 1 minus the +integrand is a strictly positive measurable function under a probability law. +Thus, throughout the fixed dimension/frame scope of sections 1--5, + + 0 < B_sign < 11 U_abs/(2I) = 4587821/876544. (M9) + +This bound does not use a sign of D_a G0 and does not require the aligned +planar factorization of section 7. The author's comment 5903080058 states the +bound after an aligned-planar discussion using the word "Likewise"; (M9) +makes its valid broader scope explicit. By contrast, the offered strict bound +`c>11U_2/(2I)=4587/856` does use section 7's D_a G0<=0, with B<0. In that +case the first geometric integrand ratio `(1+12A^2)/(1+A^2)` is >1 whenever +a!=0, which proves strictness already from the positive Gaussian mass. +Neither coefficient inequality supplies a parameter-uniform remainder. + +## Source and verification boundary + +`SOURCES.json` is the exact source map from #176. The cached `R.md`, `SC.md`, +and `P.md` were each checked against its declared size, SHA-256 and Git blob. +R and P reused matching local source bytes; SC was fetched at its explicit +historical commit/path. No author program was used for the calculations. + +`moment_consequence_checks.py` independently compares tail- and density-based +rational integrals, fixed hand-derived cases, the divergence threshold and log +coefficient. A genuine mixture of Pareto(11), Pareto(13), Pareto(15) validates +the quantitative remainder bound and supplies the nonuniform-in-p counterexample. +Wrong normalization, sign, omitted p, and missing factor-two controls are rejected. +Additional exact mixture checks compare the all-order normalized numerator +against direct component moments. Log-power integrals, quantile coefficients, +and the two rational coefficient factors are checked separately. +These finite checks support arithmetic only; they do not replace (M1), its +Gaussian assumptions, or the pending source-bound A/B analytic reviews. + +Preliminary reading of #176 §§4–7 found no blocking error before coordination +redirected this reviewer; that preliminary observation is not a completed B +verdict. The results here are conditional corollary validation (M2)--(M9) and +the positive-tail divergence implication, not acceptance of the consumed +Gaussian theorem or the full PR. Scientific effect: NONE. diff --git a/reviews/radial_moment_corollaries_20260930/NONAUTHOR_REVIEW.md b/reviews/radial_moment_corollaries_20260930/NONAUTHOR_REVIEW.md new file mode 100644 index 000000000..fcd55a978 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/NONAUTHOR_REVIEW.md @@ -0,0 +1,40 @@ +# Nonauthor review of the radial moment consequences + +Reviewer: OpenAI / Codex subagent `/root/workflow_bottleneck_audit`. +Date: 30 September 2026. + +**VERDICT: ACCEPT (M2)–(M9), including the positive-tail divergence statement, as conditional consequences of the exact specified PR176 tail, smoothness and coefficient inputs. No blocking amendment found. This is not another Slice B verdict.** + +I did not draft or amend this consequence manuscript or its control script. I read the frozen consequence, the exact PR176 proof, the source/custody map, and the exact controls after the derivation was delivered. Same provider and shared-account/source exposure imply zero organizational-independence credit. The author's prior power/log/quantile formulas are expressly credited by the manuscript; I make no novelty claim for them. + +## Exact object + +- `MOMENT_CONSEQUENCE.md`: 11,110 bytes; SHA256 `29bba56b56fd3223a2ea71d012f9abe04eb30240bc5bbfe8f5b131e0bba6f4dd`; Git blob `fb3bef3a9a46cc0135a88bae6c26e1bade4e7384`. +- `COROLLARY_CUSTODY.json`: SHA256 `b532794b61e6ffd5539ce9f7dc5509b44d04d2c238a14f33e214c8e387532a64`. +- Consumed PR176 proof: head `71772a8906cf6663d16be2cd5f63e5d4ebdcee2b`, blob `65f24b1f9a7b80522a2ee059f5f6877c9f970f0d`, SHA256 `a3693664900e23e1e763d88a0b5609936a6f31d844cbe6f18c7d0379b94ba710`. + +All nine custody-listed artifacts and all four declared source bodies matched their recorded sizes and hashes; source Git blob identities also matched. The observation file is `IDENTITY_AND_CONTROL_OBSERVATIONS.json`. + +## Analytic reconstruction + +1. The consumed remainder really has the strength used here: PR176(17) is uniform for q≥1 with an O(t^-4 q^-11) tail envelope. The consequence does not derive unbounded moments from unweighted total variation. It keeps the already-formed microscopic point-intensity law, conditioning at R>t, and the original source normalization. + +2. For a positive power p<11, nonnegative tail integration yields 1+p∫q^(p-1)S_t(q)dq. For p<0, absolute integrability follows already from S_t≤1 and ∫_1^∞q^(p-1)dq<∞. Thus the same signed identity holds throughout p<11, including the separate exact p=0 case. Inserting the uniform remainder gives a bound |p|K/(11-p)t^-4. The displayed correction is p c[1/(13-p)-1/(11-p)] = -2pc/[(13-p)(11-p)]. All special cases and sign statements follow. + +3. The range is sharp for this actual limiting law, not merely for a formal approximation. F(tq)/F(t) has a positive constant times q^-11 tail for every finite t>0 with F(t)>0. Therefore every p≥11 moment diverges, logarithmically at eleven and as a power above it. This is distinct from the capped eleventh moment and asserts nothing about finite-r field moments. + +4. The logarithmic correction can be obtained directly from ∫S_t(q)dq/q. This gives 1/11 and c(1/13-1/11) = -2c/143, with absolute error at most Kt^-4/11. No differentiation in p of a remainder is required. + +5. For the finite-order Mellin expansion, applying the source scalar expansion at s=tq gives a uniform remainder O(t^(-2N-2)q^(-13-2N)) after multiplying by t^11. Its weighted tail integral converges for every fixed p<11. The numerator is D(t)+p times that integral, so expanding D(t) at the same order yields (11+2j)/(11+2j-p), exactly the proposed coefficient. Retaining the exact denominator C(1/t) preserves normalization; its positive limit C0 keeps division stable. The stated remainder bound and p=0 convention are consistent. There is no claim that the infinite Taylor series converges. + +6. For log powers, the change of variable y=log q reduces each term to an exponential Gamma integral. Multiplication by m gives m!/(11+2j)^m. The remainder integral is bounded by m!K_N/(13+2N)^m at the stated t order. Every fixed integer m≥1 is covered; there is no uniform-in-m claim. + +7. The quantile expansion uses more than the CDF error: the exact scalar representation is smooth and even in delta, and its q derivative at delta=0 is -11q0^-12≠0. The implicit function theorem gives the local even expansion. The separately supplied differentiated tail makes F strictly decreasing for all sufficiently large arguments, identifying the local root with the unique conditional upper-tail quantile. Substitution gives coefficient c(alpha^(2/11)-1)/11. The restriction to fixed alpha in (0,1) and fixed source model is essential and present. + +8. The signed coefficient bound follows directly from the source C0 and B_sign identities. After normalization by the positive measure with density proportional to H gamma^11 G0, its remaining ratio is E[|A|/sqrt(1+A²)]. It is strictly between zero and one because the source density gives positive mass at nonzero a and A is finite. This proves 0 < B_sign < 11U_abs/(2I) at the stated general fixed-dimension/frame scope. It needs no sign for D_a G0. The stricter lower bound on c uses the aligned planar sign from source §7, as the consequence correctly states: its second contribution is nonnegative and its geometric ratio (1+12A²)/(1+A²) exceeds one on positive mass. + +## Verification and limits + +I inspected and replayed `moment_consequence_checks.py` once with `python3 -B -S`; all eleven test methods passed. The finite Pareto mixtures independently exercise tail-versus-density normalization, powers of both signs, threshold divergence, log coefficients, finite-order numerator normalization, quantile brackets and the rational coefficient factors. They support arithmetic and the claimed failure of a uniform-in-p remainder; they do not certify the source Gaussian theorem. + +No defect was found in these conditional implications. Their premise remains the exact source smooth expansion and retained Gaussian coefficient identities, whose Slice A/B reviews are separate. Constants need not be uniform as p approaches eleven, alpha approaches zero or one, or physical parameters leave fixed compact scopes. The original r limit precedes the t limit. No whole-cluster maximum law, simultaneous t(r) theorem, source acceptance or scientific-register change follows from this review. diff --git a/reviews/radial_moment_corollaries_20260930/P.md b/reviews/radial_moment_corollaries_20260930/P.md new file mode 100644 index 000000000..dfed3b8d3 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/P.md @@ -0,0 +1,494 @@ +# Uniform matrix-cap selection and unrestricted short-lifetime asymptotics + +**Object:** UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1. +**Author:** OpenAI / ChatGPT, 24 September 2026. +**Disposition:** author-side mathematical candidate; no nonauthor acceptance, formal verification, numerical finite-band certificate, or program-level closure is asserted. + +This is an additive continuation of MARKED-CYLINDER-CAP-20260924-v1, SHA256 `0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc`, reproduced unchanged in `sources/`. It addresses the matrix-probability, angular-uniformity, compact-mark, and lifetime-composition interfaces of campaign #61. It does **not** extend the predecessor's numerical constants to new parameters. Instead it derives new, non-numerically-evaluated constants by a uniform Gaussian argument. + +## 1. Model, convention, and the three theorems + +Fix a dimension d >= 2 and a side length L > 0. On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance + + K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2) + / sum_{n in Z^d} exp(-|Ln|^2/2). + +In particular d=3, L=24 is an actual three-dimensional SIDE24 application. No replacement by a nonperiodic covariance is made. Fix compact intervals B=[b_-,b_+] and K=[k_-,k_+] with 00, use the embedded local cylinder centered at the origin with pins + + M=-(r/2)u, S=(r/2)u, + f(M)=b, f(S)=s=b-k r^3, + grad f(M)=grad f(S)=0. + +Q_{r,b,k,R} is the continuous Gaussian regression law at these 2(d+1) observations, not a conditioning on a positive-probability event. A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue. Set + + W_r = |det H_M det H_S| 1{H_M<0, index(H_S)=d-1}, + Z_r = E_Q W_r, dQ^W = (W_r/Z_r)dQ. + +The index counts negative eigenvalues. The FULL normalizer Z_r has no extra pin-density, coordinate Jacobian, or adjacency condition. Let p_r(b,k,R) be the Q^W probability that the **global ordinary superlevel elder death partner** of M is S. + +**Theorem A (uniform selection candidate).** There exist r_*>0 and C=z_* r^2 and the full transverse-matrix boundary layer, including simultaneous small transverse eigenvalues. It does not presume isotropy under arbitrary rotations of the torus. + +**Theorem B (compact-mark density candidate).** Suppose B and K have positive lengths. Count ordered maximum/saddle pairs with 00, + 0 <= nu_cand(ell)-nu_eld(ell) <= C_{B,K} ell^(2/3), (1.2) + +as ell decreases to zero. An exact finite-dimensional Gaussian integral for c_{B,K} appears in Section 11. + +Theorem B is a COMPACT MARK WINDOW result, especially its quantitative difference estimate. Theorem C below separately removes the mark and distance restrictions for the LEADING asymptotic. There is no stated numerical value of C, r_*, z_*, or c_{B,K}, and no claim that r_*=1/20. The new asymptotic normalizer proof does not replace or re-certify the historical H3 finite-band floor. + +**Theorem C (unrestricted finite-lifetime leading density candidate).