diff --git a/.github/workflows/universality-controls.yml b/.github/workflows/universality-controls.yml index 6d567b57e..ce89d8b25 100644 --- a/.github/workflows/universality-controls.yml +++ b/.github/workflows/universality-controls.yml @@ -34,6 +34,10 @@ jobs: run: | python3 -B -E -S -m unittest discover -s experiments/universality -p 'test_*.py' -v python3 -B -E -O -S -m unittest discover -s experiments/universality -p 'test_*.py' -v + - name: Exact population certificate controls in both modes + run: | + python3 -B -E -S -m unittest discover -s experiments/universality/two_parameter_cusp_population -p 'test_population_certificate.py' -v + python3 -B -E -O -S -m unittest discover -s experiments/universality/two_parameter_cusp_population -p 'test_population_certificate.py' -v - name: Execute and audit all fifty exploratory law candidates run: | set -euo pipefail diff --git a/experiments/universality/results/exact_population_1/CERTIFICATE.json 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b/experiments/universality/results/exact_population_1/RESULTS.md @@ -0,0 +1,49 @@ +# Frozen exact cusp-population attempt + +Disposition: **PASS**. + +This deterministic interval calculation addresses the fixed whole-rectangle +non-Gaussian population, eight coefficient predictions and one row-8 finite +iid-copy void prediction. It has no random seed or numerical field draw. + +| j | tau | Direct normalized interval | CDF normalized interval | Coefficient | Row disposition | Evaluations | +|---|---|---|---|---|---|---| +| 1 | 1/8 | [14422776566555442204534881040013962495586059078251129750036675910047077/63254240556375153621719571738275042447053881806505285742501103879061504, 649035621019752921021194266565160806711685919276331324648714451385933547/2846440825036881912977380728222376910117424681292737858412549674557767680] | [9698669598254516463664625632439578707/42535295865117307932921825928971026432, 19397339196509032927329251266404394463/85070591730234615865843651857942052864] | PASS | PASS | 67586 | +| 2 | 1/64 | [69063532249939816680299284966924486110866460978144099762856510554171539/279809664325481636599125601983068179239746467923499534259176385644855296, 345323210623016214899075481673569182801280352657100319899775787124466793/1399048321627408182995628009915340896198732339617497671295881928224276480] | [2624695468104673502443769142824657795/10633823966279326983230456482242756608, 328086933513084187805471142903044919/1329227995784915872903807060280344576] | PASS | PASS | 68610 | +| 3 | 1/512 | [1099744642466359519050550468585749845936785800957704611625351426120293/4353833339816674240405954844637321193950080192401449130728981648113664, 29326990956828271327196906113879255436482306452109711088548371232239411/116102222395111313077492129190328565172002138464038643486106177283031040] | [167877650387528921876409474825253669/664613997892457936451903530140172288, 335755300775057843752818949751585171/1329227995784915872903807060280344576] | PASS | PASS | 69634 | +| 4 | 1/4096 | [2212512643052213608732078042821547353419453798149232348664297882661491/8698652860170118243348078016966483943736355755760407594072928904282112, 88501911054845268570143314552842024415650056226776993393194746411414143/347946114406804729733923120678659357749454230230416303762917156171284480] | [338093306672378950862883161169884765/1329227995784915872903807060280344576, 5282707916755921107232549396447141/20769187434139310514121985316880384] | PASS | PASS | 70658 | +| 5 | 1/32768 | [443316666850063038125533262144600182749065810216265631464128298290799/1739280900730346182407387488830854941301851529748465835612852564525056, 8866473976892855723515037223612054742126867023037191412255153888348613/34785618014606923648147749776617098826037030594969316712257051290501120] | [84700728865203967454530889197013491/332306998946228968225951765070086144, 84700728865203967454530889298451707/332306998946228968225951765070086144] | PASS | PASS | 71682 | +| 6 | 1/262144 | [4435688056440540262220611493978282333857958089339752792828368045804885/17391685464525725299880151421019767972798346522113031010804282765082624, 5544697989040494015026100807255246405076691392371238866068635554847973/21739606830657156624850189276274709965997933152641288763505353456353280] | [21188594396515301654298523599511523/83076749736557242056487941267521536, 21188594396515301654298523650239643/83076749736557242056487941267521536] | PASS | PASS | 72706 | +| 7 | 1/2097152 | [138639469657658427518066957549669311518558785013463409898157722351957/543481394328770614045606060594003138464877488297403086347768522866688, 693208338750157696049095105321453503803600747579462653533407232692291/2717406971643853070228030302970015692324387441487015431738842614333440] | [5298159631183492748335016824718461/20769187434139310514121985316880384, 82783744237242074192734638282557/324518553658426726783156020576256] | PASS | PASS | 73730 | +| 8 | 1/16777216 | [19806695444553867896617565311841695870669965724049474202865749429903/77639885756694366683663660105959247631155869657362992369885633314816, 1386490662409888401041501359429452679860734656841473750976833430761957/5434792002968605667856456207417147334180910876015409465891994332037120] | [165576992316951126040846910950015/649037107316853453566312041152512, 1324615938535609008326775300282855/5192296858534827628530496329220096] | PASS | PASS | 74754 | + +All mathematical endpoints and exact arithmetic, discretization and +root-sliver widths are retained in CERTIFICATE.json. A partial sum is +reported as partial work and never used as a probability enclosure. + +Row-8 void interval: [2803722782782673251877380393776657242403798367195026522469331952123672246895/3618502788666131106986593281521497120414687020801267626233049500247285301248, 89719129049045544060076172820002359990801435547911798031414733543146336604957/115792089237316195423570985008687907853269984665640564039457584007913129639936]. +Target exp(-B): [350463322308452862342756000390952727903421288082416376725075231759535363689/452312848583266388373324160190187140051835877600158453279131187530910662656, 44859305255481966379872768050041949171691312537290570488888100841330813686635/57896044618658097711785492504343953926634992332820282019728792003956564819968]. +Maximum cross-endpoint distance: 518538081611300330636719918461647525585798813205589795474212705283500573/115792089237316195423570985008687907853269984665640564039457584007913129639936; strict threshold 1/8192; disposition PASS. + +Scope statuses: + +- exact_non_gaussian_population: PASS +- rational_guard_consequents: PASS_RATIONAL_GUARDS +- analytic_cutoff_derivative_premise: REVIEW_REQUIRED +- complete_surgery_field: NOT_RUN +- independent_global_h0_selector: NOT_RUN +- gaussian_universality: NOT_RUN +- blind_50_campaign: NOT_RUN +- higher_homology: NOT_RUN +- same_field_spatial_poisson: NOT_RUN +- lean_kernel: NOT_RUN +- scientific_promotion: NOT_RUN +- organizational_independence_credit: 0 + +The rational guards check consequences of analytic cutoff derivative +premises. Construction/evaluation of the complete surgery field and an +independent global H0 selector remain NOT_RUN. Shared outward elementary +primitives and source-exposed same-provider reviews provide zero +organizational independence credit. PASS does not admit a theorem or +promote scientific status. Source drift, failure or budget exhaustion +remains attached to this original attempt. diff --git a/experiments/universality/results/exact_population_1/RUN.json b/experiments/universality/results/exact_population_1/RUN.json new file mode 100644 index 000000000..ea8b5d94b --- /dev/null +++ b/experiments/universality/results/exact_population_1/RUN.json @@ -0,0 +1,127 @@ +{ + "actual_route_evaluations": 569360, + "command": "python3 -B -S experiments/universality/two_parameter_cusp_population/run_population_certificate.py", + "end_utc": "2026-10-09T18:20:30.645833+00:00", + "exit_code": 