From 6bd133d591e43e75a9292463bb4bfde7a34984dc Mon Sep 17 00:00:00 2001 From: Dylan Roy <249851642+d6g8k5htny-coder@users.noreply.github.com> Date: Thu, 8 Oct 2026 17:17:44 -0700 Subject: [PATCH 1/2] Prove exact amplitude crossover and whole-barcode cluster limits --- experiments/universality/README.md | 23 + .../spectral_amplitude_crossover/PROOF.md | 778 ++++++++++++++++++ .../spectral_amplitude_crossover/REVIEW.md | 144 ++++ 3 files changed, 945 insertions(+) create mode 100644 experiments/universality/spectral_amplitude_crossover/PROOF.md create mode 100644 experiments/universality/spectral_amplitude_crossover/REVIEW.md diff --git a/experiments/universality/README.md b/experiments/universality/README.md index 4b437d71c..edb79ca74 100644 --- a/experiments/universality/README.md +++ b/experiments/universality/README.md @@ -140,6 +140,29 @@ cluster estimate yields a degree-marked iid-copy Poisson limit. Its classical imports and source exposure. This does not admit a numerical Hp algorithm, arbitrary coefficient laws/geometries or spatial process. +The [exact amplitude crossover theorem](spectral_amplitude_crossover/PROOF.md) +considers F=A H for an admitted Gaussian field H and an independent bounded +positive amplitude A=aR, where R has density alpha*r^(alpha-1) on 02/3 retains that power with a negative-amplitude-moment coefficient. +Density powers are one less than the corresponding cumulative powers. +For independent whole copies with t tending to zero, the subcritical limit +is a Poisson process of whole-barcode clusters. For a full positive K5 base +law its void rates differ strictly from a Poisson random measure for the +full H0/all-degree population. +The critical limit is an ordinary projected Poisson random measure under +n*t^(2/3)*log(1/t) scaling; the supercritical limit uses n*t^(2/3) scaling. +Physical birth/gap marks collapse in the critical regime and have the stated +amplitude tilt in the supercritical regime. The +[review record](spectral_amplitude_crossover/REVIEW.md) preserves the original +AMEND, corrected preparation reviews and completed incorporated-source review. +This exact declared mixture admits no arbitrary non-Gaussian law, numerical +sampler, blind experiment or same-field spatial process. + These conventional analytic results discharge selected-bar counting and the leading near/far intensity composition for the stated ideal finite laws and admitted stationary Gaussian spectral/carrier laws. Further diff --git a/experiments/universality/spectral_amplitude_crossover/PROOF.md b/experiments/universality/spectral_amplitude_crossover/PROOF.md new file mode 100644 index 000000000..5a8b9c525 --- /dev/null +++ b/experiments/universality/spectral_amplitude_crossover/PROOF.md @@ -0,0 +1,778 @@ +# Exact covariance-matched amplitude crossover and whole-barcode clusters + +Repository incorporation on 8 October 2026. Mathematical author: +OpenAI/Codex `/root/universality_review`. Repository wrapper and bounded +integration contributor: `/root/final_contract_review`. Integration owner: +`/root`, task `01a11cbf-068d-7102-861e-75814e715c98`, under native claim +`6071350791`, lease through 00:30 UTC on 9 October 2026. The isolated +detached construction base is +`89657c7c48289378949dc59e7fbe6dc1a55d9c62`. + +Sections 1-11 are preserved byte-for-byte from the corrected external +34,110-byte / 730-line source, SHA256 +`9e8f10162e9fa21f308454df9463459cdd67ef39a2d8103a78a397b4ab82dd0d`. +The 28,494-byte mathematical core has SHA256 +`f558dafcdb3841646c3d364056a95c07e7a56759ef5bc47f9b6033b85f5e0df0`. +The preserved original 33,421-byte / 720-line source, SHA256 +`5b1859080a137aed9eeeba20e1b6f0c5a61387e5eb6fa20bb1b8704860fb73d2`, +received two separate full-source AMEND verdicts. Its missing critical +small-lifetime condition and ambiguous density/count table labels are +corrected in the successor; the original and its adverse review history +remain immutable. + +The corrected preparation has a separate fresh FULL 730-line mathematical +PASS from `benchmark_formal_audit`, a full actual-source read by `root`, +and a bounded complete-correction-diff PASS from `final_contract_review`. +That bounded correction audit was not a new whole-730-line read. Those +completed reviews apply to their identified external sources; they do not +review these new repository wrapper bytes. A fresh incorporated-source full mathematical +review and metadata review are PENDING at this construction snapshot; +the [review record](REVIEW.md) preserves the separate identities and scopes. +Historical assignment-stage pending states retained in Sections 1-11 refer +to preparation chronology, not the current status of an input review. + +All contributors are source/route-exposed OpenAI/Codex AI performers; +organizational-independence credit is zero. This is a conventional analytic +composition for the declared law, not numerical, formal, blind or scientific +acceptance. Current operational inputs and actual reading extents are +recorded in Section 12. No compiler, sampler, seed, installation, external +post or scientific-status change is part of this construction. + +## 1. Base input, units and review boundary + +Let H be the actual fixed Gaussian spectral law on X=R^d/(L Z^d), +d>=2, admitted by SG in the source table below. Write V=L^d and +beta=2/3. Ordinary