** For the same fixed torus and dimension, let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume. There is a positive finite constant c_{d,L} such that + + nu_cand^all(ell) ~ c_{d,L} ell^(-1/3), + nu_eld^all(ell) ~ c_{d,L} ell^(-1/3). (1.3) + +All birth heights, all positive scaled-gap marks and all spatial separations on the torus are included in (1.3). The essential global-maximum class is excluded from the finite-lifetime count. The constant is an explicitly identified finite Gaussian/angular integral, with the birth and gap integrals analytically eliminated in Section 15. No elementary closed form or numerical enclosure is claimed. Crucially, the compact-window O(ell^(2/3)) DIFFERENCE estimate in (1.2) is not asserted for the unrestricted densities. Theorem C has a separate domination and off-diagonal proof, not a change of label on Theorem B. + +## 2. Positive Fourier spectrum and finite-jet rank + +Poisson summation gives + + K_L(z)=sum_{n in Z^d} a_n exp(2 pi i n.z/L), + a_n=exp(-2 pi^2 |n|^2/L^2) / sum_j exp(-2 pi^2 |j|^2/L^2)>0. + +Thus every Fourier mode has positive variance, and sum_n sqrt(a_n)(1+|n|)^q is finite for every q. A real sine/cosine expansion therefore converges in every C^q norm in every finite L^p norm. In particular the field has a smooth version and all finite moments of its global derivative suprema exist. + +Any finite list of distinct derivative evaluation functionals at distinct sites is linearly independent. Indeed a vanishing-variance linear combination must annihilate every Fourier mode because every a_n is positive. As a distribution it is a finite sum of derivatives of point masses with every Fourier coefficient zero, so it is zero. Smooth test functions supported at one site and with arbitrarily prescribed finite jets show that every coefficient is zero. At one site in a rotated frame the same assertion also follows from a polynomial P(R^T n) vanishing on all n in Z^d: the invertible rotation reduces this to a polynomial vanishing on the full integer lattice, hence the polynomial is zero. + +Only lists without repeated derivative functionals are called distinct. A Hessian is parametrized by its independent symmetric entries, not by d^2 allegedly independent entries. + +These arguments justify positive covariance for the original pins at each pair of distinct points and for every residual jet list modulo the pins used below. They also explain why finitely many observed modes or a numerically positive sampled covariance would not suffice for this proof. + +## 3. An exact nonsingular contact frame for all orientations + +Write x for the axial coordinate and y_1,...,y_m for the transverse coordinates, m=d-1. On the line y=0 set a=-r/2, c=r/2. Order the original observations as + + O_r=(f(a), f_x(a), f(c), f_x(c), + f_y1(a),f_y1(c), ..., f_ym(a),f_ym(c)). + +This is a permutation and orthogonal re-expression of the specified physical pins. Define U_r=T_r O_r by four axial rows + + U0 = (f(a)+f(c))/2, + U1 = (f(c)-f(a))/r, + U2 = (f_x(c)-f_x(a))/r, + U3 = (6/r^2)[f_x(a)+f_x(c)-2(f(c)-f(a))/r], (3.1) + +and, for every transverse direction j, two rows + + Vj0=(f_yj(a)+f_yj(c))/2, + Vj1=(f_yj(c)-f_yj(a))/r. + +The transformation is invertible for r>0 and has exact absolute determinant + + |det T_r| = (12/r^4)(1/r)^m = 12 r^(-(d+3)). (3.2) + +The prescribed target becomes exactly + + v_r=(b-k r^3/2, -k r^2, 0, 12k, 0,...,0). (3.3) + +This is not an approximate transformation, and the factor 12 in (3.2) is retained in the lifetime ledger. + +Taylor formulas with integral remainders give mean-square convergence, uniformly over frames, of U_r to the contact jet + + U0*=(f, f_x, f_xx, f_xxx, + f_y1,f_xy1,...,f_ym,f_xym) at 0. (3.4) + +The potentially dangerous U3 cancellation is an averaged third derivative, not a subtraction of uncontrolled random values. Uniform convergence of its covariances, and of cross-covariances with any fixed finite derivative order, follows either from those integral formulas or from the rapidly summable Fourier series. The target v_r converges uniformly on B x K to v_0=(b,0,0,12k,0,...,0). + +The functionals in (3.4) are distinct. Hence their covariance Sigma_0(R) is positive definite by Section 2. Its entries are continuous in R. The orthogonal group is compact, so its least eigenvalue has a positive uniform minimum. Consequently Sigma_r(R) remains uniformly positive definite for all sufficiently small r; Sigma_r and Sigma_r^{-1} converge uniformly. + +This compactness argument establishes existence, not an evaluated interval enclosure. It never equates Sigma_r at different orientations. Full torus rotational invariance is neither assumed nor true in general. + +Let A_r=D_y^2 f(M), represented by its m(m+1)/2 independent entries. Append these entries to U_r. At r=0 the extra functionals are f_yi_yj, i<=j; none duplicates a functional in (3.4). The enlarged covariance is therefore uniformly positive definite at contact and for small r. Its Schur complement shows that the Q-law of A_r has covariance uniformly bounded above and uniformly bounded away from zero. Its mean is uniformly bounded. In symmetric-matrix Lebesgue coordinates it has a density satisfying + + density_Q(A) <= C_0 exp(-c_0 ||A||_F^2), (3.5) + +for fixed positive C_0,c_0. This remains valid after any change of eigenvector orientation. Orthogonal invariance of the density is not required. + +## 4. Uniform conditional derivative moments and the independent residual + +Because Sigma_r^{-1} and v_r are uniformly bounded, the regression energy v_r^T Sigma_r^{-1}v_r is uniformly bounded. For each unconditioned standardized Fourier coefficient xi, covariance Cauchy-Schwarz bounds its conditional mean by the square root of this energy, and its conditional centered variance by one. Minkowski's inequality and the summable Fourier coefficients then give, for every finite p and derivative order q, + + sup_{r,b,k,R} || ||f||_{C^q(X)} ||_{L^p(Q)} < infinity. (4.1) + +The same proof applies to a centered residual after further conditioning on A_r. Write, in one full-field Gaussian coupling, + + f(z)=mu_r(z)+B_r(z).(A_r-E_Q A_r)+g_r(z), (4.2) + +where g_r is jointly independent of the entire symmetric matrix A_r. The regression coefficient B_r and its derivatives through order four are uniformly bounded, by the uniform inverse covariance from Section 3 and bounded derivative cross-covariances. The residual has uniformly bounded C^4 moments of every finite order. + +Independence in (4.2) is not asserted separately for Hessian entries, endpoint variables, or sampled derivative maxima. All residual evaluations are orthogonal to A_r; Gaussian independence on a countable dense collection and continuity give independence of the random smooth field. The residual's internal correlations are retained. + +Let J_r=1+||g_r||_{C^4(X)}, with a fixed coordinate norm. The partial-block operator norms of the cap theorem differ from this coordinate norm by finite constants depending only on d. Hence, for M3,M4 on the local cylinder, + + M3 <= K_0(J_r+||A_r||_F), + M4 <= K_0(J_r+||A_r||_F), + J_r >= 1, J_r independent of A_r, + sup ||J_r||_p < infinity for every finite p. (4.3) + +No inverse Hessian moment is used. Gaussian pin covariance and random field Hessians are different objects. + +## 5. Endpoint identities and the actual full normalizer + +For either pin i in {M,S}, put + + alpha_i=f_xx(i)/r, beta_i=grad_y f_x(i)/r, + A_i=D_y^2 f(i), + H_i=[[r alpha_i, r beta_i^T], [r beta_i,A_i]]. + +The two gradient pins imply the vector and scalar average bounds + + |alpha_i| <= M3/2, ||beta_i|| <= M3/2, + ||A_S-A_M||_op <= r M3. (5.1) + +For example integral_a^c f_xx=0; subtract this average at an endpoint and integrate the M3-Lipschitz bound. For beta use the average of grad_y f_x and its vector derivative. This does not invoke a simultaneous vector Rolle zero. + +The exact height identity gives a weighted mean of f_xxx equal to 12k on [a,c]. Variation of f_xxx over that interval is at most rM4. The same average calculation yields + + |alpha_M+6k| <= r M4/2, + |alpha_S-6k| <= r M4/2. (5.2) + +The block determinant identity, valid even when alpha_i or A_i is zero, is + + det H_i/r = alpha_i det A_i + -r beta_i^T adj(A_i) beta_i. (5.3) + +By (4.1),(5.1), det H_i/r and W_r/r^2 have uniformly bounded moments of every finite order. + +Let A_0 be D_y^2 f(0) under regression on U0*=v_0. Its law is a full-support nondegenerate Gaussian on symmetric matrices. Finite-dimensional Gaussian regression gives uniform convergence of the means and covariances of A_M to those of A_0. Equations (5.1),(5.2) imply that the congruence-scaled endpoint Hessians converge, in probability uniformly over parameters, to + + diag(-6k,A_0) at M, diag(6k,A_0) at S. + +Here the congruence uses diag(sqrt(r),I), so its off-diagonal entries are sqrt(r) beta_i. The limiting matrices are nonsingular almost surely. This follows from the full matrix density and k>=k_->0. Thus the type indicator converges to 1{A_0<0}. Uniform integrability from (5.3) gives + + Z_r/r^2 -> z_0(b,k,R) + := (6k)^2 E[det(A_0)^2 1{A_0<0}], (5.4) + +uniformly on the compact parameter set. + +For completeness, uniformity can be checked by a subsequence argument: any allegedly bad sequence of parameters has a convergent subsequence; continuous Gaussian means/covariances, (5.2), and uniform integrability give the stated limit on that subsequence. The discontinuity set of the type indicator has Gaussian probability zero. Equation (3.5) supplies a common integrable majorant. + +The right side of (5.4) is continuous and strictly positive: the Gaussian law has positive density on an open negative-definite matrix ball. Compactness therefore yields + + 00, with eigenvalues + + 00. + +Determinant monotonicity on positive-definite matrices therefore gives + + |det H_M| <= r (h/2) det B. (6.1) + +This is a canceled-pivot bound: no estimate of 1/alpha_M is taken. + +From ||A_S+B||<=rh, the ordered singular values of A_S are at most lambda_j+rh. The adjugate norm is at most the product of the largest m-1 such bounds. With the empty product equal to one when m=1, (5.3) gives + + |det H_S| <= r(h/2) [lambda_1+(3/2)rh] + product_{j=2}^m(lambda_j+rh). + +Hence the actual typed weight satisfies + + W_r <= (r^2 h^2/4) lambda_1[lambda_1+(3/2)rh] + product_{j=2}^m lambda_j(lambda_j+rh). (6.2) + +The second soft factor includes the saddle mixed-square term. Omitting it, or replacing it by an unrelated order-one factor, changes the radius power. No Gaussian-entry independence is needed for (6.2). + +## 7. Full matrix boundary-layer integration, including corank intersections + +By the predecessor's deterministic marked-cylinder theorem, pairing and the one-branch conclusion hold on + + G_r={lambda_min(-A_M)>[4/(3k)]rM3^2, + rM4<=3k/10}. + +On typed support A_M<0 automatically. It suffices to control G_r^c directly, not infer a converse from a local path-event bound. + +First suppose m>=2. In (4.3) absorb sqrt(m) into K and set U=J_r+Lambda>=1. Then h<=KU. Failure of the depth condition is contained in + + 00 and use h<=K(J_r+lambda). Depth failure implies + + lambda<=Dr(J_r+lambda)^2 + <=2DrJ_r^2+2Dr lambda^2. + +It is contained in the union + + 01/(4Dr). (7.5) + +The large-lambda branch is retained. On the near branch, h<=K(1+4D)J_r^2 for r<=1, J_r>=1. The bounded scalar density, (6.2), and direct integration in lambda give a numerator <=C r^5 E J_r^10. On the far branch, direct fourth-moment Markov with the actual weight gives + + E_Q[(W_r/r^2)1_far] + <=(4Dr)^4 E_Q[(W_r/r^2)lambda^4] <=C r^4. (7.6) + +Thus the scalar numerator is <=C r^5+C' r^6. This separate argument is why the all-dimension statement does not silently drop the scalar far branch. + +Finally, for every dimension, the fourth-derivative exception is bounded directly by + + E_Q[(W_r/r^2)1{rM4>3k_-/10}] + <=[10r/(3k_-)]^4 E_Q[(W_r/r^2)M4^4] <=C r^4. (7.7) + +This is a pointwise Markov bound followed by a joint moment estimate. It is not Cauchy-Schwarz with a missing square root. All joint moments in (7.6),(7.7) are finite by Sections 4-5. + +Divide the numerators by the FULL lower bound (5.5). The result is + + Q^W(G_r^c)<=C_3 r^3+C_4 r^4<=C r^3. (7.8) + +## 8. Pinned genericity and global event measurability + +The following argument applies at every fixed positive r and parameter, not on an asserted common probability-one set for uncountably many laws. + +Away from the pins, consider (grad f(x),Hf(x)v), with ||v||=1. Its range has dimension 2d; its parameter manifold has dimension 2d-1. Modulo the pin observations, these functionals have full covariance rank: gradient and Hessian orders are distinct, and the map H -> Hv from symmetric matrices onto R^d is surjective. For two distinct unpinned points, the map + + (grad f(x),grad f(y),f(x)-f(y)) + +has range dimension 2d+1 and parameter dimension 2d. For a tie to a pinned height use (grad f(x),f(x)-b), or the analogous s map, with dimensions d+1 and d. + +Exhaust the relevant domains by compact sets separated from pins and from coincident points; use finitely many sphere charts when v occurs. Covariance positivity from Section 2 and compactness give bounded Gaussian densities on each