#!/usr/bin/env python3 """ SCOPE NOTE, 2026-10-11 (comp 202). Where this file quotes lambda_SM(M_*) ~ 0.0927 (Buttazzo et al. 2013), the 0.8 percent agreement, the one-sigma Higgs match or comp 176's 0.4 sigma, that is the anchor of its day and is kept as found. comp 202 computes the number at three loops on current inputs: 0.0932 +- 0.0008, 1.6 sigma from e^-1/4, which is what the book states from v32.137. Nothing here is re-run; the substrate side is unaffected. PROVENANCE: PROOF Computation 95 -- Bridge Premise (B) via substrate-Wilsonian RG flow ==================================================================== Open-research item 1.1 (Bridge Premise B for the Z^2 closure). Comp 94's fluctuation-matching attempt had dimensional issues (X_bar dimensionless vs h with GeV units; T_KO units ambiguous) and yielded a numerically suspect prediction mu^2(M_*) = 2D * M_*^2. This computation takes a different route: Formulation (I) Wilsonian effective-coupling RG, made STRUCTURAL by identifying the partition- function-level RG flow generator on the substrate. THE FRAMEWORK ============= Bridge Premise (B): lambda_SM(M_*) = b * Z_H(beta_KO) with b = 1/4, beta_KO = 2 (KO mod 8), Z_H(beta_KO) = e^(-1). The Wilsonian matching reads: lambda_SM(M_*) = Pi(lambda_substrate) where Pi is the matched-scaling projection (sec:mosco-conditional) and lambda_substrate is the LG quartic at the substrate scale Lambda = sqrt(D). At the substrate scale, the bare LG quartic is b = 1/4 (P3 postulate). The Wilsonian RG flow takes lambda from Lambda = sqrt(D) down to M_*: lambda_substrate(M_*) = b * R where R is the "running renormalisation" from substrate scale to M_*. THE STRUCTURAL IDENTIFICATION ============================= Claim: R = Z_H(beta_KO) = e^(-1). Argument: the Wilsonian RG flow on the substrate is generated by integrating out substrate modes with weighting beta_KO * H_Higgs(C). The partition function over integrated-out modes is: R = E_mu[exp(-beta_KO * H_Higgs(C))] = E_mu[exp(-beta_KO * X_bar(C))] (Comp 89: H_Higgs = X_bar) = Z_H(beta_KO). This is the PARTITION-FUNCTION-LEVEL analogue of the heat-kernel RG flow generator used in standard Chamseddine-Connes machinery. For the continuum spectral triple, the heat-kernel exp(-t D^2) at flow time t generates the RG flow. For the discrete substrate spectral triple (KO-dim 10 = 2 mod 8), the analogous flow generator is exp(-beta_KO * H_Higgs(C)) with beta_KO = KO_total mod 8 = 2. The asymptotic limit of R as D -> infinity: R = E_mu[exp(-2 X_bar)] -> e^(-1) (Bernoulli MGF, Comp 88) Combining with lambda_SM(M_*) = Pi(lambda_substrate(M_*)) = lambda_substrate(M_*) (matched-scaling Pi acts identically on couplings at matched scaling): lambda_SM(M_*) = b * R = b * e^(-1) = 0.0920. This matches observed lambda_SM(M_*) ~ 0.0927 (Buttazzo 2013) at the same 0.8% level. THE STRUCTURAL PROOF OF (B) =========================== Theorem (Bridge Premise B, structural derivation). Under the partition-function-level Chamseddine-Connes correspondence, the matched-scaling projection Pi delivers: lambda_SM(M_*) = lambda_substrate(M_*) = b * R where R is the Wilsonian RG flow generator from the substrate scale Lambda = sqrt(D) to the matched scale M_*, given by: R = E_mu[exp(-beta_KO * H_Higgs(C))] = Z_H(beta_KO) For PST's KO-total = 10 (Bott periodicity), beta_KO = 10 mod 8 = 2; combined with H_Higgs = X_bar (Comp 89, structural derivation from additivity + matched scaling + uniformity), R = E_mu[exp(-2 X_bar)] -> e^(-1) as D -> infinity. Therefore: lambda_SM(M_*) = b * e^(-1) asymptotically. Identifying Z^2 = lambda_SM(M_*)/b: Z^2 = e^(-1) STRUCTURALLY. CONDITIONS REQUIRED FOR THE THEOREM ==================================== The theorem rests on the IDENTIFICATION: Wilsonian RG flow generator on the substrate = exp(-beta_KO * H_Higgs(C)) (***) This is the partition-function-level analogue of the heat-kernel flow generator exp(-t D^2) for the standard CC continuum triple. For the discrete substrate triple (where the heat-kernel expansion collapses per Comps 85, 86), the partition-function flow generator is the natural substitute. To make (***) a rigorous theorem, the following are required: (a) The substrate Wilsonian RG flow is well-defined and exists. (b) The flow generator equals exp(-beta * H) for a specific Hamiltonian H and inverse temperature beta. (c) For PST's substrate, H = H_Higgs = X_bar and beta = beta_KO = 2. (a) is established by the matched-scaling A1 of sec:mosco-conditional together with the standard Wilsonian framework (Polchinski 1984). (b) is the standard Boltzmann-distribution form of the partition function over RG-integrated-out modes. (c) is the structural identification of H_Higgs (Comp 89) and beta_KO (Comp 88) within PST. THE REMAINING STRUCTURAL CONTENT ================================ For (***) to follow from (a)-(c) without additional input, we need: the substrate Wilsonian RG flow generator is EXACTLY the Boltzmann partition function over substrate modes at temperature beta_KO. This is essentially the statement of the partition-function-level Chamseddine-Connes correspondence. Comp 92 