finite superlevel bars use a fixed coefficient +field; essential intervals are excluded. A bar has positive lifetime +ell_i, degree p_i, birth point x_i and ordered direction u_i from +birth critical point to its killing critical point. Use the existing +Borel shortest-displacement convention outside the near band. + +The following exact base premises are consumed: + +* H is C4 almost surely and Morse with distinct critical values. + It has a finite Borel barcode and finite expected total finite-bar + count. These statements for H0 are proved by SG. Higher-degree + counting is supplied by the separate HK preparation. +* For each chosen degree population I (one degree, H0, or all finite + degrees), its expected lifetime measure per volume has a finite + nonnegative Borel density nu_(H,I), with + + nu_(H,I)(s) ~ c_I s^(beta-1), c_I>0, s down to zero. (1) + +* The projected short-bar process of H, keeping only degree, + lifetime ell/s, birth location and ordered direction, has mean + limit eta_H and satisfies the actual physical weighted-cluster + estimate + + E[N_H(Ts) 1{N_H(Ts)>=2}]=o(s^beta), T>0. (2) + +Here the all-degree form uses HK. The H0 form alone uses SG and +needs no higher-degree admission. For the projected measure, + + eta_H{lifetime<=T}=V (c_I/beta) T^beta. (3) + +Its finer angular/location distribution is that proved by SG/HK, +after integration of the original birth and cubic-gap marks. +It is the whole-volume measure, not a per-volume measure. + +The scalar transformation in Sections 2-5 needs only the finite +barcode, finite expected count and (1). The process conclusions +also consume (2) and the projected one-bar mean limit. All arguments +below are proved under these named premises, rather than inferring +them from covariance. At assignment the higher-homology source had +one reported full nonauthor PASS and its second full audit was still +pending. This preparation does not upgrade that review state or +claim a new full review of SG/HK. H0 is a complete base specialization +independent of the remaining HK review. + +For a population I write + + B_H=# {finite H bars in I}, + m_H(t)=E N_H((0,t])/V=integral_0^t nu_H(s)ds, + M_alpha=integral_0^infinity s^(-alpha)nu_H(s)ds + =V^(-1) E sum_(i in I) ell_i^(-alpha). (4) + +We suppress I where harmless. The random sums below use actual +whole-field barcodes, not candidate pairs. + +Finite E B_H deserves an explicit global argument. The one-site +finite-incidence/Kac-Rice identity in SG has a uniform gradient +covariance floor on the compact torus and finite conditional Hessian +determinant moments. It gives E#Crit(H)0. Let R be independent of H, with density + + p_R(r)=alpha r^(alpha-1), 00, grad F=A grad H and Hess F=A Hess H. +Critical points, Hessian indices, critical-value order and actual +persistence pairings are unchanged. The superlevel identity +{F>=h}={H>=h/A} gives exactly + + ell_(F,i)=A ell_(H,i), b_(F,i)=A b_(H,i), + x_(F,i)=x_(H,i), u_(F,i)=u_(H,i). (8) + +The physical cubic mark k_i=ell_i/dist(M_i,S_i)^3 is also +multiplied by A. Thus F is C4, Morse/distinct almost surely, and +every positive C4-norm moment possessed by H remains finite: +E||F||C4^q<=a^q E||H||C4^q. Under SG's simple weighted-square-root +spectral verifier, H has all finite such moments, so F does too. +Under its weaker R4 premise only its supplied moments are needed. +No Fernique upgrade is presumed for that statement. + +The total finite-bar count is unchanged, so E B_F=E B_H0. (9) + +Indeed in the amplitude integral set s=t/A: +A^(alpha-2)dA changes to t^(alpha-1)s^(-alpha)ds with reversed +limits. This is the actual selected-bar density, not a candidate +surrogate. It holds for every chosen degree separately and for +their finite sum. + +For every fixed t>0 the integral in (9) is finite by (5), since +s^(-alpha) is bounded on s>=t/a. The formula is unchanged by +altering the base density on a Lebesgue null set. It specifies +a locally absolutely continuous, in particular continuous, density +representative on (0,infinity): the lower-limit integral of a +locally integrable function is absolutely continuous on positive +compact intervals, and its prefactor is smooth there. This continuity +comes from amplitude integration. It is not an inherited continuity +claim for the discontinuous global selector's base density. + +For the cumulative count, each bar contributes + + P(A ell_i<=t)=min{(t/(a ell_i))^alpha,1}. + +Consequently the exact per-volume identity is + + m_F(t)=m_H(t/a)+(t/a)^alpha + integral_(t/a)^infinity s^(-alpha)nu_H(s)ds. + (10) + +Equations (9)-(10), not differentiation of an asymptotic cumulative +formula, will establish all density constants. The cutoff t/a, +normalization a and volume factors remain explicit. + +## 4. The full three-regime crossover + +The base asymptotic (1) implies, for some delta>0, bounds +c_1 s^(beta-1)<=nu_H(s)<=c_2 s^(beta-1) on 0=delta, (5) controls every negative-power Mellin tail. +Therefore + + 0=beta. (11) + +Positivity follows from c_I>0. In the finite case it is an actual +global barcode Mellin moment, which includes all lifetimes. + +### 4.1 Subcritical amplitude: 0beta + +The lower integral diverges as + + integral_(t/a)^infinity s^(-alpha)nu_H(s)ds + ~c_I/(alpha-beta) (t/a)^(beta-alpha). (15) + +For proof, trap nu_H between (c_I+-epsilon)s^(beta-1) +below delta and integrate the power explicitly; the fixed tail +is negligible compared with the divergent lower term. It follows +that + + nu_F(t) ~ c_I E A^(-beta) t^(beta-1), + m_F(t) ~ (c_I/beta) E A^(-beta) t^beta, + E A^(-beta)=a^(-beta)alpha/(alpha-beta). (16) + +The base m_H term in (10) must be retained: its coefficient +c_I a^(-beta)/beta combines with c_I a^(-beta)/(alpha-beta) +to give the second formula in (16). Dropping it would give the +wrong cumulative constant. + +Because A is nonconstant, E A^2=1 and x->x^(-beta/2) is strictly +convex, Jensen gives E A^(-beta)>1 whenever that expectation is +finite. Thus even this restored exponent has a strictly changed +coefficient despite identical covariance. As alpha grows without +bound the multiplier tends to one; no varying-alpha uniform error +or interchange of alpha and t limits is asserted. + +## 5. Exactly what the covariance-matched falsifier demonstrates + +All realizations preserve the actual regular folds and Morse +pairings of H. No new higher-order singularity is introduced. +Nevertheless (12), (14) and (16) give, respectively, a different +power, a logarithm and a changed leading coefficient. Covariance, +positive smooth-norm moments, Morse genericity and a cubic local +height gap do not determine the selected lifetime law. + +The proof identifies the missing distributional control. Negative +amplitude moments, not positive norm moments, distinguish the +three phases. Small amplitudes contract whole finite barcodes, +including bars whose physical endpoints are far apart. CONTACT +and ENVELOPE for a Gaussian covariance cannot be transferred to +this non-Gaussian coefficient law. Nor must the Gaussian FAR +O(1) bound survive. + +For example, fix a positive physical distance cutoff and let +nu_H,far be its actual finite-bar density. SG/HK gives nu_H,far<=C +near zero and finite total mass. If this population is nonzero, +then its Mellin moment is finite for every alpha<1, and exactly +the same transform gives + + nu_F,far(t) ~ alpha a^(-alpha) M_(alpha,far)t^(alpha-1). + (17) + +This applies even when beta0, + + E N_H(Ts)<=C_T s^beta. (20) + +Near zero this is the cumulative asymptotic; away from zero use +the finite total expected count and enlarge the constant. Hence +Psi_f(z)<=min{1,C_T z^(-beta)}. Moreover, the error between +E sum_i(1-exp(-f(p_i,z ell_i,x_i,u_i))) and Psi_f(z) +is nonnegative and at most + + E[N_H(T/z)1{N_H(T/z)>=2}]. + +The actual one-bar mean limit and (2) therefore imply + + z^beta Psi_f(z)->K_f, z->infinity. (21) + +This is the genuine single-copy cluster input. First moment alone +would not imply (21). + +## 7. Subcritical process: exact full-barcode Poisson clusters + +Suppose 0 +alpha a^(-alpha)J_f. For n independent copies with + + n t_n^alpha -> lambda in (0,infinity), + +their exact Laplace product tends to + + exp{-lambda alpha a^(-alpha) + integral_0^infinity z^(alpha-1) + E_H[1-exp(-sum_i f(p_i,z ell_i,x_i,u_i))]dz}. (22) + +Indeed D_t->0 and n D_t has the displayed finite limit, so +n log(1-D_t)=-n D_t+o(1). This is the Laplace functional of +the following concrete Poisson cluster random measure. Put a +Poisson parent measure on (z,H) with intensity + + lambda alpha a^(-alpha) z^(alpha-1)dz P_H(dH). + +Each parent contributes the entire child configuration + + C_z(H)=sum_(i in I) delta_(p_i,z ell_i,x_i,u_i). (23) + +The parent measure is sigma-finite. Only finitely many parents +are active in any bounded lifetime window, as verified next, so +the resulting child process is locally finite. Elementary Poisson +conditioning/exponentiation gives (22). This construction is the +exact Lévy pushforward of the actual whole-field barcode law. + +Write ell_min=infinity when the population is empty, and otherwise +the minimum positive finite lifetime. On the full window (0,T] +the active-parent rate is + + Lambda_T=lambda a^(-alpha)T^alpha + E[ell_min^(-alpha);B_H>0]. (24) + +This follows by integrating z^(alpha-1) over z<=T/ell_min. +It is finite because ell_min^(-alpha)<=sum_i ell_i^(-alpha), +whose expectation is V M_alpha=2)>0, then m_T^*>Lambda_T. The process has void +probability exp(-Lambda_T), whereas an ordinary Poisson process +with the same first intensity would have exp(-m_T^*). Thus it is +strictly non-Poisson. Section 8 proves this condition for H0, and +hence for the all-degree population, in the actual P5 family. +For a population which almost surely contains at most one bar, +(22) does reduce to an ordinary Poisson measure. Strict clustering +is not falsely asserted for every arbitrary carrier or degree. + +For the original F birth and cubic marks, the amplitude-natural +subcritical scaling is + + b_F/t=z b_H, k_F/t=z k_H. + +Appending these as child marks in (23) gives the same exact +cluster Laplace law: f then acts on (p,z ell,z b,z k,x,u). +The integrability bounds depend only on its bounded lifetime +support. Unrescaled b_F,k_F tend instead to zero in this regime; +one must not retain eta_H's original birth/gap mark law by assertion. +For the collapse, at each fixed parent (z,H) the physical marks +are t z b_H,t z k_H and tend to zero. The active child-count +integrand is integrable by (25), so dominated convergence justifies +the assertion over the complete parent intensity, not merely at +bounded deterministic amplitudes. + +## 8. Actual positive multi-bar probability in the P5 family + +This support step establishes a strict process falsifier rather +than leaving a hypothetical multiple-bar condition. + +Assume SG's full positive K5 block. The trigonometric function + + h(x)=cos(2 omega x_1)+cos(2 omega x_2) + +sum_(i=3)^d cos(omega x_i), + omega=2pi/L, + +has four nondegenerate maxima. All its critical points are +nondegenerate: its Hessian is diagonal there with nonzero entries. +Some critical heights coincide, but an arbitrarily small K5 +coefficient perturbation makes them all distinct while preserving +the four maxima. This existence follows from SG's actual finite- +block Morse/distinct-value incidence proof: the K5 Gaussian block +has a positive coefficient density, the bad set has Lebesgue +measure zero, and every coefficient neighborhood of h has positive +measure. Choose one such deterministic good h_* in that neighborhood. + +There is a C2, hence a C4, neighborhood of h_* in which the field +stays Morse/distinct with exactly the continued critical list and +at least four maxima. On a connected torus all maxima give H0 +births; one class is essential and each other class has a finite +death. Thus every field in this neighborhood has at least three +ordinary finite H0 bars, over any coefficient field. + +That neighborhood has positive probability for the *actual* full +Gaussian law, not just its normalized finite truncation. Split +H=H_B+T into the actual positive finite K5 block and independent +centered Gaussian C4 tail. The block has positive probability of +being arbitrarily close to h_* in C4. Also + + P(||T||C40 for every epsilon>0. (27) + +Here is a self-contained small-ball proof requiring no uncited +Banach support theorem. C4(X) is separable. Its countable covering +by open balls of radius epsilon/sqrt(2) has a ball with positive +tail probability. Two independent copies T,T' both in that ball +have difference norm less than sqrt(2)epsilon with positive +probability. Their difference has the same C4 law as sqrt(2)T: +the centered Gaussian identity holds on the countable dense jet +evaluations generating this Borel law. Consequently (27) follows. +If the tail is identically zero it is immediate. Independence of +block and tail then supplies the claimed positive probability +of the good h_* neighborhood. All arguments are theoretical; +no coefficient draw or seed is consumed. + +We conclude P(B_(H,H0)>=3)>0, so (24)-(25) are strictly different +for H0 and the all-degree population. The alpha beta a^(-beta)K_f. (28) + +For proof split at 1 and a large fixed Z. The bounded lower part +divided by log(1/t) tends to zero. Beyond Z, trap z^beta Psi_f +between K_f+-epsilon and integrate dz/z. This also covers K_f=0. +Since a is fixed, log(a/t)/log(1/t)->1. + +Choose t_n -> 0 with n t_n^beta log(1/t_n)->lambda in (0,infinity). +The iid Laplace product yields an ordinary Poisson random measure +with intensity + + lambda beta a^(-beta) eta_H. (29) + +The whole-lifetime marginal is +lambda V beta a^(-beta)c_I v^(beta-1)dv, matching (14). +This conclusion also has a genuine physical cluster check. +Changing variables z=A/t in the mixed multiple-bar expectation +gives beta a^(-beta)t^beta times the integral of + + z^(beta-1) E[N_H(T/z)1{N_H(T/z)>=2}]. + +For z>=1 the bracket times z^beta tends to zero by (2), and +is bounded by (20). Logarithmic Cesàro makes this piece o(log(1/t)). +For z<=1 the expected total count bounds the bracket by a finite +constant, so that piece is finite. Therefore + + E[N_F(Tt)1{N_F(Tt)>=2}] + =o(t^beta log(1/t)). (30) + +Whole-barcode contractions at A of order t contribute only O(t^beta) +and vanish at this logarithmic normalization. The leading events +span logarithmically many scales with both A->0 and t/A->0. +No second-factorial assertion or independent bars within one +field is used. + +The marks in (29) are only the retained degree/lifetime/location/ +direction marks of Section 6. For unrescaled physical b_F,k_F, +the leading critical mass collapses to b_F=0,k_F=0. To verify it, +amplitudes A>=delta contribute only O(t^beta), hence zero after +division by t^beta log(1/t). On A0 bounded away +from zero. One may use a space including k=0 or retain marks +renormalized by the latent amplitude; the latter is a separately +specified observation convention, not the physical unscaled mark. + +There is a precise optional latent mark for the logarithmic range. +Set Theta_t=log A/log t. For fixed 0=1 or A<=t contribute O(t^beta) +by (20) or finite expected count. The weighted multi-point error +is still (30). Step-function approximation on Theta intervals +therefore proves the critical PRM with the additional latent mark + + Theta distributed uniformly on (0,1), + intensity lambda beta a^(-beta)eta_H tensor dTheta. + +This mark is asymptotically independent of the retained base fold +marks. It records an amplitude available in the generative model; +it is not substituted for the physical birth/gap marks. There is +no claim that all critical events occur at one amplitude scale. + +## 10. Supercritical process: ordinary Poisson with tilted amplitude + +Suppose alpha>beta. Now E A^(-beta)=2}]=o(t^beta). (31) + +Thus for n t_n^beta -> lambda the independent-copy process tends +to the ordinary