such set. An elementary overdetermined-zero lemma applies: on a mesh of size epsilon, a random C^1 map from dimension q to dimension p>q, with derivative norm bounded by N, could have a zero only if one of O(epsilon^-q) mesh values has norm O(N epsilon). The density bound makes the probability O(N^p epsilon^(p-q)), tending to zero. First fix N, then let N increase; the random derivative norm is almost surely finite. This proves absence of all these zeros. + +Pinned Hessians have a full density on their symmetric entries for fixed distinct pins, so their determinants are nonzero almost surely. The pinned heights differ because k,r>0. Countable exhaustion now proves that Q is almost surely Morse with distinct critical values. The transfer to Q^W follows by absolute continuity and 0f(M)} min_t f(gamma(t)), + +with supremum of an empty set equal to -infinity. The event d_f(M)>h can be represented by a countable union of polygonal-path tests, with rational interior vertices and endpoints, all staying strictly above h and with endpoint value above f(M). Strict clearances allow approximation by such paths. For variable M, use countably many coordinate charts and an initial segment depending continuously on M. These are Borel tests in (M,f). Thus equality d_f(M)=f(S) is Borel. On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint. + +On G_r the unchanged deterministic cap theorem controls every boundary exit and supplies a connection through S to a point higher than b. Therefore the GLOBAL ordinary elder partner is S, not merely a local adjacency candidate. Equations (7.8) and the full-probability locus prove Theorem A. The cap theorem also gives exactly one actual ascending branch to M. It does not identify that branch with the transverse ridge. + +## 9. A source-bound weighted Kac-Rice interface + +For clarity, this section specifies why a global elder mark is admissible; a formula for local Hessian index alone would not suffice. + +Apply Kac-Rice first on compact spatial pair domains separated from the diagonal, to the Gaussian vector field (grad f(x),grad f(y)). It is C^1 and has positive definite point covariance for x!=y by Section 2. Its derivative is block diagonal with Jacobian |det H_x det H_y|. Conditional laws of the whole smooth field depend continuously on the imposed gradient values by finite-dimensional Gaussian regression. Derivative moments are locally bounded. + +The expected-integral extension in Armentano-Azais-Leon, *On a general Kac-Rice formula for the measure of a level set*, arXiv:2304.07424v3, Theorem 7.1, gives the identity for bounded nonnegative continuous weights in the location and an additional Gaussian field mark. It is enough first to use cylinder weights depending on finitely many fixed evaluations of f and its derivatives through order two. Their conditional distributions depend continuously on the imposed gradients by Gaussian regression, so the paper's continuity requirements hold. The field is a random element of the separable C^2 space on the torus. On each compact pair domain the two measures on location x C^2 are finite, by the unweighted formula. Cylinder evaluations on a countable dense set, together with location coordinates, generate this Borel sigma-algebra. Equality on their bounded continuous test-function algebra gives equality of the Borel measures by the monotone-class theorem. This extends the identity to bounded Borel global marks, rather than pretending the elder indicator is continuous or lower semicontinuous. No independence between the mark and gradient field is assumed. + +Use the Borel elder mark from Section 8 and the type indicators. Disintegrate the two heights using their nondegenerate joint Gaussian density conditional on the gradients. This gives the full-pin density times E_Q[W] for candidates, and the same quantity times p_r for selected pairs. The continuous Gaussian regression kernel defines the canonical density version; the underlying measure identity is unaffected by height-null-set choices. + +Exhaust the off-diagonal spatial domain toward r=0. The r dr bound proved in the next section makes the expectations locally integrable, and nonnegative monotone convergence completes the near-diagonal formula. No finite-grid approximation of the continuum field is substituted. + +## 10. Exact radial ledger: the dimension cancels + +Let pi_r(R;v_r) denote the density of U_r at (3.3). Since U_r=T_r O_r, + + density_{O_r}(specified pins) + =12 r^(-(d+3)) pi_r(R;v_r). (10.1) + +Use midpoint z, directed separation h=r u, birth b, and gap mark k. The change from the ordered pair (M,S) to (z,h) has absolute determinant one. Polar separation contributes r^(d-1) dr d sigma(u), where sigma is ordinary, NOT probability-normalized, surface area on S^(d-1). The height change (f(M),f(S))=(b,b-k r^3), at fixed r, contributes r^3 db dk. There is no factor 1/2: maximum/saddle pairs are ordered by their different roles. + +Consequently the candidate intensity per unit midpoint volume is + + r A_r(b,k,u) dr db dk d sigma(u), + A_r=12 pi_r(R;v_r) (Z_r/r^2). (10.2) + +The powers are + + spatial (d-1) + height-mark 3 + pin -(d+3) + + determinant-normalizer 2 = 1. + +By Sections 3 and 5, + + A_r -> A_0=12 pi_0(R;v_0) z_0(b,k,R)>0 (10.3) + +uniformly and with a uniform finite upper bound. Although frames were used in the proof, (10.2) is invariant under a rotation of the transverse frame: the original zero-gradient law and determinant weight are unchanged by that orthogonal coordinate change. Thus it is a function of u alone. Uniformity was proved on the compact orthogonal group and descends to the sphere; no nonexistent global continuous transverse frame on the sphere is assumed. + +The selected intensity is r A_r p_r with the same measures. This supplies the domination needed below, rather than assuming a fixed-axis statement can be integrated. + +## 11. Lifetime pushforward and the exact compact coefficient + +For ell=k_->0. This proves (1.2). It is an O(ell^(2/3)) difference, not a claim that either density separately has a second-order expansion with that remainder. Convergence of A_r to A_0 was not assigned a numerical rate. + +## 12. Short bars, moments, and the exact remaining boundary + +Integrating the density versions gives the following expected per-unit-volume results for the same spatial/mark-restricted populations: + + E N_eld(0,t] ~ (3/2)c_{B,K} t^(2/3), + 0<=E[N_cand(0,t]-N_eld(0,t]]<=C t^(5/3). (12.1) + +For any real q>-2/3, + + E sum_{selected, ell<=t} ell^q + ~ c_{B,K} t^(q+2/3)/(q+2/3). (12.2) + +The same leading expression holds for candidates. For q>-5/3 the nonselected difference alone satisfies + + E sum_{candidate but not selected, ell<=t} ell^q + <= C_q t^(q+5/3). (12.3) + +For q<=-2/3 the two individual expectations in (12.2) are infinite for every sufficiently small positive t, because their densities have a positive ell^(-1/3) leading term. Equation (12.3) concerns the nonselected counting measure directly, not subtraction of two infinities. In particular the compact-window inverse-lifetime threshold is p<2/3 for integrability of ell^(-p), with logarithmic divergence at p=2/3. This is unrelated to the inverse-Hessian/Schur thresholds in issue #56. + +The corresponding unrestricted cumulative and individual-moment corollaries follow from Theorem C with c_{d,L} in place of c_{B,K}, by the same integrable-power argument. The compact-window nonselected difference rates (12.1),(12.3) are not automatically unrestricted rates. + +## 13. Removing all birth and gap cutoffs by a direct intensity majorant + +The covariance Sigma_r of U_r depends on radius and frame, NOT on the imposed target. Section 3 therefore supplies a fixed r_0>0, independent of b and k, on which its eigenvalues are bounded above and away from zero. Reduce r_0 below the torus injectivity scale and below one. Do NOT require a globally uniform lower bound for Z_r or a globally uniform constant in (1.1). + +For every b in R, k>0 and 0= c(b^2+k^2), (13.1) + +with fixed c,C>0. For the second inequality it suffices to use its two coordinates b-k r^3/2 and 12k: if r<=1, then b^2<=2(b-k r^3/2)^2+k^2/2. Thus the invertible covariance bound yields + + pi_r(R;v_r) <= C exp[-c(b^2+k^2)]. (13.2) + +Conditional Fourier-coefficient means are bounded by C|v_r| and their centered variances by one. The same summability/Minkowski proof as Section 4 now gives the TARGET-GROWTH bound + + E_Q (1+||f||_{C^3})^p <= C_p(1+|b|+k)^p (13.3) + +for every fixed finite p, uniformly over r and frames on this r_0 band. This is not the compact-target bound (4.1) applied outside its scope; it is its explicit polynomial-growth version. + +By (5.1),(5.3), for r<=1, + + |det H_i|/r <= C (1+||f||_{C^3})^d, + 0<=W_r/r^2<=C(1+||f||_{C^3})^(2d). + +These inequalities remain true when k is arbitrarily small; no division by k or by Z_r has occurred. Combining with (13.2),(13.3) gives the full unnormalized-intensity majorant + + 0<=A_r(b,k,u)=12 pi_r(R;v_r) Z_r/r^2 + <= C(1+|b|+k)^(2d) exp[-c(b^2+k^2)] =: H(b,k). (13.4) + +For every fixed positive r and every b,k>0, Z_r is positive and finite: append BOTH endpoint Hessians to the distinct-site pins; the full residual Gaussian density is positive on a product of an open maximum cone and an open index-(d-1) cone. Thus p_r is defined, even though no globally uniform lower floor has been used. For every fixed b,k>0, Theorem A on a compact neighborhood implies p_r->1 as r->0. + +At a fixed lifetime ell, the near-pair restriction r<=r_0 is exactly k>=ell/r_0^3. After the exact change (11.1), + + ell^(1/3) nu_cand^near(ell) + = integral_{R x (0,infinity) x S^(d-1)} + 1{k>=ell/r_0^3} A_{(ell/k)^(1/3)}(b,k,u) + /(3 k^(2/3)) db dk d sigma, (13.5) + +with the integrand defined to be zero where the indicator is zero. The selected near density has the additional factor p_r. Both integrands are dominated by H(b,k)/(3 k^(2/3)). This majorant is integrable: the k exponent -2/3 is strictly greater than -1 at zero, and Gaussian tails absorb the polynomial at infinity and in b. This explicitly handles the shrinking-k boundary where a compact-window selection constant can blow up. + +For each fixed b,k>0,u, the indicator tends to one, A_r->A_0 and p_r->1. Dominated convergence therefore gives the same limit for the two near densities: + + c_{d,L}=4 integral_{R x (0,infinity) x S^(d-1)} + k^(-2/3) pi_0(R;v_0) z_0(b,k,R) + db dk d sigma(u), (13.6) + 00, + D_u=E[(det A_u)^2 1{A_u<0} | V_u=0]. (15.1) + +Here A_u conditional on V_u=0 is an actual centered nondegenerate Gaussian symmetric matrix, not automatically GOE. Finite-jet rank proves tau_u>0, positive densities p_G(0),p_Vu(0), and D_u>0. All are continuous in the frame and invariant under transverse-frame changes. + +The contact density and cone moment factor as + + pi_0(R;v_0)=p_(f,V_u)(b,0) p_G(0) phi_tau_u(12k), + z_0=(6k)^2 E[(det A_u)^2 1{A_u<0} | f=b,V_u=0], + +where phi_tau is the centered scalar Gaussian density of standard deviation tau. Integrating b by ordinary Gaussian disintegration gives p_Vu(0) D_u. Thus (13.6) becomes + + c_{d,L}=144 integral_{S^(d-1)} p_G(0) p_Vu(0) D_u + [integral_0^infinity k^(4/3) + phi_tau_u(12k) dk] d sigma. + +The scalar integral can be evaluated exactly. With t=12k and the elementary gamma integral, + + 144 integral_0^infinity k^(4/3) phi_tau(12k) dk + = Gamma(7/6) tau^(4/3) / [24^(1/3) sqrt(pi)]. + +Consequently the full leading coefficient is + + c_{d,L}= Gamma(7/6) / [24^(1/3) sqrt(pi)] + * integral_{S^(d-1)} p_G(0) p_Vu(0) tau_u^(4/3) + D_u d sigma(u). (15.2) + +This eliminates both unbounded mark integrals. The remaining angular and negative-definite-cone expectation are finite, strictly positive and specified by derivatives of the EXACT periodized covariance. Formula (15.2) is an explicit finite-dimensional Gaussian representation, not a reported elementary closed form or numerical enclosure. The factor24 is a numerical Jacobian/gamma factor, NOT the side length L; it is the same factor for every L. Changing the field amplitude by a>0 scales the coefficient by a^(-2/3), consistent with nu_(af)(ell)=a^(-1)nu_f(ell/a); this is an additional normalization check. + +## 16. Supplied interfaces, retained limits, and review + +**Interfaces supplied by this candidate:** the actual transverse matrix boundary argument (including d=3); a uniform compact-mark asymptotic full normalizer; all orientations without false isotropy; the global Borel marked Kac-Rice interface; exact pin/radial/lifetime factors; compact-window leading density