0, + "exposure": "Source-exposed same provider/team", + "final_runtime_budget_gate": "PASS", + "full_declared_campaign": true, + "invocation_scope": "declared full attempt", + "organizational_independence_credit": 0, + "outputs": { + "CERTIFICATE.json": { + "bytes": 71881, + "sha256": "8df03e0bfc338f27796c25816e319ee2304222d325540fe3372c2308a2d1a015" + }, + "RESULTS.md": { + "bytes": 6290, + "sha256": "8fe63fb42cd05fc7438e54e235b46e7fc74bb42b2ca8d5007d0011e62808c4c9" + } + }, + "parameters": { + "A": "1/65536", + "L": "1", + "d": 2, + "evaluation_budget_per_row": 80000, + "rectangle_density": "151115727451828646838272/3", + "rng": "NONE", + "rows": [ + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8 + ], + "u0": "3/4294967296", + "v0": "1/140737488355328", + "wall_clock_budget_seconds": 900, + "whole_rectangle_bar_mass": "1/5" + }, + "performer": "OpenAI/Codex population certificate runner", + "python": "3.9.6 (default, Dec 2 2025, 07:27:58) \n[Clang 17.0.0 (clang-1700.6.3.2)]", + "runtime_seconds": "29.981752", + "schema_version": 1, + "source_drift": [], + "sources_after": { + "experiments/periodic_h0/finite_certificate.py": { + "bytes": 14843, + "sha256": "0674a2621dff6c040cbb24cc0b6e6df7847541ebc171ebdb28070afa05e04e6b" + }, + "experiments/periodic_h0/finite_count_loss.py": { + "bytes": 2698, + "sha256": "85fca300f2dcb1af6102a3d82205e80cf5cb7f345b945186f2deb14d6b7f4317" + }, + 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+ "bytes": 15296, + "sha256": "a23f88801e678dd1af484cac8f5564801d9d410d835b560a280c0ea6dfc8996e" + }, + "experiments/universality/two_parameter_cusp_population/PROOF.md": { + "bytes": 38350, + "sha256": "9e8caa52b6448e23e465e646ca98d6e42e0cd71442ad4118b59145ee849e1326" + }, + "experiments/universality/two_parameter_cusp_population/population_certificate.py": { + "bytes": 20165, + "sha256": "314cd46ebe3b60055d01f47fe0d34db450b885f8757fe071fd715d4ccfd2a38e" + }, + "experiments/universality/two_parameter_cusp_population/run_population_certificate.py": { + "bytes": 23196, + "sha256": "3799f53e2066ae3f47a6b65ca93a130f2531ee78df1e0f4a094077cd8eabfb4e" + }, + "experiments/universality/two_parameter_cusp_population/test_population_certificate.py": { + "bytes": 15086, + "sha256": "3e1da0372f4cb3e6778a28f681096c2bc47bd64ef8317bbb286fee51184679e1" + } + }, + "start_utc": "2026-10-09T18:20:00.664198+00:00", + "status": "PASS", + "timer_scope": "Acceptance deadline covers source freezing, import, campaign, post-source hashing and serialization preparation of all three artifacts. runtime_seconds is sampled before final RUN encoding; exclusive artifact writes and flush are excluded. Final deadline is checked after RUN encoding." +} diff --git a/experiments/universality/two_parameter_cusp_population/CONTROLLED_FALSIFICATION.md b/experiments/universality/two_parameter_cusp_population/CONTROLLED_FALSIFICATION.md new file mode 100644 index 000000000..e6cf2677e --- /dev/null +++ b/experiments/universality/two_parameter_cusp_population/CONTROLLED_FALSIFICATION.md @@ -0,0 +1,270 @@ +# Predeclared exact cusp-population falsification contract + +Draft prepared outside the repository by OpenAI/Codex +`population_contract_design`, under integration owner `root`. The mathematical +author of the consumed two-control source is `universality_review`; this agent +contributes the finite test contract and has read the source and supplied route. +All are source-exposed members of the same provider/team. Organizational +independence credit is zero. This is a deterministic interval experiment with +no RNG, seed, numerical field draw, blind review or new theorem admission. + +## Source and object frozen before execution + +Consume the actual 782-line +`experiments/universality/two_parameter_cusp_population/PROOF.md`, SHA256 +`9e8caa52b6448e23e465e646ca98d6e42e0cd71442ad4118b59145ee849e1326`. +The contract designer freshly reproduced this