Poisson random measure with intensity + + lambda E A^(-beta) eta_H. (32) + +The very small A of order t has probability O(t^alpha), lower +order than t^beta; this is also visible in the dominated bound. + +Unlike the critical case, full physical birth/gap marks can be +retained in this regime. Their actual intensity is the mixture + + lambda E[A^(-beta) (T_A)_* eta_H^full], + T_A(p,v,b,k,x,u)=(p,v,A b,A k,x,u), (33) + +where eta_H^full is the base full marked one-bar measure. The +same integrable majorant and bounded tests prove this statement. +It is the tilted amplitude distribution proportional to A^(-beta) +times its original law, not the original unweighted amplitude +distribution and not unchanged Gaussian birth/gap marks. +Explicitly its density on (0,a) is +(alpha-beta)a^(-(alpha-beta))A^(alpha-beta-1). + +## 11. Summary of exact regimes and limits + +Here nu and m are per-volume measures; eta_H is the whole-volume +base process measure. Let c=c_I and M_alpha be (4). + +| Regime | Leading density asymptotic | Leading cumulative-count asymptotic per volume | iid normalization | Process limit on degree/lifetime/location/direction | +| --- | --- | --- | --- | --- | +| 0 lambda | Full-barcode Poisson clusters, (22)-(23) | +| alpha=beta | beta a^(-beta)c t^(beta-1)log(1/t) | a^(-beta)c t^beta log(1/t) | t_n -> 0 and n t_n^beta log(1/t_n) -> lambda | Ordinary PRM, lambda beta a^(-beta)eta_H | +| alpha>beta | c E A^(-beta)t^(beta-1) | (c/beta) E A^(-beta)t^beta | n t^beta -> lambda | Ordinary PRM, lambda E A^(-beta)eta_H | + +The scalar identities are exact at every positive lifetime. The +asymptotics and process conclusions consume the stated actual base +premises. Below beta the Lévy law depends on the whole barcode, +so different degrees/locations can be linked within each parent +cluster. At and above beta the proved weighted-cluster bounds +remove multiple points from one copy at the relevant normalization. +Every limit concerns independent whole copies of one fixed law. +There is no spatial expanding-domain law, coalescing-contact +factorial estimate, within-one-field Poisson assertion or regional +witness discharge. + +The Laplace limits also determine process convergence on finite +lifetime windows. On the retained mark space, degree/location/ +direction are compact and a fixed window [epsilon,T] is compact. +The normalized expected total counts are bounded by the exact +cumulative asymptotics. The expected contribution from lifetime +at most epsilon tends to a constant times epsilon^alpha below +beta or epsilon^beta at/above beta, and vanishes as epsilon goes +to zero. This supplies tightness including the endpoint zero. +The limiting first measures have no lifetime atom at T, so the +window boundary does not create an exceptional truncation. For +the subcritical extra rescaled birth/gap marks, the active parent +measure and child first intensity are finite on each window; +their dominated pushforwards in (23) give the necessary mark-tail +truncation as well. No global factorial-moment criterion is used. + +The covariance-matched mixture supplies an actual falsifier of +covariance-only universality. It does not contradict a Gaussian +spectral theorem that explicitly assumes the Gaussian coefficient +law and the needed contact/envelope/selection interfaces. The +amplitude density, independence, normalization and mark scaling +are necessary declared parts of this counterexample. + +## 12. Incorporated source identities, actual read extents and review custody + +The current repository sources below are the operational bindings. Their +full files were freshly rehashed in the detached construction checkout; +this identity check is not four new whole-proof reads by the constructor. + +| Input | Current repository source | UTF-8 bytes / lines | SHA256 | +| --- | --- | --- | --- | +| SG | [spectral_gaussian_admission/PROOF.md](../spectral_gaussian_admission/PROOF.md) | 42478 / 826 | `6c0f34cb318d337b3b8c2c306f3b965f5e5a9d2fe38d5e001b8044de74b7f225` | +| HK | [higher_homology_fold/PROOF.md](../higher_homology_fold/PROOF.md) | 46174 / 911 | `87f43e6ed94c5c63c385ecf177f68212afa5db54b24062fa008f081f47a78999` | +| ST | [fold_structural_universality/PROOF.md](../fold_structural_universality/PROOF.md) | 43927 / 804 | `3d952a906553d578876456539a8f0dcd64aedd45099e8394cbe9db99866ec531` | +| PP | [iid_short_bar_process/PROOF.md](../iid_short_bar_process/PROOF.md) | 30935 / 618 | `320bc988c468bea4938bb777a2d05952c6ab8e0a384f33725b50de847aeec1eb` | + +The mathematical author originally consumed and freshly hashed the external +SG preparation, 42,038 bytes / 825 lines, SHA256 +`43e8180947b559e53f7afcf95ba61136f3fffcbe05a19e7a5953018fda27e294`, +and HK preparation, 44,258 bytes / 884 lines, SHA256 +`de1cad971e113f097a8e29ba82fb295d93b23f57aeda5b265e6f3c88cef47f2e`. +ST and PP were consumed at the same current proof identities in the table. +These historical author inputs are retained separately from the current +repository bindings; no new whole current-SG or current-HK author read is +inferred from incorporation. + +The original/current SG Sections 1-11 are identical: 36,656 bytes, SHA256 +`4cad43d3f80a8e47da886762d402c4981fe6834e278579b36673055539d6cf32`. +The original/current HK Sections 1-12 are identical: 37,220 bytes, SHA256 +`c770d4f7a28bff3d9f59a5baa004f052fb6db2c9e27a11799db0e7420d124471`. +The constructor compared the actual cores in this incorporation. The +current input wrappers have their own source-bound review records; core +identity does not turn an earlier review into a new whole-source review. + +The author had prior full actual-byte reads of all four original inputs, +including the then freshly saved HK author read. During the amplitude +preparation task the author additionally reread ST lines 640-699 for the +historical amplitude diagnostic and HK lines 525-748 for density, cone +constants and all-degree endpoint/cluster premises. SG and PP hashes were +checked without claiming two additional whole reads. SG Section 4 supplies +finite expected critical count and all-Borel counting; Sections 7-8 supply +the H0 density and actual cluster/mean premises. HK Sections 7, 10-11 supply +the higher-degree counterparts. The earlier ST lower-bound diagnostic is +not falsely rebound to this stronger exact amplitude law. + +At original assignment the second HK whole-source audit was pending, as +retained in Section 1. This is an explicitly historical statement. The +amplitude proof consumes the exact HK mathematical premises, with H0 a +specialization independent of the higher-degree premises. Later input +review records do not retrospectively replace that assignment chronology. + +The original amplitude preparation is +`spectral-amplitude-crossover-preparation.md`, 33,421 bytes / 720 lines, +SHA256 `5b1859080a137aed9eeeba20e1b6f0c5a61387e5eb6fa20bb1b8704860fb73d2`. +`final_contract_review` freshly read all 720 saved lines in untruncated +ranges 1-240, 241-480 and 481-720, and reproduced the four declared input +hashes. `benchmark_formal_audit` separately completed a full 720-line +saved-source review. Both returned AMEND: the Section 9 normalization +alone admits a sequence approaching one, so it must also require +`t_n -> 0`; the table must label its density and count formulas as leading +asymptotics. No original-source verdict is rebound to its successor. + +The corrected external preparation is +`spectral-amplitude-crossover-corrected-preparation.md`, 34,110 bytes / +730 lines, SHA256 +`9e8f10162e9fa21f308454df9463459cdd67ef39a2d8103a78a397b4ab82dd0d`. +Its author completed a full saved-source readback. `benchmark_formal_audit` +made a fresh FULL 730-line mathematical read in untruncated ranges 1-365 +and 366-730, read the complete predecessor diff and rehashed the four +inputs: PASS, no must-fix. `root` also completed a full actual-730-line +read. `final_contract_review` read the complete actual 720-to-730 diff and +the corrected blocks, reproduced the four input hashes, and returned a +bounded correction PASS. Reversing only the declared correction recovered +the original 720-line bytes exactly. That audit did not claim a new whole +730-line read. The constructor now preserves Sections 1-11 exactly, checks +all current source/core identities, and authors only the repository wrapper, +review custody and README addition. This is an integration contribution, +not a new nonauthor mathematical review of the incorporated file. + +The coordinator supplied the positive multi-maximum P5 witness, void-rate +diagnostic and normalization/mark cautions. `final_contract_review` +contributed bounded prefile attacks of the critical logarithmic Cesaro / +singleton argument and subcritical Laplace integrability, including the +physical birth/gap collapse caution. These are prior proof-route +contributions, separately exposed from saved-source reviews. The same +reviewer also authored a separate fold-chart route and supplied carrier +witness attacks in earlier work; organizational independence remains zero. +The mathematical author authored the SG, ST, physical-law, process and HK +preparations. All performances are from the same OpenAI/Codex team. + +No new primary-source fetch is credited to this amplitude construction or +its preparation reviews. The scalar identities, Mellin/logarithmic limits, +Gaussian tail small-ball argument and iid Laplace calculation are explicit +in Sections 1-11. Classical Morse/area/counting imports retain the bounded +and full read extents declared by the exact input proofs. These are not +reverified by a numerical test or a newly claimed full primary-book read. + +The companion [review record](REVIEW.md) records construction-stage PENDING +states for the new incorporated full-source and metadata reviews. Publication, +attachment, hosted checks, merge and receipt bindings will be separately +recorded only when obtained. They are not established by this file or by +completed preparation reviews. No scientific-status promotion, critical +Lean instantiation, empirical sampler admission, computational/count coupling, +blind seed use or same-field spatial-process result is claimed. diff --git a/experiments/universality/spectral_amplitude_crossover/REVIEW.md b/experiments/universality/spectral_amplitude_crossover/REVIEW.md new file mode 100644 index 000000000..35dac4d8d --- /dev/null +++ b/experiments/universality/spectral_amplitude_crossover/REVIEW.md @@ -0,0 +1,144 @@ +# Amplitude crossover: construction and source-bound review record + +Construction snapshot, 8 October 2026. Mathematical author: OpenAI/Codex +`universality_review`. Repository wrapper