and quantitative selection difference; a separate integrable unnormalized Gaussian majorant removing ALL birth/gap cutoffs; an off-diagonal bound completing ALL spatial separations; the unrestricted finite-bar leading density and its cumulative/moment consequences; and analytic elimination of birth and gap in its coefficient. + +**Not supplied:** a numerical r_* or C on a prescribed band; a globally uniform O(r^3) probability constant over unbounded b,k; the compact O(ell^(2/3)) difference as an unrestricted remainder; an elementary closed form or numerical enclosure of (15.2); a rigorously evaluated lower coefficient for pairing failure; the original numerical H3/LPW replay; a shrinking-annulus RN certificate; a24-jet displacement enclosure; formal proof verification; nonauthor mathematical acceptance; or unrestricted P15 prize closure. No historical scientific status is changed by this author-side proof. + +The exact-arithmetic program checks pin transforms/targets, block determinants, finite typed-matrix envelopes, spectral Jacobians, radius/lifetime exponents, explicit small-eigenvalue integral models, global target coercivity and coefficient scaling. Semantic mutations challenge the load-bearing factors. These are finite algebra/implementation controls, NOT Gaussian simulation or proof of continuum estimates. + +Priority nonauthor review: (i) desingularized full residual covariance; (ii) uniform all-field moments and independent-matrix regression; (iii) both soft determinant factors; (iv) eigenvalue integration retaining the random largest-eigenvalue width and all corank strata; (v) scalar far branch; (vi) full-normalizer type convergence; (vii) continuous-to-Borel marked Kac-Rice extension; (viii) every factor in the pin/lifetime ledger; (ix) target-growth bound (13.3) and integrability at k=0 in (13.5); (x) distinct-site compact covariance in the far domain; and (xi) parity, b-disintegration and the gamma factor in (15.2). + +### Primary-source attribution + +Armentano, Azais and Leon, arXiv:2304.07424v3, Sections 2 and 7, supply established Gaussian Kac-Rice/expected-integral background. The precise full-field/Borel and pair application is explained in Section 9, not attributed wholesale to that paper. Curry, *The Fiber of the Persistence Map for Functions on the Interval*, arXiv:1706.06059v2, Section 3, supplies elder-rule background; the all-dimensional separating cap is the explicitly identified in-project predecessor. Beliaev, Cammarota and Wigman, *No repulsion between critical points for planar Gaussian random fields*, arXiv:1911.03455, is neighboring two-point/index-asymptotic work, not a source of our elder-selection conclusion. No priority or novelty claim follows from this bounded reconnaissance. diff --git a/reviews/radial_moment_corollaries_20260930/PROOF.md b/reviews/radial_moment_corollaries_20260930/PROOF.md new file mode 100644 index 000000000..65f24b1f9 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/PROOF.md @@ -0,0 +1,334 @@ +# Smooth inverse-radius expansion and sharp rates for microscopic extreme marks + +Object: OA-RADIAL-RATE-20260930-v1. Author: OpenAI / GPT-6 Astra Pro. +Disposition: AUTHOR-SIDE CONDITIONAL PROOF CANDIDATE; NONAUTHOR REVIEW OPEN. +Scientific effect NONE. No parent proof, register, Boolean, prize, or earlier +review is changed. This conversation authored SC/P and contributed to the +related cluster line; no independent revalidation of those premises is implied. + +## 1. Exact object and scope + +Fix d>=2, L>0, b real, k>0 and an orthonormal frame, under the original model +and full endpoint normalizer. This paper concerns the ALREADY-FORMED limiting +microscopic POINT intensity of [R]. First the original field radius r tends to +zero; only then does the microscopic distance threshold t tend to infinity. +No finite-r convergence rate is proved. Whole-cluster maximum selection and +its moving tie boundaries are not consumed or treated here. + +Exact source identities are in SOURCES.json. [R] is PR166 RADIAL_TAIL.md at +80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337, blob0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4: +R7-R13 supply the root-resolved integral, Gaussian domination and positive leading +coefficient. [SC] is PR162 PROOF.md at382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf, +blob16c56821b52fd76b0be791622b9c3809eafde75a: the nondegenerate raw Gaussian contact +jet, orthogonal spectral convention and finite nonempty microscopic measure. +[P] is imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md at +7a1cb09a9d58d179393e6d29146252f714b9c9bf, blobdfed3b8d318a3ab1950957f393307733a4bef3f2: +only the exact periodized covariance and contact conditioning, used in Section7. +Neither PR169's whole-cluster candidate, its companion-ratio addendum, nor any +numerical coefficient note is a premise. Source reviews are not reviews here. + +Use [R]'s variables lambda=(a,h,O,xi), with ordered positive hard curvatures h, +normalized Haar O, and remaining cubic entries xi. Put + + A=a/(12k), gamma=sqrt(1+A^2), + beta0=a^2/(12k), c0=a^3/(144k^2), + A0=O diag(0,-h) O^T, + H(h)=product h_j^3 product_(i0 on S. + +For y=1/Z define + + g_lambda,u,v(y)=G_lambda(beta0+k b0 y^2, + c0+(a b0/4)y^2+2k(v-b0u)y^3). (2) + +[R, R11] says exactly that the tail F(t)=integral sum_i 1{|z_i|>t}dM is + + F(t)=K0 integral_(S,lambda) Q H + integral_R |Z|^-12 g(1/Z) + 1{(u-AZ)^2+Z^2>t^2}dZ du dv d lambda. (3) + +The multiplicity of roots is already correct in this formula. No extra factor +for a double cluster, pin density, or endpoint normalizer is introduced. + +## 2. A fixed-domain smooth even representation + +**Theorem 1.** There is a smooth even function C(delta), defined for +|delta|<2/3 and with every fixed derivative bounded on |delta|<=1/3, such that + + F(t)=t^-11 C(1/t), t>=3. (4) + +Smoothness also holds in L1 for the underlying UNSIGNED shape/parameter measures. +The claim is C-infinity, not analyticity or convergence of an infinite Taylor +series. Consequently for every fixed N>=0 there are finite coefficients C_(2j) +with + + F(t)=sum_(j=0..N) C_(2j) t^(-11-2j)+O_N(t^(-13-2N)), + -F'(t)=sum_(j=0..N)(11+2j)C_(2j)t^(-12-2j) + +O_N(t^(-14-2N)). (5) + +C_(0)>0 is [R]'s C_star. Constants are at the fixed source parameters; no growing +dimension, volume, vanishing-gap, or parameter-uniform claim is made. + +**Proof.** The reciprocal substitution in (3) gives |Z|^-12 dZ=|y|^10 dy on +both signs. For t>|u| the physical radius condition is + + (t^2-u^2)y^2+2uAy-gamma^2<0. + +Set delta=1/t and y=delta w. Its two endpoints in w are + + w_plus/minus(delta)=m(delta) +/- d(delta), + m(delta)=-uA delta/(1-u^2 delta^2), + d(delta)=sqrt(gamma^2-u^2 delta^2)/(1-u^2 delta^2). (6) + +Thus (4) holds with + + J_delta(u,v,lambda)=d(delta) integral_(-1)^1 + [m(delta)+d(delta)x]^10 + g(delta[m(delta)+d(delta)x]) dx, + C(delta)=K0 integral Q H J_delta du dv d lambda. (7) + +There is no moving integration boundary left. On |delta|<=delta0<2/3 the +denominator is uniformly positive and the square-root radicand is bounded +below by a positive constant, since |u|<=3/2 and gamma>=1. Each fixed derivative +of m,d is bounded by a polynomial in1+|a| with constants depending on delta0,k. + +For every fixed j the nondegenerate Gaussian contact density satisfies + + sup_(beta,c) |partial_(beta,c)^j G_lambda(beta,c)| + <= C_j exp[-c_j(|h|^2+a^2+|xi|^2)] (8) + +uniformly in O. Indeed each derivative is a polynomial times the same Gaussian; +absorb that polynomial into a weaker Gaussian exponent and then discard beta,c +from its quadratic lower bound. A bounded mean changes the constants only. +Uniform tensor norm equivalence under O supplies the displayed retained variables. +No inverse hard eigenvalue is used. This is an elementary derivative extension +of [R]'s raw Gaussian bound, not an extra stochastic assumption. + +The chain rule in (7), for any fixed number of delta derivatives, now gives an +integrable majorant of the form + + C Q H (1+|a|)^M exp[-c(|h|^2+a^2+|xi|^2)], (9) + +on the bounded shape domain and x in[-1,1]. The actual shifted beta,c values +may be large: (8) is a supremum over them, not an invalid compactness assumption. +This proves differentiation under all integrals and smoothness in L1 as well +as for the total mass. The exact identities m(-delta)=-m(delta), +d(-delta)=d(delta), followed by x->-x, show J_(-delta)=J_delta. This cancellation +holds even when g(y) is not even; its cubic y^3 term is retained. + +Taylor's theorem with the integrated derivative bounds proves (5), including +its differentiated remainder. It is NOT obtained by formally differentiating +an unspecified big-O estimate. C(0) is the positive source coefficient. QED. + +## 3. Explicit second coefficient + +Write C_0=C_(0), C_2=C_(2), to distinguish coefficients from their tail powers11,13. +Set the differential operator acting on G_lambda at fixed a,h,O,xi + + D_a=k partial_beta+(a/4)partial_c. + +The pointwise density has expansion g(y)=G0+g2 y^2+g3 y^3+O(y^4), with + + g2=b0 D_a G0. (10) + +There is no linear y term. Expanding the exact endpoints, or differentiating +(7), gives + + j0=(2/11)gamma^11 G0, + j2=u^2 gamma^9(1+12A^2)G0+(2/13)b0 gamma^13 D_a G0, + J_delta=j0+j2 delta^2+O_L1(delta^4). (11) + +For the geometric part, write w_+=gamma-uA delta+ +[u^2 gamma-u^2/(2gamma)]delta^2+O(delta^3), with the corresponding reflected +negative endpoint. Then (w_+^11-w_-^11)/11 equals +2gamma^11/11+u^2 gamma^9(1+12A^2)delta^2+O(delta^4). +The g2 term contributes2g2 gamma^13 delta^2/13. The odd g3 term first contributes +at order delta^4 after the two physical sides are summed; it is not dropped +from the all-order argument. + +The necessary exact shape integrals are + + I=integral_S Q=246528/35, + U2=integral_S u^2 Q=240192/35, + B=integral_S b0 Q=-428544/7=3I-12U2, + Uabs=integral_S |u|Q=5898627/880. (12) + +They follow by integrating the polynomial Q in v and then on the strips cut +at u=-1/2 and u=0. The companion computes them in exact Fraction arithmetic. +In particular + + C_0=K0*(2I/11) integral H gamma^11 G0 d lambda, + C_2=K0 integral H [U2 gamma^9(1+12A^2)G0 + +(2B/13)gamma^13 D_a G0]d lambda. (13) + +Both integrals are finite by (8). C_0>0. No general sign or nonzero assertion +for C_2 is made in arbitrary dimension/frame; Section7 supplies an exact +noncancellation result at a narrower source scope. No numerical evaluation of +either Gaussian integral is claimed. + +## 4. Rates for the radial ratio and unsigned marks + +Let P_t be point-INTENSITY sampling conditioned on physical radius R>t. Define +q=R/t>=1 and retain Y=(u,v,lambda), but FORGET sign Z. These are the root-resolved +latent coordinates of [R], not a configuration-uniform selection. Define the +finite measures with densities + + dmu0=K0 Q H j0 dY, dmu2=K0 Q H j2 dY, + c=C_2/C_0. (14) + +mu0 has mass C_0 and mu2 mass C_2. The joint law of (q,Y) has the L1 expansion + + dP_t(q,Y)=11q^-12 dq dmu0(Y)/C_0 + +t^-2 [13q^-14 dq dmu2(Y)/C_0 + -11c q^-12 dq dmu0(Y)/C_0]+O_L1(t^-4). (15) + +In particular its total-variation distance to the stated independent limiting +product is O(t^-2). Projection to Y, or to the downward height +eta=u+1/2+v/6, retains this rate. This supplies an O(t^-2) TV rate for the +intensity-selected height law H_eta/I in [R, H1-H2]. No spatial-configuration +TV claim is made: the map to actual positions depends on t. + +Proof: for fixed Y, the radius tail density is t^-11 K0 QH J_(1/t). +Differentiate the smooth expression already justified in Section2; at tq its +radial density times t equals + +t^-11 K0 QH q^-12[11J_(1/(tq))+(1/(tq))J'_(1/(tq))]. + +Divide by F(t)=t^-11 C(1/t) and use the L1 Taylor estimates. The remainder is +bounded after integration by a constant times t^-4 q^-12, integrable on q>=1. +This proves (15) without differentiating a merely pointwise asymptotic. QED. + +Let Pareto_11 have density11q^-12 on q>=1. For the scalar radial ratio, (15) gives + + p_t(q)=11q^-12[1+c t^-2((13/11)q^-2-1)]+O_L1(t^-4). + +**Sharp scalar coefficient:** + + dTV(Law_(P_t)(R/t),Pareto_11) + =kappa |c| t^-2+O(t^-4), + kappa=(2/13)(11/13)^(11/2). (16) + +The same leading coefficient holds for the supremum distance of their CDFs. +The correction changes sign only at q0=sqrt(13/11). Its positive integral is +q0^-11-q0^-13=kappa; its total integral is zero. Half its L1 norm is therefore +kappa, proving (16). The uniform CDF remainder follows from the L1 remainder; +the maximum of q^-11(1-q^-2) is also kappa. If C_2=0 the stated error is O(t^-4), +not an unjustified claim of a nonzero second-order term. + +As another quantitative consequence, uniformly over q>=1, + + F(tq)/F(t)=q^-11[1+c t^-2(q^-2-1)]+O(t^-4 q^-11). (17) + +This is a source-specific second-order regular-variation statement. Generic +second-order tail theory is not claimed novel. + +## 5. Retaining the sign has a different, strictly slower sharp rate + +The cancellation in Section2 used both signs. Keep epsilon=sign Z and the +same Y. The two half integrals are exactly + + J_delta,epsilon=integral_0^(d+epsilon m) w^10 g(epsilon delta w)dw, + J_delta,epsilon=j0/2+epsilon h1 delta+O_L1(delta^2), + h1=-uA gamma^10 G0. (18) + +The linear term comes from the physical cutoff shift m. g'(0)=0, so no density +term cancels it. Applying the differentiated density argument to each half yields + + dTV(Law_(P_t)(epsilon,R/t,Y), + UniformSign x Pareto_11 x mu0/C_0) + =B_sign t^-1+O(t^-2), + B_sign=(K0/C_0) Uabs integral H |A|gamma^10 G0 d lambda>0. (19) + +Indeed the first signed correction has density +12 epsilon h1 q^-13 K0 QH/C_0; its half-L1 norm, summed over the two signs and +integrated over q, is the displayed B_sign because integral_1^infinity12q^-13dq=1. +Finiteness follows from (8); strict positivity follows from the full positive +Gaussian density on open sets with a!