identity and read actual lines +1-260, 261-520 and 521-782. Historical tests and reviews earn no execution +credit in this run. Record this contract's hash and all consumed Python source +hashes in the run before evaluating the population. If these frozen inputs +change during execution, retain the attempt as `INCONCLUSIVE_SOURCE_DRIFT`. + +Freeze these existing interval-source hashes as well: + +| Repository source | SHA256 | +| --- | --- | +| `experiments/periodic_h0/finite_certificate.py` | `0674a2621dff6c040cbb24cc0b6e6df7847541ebc171ebdb28070afa05e04e6b` | +| `experiments/periodic_h0/gaussian_tail.py` | `07e55a62ed0bb6da6a9395762e2e0c033126f83b46abd48db4c24df710779db8` | +| `experiments/periodic_h0/mean_adaptive_confidence.py` | `d83b4d530d00aafab1ebf5aa262869911db94d3b088309df3d878089285922a5` | +| `experiments/periodic_h0/finite_count_loss.py` (imported dependency) | `85fca300f2dcb1af6102a3d82205e80cf5cb7f345b945186f2deb14d6b7f4317` | + +Before the full run, freeze a manifest of the four new source files below, +this proof and these four interval dependencies. Record the actual hashes in +`RUN.json` and check them again at the end. The contract hash lives in the +manifest, avoiding a self-referential hash in this document. No claim is made +that the not-yet-written implementation already has a frozen identity. + +Fix spatial dimension `d=2`, period `L=1`, `w_i=3^-i`, unnormalized field +height, the superlevel H0 elder rule, and exclusion of the essential class. +Haar translation is part of the analytic field law and is integrated out; it +is not simulated. Fix `A=2^-16`, `u0=3*A^2`, `v0=2*A^3`, and full symmetric +rectangle density `c0=1/(24*A^5)`. Set counts to zero on the null discriminant +and tie controls, as the consumed proof does. Full-rectangle probabilities +must retain their mass; do not condition on the existence of a finite bar. + +For the positive-control chart put `a=A*z`, `h=a*theta`, `c=-a`. Its three +roots are `c-h`, `c+h`, `-2*c`; `u=3*a^2+h^2`, +`v=2*a*(a^2-h^2)`. Freeze `z>0`, `0 0` and selected gap +`P(-a-h)-P(-a+h)=4*a*h^3`. Reflection must exchange the younger side and +preserve the gap. This finite critical-height selector consumes the source's +global-selection proof; it is not an independent whole-field H0 algorithm. + +## Quantities and two separately derived interval routes + +Define `m(tau)=P(0=9/4`. This saturation also checks +the sign multiplicity, rectangle normalization and support convention. + +Use only `fractions.Fraction` for mathematical values and interval endpoints. +Reuse the existing `finite_certificate.sqrt_bounds`, +`gaussian_tail.exp_neg_bounds`, and `mean_adaptive_confidence.log_bounds` +with their actual source identities. Any required cube-root enclosure uses +rational bisection of a monotone cubic. Nested outward square-root enclosures +supply quarter powers; do not use float powers, fitted slopes or numerical +quadrature to certify an endpoint. `pi_bounds` is available for separately +scoped physical-coordinate work, but this contract does not evaluate physical +positions or claim that it used that function. +The two routes have separate analytic derivations but may share the root +bracket and outward elementary primitives. Their agreement earns no blindness, +organizational independence or independence of the shared interval engine. + +## Fixed schedule and numerical budgets + +Freeze eight rows, `j=1,...,8`, with `tau_j=2^(-3*j)` and `n_j=4^j`. +Then `n_j*tau_j^(2/3)=1` exactly. For each row report rational intervals for +both routes' `m(tau_j)`, both normalized masses `n_j*m(tau_j)`, their +intersection, and the exact iid void probability +`V_j=(1-m(tau_j))^n_j`. The iid-copy law is an analytic premise; no iid +samples are produced. Enclose `V_j` using outward log/exponential arithmetic, +or exact rational exponentiation if it stays within the same fixed budget. + +Bracket the split root in a dyadic interval of width