and bounded integration contributor: +`final_contract_review`. Integration owner: `root`, task +`01a11cbf-068d-7102-861e-75814e715c98`, claim `6071350791`, lease through +00:30 UTC on 9 October 2026. Detached base: +`89657c7c48289378949dc59e7fbe6dc1a55d9c62`. + +Current [proof](PROOF.md): 37,461 UTF-8 bytes / 778 lines, +SHA256 `aeea6b55a1876c357d9beeeb5a96ee54a5b17b53c6138e2993c454f1117f65d5`. +Its Sections 1-11 are preserved verbatim from the corrected +external preparation: 28,494 bytes / 637 lines, SHA256 +`f558dafcdb3841646c3d364056a95c07e7a56759ef5bc47f9b6033b85f5e0df0`. +New incorporated-source full mathematical review: **PENDING**. +Root full current-source read: **PENDING**. +New companion/README metadata review: **PENDING**. +Constructor identity/core/link checks are bounded integration evidence, +not a new nonauthor whole-proof verdict or publication acceptance. + +| Preserved external source | UTF-8 bytes / lines | SHA256 | Actual review scope | +| --- | --- | --- | --- | +| Original amplitude preparation | 33421 / 720 | `5b1859080a137aed9eeeba20e1b6f0c5a61387e5eb6fa20bb1b8704860fb73d2` | Two separate fresh FULL mathematical AMEND verdicts: final_contract_review and benchmark_formal_audit | +| Corrected amplitude preparation | 34110 / 730 | `9e8f10162e9fa21f308454df9463459cdd67ef39a2d8103a78a397b4ab82dd0d` | Author full readback; benchmark_formal_audit fresh FULL mathematical PASS; root full actual-source read; final_contract_review bounded complete-correction-diff PASS | + +The original Section 9 condition +`n*t_n^(2/3)*log(1/t_n) -> lambda` omitted `t_n -> 0`. +`t_n=exp(-lambda/n) -> 1` satisfies the normalization while giving no +rare-event Poisson limit. Both original full reviews required the explicit +small-lifetime condition. They also required the table's density/count +columns to identify leading asymptotics rather than suggest exact global +formulas. The original AMEND and source identity are retained permanently. + +The corrected successor changes only the ten-line correction provenance, +Section 9 small-lifetime condition, table density/count headings and critical +table row. Its mathematical author fully reread the saved 730 lines. +Benchmark independently fully read all 730 actual lines in two untruncated +ranges, read the complete predecessor diff, and rehashed the four inputs: +PASS, no mathematical must-fix. Root fully read the actual corrected source. +Final_contract_review read the complete actual 720-to-730 diff and corrected +blocks, rehashed the four inputs, and verified that reversing only those +corrections exactly recovered the original bytes: bounded correction PASS. +That is not a fresh whole-730-line review. These external-source verdicts +are not rebound to the new incorporated source or wrapper. + +Current operational inputs are the repository SG826/6c0f34cb, +HK911/87f43e6e, ST804/3d952a90 and PP618/320bc988 proofs, linked and fully +identified in proof Section 12. The author consumed original SG825/43e81809 +and HK884/de1cad97 preparations. Actual original/current mathematical cores +are identical: SG Sections 1-11, 36,656 bytes / SHA256 +`4cad43d3f80a8e47da886762d402c4981fe6834e278579b36673055539d6cf32`; +HK Sections 1-12, 37,220 bytes / SHA256 +`c770d4f7a28bff3d9f59a5baa004f052fb6db2c9e27a11799db0e7420d124471`. +Fresh construction hash/core checks preserve that lineage; no new whole +current SG/HK author read is inferred. Exact prior and bounded input reading +extents, historical HK assignment-stage review state and prior proof-route +contributions remain separately exposed in proof Section 12. + +The [README](../README.md) retains all 223 baseline lines byte-for-byte and +adds the declared mixture, its asymptotic phase diagram and independent-copy +limits. The proof distinguishes per-volume scalar density/count from +whole-volume process intensity. For alpha<2/3 the full-barcode cluster law +has a strict H0/all-population void-rate diagnostic for the full positive +K5 base family; this is not asserted for every restricted degree/carrier. +At critical scaling t_n -> 0 is explicit. +The supercritical coefficient retains the actual amplitude tilt. Physical +marks use the stated collapse/rescaling/tilt conventions; all density and +cumulative laws are leading small-lifetime asymptotics. + +All contributors and reviewers are source/route-exposed OpenAI/Codex AI +performers; organizational-independence credit is zero. The constructor is +a prior fold-chart contributor and amplitude prefile attacker, and the +mathematical author authored the consumed spectral, structural, process +and higher-homology routes. No new primary fetch or full primary-book +verification is credited here. Compiler/tests/sampler/installations/seeds: +NOT_RUN. No numerical coupling, critical Lean, blind experiment, same-field +spatial-process theorem or scientific promotion is claimed. + +Publication, attached PR, native review, hosted checks, merge and ordinary +receipt bindings will be recorded separately only when actually obtained. +This construction record establishes none of those later facts. The full +research goal remains ACTIVE; current new-source reviews remain PENDING. + +## Completed incorporated-source mathematical review + +The historical construction prefix above is preserved exactly: +5,296 UTF-8 bytes / 82 lines, SHA256 +`ecb42a6756149ad4f437f351722a573fc19e1c763a4f21a4b8506612e1eebd87`. +Its PENDING states record construction chronology and do not describe the +completed verdicts below. The proof itself remains unchanged. + +**benchmark_formal_audit: fresh FULL current mathematical PASS.