=0 and positive hard spectrum. The gauge +convention for O is exactly that in [R]; (19) is a result for those retained +latent variables, not a statement that a sign alone has a nonzero marginal bias. +Marginalizing can cancel a correction, and specifically forgetting epsilon +cancels this one. Thus signed and unsigned rates must not be conflated. + +## 6. A finite part for the critical eleventh moment + +Let M11(t)=integral sum_i min(R_i,t)^11 dM. The prior theorem gave its leading +logarithm. There is now a finite constant L11 such that + + M11(t)=11 C_0 log t+L11-(11/2)C_2 t^-2+O(t^-4). (20) + +One exact definition is + + L11=11 integral_0^1 x^10 F(x)dx + +11 integral_1^infinity[x^10 F(x)-C_0/x]dx. + +The first integral is finite by finite total point mass. The second converges +by (5) at N=1. Tonelli gives M11(t)=11 integral_0^t x^10 F(x)dx; subtracting +its limiting finite part and integrating the controlled remainder proves (20). +No numerical value for L11 is asserted. This is again after the r limit only. + +## 7. Exact positive second correction in the aligned planar periodic model + +**Theorem 2.** In d=2, with the axial coordinate aligned with one of the lattice +axes of [P]'s EXACT periodized kernel, C_2>0 for every fixed L,k>0 and b real. +Consequently the radial TV and CDF errors in (16) are genuinely Theta(t^-2), +with the positive coefficient in (16); not merely an upper bound. + +Proof: [P]'s kernel factors as K_L(x,z)=k_L(x)k_L(z). Let m2,m4,m6 be the even +spectral moments of the one-dimensional variance-one factor; m2>0 and the full +positive lattice spectrum gives m4-m2^2>0 and m6-m4^2/m2>0. At contact condition +on (f,fx,fxx,fxxx,fz,fxz)=(b,0,0,12k,0,0). + +Even-total-order jets and odd-total-order jets are independent. For the odd +soft cubic jets (a,beta,c)=(fxxz,fxzz,fzzz), separability and reflection in each +axis give a centered DIAGONAL conditional Gaussian with variances + + va=vbeta=m2(m4-m2^2), vc=m6-m4^2/m2. (21) + +For transparency: Cov(beta,(fx,fxxx))=(-m2^2,m2 m4) is -m2 times the first row +of Cov(fx,fxxx), so its conditional mean is -m2 fx=0 even though fxxx=12k. +a and c correlate only with fz in that odd conditioning block, with covariances +-m2^2 and -m4; their raw covariance m2 m4 is removed by that regression. Their +conditional covariance with beta vanishes by coordinate reflection. The diagonal +variances follow by subtracting these rank-one regressions. The even soft +curvature has its own strictly positive conditional density at zero and merely +multiplies this cubic Gaussian by a positive b-dependent scalar. + +Therefore on the cusp + + D_a G0=-[a^2/(12vbeta)+a^4/(576k^2 vc)]G0<=0. (22) + +In (13), U2>0, B<0 and G0>0. The geometric term is strictly positive and the +second term is nonnegative. Thus C_2>0, without replacing finite L by a continuum +covariance or numerically approximating a lattice sum. The same common even +b-dependent factor cancels from c=C_2/C_0 and B_sign at this planar aligned +scope. We do not extend this diagonal regression or noncancellation assertion +to arbitrary rotations or d>=3 without a separate argument. QED. + +## 8. Verification and review boundary + +A: reciprocal physical cutoff, fixed-domain map, C-infinity L1 Gaussian derivative +domination, exact parity and differentiated all-order asymptotic expansion. +B: j0/j2 coefficients, all four exact shape integrals, signed/unsigned TV and CDF +constants, finite-part moment, and aligned planar regression/noncancellation. + +The stdlib companion checks shape integrals, formal endpoint equations, exact +Taylor coefficients, cancellation and signed bias, independent scalar controls +and deliberately false variants. These are finite checks, not a formal proof +of any Gaussian input or differentiation under an infinite-dimensional integral. +No Lean formalization, global status change, coefficient enclosure or self-merge. + +No rate for r->0, no uniform simultaneous t(r)->infinity statement, no claim +about the remote singleton population, no convergence rate for whole-cluster +maximum selectors across ties, and no spatial independence/elder pairing is +added by this packet. Sign-forgetting is part of (15), not an optional omission. diff --git a/reviews/radial_moment_corollaries_20260930/R.md b/reviews/radial_moment_corollaries_20260930/R.md new file mode 100644 index 000000000..0f14417ef --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/R.md @@ -0,0 +1,277 @@ +# An eleven-power microscopic radius tail and its universal extreme-height law + +Object: OA-TWO-SCALE-CLUSTER-GEOMETRY-20260929-R1. +Author: OpenAI / GPT-6 Astra Pro, foreground recovery, 29 September 2026 (Central). +Disposition: AUTHOR-SIDE CONDITIONAL PROOF; NONAUTHOR REVIEW REQUIRED. +Scientific effect NONE. No scientific-status transition or numerical Gaussian +coefficient is asserted. This is an analytic calculation for the explicit +limiting MICROSCOPIC measure M of TWO_SCALE_LAW.md (M1), not for the entire +finite-r global field or its remote singleton component. + +## 1. Which radius and which sampling law? + +Use the exact source conventions and fixed d,L,b,k,frame of TWO_SCALE_LAW.md. +For each additional root (X_i,Z_i) of the unsheared planar cubic, its physical +rescaled position is + + v_i=X_i e_axis+Z_i e_soft(O), R_i=|v_i|. + +Define the finite point-intensity measure by integrating the sum over these roots +against M. Its tail mass is + + F(t)=integral sum_i 1{R_i>t} dM. (R1) + +This is point-intensity sampling: a cluster with two qualifying points contributes +two, not one. Every assertion below is a statement about the limiting microscopic +measure. In an application to the Gaussian field, r->0 is taken before t->infinity. +No finite-r uniform spatial moment or rate is inferred. + +**Theorem R.** There is an explicit finite positive C_star such that + + F(t) ~ C_star t^-11. (R2) + +For p>=0 the microscopic intensity moment integral sum_i R_i^p dM is finite +exactly when p<11. The same threshold holds for the maximum radius of a nonempty +cluster under its normalized limiting law. At the boundary, the capped intensity +moment satisfies + + integral sum_i min(R_i,t)^11 dM ~ 11 C_star log t. (R3) + +A full formula for C_star is given in (R13). No numerical evaluation of its +Gaussian integral is claimed. + +## 2. Resolve one stationary point by an exact change of variables + +Retain the canonical shear invariants of CUB: + + u=X+aZ/(12k), B=beta-a^2/(12k), + D=(c-a beta/(4k)+a^3/(72k^2))/2, x=-s>0. + +At fixed a the inverse raw-jet map is + + beta=a^2/(12k)+B, + c=a^3/(144k^2)+aB/(4k)+2D, + |d(beta,c)/d(B,D)|=2. (R4) + +The factor 2 is essential. This is a change in the JET coordinates, separate +from the determinant-one spatial shear. The remaining third-jet entries, denoted +xi, are unchanged. + +Let + + b0(u)=3-12u^2, v=xZ^2/k. + +For an extra root, the canonical conic and line give the inverse parametrization + + x=kv/Z^2, + B=k b0(u)/Z^2, + D=k(v-b0(u)u)/Z^3, Z!=0. (R5) + +Conversely, every parameter in the region defined below with Z!=0 gives a strict +window root at (X,Z)=(u-aZ/(12k),Z). The normalized DOWNWARD height is + + eta=-P/k=u+1/2+v/6. (R6) + +Typed endpoints require v>|b0(u)|/2. The open height window requires 0-6u-3 is automatic on R7: +for -1/2<=u<1/2 its right side is nonpositive; for -3/2=0. The upper height inequality and v>|b0|/2 are +compatible exactly on the displayed u interval. Boundaries are excluded. + +An explicit determinant computation gives the ORIENTED Jacobian + + det d(x,B,D)/d(v,u,Z) + =6k^3(12u^2+4uv-3)/Z^8 + =-6k^3[b0(u)-4uv]/Z^8. (R8) + +On R7 both b0(u)-4uv and 4v^2-b0(u)^2 are strictly positive. The first is also +the saddle determinant sign from CUB; it can be checked directly: for u<-1/2, +use v>|b0|/2, and for u>=0 use v<3-6u, giving a lower bound +12(u-1/2)^2, with the remaining interval -1/2<=u<0 immediate. The second follows +from v>|b0|/2. Thus the Jacobian is nonzero in the open region. + +For each fixed (x,B,D) in the typed domain there are at most two extra window +roots. The usual change of variables with multiplicity is legitimate here: +cover R7 times (R minus {0}) by countably many neighborhoods where R5 is a +diffeomorphism, make them disjoint measurably, apply the ordinary change of +variables to each, and sum the resulting nonnegative integrals. The multiplicity +is exactly the number of counted roots, rather than an omitted or extra factor +of two. Degenerate/boundary jets are algebraically null and not in the strict +window. This justifies applying R5 to a point-intensity sum. + +Define + + Q(u,v)=[4v^2-b0(u)^2][b0(u)-4uv]. (R9) + +The endpoint weight is w=9k^4[4v^2-b0(u)^2]/Z^4. Combining this with the absolute +Jacobian R8 and the raw-jet Jacobian 2 of R4 gives exactly + + w ds d beta dc =108 k^7 Q(u,v)|Z|^-12 du dv dZ, (R10) + +in the point-counted integral, with a and all other spectral/tensor variables +retained. This identity incorporates the root multiplicity through the map, +not by an additional constant. The two signs of Z will be integrated next. + +## 3. Tail scaling, domination and the complete coefficient + +Let h,O,xi be the remaining variables of M, and let tau(a,beta,c,xi) denote the +full rotated cubic tensor in those coordinates. Its raw contact Gaussian density +remains evaluated at + + h0(A0,Rot_O(tau)), A0=O diag(0,-h)O^t. + +It is NOT replaced by an isotropic density. Combining M1 and R10 gives the exact +representation of F(t) as the integral over R7 and Z!=0 of + + (108 k^7 c_m/z0) Q(u,v) H(h) |Z|^-12 + h0(A0,Rot_O tau(a, a^2/(12k)+k b0/Z^2, + a^3/(144k^2)+a b0/(4 Z^2)+2k(v-b0 u)/Z^3, xi)) + *1{ (u-aZ/(12k))^2+Z^2 > t^2 }, (R11) + +with measure du dv dZ da dxi dh dO, ordered positive h, and the empty-product +conventions for d=2. The coefficient a b0/(4Z^2) in c includes the k cancellation +in aB/(4k). Formula R11 is often where confusing the spatial and jet Jacobians, +or integrating out a before testing the actual radius, gives a wrong exponent +or coefficient. + +Put Z=t z and gamma(a)=sqrt(1+(a/(12k))^2). Then |Z|^-12 dZ=t^-11 |z|^-12 dz. +At each fixed nonzero z the raw jet tends to the cusp slice + + beta_cusp=a^2/(12k), c_cusp=a^3/(144k^2), + +and the radius indicator tends almost everywhere to 1{gamma(a)|z|>1}. +For t>=3 and |u|<=3/2, the indicator in R11 implies + + |z|>1/[2gamma(a)] + +by the triangle inequality t<|u|+gamma|Z|. Orthogonal norm equivalence and SC's +Gaussian density bound imply, uniformly in t,z,u,v,O, + + h0(A0,Rot_O tau(a,beta,c,xi)) + <= C exp[-c(|h|^2+a^2+|xi|^2)]. + +Dropping beta and c from this upper bound is legitimate; it never introduces +an inverse hard eigenvalue. Integration in z over |z|>1/(2gamma) costs at most +a constant times gamma^11. Q is bounded on the bounded region R7, H(h) is a +nonnegative polynomial, and Gaussian integration in h,a,xi dominates gamma^11. +Dominated convergence therefore applies to t^11 F(t). + +Let + + J_cusp=integral H(h) gamma(a)^11 + h0(A0,Rot_O tau(a,a^2/(12k),a^3/(144k^2),xi)) + da dxi dh dO. (R12) + +This is finite by the same majorant and positive by positivity of the Gaussian +density and H on open sets. Integrating both signs of z gives +integral_(gamma|z|>1)|z|^-12 dz=2 gamma^11/11. Finally, + + I_shape=integral_Dshape Q(u,v) du dv=246528/35, + + C_star=(216/11) k^7 (c_m/z0) I_shape J_cusp + =(53250048/385) k^7 (c_m/z0) J_cusp. (R13) + +For an exact check of I_shape, integrate v first using + + (4b0/3)v^3 -4u v^4 -b0^3 v +2u b0^2 v^2. + +The strips -3/2=2 under the source interface. The exponent 11 +is dimension-independent here because the hard directions affect only the +finite integral J_cusp; the soft jet weights are s,B of degree -2, D of degree -3, +and endpoint weight of degree -4, for total 2+2+3+4=11. This explanatory homogeneity +is not substituted for the dominated-convergence proof. + +For p>0, Tonelli gives the moment identity integral R^p=p integral_0^infinity +t^(p-1)F(t)dt. Finite total point mass handles t<=1 and R2 handles infinity. +This proves the threshold in Theorem R and the boundary formula R3. If R_max +is the maximum radius of a nonempty cluster, its unnormalized tail lies between +F(t)/2 and F(t), since there are at most two points. Its normalized law therefore +has the same exact moment threshold, although no exact maximum-tail coefficient +is claimed. QED. + +## 4. A universal law for an intensity-selected far local point + +Normalize the point-intensity measure restricted to R_i>t. For that selected +point retain its canonical shape variables (u,v), downward height eta, radial +ratio q=R_i/t, sign epsilon=sign Z, and the remaining variables (a,h,O,xi). + +**Theorem H (extreme local marks).