at most `2^-96` by +96-step rational bisection of `g(r)=36*r^3/(3+r^2)^2`. Use outward elementary +operations with at least 128 bits and enforce total arithmetic inflation +at most `2^-22` for each normalized direct integral. For these rows the +split has `rhi<1/2`. A failed bracket, positivity or arithmetic-width gate +is inconclusive; no uncertified endpoint enters an acceptance predicate. +The `2^-22` arithmetic gate is an actual accumulated exact rational bound: +sum the positive quadrature weights times point/coefficient interval widths +for both the lower and upper quadrature contributions, including root-power +and coefficient uncertainty. Record that sum separately from discretization +and root-sliver widths and replay-check it. A precision setting alone is not +evidence for this gate. A sufficient auxiliary check is `rhi>2^-9`, giving +combined lower/upper quadrature weights at most `2/rhi<1024`; intervals of +full width at most `2^-40` at every weighted evaluation then contribute less +than `2^-30`. Actual accumulated arithmetic width remains the acceptance gate. + +Direct core: put `theta=rlo*x` on `x in [0,1]`; use exactly `2^16` +Darboux panels. The normalized integrand is increasing, with range at most +`3/10`, since `n_j*rlo^2<=1/2`. Its enclosure width is at most +`3/(10*2^16)`. Record the monotonicity and range gates. + +Direct tail: put `theta=rhi*t`. On each octave +`[2^k,min(2^(k+1),1/rhi)]`, use exactly 512 panels with midpoint lower and +trapezoid upper sums. Its normalized integrand is +`C*(9*t^(-11/4)-rhi^2*t^(-3/4))`, where +`C=n_j*tau_j^(5/4)/(15*4^(5/4))*rhi^(-7/4)<=1/30`. +It is convex; `f''<=99/32*t^(-19/4)`. The summed discretization width is +at most `297/(512*512^2)`. Enclose the intervening root sliver by +`[0,(3/5)*n_j*(rhi-rlo)]`. Total direct normalized width must be +strictly less than `2^-17`. + +Declare at most 80,000 direct integrand evaluations per row, with midpoint +and endpoint enclosure computations counted explicitly. Compute each unique +node once and reuse its enclosure in adjacent panels: a cached use is not +another evaluation, while any actual repeated computation counts again. +At row 8 the plan uses at most `65,537+9*(513+512)=74,762` evaluations. +This function-evaluation count is separate from arithmetic inflation, which +counts both weighted lower/upper uses of every enclosure. The closed-CDF +normalized enclosure must have width at most `2^-16`. The direct/CDF +intervals must overlap. The reported void enclosure and target Poisson +enclosure must each have full width at most `2^-16`. + +No adaptive extra rows, changed thresholds, increased panels, increased +root iterations or relaxed acceptance threshold belong to this run. Any +exhausted evaluation/precision budget is `INCONCLUSIVE_BUDGET`, preserving +partial intervals and counts. A successor contract may predeclare a changed +budget and link this result; it must not relabel the failed attempt as PASS. +The full eight-row runner has a 900-second wall-clock budget, enforced with +a monotonic timer. Runtime exhaustion is `INCONCLUSIVE_BUDGET`; record the +completed work and elapsed time without calling that attempt a full run. + +## Predictions and predeclared falsification predicates + +Let `B=9/(28*cuberoot(2))`, enclosed by rational bisection, and +`E(tau)=(9/8)*tau^(2/3)+tau^(7/12)`. All eight rows test the prediction +`abs(m(tau)/tau^(2/3)-B)<=E(tau)`. Record the complete predicted interval, +the observed certified interval and their distance; no regression estimates +replace this declared coefficient and error envelope. +For certified intervals `B in [Blo,Bhi]`, `E in [Elo,Ehi]` and normalized +mass `Lambda in [Llo,Lhi]`, PASS requires containment in the common inner +envelope: `Llo>=Bhi-Elo` and `Lhi<=Blo+Elo`. A certified falsifier requires +disjointness from the outer envelope `[Blo-Ehi,Bhi+Ehi]`. Neither containment +in that outer interval nor interval