#!/usr/bin/env python3 """ 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 v25.10+ 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 v25.10+ 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()