elevates the Z^2 closure to a 'structural proof sketch' conditional on Bridge Premise (B); Comp 95 (this) identifies (***) as the PHYSICAL CONTENT of (B). (***) is plausible because: - The Boltzmann-distribution form of the partition function over integrated-out modes is the universal Wilsonian-RG result. - The PST substrate has discrete spectrum, so the partition function is the natural integration measure (no continuous-spectrum a_4 coefficient to recover). - The KO-tempered beta_KO = 2 is structurally forced by the foundational object's KO-dim mod 8 (sec:foundational-object). CLOSURE STATUS ============== Comp 95 reduces Bridge Premise (B) to the SINGLE structural claim (***): the substrate Wilsonian RG flow generator equals the Boltzmann partition function at KO temperature. If (***) is granted, the bridge identification follows structurally: lambda_SM(M_*) = b * Z_H(beta_KO) = (1/4) * e^(-1) = 0.0920. If (***) is not granted, alternative derivations (e.g., Comp 94's fluctuation-matching attempt, or formulation II's spectral-action variation) remain candidate routes. (***) is the partition-function-level analogue of the standard heat-kernel RG flow generator. Establishing it as a formal theorem within the PST framework is the active research direction. Comp 95 status: Bridge Premise (B) REDUCED to a single explicit structural claim (***). This is a sharper-than-Comp-92 statement of the open content: not 'derive (B)' generally, but specifically 'show that the substrate Wilsonian RG flow generator equals the Boltzmann partition function at the KO-tempered temperature'. """ import math def main(): print("=" * 100) print(" Computation 95 -- Bridge Premise (B) via substrate-Wilsonian RG flow") print("=" * 100) print() print("THE STRUCTURAL IDENTIFICATION") print("-" * 100) print() print(" Wilsonian matching:") print(" lambda_SM(M_*) = lambda_substrate(M_*) (matched scaling Pi identity)") print() print(" Wilsonian RG flow from substrate scale Lambda = sqrt(D) to M_*:") print(" lambda_substrate(M_*) = b * R") print(" where R is the RG-flow renormalisation factor.") print() print(" Identification (Comp 95 conjecture):") print(" R = E_mu[exp(-beta_KO * H_Higgs(C))]") print(" = E_mu[exp(-beta_KO * X_bar(C))] (Comp 89: H_Higgs = X_bar)") print(" = Z_H(beta_KO)") print() print(" Combining:") print(" lambda_SM(M_*) = b * Z_H(beta_KO)") print() print("ASYMPTOTIC EVALUATION") print("-" * 100) print() beta_KO = 2 R_asymptotic = math.exp(-1) b = 0.25 print(f" beta_KO = {beta_KO} (KO_total mod 8, Comp 88)") print(f" R = E_mu[exp(-{beta_KO} X_bar)] -> exp(-1) = {R_asymptotic:.6f}") print(f" lambda_SM(M_*) = b * R = {b} * {R_asymptotic:.4f} = {b * R_asymptotic:.6f}") print() print(f" Observed (Buttazzo 2013): lambda_SM(M_*) ~ 0.0927") print(f" Match: {b * R_asymptotic / 0.0927 * 100:.2f}% (0.8% deviation)") print() print("FINITE-D CONVERGENCE") print("-" * 100) print() print(f" {'D':>8} {'R(beta_KO)':>15} {'b * R':>15} {'error vs e^-1/4':>20}") for D in [10, 100, 1000, 10000]: R_D = ((1 + math.exp(-beta_KO / D)) / 2) ** D lam = b * R_D err = abs(lam - b * math.exp(-1)) print(f" {D:>8} {R_D:>15.8f} {lam:>15.8f} {err:>20.2e}") print() print("THE STRUCTURAL CLAIM TO BE PROVEN") print("-" * 100) print() print(" (***) The substrate Wilsonian RG flow generator equals the") print(" Boltzmann partition function exp(-beta_KO * H_Higgs(C))") print(" on substrate configurations.") print() print(" Comp 95 status: Bridge Premise (B) REDUCED to (***).") print() print(" Plausibility:") print(" - Boltzmann-distribution form is the universal Wilsonian RG result") print(" (Polchinski 1984).") print(" - The PST substrate has discrete spectrum; the partition function") print(" is the natural integration measure (no continuous-spectrum a_4") print(" coefficient to recover -- Comps 85, 86).") print(" - beta_KO = KO_total mod 8 = 2 is structurally forced by the") print(" foundational object (sec:foundational-object).") print(" - H_Higgs = X_bar is structurally derived from additivity + matched") print(" scaling + uniformity (Comp 89, no free parameters).") print() print(" Establishing (***) as a formal theorem within the PST framework") print(" CLOSES Z^2 = e^(-1) at full proof level.") print() print("CLOSURE STATUS") print("-" * 100) print() print(" Before Comp 95: Bridge Premise (B) was the 'single explicit premise'") print(" reducing Z^2 closure to one undefined structural") print(" claim.") print() print(" After Comp 95: Bridge Premise (B) is REDUCED to a SHARPER, MORE") print(" CONCRETE claim (***): the substrate Wilsonian RG flow") print(" generator equals the Boltzmann partition function at") print(" beta_KO. This is testable / provable within standard") print(" Wilsonian-RG machinery applied to the discrete substrate.") print() print(" The active research direction is to establish (***) as a formal") print(" theorem. This is the partition-function-level analogue of the") print(" heat-kernel RG flow generator for the continuum spectral triple,") print(" adapted to the discrete substrate where the heat-kernel collapses.") print() if __name__ == "__main__": main()