overlap alone proves the prediction. + +At `j=8`, `E(tau_8)<=41/524288`. Verify the mass gate +`n_8*m(tau_8)<1/3`. It gives +`abs((1-m)^n_8-exp(-n_8*m))<1/(16*n_8)=1/1048576`. +Let the target be `exp(-B)`. The declared endpoint error allowance is + +``` +41/524288 + 1/1048576 + 2*(2^-16) + = 115/1048576 < 2^-13. +``` + +The row-8 acceptance predicate is the explicit rational endpoint test +`max(abs(Vlo-target_hi),abs(Vhi-target_lo))<2^-13`. +Interval overlap alone is not enough. If certified intervals have minimum +distance greater than `2^-13`, the declared finite void prediction is +`FALSIFIED`. If intervals straddle that boundary or fail their width gate, +the result is `INCONCLUSIVE_PRECISION`. + +If a row's certified normalized-mass interval is disjoint from the outer +prediction envelope, label that coefficient prediction `FALSIFIED` +after source, soundness and negative-control gates pass. Freeze new-lemma +work on the affected assertion and retain the counterinterval. If the +observed interval is neither disjoint from the outer envelope nor contained +in the common inner envelope, report `INCONCLUSIVE_PRECISION`. The explicit +inner endpoint inequalities yield that row's PASS. + +Disjoint CDF and direct intervals are `FAIL_ROUTE_CONSISTENCY`, not a PASS +for either route. Invalid input, a failed arithmetic guarantee or an +undetected negative control is `FAIL_IMPLEMENTATION`. Such results provide +no certified scientific counterexample until their soundness issue is +resolved. A PASS means this fixed non-Gaussian population survived these +eight declared coefficient tests and the one declared void test. It does +not test general Gaussian one-third universality, all A3 populations, Hk, +same-field spatial Poisson statistics or scientific acceptance. + +## Required negative controls and independent scope statuses + +Freeze exact controls `a=A/4` with `theta=1/4,1/2,3/4` and their reflected +signs. They check root stationarity, order, Hessian signs, actual elder +selection and the displayed gap. Boundary controls `theta=0,1,3` must be +classified as discriminant or tie and excluded under the declared convention. +The adversarial control `a=A/4,theta=2` has ordered roots `(-3*a,a,2*a)` +and negative `v`. All these fixtures lie inside the frozen control rectangle. +The actual younger-right lifetime divided by `a^4` is `3/4`; the fixed-left +formula divided by `a^4` falsely gives `32`. The critical-height selector +must reject that wrong formula. These fixtures still earn no independent +whole-field H0 selector execution credit. +Also detect a missing factor two by comparing its total mass `1/10` with +the exact `1/5`, and reject bar-conditioned mass `1` as a rectangle mass. +Reject non-Fraction mathematical inputs and reversed/nonenclosing intervals. + +Check rational analytic guard witnesses separately: `R=1/8`, `r1=1/32`, +`a*=r1/4`, `delta=r1^4/64`, `B0<=r1^2/2`, `D0<=r1`, +`B1<=33*r1`, `D1<=65`, `g*>=r1^3/64`. Use the proposed C-infinity +cutoff defined by `q(t)=exp(-1/t)` for `t>0` and zero otherwise, +`psi(t)=q(1-t)/(q(t)+q(1-t))`, flat beyond 0 and 1, with +`abs(psi')<=8`; the radial cutoff gives `abs(grad chi)<=64/r1`. +The monotone surgery uses +`c(r)=r/4+(1-r/4)*(1-psi((r-r1^2)/(R^2/2-r1^2)))`. +These analytic premises require substantive review; exact arithmetic only +checks their consequent rational inequalities. In particular + +``` +B1*u0+D1*v0 <= 101441/2^47 < 2^-22 <= g*/2, +B0*u0+D0*v0 <= 1537/2^52 < 2^-28 = delta/4. +``` + +Also check `R^2=2, L>0, X=(R/LZ)^d, spatial volume V=L^d, and +omega=2 pi/L. Persistent homology is ordinary superlevel H0 over a fixed +field K, with the elder rule and the essential global-maximum class excluded. +Use the explicit smooth seed and quartic surgery of Section 2. The family is + + F_(u,v)=F_0+chi(s,z_perp)[u s^2/2+v s], (1) + +on the full control rectangle + + -u_0=eta_c u_0^(3/2), eta_c=2/(3 sqrt(3)). (2) + +Section 2 supplies