** As t->infinity, the joint law converges +weakly to the product of: + +1. shape density Q(u,v)/I_shape on R7; +2. a Pareto radial ratio q>=1 with density 11q^-12; +3. a symmetric sign epsilon in {-1,1}; +4. the parameter measure with density H(h) gamma(a)^11 times the cusp Gaussian + in R12, divided by J_cusp. + +In particular eta=u+1/2+v/6 is independent of the limiting radial ratio and the +remaining parameter measure. Its probability density on 01. The limiting Gaussian depends only on (a,h,O,xi), while +Q depends only on (u,v). Division by R13 gives exactly the four factors above. +No claim of spatial independence in the original field is involved. + +To obtain H1, substitute v=6e-6u-3, with dv=6de. Conditions R7 become + + -1/2-sqrt(e) < u < 1/2-sqrt(1-e), 0=0 a reproducible rational formula before dividing by I_shape is + + integral e^j H_eta(e) de + =(110592/7)[1/(j+9/2)-B(j+1,9/2)+1/(j+1)] + +13824[1/(j+5)-2/(j+4)+5/(j+3)-4/(j+2)], (H4) + +with B(j+1,9/2)=j!/product_(l=0..j)(9/2+l). No special-function numeric library +is needed to check these moments. Seven such moments are checked independently +against exact double polynomial integration on R7. QED. + +## 5. Evidence boundary and useful falsifiers + +The finite companion independently checks the oriented root Jacobian, the inverse +raw-jet map and its factor 2, stationary heights and gradients at 648 rational +parameter choices, the two exact shape integrals, H4 versus direct integration, +and the k^7/radius-power ledgers. Eight explicit false variants must fail. +These controls do not prove the Gaussian bound, change-of-variables multiplicity, +weak convergence or domination; those are the written analytic obligations above. + +The result would fail if the contact slice lost full positive Gaussian density, +if the physical radius were incorrectly replaced by |Z| before retaining a, +or if the invariant density were assumed isotropic rather than kept under Haar. +A finite-r spatial-moment convergence claim, a global tail including remote +singletons, an ordinary configuration-selected version of H1, or a numerical +Gaussian value of C_star would require new work and is not supplied here. diff --git a/reviews/radial_moment_corollaries_20260930/README.md b/reviews/radial_moment_corollaries_20260930/README.md new file mode 100644 index 000000000..3e161ed95 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/README.md @@ -0,0 +1,20 @@ +# Moments of the radial tail + +For the microscopic point-intensity law, let `Q_t=R/t` conditional on `R>t`. The tail exponent **11** is the sharp moment boundary: every fixed real power below 11 is finite, and every power at or above 11 diverges in the actual conditional microscopic law. + +If `c=C₂/C₀`, the source's uniform tail expansion gives + +```text +E Q_t^p = 11/(11-p) − 2pc/((13-p)(11-p)) t^-2 + O_p(t^-4), p<11 +E log Q_t = 1/11 − 2c/143 t^-2 + O(t^-4). +``` + +[The derivation](MOMENT_CONSEQUENCE.md) also supplies every fixed finite-order power/log-moment expansion, checks the author's fixed quantile formula, and shows that `0=2, m=d-1, torus side L>0, birth b, gap mark k>0, and an orthonormal +axial/transverse frame. Use exactly [P]'s variance-one periodized Gaussian +covariance, original endpoint observations + + M=(-r/2,0), S=(r/2,0), + f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0, + +and endpoint regression Q_r. Let W_r=F_d(H_M)F_(d-1)(H_S), Z_r=E_Q W_r and +Q_r^W=(W_r/Z_r)Q_r. N_r counts ALL additional critical points with heights in +(b-k r^3,b), excluding the two pins. Constants below depend on these fixed +parameters, d, and L; where stated they also depend on R or rho. No uniformity +in varying dimension, volume, or unbounded/vanishing marks is claimed. + +Exact source commits, paths, sizes and hashes are in SOURCES.json. The inputs are: + +[P] parent Sections2-5: positive Fourier spectrum, nonsingular contact frame U_r, +its target v_r, smooth finite-jet regression, every fixed C^q moment, and +Z_r/r^2 -> z0>0. The endpoint-gradient identity gives W_r/r^2 <= C K^(2d), where +K>=1 is a fixed multiple of 1+||f||C4 bounding operator derivatives through four. + +[D5] the fixed-d global first moment and height-window shell estimate + + E_W N_r <= C r^3, + E_W N({Rr<|x|=4. (1) + +[RM,RC] for fixed rho>0, on D_rho={|x|>=rho}, with Lambda the SUM of the +index-specific normalized contact kernels, + + E_W N(D_rho)=k r^3 integral_Drho Lambda+O_rho(r^4), + E_W (N(D_rho))_2=O_rho(r^5). (2) + +[C6] E_W (N_r)_q<=C_q r^3 for every fixed q>=2. With (1), Stirling's identity +also gives E_W N_r^q=O_q(r^3). We use this only for moment uniform integrability +and consequences, not for the new near-event domination or mixed-event estimate. + +[CUB] the exact deterministic classifier in #158, Theorem C, at its xAI-reviewed +algebraic scope. Its Gaussian sector theorem is NOT consumed. [DEEP] is credited +for the planar exclusion construction; Section3 proves the vector version here. +[RCL] the previously reviewed generic Poisson/compound-Poisson consequences. +The defective global proof in #159 is NOT a premise. This note gives an alternative +spectral-coordinate proof, not a retrospective acceptance of its shear argument. + +Theorem (conditional global cluster law). There are explicitly specified finite +positive numbers a1,a2,beta_far such that, putting + + nu1=a1+beta_far, nu2=a2, + +for every fixed nonnegative integer q, + + sum_(n>=1) n^q |r^-3 Q_r^W(N_r=n)-nu1*1{n=1}-nu2*1{n=2}| -> 0. (3) + +The coefficients are (17) and (22) below. Thus the full limiting nonempty count +measure is unique and supported on {1,2}. This is not a joint law of clusters +from distinct pin pairs, and not a spatial independence statement. Unscaled +Law(N_r) tends to delta_0. The zero mass must be CENTERED before multiplying +by r^-3; r^-3 Q_r^W(N_r=0) diverges. + +## 2. Physical jets, residual coupling, and exact orthogonal coordinates + +Let A=D_y^2 f(0) be the m-by-m midpoint transverse Hessian and T the list of +independent third derivatives at 0 EXCEPT f_xxx. Set J=(A,T). At contact, +(U_0,J) is the complete nonrepeated jet list through degree three: U_0 supplies +f, all first derivatives, f_xx, f_xyi, and f_xxx. Positive Fourier spectrum +therefore gives a positive-definite covariance. For sufficiently small r, the +joint covariance and its inverse are uniformly bounded. The conditional density +h_r(A,T) under Q_r converges locally uniformly to h_0 and obeys + + h_r(A,T)<=C exp[-c(||A||F^2+|T|^2)]. (4) + +The bounded conditional means have been absorbed by reducing c. + +Take an independent copy F of the unconditioned smooth field. For each raw target +j, construct the conditional field by regression on V_r=(U_r,J): + + f_r^j=F+Cov(F,V_r)Cov(V_r)^-1((v_r,j)-V_r(F)). (5) + +All coefficient functions are bounded in C4 and converge there. Averaged contact +functionals in U_r are bounded by a constant times ||F||C3; this uses the integral +forms of the pin divided differences, not separate uncontrolled r^-p terms. +Consequently, with S=C(1+||F||C4), independent of the target j, + + K(f_r^j)<=C(S+|j|), E S^p=1, and let K bound the operator norms of +all derivatives through four. Let lambda=lambda_min(-A). If + + rK<=k/(4R), lambda>(8RK+56RK^2/k)r, (11) + +there are only the two pinned critical points in the cylinder. + +Proof. Throughout the cylinder f_yy<=-lambda/2, since its variation from A is at +most 2RKr. Put g(x)=f_y(x,0). The pin gradients give g(-r/2)=g(r/2)=0. Applying +the scalar two-node interpolation estimate to every unit projection of g gives +||g(x)||<=KR^2r^2, also outside the pin interval. No vector-valued Rolle zero is +assumed. Likewise + + g'(x)=r^-1 integral_(-r/2)^(r/2)[g'(x)-g'(t)]dt + +implies ||g'(x)||<=K(R+1/2)r. For rho=4R^2Kr^2/lambda=11k>0. +The strictly convex function psi' already has the two pinned zeros and cannot +have a third. All critical points in the cylinder lie on its graph. This proves +(11) in every fixed dimension. + +On the typed support H_M<0, A has largest eigenvalue less than rK/2 by the +Hessian Lipschitz bound, so sk/(4R), + + |s|<=C_R K^2<=a_R U^2+a_R r^2 |s|^2, a_R>=1. (12) + +Here K>=1, k is fixed, and (10) was used. The elementary absorption dichotomy is + + x<=a U^2+a r^2 x^2 implies + x<=2a U^2 OR x>=1/(2a r^2), x>=0. (13) + +If x>2aU^2, subtracting aU^2 from (12) proves the second alternative. +On the second alternative |A|op>=r|s|>=c_R/r, hence K>=c_R/r. The first excluded +event also implies K>=c_R/r. All exceptional mass can therefore be bounded in +ORIGINAL Q_r coordinates, before any spectral rescaling. For every p,M>=0, + + E_W[K^p;exceptional] <= C E_Q[K^(p+2d);K>=c_R/r] + <= C_(p,M,R) r^M. (14) + +The last bound uses one higher fixed Q_r moment. The normalizer is the full +Z_r>=z_*r^2 and W_r<=Cr^2K^(2d). There is no need to posit a conditional Gaussian +tail that is uniform in an unbounded scaled target. The exceptional branch has +not been discarded; it is negligible to every fixed power of r. + +## 4. One global majorant, including multiple soft eigenvalues + +At each pin the axial Hessian column has norm <=rK/2 by the vector gradient +average. The TRANSVERSE part of the midpoint column in the soft eigenvector direction is +r s times that eigenvector; its endpoint transverse perturbation is bounded by +rK/2. Its axial entry at a pin is already bounded by the short axial column. +Thus the full soft column is bounded by r(|s|+K), up to a fixed harmless constant. +The axial and soft columns correspond to orthogonal coordinate directions. Hadamard therefore gives + + W_r/r^4 <= C K^(2d-2)(|s|+K)^2. (15) + +This does not divide by any hard eigenvalue. On the regular near-occurrence +branch |s|<=2a_R U^2, (10) gives K<=C_R U^2 (r<=1), so (15)<=C_R U^(4d). +Writing v=m(m-1)/2, (9) gives |J_r|<=C_R U^(2v). Thus the density of the rescaled +NONEMPTY near measure, in spectral variables and the coupling variable F, is +bounded by + + G_R=C_R exp[-c(|h|^2+|tau|^2)] + U^(4d+2v) 1{|s|<=2a_R U^2}. (16) + +Indeed disintegration gives r^-3 times r from (9), times r^4 from the weight, +divided by Z_r: the prefactor is c_m r^2/Z_r, which is bounded. Integrating s in +(16) contributes only another U^2. Gaussian h,tau integration and every fixed +moment of S make G_R integrable. Multiplication by any fixed polynomial in +K, |s|, |h|, |tau| remains integrable on the regular branch. Formula (14) controls +the weighted exceptional branch as well: for a scaled-jet power, use |s|<=K/r +and choose the arbitrarily large power M in (14) to absorb that extra r^-p. + +Consequences: the r^-3 near-occurrence measures are tight in all the spectral +jet variables; their tails outside increasing compact sets tend to zero uniformly +as r->0. Their occurrence-weighted K moments are O(r^3). If m>=2, the limsup of +scaled near-occurrence mass with |h2|<=eta tends to zero as eta->0, directly by +integrating G_R over that strip. The same holds near repeated hard spectra. No +inverse determinant or 1/eta bound appears. This proves the needed multiple-soft +exclusion IN MEASURE, without