diff --git a/experiments/universality/README.md b/experiments/universality/README.md index 4b437d71c..a59b44189 100644 --- a/experiments/universality/README.md +++ b/experiments/universality/README.md @@ -140,6 +140,34 @@ 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. +For 2/3=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, choose a fixed positive physical distance cutoff for which +the actual finite-bar density nu_H,far is bounded near zero and has finite +total mass, and suppose this population is nonzero. SG/HK supplies the +far bound at its declared near-band cutoff; applying it at another cutoff +requires that bound there as well. Under these stated premises the Mellin +moment M_(alpha,far) is positive and finite for 0 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, + + 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..7c96e2287 --- /dev/null +++ b/experiments/universality/spectral_amplitude_crossover/REVIEW.md @@ -0,0 +1,224 @@ +# 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. + +## 25 October 2026 author correction and current source bindings + +The preceding 144-line / 8,932-byte record is retained byte-for-byte from +Main334 parent `6bd133d591e43e75a9292463bb4bfde7a34984dc`, SHA256 +`351ba1e9b866bae880da2042b24f827a1f0aef7613f728d572a6722669a4c5b1`. +Its construction-stage current/PENDING labels and completed review claims +refer to their dated source identities. This append supersedes their use +as statements about the corrected proof or this extended review record. + +Actual correction author is OpenAI/Codex Root01a11cbf, under new finite +claim2837 / `88cdcb4d-4fdc-4b18-8a2e-3a38d6bfbd8a`, issued25October2026 +at14:40:45UTC. The [current author pickup](https://github.com/d6g8k5htny-coder/main/pull/334#issuecomment-6399017462) +retains the exact scope and deadlines. No old lease or reviewer claim is +revived. The parent proof37461B retains SHA256 +`aeea6b55a1876c357d9beeeb5a96ee54a5b17b53c6138e2993c454f1117f65d5`; +the parent README16387B retains SHA256 +`1484efcd0704727dbb2d3dcfcb76ad833fd989152c6004d82f77e3da2acea636`. +Their original native objects and actual review history remain unchanged. + +### Mathematical disposition of A1 + +The original Anthropic Claude S [scoped adverse review](https://github.com/d6g8k5htny-coder/main/pull/334#issuecomment-6139276725) +identified A1 and D1 on13October; the later Copilot findings +[4365128840](https://github.com/d6g8k5htny-coder/main/pull/334#discussion_r4365128840) +and [4365128873](https://github.com/d6g8k5htny-coder/main/pull/334#discussion_r4365128873) +remain separate native records. Muse Spark's +[documentary delta read](https://github.com/d6g8k5htny-coder/main/pull/334#issuecomment-6154507888) +checked the recorded bindings; it was not a new derivation of Section5. +Those performances keep their authors, scopes, dates and zero claimed +organizational credit. This amendment does not adopt the separate stronger +supplements or claim their review. + +ST's actual FAR interface is `t^(1/3) nu_far(t)->0`; 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