compatible explicit smallness conditions, including +root containment, annular exclusion and a global high-level barrier. +All constants and rectangle sides are fixed before taking any lifetime limit. +Choose (u,v) uniformly in this full rectangle, with density + + c_0=1/(4u_0 v_0), (3) + +and independently choose Haar-uniform Y in X. The random field is + + G(x)=F_(u,v)(x-Y). + +We leave its scale unnormalized in the main theorem. This law is stationary, +has smooth sample fields and uniform bounds on every spatial C^q norm, and +is almost surely Morse with distinct critical values. The density (3) is +positive and smooth in the interior, including at the control origin; +its extension by zero across the rectangle boundary is not claimed smooth. + +The only controls producing a finite H0 bar are the open cusp wedge + + W={0=u_0^2/4. This is derived by a direct pushforward of +control measure, not by differentiating a cumulative asymptotic. Write + + K_c=4/3^(5/4), + Q_c=(2/3) K_c^(-2/3), + C_edge=(8c_0/7) K_c^(-2/3) u_0^(7/6). (7) + +Then + + nu(t) ~ C_edge t^(-1/3), + E N_t ~ (3/2) C_edge t^(2/3), t down to zero. (8) + +These are whole-field coefficients. Per unit spatial volume divide by V. +For any fixed 00 and n t_n^(2/3)->lambda +in (0,infinity), their actual short-H0 points with lifetime mark ell/t_n +and birth position x converge to a Poisson random measure with intensity + + lambda C_edge a^(-1/3) da dx/V. (10) + +The full latent-control, sign, physical birth and cubic marks are specified +in Sections 7--8. This uses the exact single-field bound N_t in {0,1}. +It is not a same-field spatial or expanding-domain process claim. + +## 2. Actual global construction and uniform rectangle bounds + +Here we reproduce the needed global surgery, rather than presuming a local +normal form has a harmless completion. Put w_i=3^(-i) and + + h(y)=sum_i w_i cos(omega y_i), b=sum_i w_i. + +Its 2^d critical points have y_i in {0,L/2}, diagonal nonzero Hessians, +and distinct values. In comparing two sign patterns, their first differing +weight dominates the sum of all later weights. The origin is the unique +maximum, and all other critical values are <=b-Delta, Delta=2w_d. + +In |y_i|b-R^2} is precisely the +chart ball D_R. Indeed each coordinate's nonnegative sine-square loss +is then [0,1] with + + c(r)=r/4 for r<=r_1^2, + c(r)=1 for r>=R^2/2, + c(r)>0 for r>0, c'(r)>=0. + +For example integrate a nonnegative smooth derivative equal to 1/4 +on [0,r_1^2], tapering to zero before R^2/2, and add an interior bump +to make its total integral exactly one. This is possible because R^2<1. +Define on D_R, retaining h outside, + + F_0=b-s^2 c(s^2+|z_perp|^2)-|z_perp|^2. (11) + +It equals h on a full boundary collar, so is globally C-infinity. +Its derivatives are + + (F_0)_s=-2s[c(r)+s^2 c'(r)], + (F_0)_(z_i)=-2z_i[1+s^2 c'(r)], r=s^2+|z_perp|^2. + +They vanish in D_R only at the origin. All old critical points outside +are unchanged. In |z|<=r_1 the exact formula is + + F_0=b-s^4/4-(1+s^2/4)|z_perp|^2. (12) + +Let chi be a fixed radial smooth cutoff, equal to one for |z|<=r_1/2, +zero for |z|>=r_1, and between zero and one elsewhere. Its zero extension +is smooth on X. Formula (1) therefore defines a global field jointly +C-infinity in x,u,v, including at u=v=0. In its exact core, + + F_(u,v)=b+P_(u,v)(s)-(1+s^2/4)|z_perp|^2, + P_(u,v)(s)=-s^4/4+u s^2/2+v s. (13) + +For explicit uniform guards, set + + a_*=r_1/4, delta=r_1^4/64, + g_*=min_(r_1/2<=|z|<=r_1)|grad_z F_0|>0, + B_j=||chi s^2/2||_(C^j), D_j=||chi s||_(C^j), j=0,1, + +where for j=1 the gradient norm alone suffices. Fix a constant +c_v>=eta_c, set v_0=c_v u_0^(3/2), and choose u_0>0 small enough that + + u_0=a_*, its root equation and (14) would give + + |s|^3<=u_0|s|+v_0<|s|^3/4+|s|^3/4, + +a contradiction. Thus all axial roots are strictly inside the region +where chi=1. On the transition annulus the perturbation gradient is +