a pointwise assertion that two soft directions +cannot occur. + +Using C6 only now, Cauchy-Schwarz also yields +E_W[N_R K^p]<=sqrt(E_W N_R^2)*sqrt(E_W[K^(2p);N_R>0])=O_(R,p)(r^3). +Thus the LOCAL derivative-weighted bound is established, not assumed. The global +analogue is not needed for the rest of this proof. + +## 5. Pointwise normal form, typed-boundary control, and the near limit + +Fix spectral variables s,h,O,tau with 0 P_s,tau(X,Z) in C2, + r^-2 grad_w f(rX,rZ,r^2V) -> -diag(h)V+Q_tau(X,Z) in C1, + +where + + P_s,tau=2kX^3-3kX/2-k/2+sZ^2/2 + +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3, + a=tau_xxz, beta=tau_xzz, c=tau_zzz. + +Q_tau contains the hard-index cubic terms. For fixed positive h the hard +Hessian stays negative definite for small r. On the full physical Rr ball, +grad_w f=-diag(h)w+O(Kr^2)+O(Kr|w|). Every critical point therefore has +|w|<=C_(h,R,F,tau) r^2. The hard implicit function is unique and smooth, and +substitution into f/r^3 changes the planar potential by O(r) in C2 on fixed +sets. Its inverse-hard-eigenvalue constants are used ONLY for pointwise +convergence, never in (16). + +At the endpoints the soft Hessian blocks divided by r converge to + + M_M=[[-6k,-a/2],[-a/2,s-beta/2]], + M_S=[[ 6k, a/2],[ a/2,s+beta/2]]. + +A Schur complement across the fixed hard block, or the determinant expansion, +gives W_r/r^4 -> product h_j^2 * w(s,a,beta), where + + w=[-6k(s-beta/2)-a^2/4]_+ [a^2/4-6k(s+beta/2)]_+. + +It is positive exactly on s<-|B|/2, B=beta-a^2/(12k), and equals +9k^2(4s^2-B^2) there. Outside this open typed domain its limiting value is zero; +on the typed boundary the determinant limit is also zero. Consequently no count +convergence is needed at a point with zero limiting weight. This is the explicit +handling of the typed boundary, not an assumed compact exclusion from it. + +Within the typed domain use [CUB]'s exact classifier n(s,a,beta,c) in {0,1,2}. +Let n_R count those additional cubic critical points in X^2+Z^2=1, +Z!=0 at any critical root, and the two gradient equations solve for (beta,s) +as smooth functions of (X,Z,a,c). A circle coordinate plus a,c is three parameters, +so its image in four-dimensional (s,a,beta,c) space is null. Cover by countably +many compact charts with |Z| bounded away from zero. Height boundary relations +are the nonzero polynomial threshold of [CUB]. The pins are nondegenerate and +remain EXACT, so they cannot split into new counted roots. + +Choose disjoint neighborhoods of relevant zeros. C2 convergence and the inverse +function theorem preserve precisely one root in each; on the rest of the compact +set in a slightly enlarged height window the gradient is bounded away from zero. +No condition on lower-height degeneracies is necessary. Hard slaving then proves +N_R -> n_R pointwise for these fixed variables. The hard coordinates vanish on +the r spatial scale. The extra points have full index d-1. + +For precision, extend the coefficient kernel by zero off ordered positive h and +the typed domain. With T=Rot_O(tau), define + + dM(s,h,O,tau) = (c_m/z0) h_0(O diag(0,-h)O^t,T) + * product_(j=2..m) h_j^3 + * product_(2<=i=1. To see this, on the regular branch sum the absolute conditional +errors over the count index j: it is bounded by the weight error plus twice +the weighted probability of a count mismatch. Pointwise convergence gives zero +and (16) dominates. Cases of zero limiting weight give zero directly. On a fixed typed jet with n_R>0 and finite S, K is bounded as r->0, so the +exceptional branch is eventually absent and the regular-branch indicator is +eventually one. No part of the proposed limit is lost by that split. The +exceptional mass is o(r^3) by (14). This proves + + r^-3 Q_W(N_R=j) -> a_j^R=integral 1{n_R=j}dM, j=1,2, + r^-3 Q_W(N_R>=3) -> 0. (18) + +There is NO finite uncentered n=0 formula. For fixed R, the right side is a +finite measure only on nonempty counts. + +## 6. Integrability at arbitrary scaled radius + +For the planar classifier put D=(c-a beta/(4k)+a^3/(72k^2))/2 and +T_*=-(s-B)^2(B+2s)/(12k). It says: s>B gives n=2 below D^2=T_* and n=1 at or above; +s<=B gives n=1 strictly above that threshold and n=0 otherwise. On n>=1, x=-s>0 +satisfies + + x<=2|B|+(48kD^2)^(1/3). + +Indeed s>B forces x<|B|. Otherwise D^2>T_*, and if x>=2|B| then +T_*>=(x/2)^2*x/(12k)=x^3/(48k). Since w<=36k^2 x^2, + + integral_s w 1{n>=1} <=384k^2 |B|^3+2304k^3 D^2. (19) + +This is a polynomial bound in the cubic coordinates at fixed k. The remaining +factor in (17) is a nonnegative polynomial in h times a uniformly Gaussian-decaying +function of h,tau, by ORTHOGONAL norm preservation. Therefore + + a_j=integral 1{n=j}dM, j=1,2 (20) + +are finite. They are strictly positive: take the open two-point cubic sector +(s,a,beta,c)=(-3k/2,0,-2k,0), or the one-point sector (-k,0,0,2k), choose distinct +positive hard h in a compact box, and arbitrary remaining cubic entries in a +small box. Gaussian densities are positive, and the spectral/weight factors are +positive on an open set. For m=1 the empty hard products are 1. + +For every fixed jet, n_R=n eventually as R->infinity. Exact-count indicators +{n_R=1} need NOT be monotone, but are bounded by 1{n>=1}; (19) supplies an +integrable majorant. Dominated convergence proves a_j^R->a_j. No inverse shear +parameter is suppressed by integrating out variables that a spatial observable +still uses. + +## 7. Remote singleton coefficient and vanishing mixed event + +By (1) and [RM]'s fixed-remote mean limit, for 00; equivalently +choose R_r=rho'/r, which is permitted because the shell constant is uniform in +R_r>=4. The kernel is nonnegative and the annulus is fixed, so its mean limit +is supplied by [RM]. Thus + + beta_far=k integral_(X\{0}) Lambda(x)dx (22) + +exists, is finite, and differs from the fixed-rho integral by O(rho^2). +It is positive by positivity on any fixed remote region of positive volume. +Corollary D of [RC] and the mean limit give +r^-3 Q_W(N_far^rho=1)->k integral_Drho Lambda and +Q_W(N_far^rho>=2)=O_rho(r^5). + +Fix R and rho. To control the mixed event, first restrict the spectral jets +(s,h,tau) to a fixed compact box H, retaining all O. Under the conditional law +(5) with these targets, the covariance of (f(x),grad f(x)), |x|>=rho, has a +uniform positive lower bound. At r=0 the conditioning is the COMPLETE degree-three +jet at the origin and x is a distinct site; positive Fourier spectrum gives +full joint rank. Compactness in x, O, and the target-independent covariance, +plus continuity through r=0, proves the bound. Conditional Hessian moments are +bounded on H and on the height window. The usual unweighted Gaussian Kac-Rice +formula, integrated over the window of length k r^3, gives + + E[N_far^rho | U_r,J]<=C_(H,rho) r^3. (23) + +On H, (15) and (6) give a uniform second moment of W_r/r^4. Conditional +Cauchy-Schwarz therefore gives + + E[(W_r/r^4)1{N_far^rho>0} | U_r,J]<=C_(H,rho) r^(3/2). + +Disintegrating over H with (9), the full normalizer, and bounded density and +Vandermonde gives Q_W(spectral jets in H,N_far^rho>0)=O_(H,rho)(r^(9/2)). +In particular this bounds the compact-jet near/far intersection. +Outside H, its probability is bounded by Q_W(N_R>0,jets outside H). By (16) +and (14), its scaled limsup tends to zero as H exhausts the spectral variables. +Thus + + Q_W(N_R>0,N_far^rho>0)=o_(R,rho)(r^3). (24) + +The full mixed term has no asserted r^(9/2) rate: that rate was proved only on +a fixed compact jet box. This proof avoids conditioning on a near WITNESS whose +observation frame degenerates. It uses fixed-origin physical jets instead. + +## 8. The full count limit and all polynomial weights + +Write N=N_R+N_mid+N_far. For fixed R,rho, the bad events N_mid>0, +{N_R>0,N_far>0}, N_far>=2, and N_R>=3 have probabilities bounded by (1), (24), +(2), and (18), respectively. On their complement N equals either the near +count in {0,1,2} or a remote singleton, but not both. Therefore, with an absolute +constant C independent of R,rho, + + limsup |r^-3 Q(N=2)-a_2^R| <= C(R^-2+rho^2), + limsup |r^-3 Q(N=1)-a_1^R-k integral_Drho Lambda| <= C(R^-2+rho^2), + limsup r^-3 Q(N>=3) <= C(R^-2+rho^2). (25) + +First r->0 at fixed R,rho, then R->infinity and rho->0, using (20)-(22). +This proves coordinate convergence to nu1=a1+beta_far and nu2=a2, and vanishing +scaled tail mass. For a fixed integer q>=0, C6 supplies E N^(q+1)<=C r^3; hence + + r^-3 E[N^q;N>M]<=C/M. + +Finite-coordinate convergence plus this uniform tail proves (3). No interchange +of a nonuniform radius limit and r limit has been assumed. No finite unscaled +polynomial moment theorem alone would imply (3); the new domination and regional +sandwich are the essential ingredients. + +In particular + + E N/r^3 -> nu1+2nu2, + E (N)_2/r^3 -> 2nu2, + E (N)_q/r^3 -> 0 for every fixed q>=3. (26) + +The positive-conditioned count converges in TV to +(nu1 delta1+nu2 delta2)/(nu1+nu2), and its size-biased version to +(nu1 delta1+2nu2 delta2)/(nu1+2nu2). The two laws are different. Their formulas +use the normalized full tilt, not a restricted normalizer. + +For independent replicas with floor(t/r^3) counts, the identified compound limit +is P1+2P2, with independent P1~Pois(t nu1), P2~Pois(t nu2), as follows either by +the exact generating function or [RCL]'s proved independent-replica bound. +There is no independence claim for regions of a single field. + +The sharp [RCL] Poisson comparison also now has coefficients: +optimal ordinary-Poisson TV error/r^3 -> nu2 and mean-matched error/r^3 ->2nu2, +so their ratio tends to 2 (not just liminf>=2). The canonical approximant using +the ACTUAL conditional mark law still has Theta(r^6) error. Replacing its mark +law by the identified LIMIT does not inherit that rate: without a convergence +rate for the coefficients, this replacement is only o(r^3) in one-count TV. + +## 9. What is solved here, conditional on review, and what is not + +The proof supplies a replacement for scaled jet tails, typed-boundary control, +nearly singular hard blocks, the near/far mixed event and the radius sandwich. +It works for every FIXED d>=2 via orthogonal spectral coordinates. It does not +require the stronger global derivative-weighted insertion estimate proposed in +#157; Section4 proves the local version as a consequence and the global law does +not need the global version. Thus that former sufficient route is bypassed, not +silently assumed true. + +The local coefficients in d=2 reduce exactly to #158's alpha1,alpha2, and its +contact birth factorization remains a local statement. In higher dimensions the +Haar/stable integration may couple birth-dependent even jets to orientation; +no higher-dimensional or global birth cancellation is asserted. + +No convergence rate, numerical coefficient enclosure, unbounded-mark theorem, +growing-volume limit, inter-pin-pair cluster independence, elder selection of the +extra points, or full Lean formalization is claimed. The input Gaussian and +regional sources remain conditional premises at their exact stated scopes. +Theorem (3) is a new author-side candidate, not accepted merely because finite +controls or existing formal gates pass. + +Review division: A, vector exclusion, absorption, and orthogonal spectral measure; +B, integrable majorant, typed/hard-boundary handling and pointwise normal form; +C, compact-jet remote conditioning, mixed event, radius sandwich and consequences. +The prior #159 diagnosis and the distinction between that proof and this replacement +are retained in AUDIT_AND_REPAIR.md. 