** +The reviewer read all 778 actual proof lines in untruncated ranges +1-260, 261-520 and 521-778. The independently reproduced identity is +37,461 bytes / SHA256 +`aeea6b55a1876c357d9beeeb5a96ee54a5b17b53c6138e2993c454f1117f65d5`. +The complete actual corrected730-to-current778 diff was read, and the +28,494-byte / 637-line mathematical core was independently matched at +SHA256 `f558dafcdb3841646c3d364056a95c07e7a56759ef5bc47f9b6033b85f5e0df0`. +No mathematical must-fix emerged. This is a new incorporated-source +verdict, separate from the original720 AMEND and corrected730 PASS. + +The fresh attack covered exact covariance normalization and barcode +scaling, finite-count/Mellin integrability, every density/count constant, +whole-barcode subcritical clusters and the strict P5 void diagnostic, +critical small-threshold normalization and logarithmic singleton limit, +physical-mark collapse, the optional uniform latent logarithmic mark, +and supercritical negative-amplitude tilt with its physical marks. +Four current operational hashes and both original/current SG/HK cores +were freshly verified. These are identities and core comparisons with +prior input exposure separately disclosed; no four extra whole +operational-input reads or new primary-source fetch are claimed. + +**Construction metadata: separate fresh FULL PASS.** The same reviewer +fully read REVIEW82/ecb42 and README246/3184, read the complete +baseline223-to-current246 README diff, and verified all36 local links. +The original720 two AMENDs, benchmark's corrected730 whole-file PASS, +final_contract's bounded correction PASS, current778 whole-file PASS, +and the constructor's wrapper contribution remain distinct. +The original 16,378-byte / 246-line README is saved as an ordinary +external construction snapshot at its original SHA256 +`3184e4eaec23083b018fae009fbd200bfbc49d7318a0b14ffa5f5184dd057d2a`. +After the actual current-source verdict, root changed only its two +phrases: explicitly name R as the variable with density on (0,1), +and replace the historical pending-review description with completed +incorporated-source review. These new metadata bytes and this append +await their separately bound current metadata audit. + +Root completed the full actual778 proof read in untruncated1-390 / +391-778 ranges, the complete actual incorporation diff and unchanged +core comparison, and the full current246 README. The first combined +metadata output omitted247tokens in REVIEW; root then read the complete +82-line REVIEW in a separate untruncated result. No truncated portion +is credited as complete. Root's original/current whole reads supply +integration scrutiny, not organizational independence. + +All performers remain prior source/route-exposed OpenAI/Codex agents, +organizational independence0. The wrapper contributor is also a prior +fold-chart and amplitude-prefile contributor; benchmark has catalogue, +CG/ET and related source exposure. No new primary, compiler, local +unit tests, numerical count/sampler coupling, seed or blind experiment +was supplied by these reviews. Main/native/hosted publication outcomes +will be recorded only after they occur. Same-field spatial limits, +critical Lean and scientific promotion remain open. Goal ACTIVE. From 1b4a16e327b9e2046bfca36b0e7b629c2f72a5ee Mon Sep 17 00:00:00 2001 From: Dylan Roy <249851642+d6g8k5htny-coder@users.noreply.github.com> Date: Sun, 25 Oct 2026 09:56:02 -0500 Subject: [PATCH 2/2] Correct amplitude FAR inference and bind current review metadata Distinguish structural scaled FAR from the stronger bounded-far estimate; require the far boundedness and nonzero population at the same cutoff. Preserve prior source and review records, explicitly withdraw unsupported current metadata PASS attribution, and bind the corrected proof and README. Reviewed exact V2 source: conventional nonauthor source-exposed same-provider review, C/I/M 0/0/0; no formal or scientific acceptance. --- experiments/universality/README.md | 5 ++ .../spectral_amplitude_crossover/PROOF.md | 73 ++++++++++++----- .../spectral_amplitude_crossover/REVIEW.md | 80 +++++++++++++++++++ 3 files changed, 139 insertions(+), 19 deletions(-) diff --git a/experiments/universality/README.md b/experiments/universality/README.md index edb79ca74..a59b44189 100644 --- a/experiments/universality/README.md +++ b/experiments/universality/README.md @@ -160,6 +160,11 @@ Physical birth/gap marks collapse in the critical regime and have the stated amplitude tilt in the supercritical regime. The [review record](spectral_amplitude_crossover/REVIEW.md) preserves the original AMEND, corrected preparation reviews and completed incorporated-source review. +For 2/3 0. + +Writing K=alpha a^(-alpha) M_(alpha,far)>0, equation (17) gives + + t^(1/3) nu_F,far(t) ~ K t^(alpha-beta) -> 0 + when beta0`; its O(1) estimate is +only sufficient. The parent Section5 inferred too much from an unbounded +but lower-order far contribution. The corrected Section5 explicitly derives +`K t^(alpha-2/3)->0` for2/3