#!/usr/bin/env python3 """ PROVENANCE: PROOF Computation 110 -- Bridge Premise (B) attack, milestones M1 + M2: discrete Wilson-Polchinski flow + wave-function renormalisation identification of the bridge factor ========================================================================= STATUS (v26.13): IN PROGRESS. This computation sets up the framework for closing Bridge Premise (B) from P1-P3 alone, via the wave-function renormalisation route, and reduces the open problem to a single sharp statement (milestone M3). It does NOT close (B). It is the scaffolding + precise target, analogous to how Comp 101 reduced (B) to two sub- questions before Comp 103 attacked one of them. THE TARGET (Bridge Premise B) ============================== The substrate-side factor is closed from P1-P3 (Comp 100): Z_H(beta_KO) = E_mu[exp(-beta_KO * X_bar)] = ((1 + e^(-beta_KO/D))/2)^D -> e^(-beta_KO/2) = e^(-1) at beta_KO = 2, D -> infty. Bridge Premise (B) is the identification lambda_SM(M_*) = b * Z_H(beta_KO) = (1/4) * e^(-1) = 0.0920 (B) between the SM Higgs quartic at M_* and the substrate bare quartic b = lambda_PST = 1/4 times the substrate partition function. WHY THE NAIVE ROUTE FAILED (Comp 103) ====================================== Standard Wilsonian threshold matching of a quartic is ADDITIVE in the loop corrections (lambda_IR = lambda_UV + finite), not a multiplicative dressing. Comp 103 showed the multiplicative form is NOT derivable as a path-integral (measure change-of-variable) Jacobian: - linear-map determinant: configuration-independent (constant); - Cramer-Bernoulli LDP: Gaussian in (X_bar - 1/2), not exp(X_bar). THE OPENING (this computation) =============================== A coupling flows MULTIPLICATIVELY under the RG -- not additively -- precisely when the flow is dominated by WAVE-FUNCTION (field-strength) renormalisation Z_phi: psi -> Z_phi^(1/2) psi => lambda -> lambda / Z_phi^2 (with Z_Gamma4 = 1). This is exactly the multiplicative structure (B) needs, and it is the reinterpretation Comp 101 proposed (Z^2 as wave-function renormalisation rather than coupling renormalisation). The sharpened conjecture: THE BRIDGE FACTOR IS THE FIELD-STRENGTH RENORMALISATION OF THE SUBSTRATE-TO-SM PROJECTION Pi, with Z_phi^2 = Z_H(beta_KO) = e^(-1). i.e. (B) is NOT a coupling-matching Jacobian (Comp 103 killed that), it is the wave-function renormalisation of the modal field as it projects to the SM Higgs. This computation does: M1 -- set up the discrete Wilson-Polchinski flow on the Boolean lattice with cutoff f(x) = e^(-x) (Comp 100) and matched scaling Lambda^2 = D; show the flow reduces to the low Walsh shells V_0 + V_1 where X_bar lives (Comp 98). M2 -- define the field-strength renormalisation Z_phi from the substrate two-point function of the order-parameter field, and show that IF the matched-scaling flow weights the field by the substrate Boltzmann factor exp(-beta_KO X_bar), then Z_phi^2 = Z_H(beta_KO) = e^(-1) numerically. M3 (NOT done here) -- prove the Polchinski flow actually GENERATES the weight exp(-beta_KO X_bar) as its field-strength kernel. This is the genuinely new mathematics and the remaining crux. ========================================================================= """ import math # ------------------------------------------------------------------------- # Substrate primitives (P1): Bernoulli measure, Walsh basis, Boolean # Laplacian. Conventions match Comp 100: Delta = sum_a (1 - tau_a), each # (1 - tau_a) has eigenvalues {0, 2}, so Delta chi_S = 2|S| chi_S. # ------------------------------------------------------------------------- def boolean_laplacian_eigenvalue(s): """Delta chi_S = 2|S| chi_S (|S| = s).""" return 2 * s def cutoff(x): """Matched-scaling cutoff f(x) = e^(-x), forced by P1's tensor-product independence (Comp 100: f(x+y) = f(x)f(y) => f = exp).""" return math.exp(-x) def propagator_mode(s, D): """Substrate propagator in Walsh mode of weight s at matched scaling Lambda^2 = D: G_S = f(2|S|/Lambda^2) / (2|S|) for |S| >= 1. The zero mode (s = 0) is the order parameter / massless direction and is handled separately (it is NOT a propagating mode integrated out).""" if s == 0: return None # order-parameter zero mode lam2 = D return cutoff(boolean_laplacian_eigenvalue(s) / lam2) / boolean_laplacian_eigenvalue(s) # ------------------------------------------------------------------------- # The Higgs Hamiltonian / order-parameter field (Comp 89, Comp 98): # X_bar(C) = (1/D) sum_a C_a, with Walsh decomposition # X_bar = (1/2) chi_emptyset - (1/(2D)) sum_{|S|=1} chi_S # supported entirely on the lowest two Walsh shells V_0 (+) V_1. # ------------------------------------------------------------------------- def xbar_walsh_support(): """Return the Walsh-shell support of X_bar (Comp 98).""" return {0: 0.5, # coefficient of chi_emptyset 1: -0.5} # coefficient of (1/D) sum_{|S|=1} chi_S def xbar_mean(): """E_mu[X_bar] = 1/2 (Bernoulli site mean).""" return 0.5 def xbar_variance(D): """Var_mu(X_bar) = (1/D^2) * D * Var(C_a) = (1/D^2)*D*(1/4) = 1/(4D).""" return 1.0 / (4 * D) # ------------------------------------------------------------------------- # KO-tempered Bernoulli partition function (Comp 87/88/100): # Z_H(beta) = E_mu[exp(-beta X_bar)] = ((1 + e^(-beta/D))/2)^D # -> e^(-beta/2) as D -> infty. # At beta_KO = 2 (one-bit Clifford eigenvalue range, Comp 100/102): e^(-1). # ------------------------------------------------------------------------- def Z_H(beta, D): """Closed-form KO-tempered partition function.""" return ((1.0 + math.exp(-beta / D)) / 2.0) ** D def Z_H_limit(beta): """D -> infty limit.""" return math.exp(-beta / 2.0) # ------------------------------------------------------------------------- # M2: wave-function renormalisation from the substrate two-point function. # # The substrate generating functional with a source J coupled to the # order-parameter field, and the matched-scaling Boltzmann weight # exp(-beta X_bar) supplied by the substrate fluctuations: # # Z[J] = E_mu[ exp(-beta X_bar + J * field) ]. # # The NORMALISATION Z[0] = Z_H(beta) is the multiplicative factor that # dresses the bare quartic. Identifying it as a FIELD-STRENGTH # renormalisation: the projected SM field phi = Z_phi^(1/2) psi is # canonically normalised only after the substrate fluctuations are # integrated with their matched-scaling weight, which rescales the field # normalisation by exactly Z[0]: # # Z_phi^2 = Z[0] / Z[0]_{beta=0} = Z_H(beta_KO) / 1 = Z_H(beta_KO). # # Then with trivial vertex renormalisation (Z_Gamma4 = 1 at tree level, # CC inner fluctuations; Comp 101 sub-question 2 satisfied): # # lambda_SM(M_*) = b * Z_phi^2 = (1/4) * Z_H(beta_KO) -> (1/4) e^(-1). # ------------------------------------------------------------------------- def wavefunction_renormalisation(beta, D): """Z_phi^2 candidate = Z_H(beta) (the field-strength rescaling under the matched-scaling projection). CONDITIONAL on M3 (that the flow generates the weight exp(-beta X_bar)).""" return Z_H(beta, D) def main(): print("=" * 72) print("Computation 110: Bridge Premise (B) attack -- M1 + M2") print("Wave-function renormalisation route (sets up the M3 target)") print("=" * 72) print() BETA_KO = 2 # one-bit Clifford Cl(1,0) eigenvalue range (Comp 100/102) b = 0.25 # lambda_PST, doublet convention # ----- M1: discrete Wilson-Polchinski flow setup ----- print("M1. DISCRETE WILSON-POLCHINSKI FLOW") print("-" * 72) print(" Cutoff f(x) = e^(-x) (Comp 100, forced by P1 independence).") print(" Boolean Laplacian Delta chi_S = 2|S| chi_S.") print(" Matched scaling Lambda^2 = D.") print(" Propagator in mode S (|S| >= 1): G_S = e^(-2|S|/D) / (2|S|).") print() D = 12 print(f" Sample propagators at D = {D}:") print(f" {'|S|':>4} {'eig 2|S|':>9} {'cutoff':>10} {'G_S':>12}") for s in range(0, 6): if s == 0: print(f" {s:>4} {'0':>9} {'1 (zero mode = order parameter)':>30}") else: eig = boolean_laplacian_eigenvalue(s) c = cutoff(eig / D) g = propagator_mode(s, D) print(f" {s:>4} {eig:>9} {c:>10.5f} {g:>12.6f}") print() print(" X_bar (the Higgs Hamiltonian, Comp 89) is supported ONLY on") print(" the lowest two Walsh shells V_0 (+) V_1 (Comp 98):") print(" X_bar = (1/2) chi_emptyset - (1/(2D)) sum_{|S|=1} chi_S.") print(" => integrating out the |S| >= 2 modes is a TRIVIAL fixed-point") print(" step for X_bar (Comp 98): the flow leaves X_bar unchanged") print(" until it reaches the low shells. The relevant field") print(" content at M_* is exactly X_bar.") print() # ----- M2: wave-function renormalisation ----- print("M2. WAVE-FUNCTION RENORMALISATION FROM THE TWO-POINT FUNCTION") print("-" * 72) print(f" Order-parameter field statistics under mu:") print(f" E_mu[X_bar] = {xbar_mean()}") for Dv in (10, 100, 1000): print(f" Var_mu(X_bar) = 1/(4D) = {xbar_variance(Dv):.6e} (D = {Dv})") print() print(" Field-strength renormalisation candidate (CONDITIONAL on M3):") print(" Z_phi^2 = Z[0] = E_mu[exp(-beta_KO X_bar)] = Z_H(beta_KO)") print() print(f" Numerical Z_phi^2 = Z_H(beta_KO = {BETA_KO}):") print(f" {'D':>6} {'Z_phi^2':>12} {'dev from e^-1':>16}") e_inv = math.exp(-1.0) for Dv in (6, 10, 100, 1000, 10000): z = wavefunction_renormalisation(BETA_KO, Dv) dev = 100.0 * (z - e_inv) / e_inv print(f" {Dv:>6} {z:>12.6f} {dev:>15.3f}%") print(f" {'limit':>6} {e_inv:>12.6f} {'0.000':>15}%") print() print(" => Z_phi^2 -> e^(-1) (D -> infty), reproducing the substrate") print(" factor of Comp 100 in the wave-function-renormalisation") print(" reading.") print() # ----- the resulting bridge value ----- print("RESULTING BRIDGE VALUE (conditional on M3)") print("-" * 72) print(" With Z_Gamma4 = 1 (trivial vertex, tree-level CC inner") print(" fluctuations, Comp 101 sub-question 2):") print(f" lambda_SM(M_*) = b * Z_phi^2 = {b} * Z_H({BETA_KO})") for Dv in (100, 10000): lam = b * Z_H(BETA_KO, Dv) print(f" D = {Dv:>5}: {lam:.6f}") lam_lim = b * e_inv print(f" limit: {lam_lim:.6f} (= e^(-1)/4)") print(f" observed lambda_SM(M_*) ~ 0.0927 (Buttazzo)") print(f" ratio: {lam_lim/0.0927:.4f}") print() # ----- honest status: what M1+M2 achieve, what M3 must prove ----- print("=" * 72) print("STATUS: what M1 + M2 achieve, and the remaining crux (M3)") print("=" * 72) print() print(" ACHIEVED (M1, M2):") print(" - The discrete Wilson-Polchinski flow is set up with the") print(" P1-forced cutoff f = exp and matched scaling Lambda^2 = D.") print(" - The flow reduces to the low Walsh shells where X_bar lives") print(" (Comp 98), so the field content at M_* is exactly X_bar.") print(" - The bridge factor is RE-CAST as a field-strength") print(" renormalisation Z_phi^2 (multiplicative by construction),") print(" side-stepping the additive-matching obstruction that") print(" Comp 103 established for the coupling-Jacobian reading.") print(" - Numerically, Z_phi^2 = Z_H(beta_KO) -> e^(-1), reproducing") print(" the Comp 100 value in the new reading.") print() print(" NOT ACHIEVED -- the remaining crux (M3):") print(" The identification Z_phi^2 = E_mu[exp(-beta_KO X_bar)] is so") print(" far IMPOSED (the field is weighted by the substrate Boltzmann") print(" factor). What must be PROVEN is that the Wilson-Polchinski") print(" flow GENERATES this weight as its field-strength kernel --") print(" i.e. that integrating the substrate fluctuations from") print(" Lambda = sqrt(D) down to M_* produces exactly the multiplier") print(" exp(-beta_KO X_bar) on the order-parameter field, with") print(" beta_KO = 2 inherited from the one-bit Clifford range.") print() print(" Until M3 is proven, this is the SAME gap as (B) in a sharper") print(" form: 'derive the multiplicative bridge factor' has become") print(" 'show the Polchinski field-strength kernel is the KO-tempered") print(" Boltzmann weight'. That is a concrete, well-posed RG-flow") print(" computation -- the natural next milestone -- and it either") print(" closes (B) or yields a Comp-103-style sharp obstruction.") print() print(" This computation does NOT close Bridge Premise (B).") if __name__ == "__main__": main()