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[{"coefficient_of_c_t_minus_2": "200/12543", "leading": "11/111", "p": "-100"}, {"coefficient_of_c_t_minus_2": "7/180", "leading": "11/18", "p": "-7"}, {"coefficient_of_c_t_minus_2": "1/84", "leading": "11/12", "p": "-1"}, {"coefficient_of_c_t_minus_2": "4/621", "leading": "22/23", "p": "-1/2"}, {"coefficient_of_c_t_minus_2": "0", "leading": "1", "p": "0"}, {"coefficient_of_c_t_minus_2": "-4/525", "leading": "22/21", "p": "1/2"}, {"coefficient_of_c_t_minus_2": "-1/60", "leading": "11/10", "p": "1"}, {"coefficient_of_c_t_minus_2": "-4/99", "leading": "11/9", "p": "2"}, {"coefficient_of_c_t_minus_2": "-20/3", "leading": "11", "p": "10"}, {"coefficient_of_c_t_minus_2": "-2180/21", "leading": "110", "p": "109/10"}, {"coefficient_of_c_t_minus_2": "-21998000/2001", "leading": "11000", "p": "10999/1000"}], "quantile_controls": "exact rational brackets for Pareto mixtures", "scientific_effect": "NONE", "scope": "PR176 equation17 conditional moment consequence only", "tests": 11} diff --git a/reviews/radial_moment_corollaries_20260930/moment-checks-normal.log b/reviews/radial_moment_corollaries_20260930/moment-checks-normal.log new file mode 100644 index 000000000..e75147417 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/moment-checks-normal.log @@ -0,0 +1,16 @@ +test_actual_probability_mixture_has_claimed_moment_remainder_bound (__main__.MomentConsequenceTests.test_actual_probability_mixture_has_claimed_moment_remainder_bound) ... ok +test_all_order_finite_tail_matches_direct_pareto_mixture (__main__.MomentConsequenceTests.test_all_order_finite_tail_matches_direct_pareto_mixture) ... ok +test_all_order_log_powers_and_normalization (__main__.MomentConsequenceTests.test_all_order_log_powers_and_normalization) ... ok +test_direct_log_tail_integral (__main__.MomentConsequenceTests.test_direct_log_tail_integral) ... ok +test_hand_checked_special_cases (__main__.MomentConsequenceTests.test_hand_checked_special_cases) ... ok +test_offered_rational_coefficient_factors (__main__.MomentConsequenceTests.test_offered_rational_coefficient_factors) ... ok +test_quantile_expansion_brackets_an_actual_conditional_law_exactly (__main__.MomentConsequenceTests.test_quantile_expansion_brackets_an_actual_conditional_law_exactly) ... ok +test_tail_and_density_derivations_agree_for_negative_zero_positive_p (__main__.MomentConsequenceTests.test_tail_and_density_derivations_agree_for_negative_zero_positive_p) ... ok +test_threshold_and_above_diverge (__main__.MomentConsequenceTests.test_threshold_and_above_diverge) ... ok +test_uniform_in_p_remainder_overclaim_has_exact_counterexample (__main__.MomentConsequenceTests.test_uniform_in_p_remainder_overclaim_has_exact_counterexample) ... ok +test_wrong_normalizer_sign_and_missing_p_are_detected (__main__.MomentConsequenceTests.test_wrong_normalizer_sign_and_missing_p_are_detected) ... ok + +---------------------------------------------------------------------- +Ran 11 tests in 0.003s + +OK diff --git a/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.json b/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.json new file mode 100644 index 000000000..d311bc520 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.json @@ -0,0 +1 @@ +{"all_order_controls": "normalized finite Pareto mixtures", "exact_rational_powers": ["-100", "-7", "-1", "-1/2", "0", "1/2", "1", "2", "10", "109/10", "10999/1000"], "log_correction": "-2/143", "log_leading": "1/11", "moment_rows": [{"coefficient_of_c_t_minus_2": "200/12543", "leading": "11/111", "p": "-100"}, {"coefficient_of_c_t_minus_2": "7/180", "leading": "11/18", "p": "-7"}, {"coefficient_of_c_t_minus_2": "1/84", "leading": "11/12", "p": "-1"}, {"coefficient_of_c_t_minus_2": "4/621", "leading": "22/23", "p": "-1/2"}, {"coefficient_of_c_t_minus_2": "0", "leading": "1", "p": "0"}, {"coefficient_of_c_t_minus_2": "-4/525", "leading": "22/21", "p": "1/2"}, {"coefficient_of_c_t_minus_2": "-1/60", "leading": "11/10", "p": "1"}, {"coefficient_of_c_t_minus_2": "-4/99", "leading": "11/9", "p": "2"}, {"coefficient_of_c_t_minus_2": "-20/3", "leading": "11", "p": "10"}, {"coefficient_of_c_t_minus_2": "-2180/21", "leading": "110", "p": "109/10"}, {"coefficient_of_c_t_minus_2": "-21998000/2001", "leading": "11000", "p": "10999/1000"}], "quantile_controls": "exact rational brackets for Pareto mixtures", "scientific_effect": "NONE", "scope": "PR176 equation17 conditional moment consequence only", "tests": 11} diff --git a/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.log b/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.log new file mode 100644 index 000000000..e75147417 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.log @@ -0,0 +1,16 @@ +test_actual_probability_mixture_has_claimed_moment_remainder_bound (__main__.MomentConsequenceTests.test_actual_probability_mixture_has_claimed_moment_remainder_bound) ... ok +test_all_order_finite_tail_matches_direct_pareto_mixture (__main__.MomentConsequenceTests.test_all_order_finite_tail_matches_direct_pareto_mixture) ... ok +test_all_order_log_powers_and_normalization (__main__.MomentConsequenceTests.test_all_order_log_powers_and_normalization) ... ok +test_direct_log_tail_integral (__main__.MomentConsequenceTests.test_direct_log_tail_integral) ... ok +test_hand_checked_special_cases (__main__.MomentConsequenceTests.test_hand_checked_special_cases) ... ok +test_offered_rational_coefficient_factors (__main__.MomentConsequenceTests.test_offered_rational_coefficient_factors) ... ok +test_quantile_expansion_brackets_an_actual_conditional_law_exactly (__main__.MomentConsequenceTests.test_quantile_expansion_brackets_an_actual_conditional_law_exactly) ... ok +test_tail_and_density_derivations_agree_for_negative_zero_positive_p (__main__.MomentConsequenceTests.test_tail_and_density_derivations_agree_for_negative_zero_positive_p) ... ok +test_threshold_and_above_diverge (__main__.MomentConsequenceTests.test_threshold_and_above_diverge) ... ok +test_uniform_in_p_remainder_overclaim_has_exact_counterexample (__main__.MomentConsequenceTests.test_uniform_in_p_remainder_overclaim_has_exact_counterexample) ... ok +test_wrong_normalizer_sign_and_missing_p_are_detected (__main__.MomentConsequenceTests.test_wrong_normalizer_sign_and_missing_p_are_detected) ... ok + +---------------------------------------------------------------------- +Ran 11 tests in 0.003s + +OK diff --git a/reviews/radial_moment_corollaries_20260930/moment_consequence_checks.py b/reviews/radial_moment_corollaries_20260930/moment_consequence_checks.py new file mode 100644 index 000000000..952759716 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/moment_consequence_checks.py @@ -0,0 +1,171 @@ +"""Independent exact controls for the consequence of PR176 equation (17). + +No author code is imported. These finite arithmetic/probability controls do not +prove the source Gaussian theorem, its remainder, or any original finite-r rate. +""" +from fractions import Fraction as F +import json +from math import factorial +import unittest + + +POWERS = [F(-100), F(-7), F(-1), F(-1, 2), F(0), F(1, 2), + F(1), F(2), F(10), F(109, 10), F(10999, 1000)] + + +def integral(exponent): + """Exact integral of q^exponent over [1,infinity), or divergent.""" + if exponent >= -1: + raise ValueError('divergent power integral') + return -1 / (exponent + 1) + + +def tail_moment(p): + return (1 + p * integral(p - 12), + p * (integral(p - 14) - integral(p - 12))) + + +def density_moment(p): + return (11 * integral(p - 12), + 13 * integral(p - 14) - 11 * integral(p - 12)) + + +def proposed_moment(p): + return F(11) / (11 - p), -2 * p / ((13 - p) * (11 - p)) + + +class MomentConsequenceTests(unittest.TestCase): + def test_tail_and_density_derivations_agree_for_negative_zero_positive_p(self): + for p in POWERS: + with self.subTest(p=p): + self.assertEqual(tail_moment(p), proposed_moment(p)) + self.assertEqual(density_moment(p), proposed_moment(p)) + + def test_hand_checked_special_cases(self): + for p, wanted in [(F(-1), (F(11, 12), F(1, 84))), + (F(0), (F(1), F(0))), + (F(1), (F(11, 10), F(-1, 60))), + (F(2), (F(11, 9), F(-4, 99)))]: + self.assertEqual(tail_moment(p), wanted) + + def test_threshold_and_above_diverge(self): + for p in (F(11), F(111, 10), F(12), F(100)): + with self.assertRaises(ValueError): + integral(p - 12) + + def test_direct_log_tail_integral(self): + self.assertEqual(integral(F(-12)), F(1, 11)) + self.assertEqual(integral(F(-14)) - integral(F(-12)), F(-2, 143)) + + def test_actual_probability_mixture_has_claimed_moment_remainder_bound(self): + # A genuine mixture of Pareto(11), Pareto(13), Pareto(15): its tail + # obeys (17) with c=1 and remainder (1/2)t^-4 q^-11(q^-4-1). + for t in (4, 8, 16): + delta, bound = F(1, t * t), F(1, 2) + weights = [1 - delta - bound * delta**2, delta, bound * delta**2] + self.assertEqual(sum(weights), 1) + self.assertTrue(all(w > 0 for w in weights)) + for p in POWERS: + exact = sum(w * F(alpha) / (alpha - p) + for w, alpha in zip(weights, (11, 13, 15))) + leading, correction = proposed_moment(p) + error = exact - leading - delta * correction + self.assertLessEqual(abs(error), abs(p) * bound * delta**2 / (11 - p)) + exact_log = sum(w / alpha for w, alpha in zip(weights, (11, 13, 15))) + self.assertLessEqual(abs(exact_log - F(1, 11) + F(2, 143) * delta), + bound * delta**2 / 11) + + def test_uniform_in_p_remainder_overclaim_has_exact_counterexample(self): + p, delta, bound = F(10999, 1000), F(1, 16), F(1, 2) + actual_error = abs(bound * delta**2 * (F(15) / (15 - p) - F(11) / (11 - p))) + self.assertGreater(actual_error, bound * delta**2) + + def test_wrong_normalizer_sign_and_missing_p_are_detected(self): + cases = [ + ('missing denominator normalization', F(0), F(1)), + ('reversed correction sign', F(1), F(1, 60)), + ('omitted p factor', F(0), F(-2, 143)), + ('omitted factor two', F(1), F(-1, 120)), + ] + for name, p, false_coefficient in cases: + with self.subTest(mutant=name): + self.assertNotEqual(tail_moment(p)[1], false_coefficient) + self.assertNotEqual(integral(F(-14)) - integral(F(-12)), F(-1, 143)) + + def test_all_order_finite_tail_matches_direct_pareto_mixture(self): + # F(s)=sum C_j s^(-11-2j), s>=1, divided by F(1), is an actual + # survival function. Conditioning at t gives component weights + # proportional to C_j t^(-2j); its conditional moments are exact. + coefficients = (F(2), F(3), F(5), F(7)) + for t in (3, 7, 13): + pieces = [c / t**(2*j) for j, c in enumerate(coefficients)] + normalizer = sum(pieces) + weights = [part / normalizer for part in pieces] + for p in POWERS: + direct = sum(w * F(11+2*j) / (11+2*j-p) + for j, w in enumerate(weights)) + from_tail = 1 + p * sum(part * integral(p-12-2*j) + for j, part in enumerate(pieces)) / normalizer + numerator = sum((11+2*j) * part / (11+2*j-p) + for j, part in enumerate(pieces)) + self.assertEqual(direct, from_tail) + self.assertEqual(direct, numerator / normalizer) + self.assertNotEqual(direct, numerator) # missing denominator + + def test_all_order_log_powers_and_normalization(self): + for m in (1, 2, 3, 4): + for alpha in (11, 13, 15, 17): + # Integration by parts in y=log(q): + # integral y^n exp(-alpha*y)dy = n/alpha times the n-1 integral. + gamma_integral = F(1, alpha) + for n in range(1, m): + gamma_integral *= F(n, alpha) + tail_log_moment = m * gamma_integral + exponential_log_moment = F(factorial(m), alpha**m) + self.assertEqual(tail_log_moment, exponential_log_moment) + if m > 1: + self.assertNotEqual(tail_log_moment, F(1, alpha**m)) + + def test_quantile_expansion_brackets_an_actual_conditional_law_exactly(self): + # The exact conditional tail q^-11(1+c t^-2 q^-2)/(1+c t^-2) + # comes from F(s)=s^-11(1+c s^-2), with an irrelevant global normalizer. + # Bracket the true quantile using only rational comparisons, avoiding + # floating point root finding or differentiation of a remainder. + for q0 in (F(5, 4), F(3, 2), F(2)): + alpha = q0**-11 + for c in (F(1), F(2)): + correction = c * (q0**-2 - 1) / 11 + for t in (8, 16, 32): + delta = F(1, t*t) + def survival(q): + return q**-11 * (1+c*delta*q**-2) / (1+c*delta) + qlo = q0*(1+correction*delta-delta**2) + qhi = q0*(1+correction*delta+delta**2) + self.assertGreater(qlo, 1) + self.assertGreater(survival(qlo), alpha) + self.assertLess(survival(qhi), alpha) + # Reversing the coefficient produces an O(delta) error + # that lies outside this O(delta^2) bracket. + wrong = q0*(1-correction*delta) + self.assertGreater(wrong, qhi) + + def test_offered_rational_coefficient_factors(self): + integral_shape, u2, uabs = F(246528, 35), F(240192, 35), F(5898627, 880) + self.assertEqual(11*u2/(2*integral_shape), F(4587, 856)) + self.assertEqual(11*uabs/(2*integral_shape), F(4587821, 876544)) + + +if __name__ == '__main__': + suite = unittest.defaultTestLoader.loadTestsFromTestCase(MomentConsequenceTests) + result = unittest.TextTestRunner(verbosity=2).run(suite) + if not result.wasSuccessful(): + raise SystemExit(1) + print(json.dumps({'scope': 'PR176 equation17 conditional moment consequence only', + 'exact_rational_powers': [str(p) for p in POWERS], + 'moment_rows': [{'p': str(p), 'leading': str(tail_moment(p)[0]), + 'coefficient_of_c_t_minus_2': str(tail_moment(p)[1])} + for p in POWERS], + 'log_leading': '1/11', 'log_correction': '-2/143', + 'quantile_controls': 'exact rational brackets for Pareto mixtures', + 'all_order_controls': 'normalized finite Pareto mixtures', + 'tests': result.testsRun, 'scientific_effect': 'NONE'}, sort_keys=True))