#!/usr/bin/env python3 """ PROVENANCE: PROOF Computation 113 -- Bridge Premise (B), milestone M4(b): the Z_phi <-> matched-scaling trace identification, attempted ab initio on the discrete substrate ========================================================================= STATUS (v26.13): M4(b) of the wave-function attack on Bridge Premise (B). M3 (Comp 111) reduced (B) to two shared inputs; M4(a) (Comp 112) closed the first (beta_KO = 2 is the binary-alphabet gap). This computation attacks the second: prove that the order-parameter field-strength renormalisation Z_phi^2 EQUALS the matched-scaling heat-kernel object e^(-1), ab initio on the lattice -- rather than asserting it via the standard QFT relation. THE TARGET ========== Bridge factor: lambda_SM = b * Z_phi^2, with Z_phi^2 =? e^(-1). On the substrate side, e^(-1) appears as the matched-scaling object (1/2^D) Tr exp(-Delta/D) = ((1+e^(-2/D))/2)^D -> e^(-1) (Comp 100) = E_mu[exp(-2 X_bar)] (Comp 111) i.e. a VACUUM AMPLITUDE (a partition function / trace over ALL substrate modes). The question M4(b) must settle: is the FIELD-STRENGTH of the order-parameter Higgs equal to this vacuum amplitude? THE TENSION THIS COMPUTATION SURFACES (the honest core) ======================================================= A field-strength renormalisation and a vacuum amplitude are a priori DIFFERENT objects: - Z_phi (field-strength) is a normalisation of ONE field's two-point function -- the residue of the order-parameter propagator. - (1/2^D) Tr exp(-Delta/D) is a vacuum amplitude -- a trace/product over ALL D modes. Computing the order-parameter two-point residue directly gives a SINGLE-MODE factor (the |S|=1 shell where X_bar lives, Comp 98), which is NOT the full product e^(-1) unless every substrate mode is folded in. So the identification holds IFF the substrate->Higgs projection Pi is the TOTAL reduction: all 2^D substrate configurations collapse onto the one surviving order parameter X_bar, so the Higgs field-strength absorbs the FULL substrate vacuum amplitude. This computation makes that precise and tests it numerically; it does NOT prove Pi is the total reduction (that is the residual M4(b) isolates). ========================================================================= """ import math import itertools def xbar(C): return sum(C) / len(C) def all_configs(D): return itertools.product((0, 1), repeat=D) def laplacian_eigenvalue(S_size): """Boolean Laplacian eigenvalue on the |S|-Walsh shell: 2|S|.""" return 2.0 * S_size def vacuum_amplitude(D): """(1/2^D) Tr exp(-Delta/D) = product over ALL modes -> e^-1.""" # Tr exp(-Delta/D) = sum_S exp(-2|S|/D) = (1 + e^{-2/D})^D return ((1.0 + math.exp(-2.0 / D)) / 2.0) ** D def single_mode_residue(D, s): """ Field-strength of the order parameter from its OWN two-point function. X_bar - 1/2 = -(1/2D) sum_a sigma_a lives on the |S|=1 shell (eigenvalue 2). Heat-kernel smoothing exp(-s Delta) scales each sigma_a by e^{-2s}, so the order-parameter field-strength from the single shell is e^{-2s}. (One factor, not a product over D modes.) """ return math.exp(-2.0 * s) def main(): print("=" * 72) print("Computation 113: M4(b) -- Z_phi^2 vs matched-scaling trace") print("=" * 72) print() e_inv = math.exp(-1.0) # ---- 1. the two candidate objects are numerically different ---- print("1. TWO CANDIDATE OBJECTS FOR Z_phi^2 -- AND THEY DIFFER") print("-" * 72) print(" (a) vacuum amplitude (trace over ALL modes) = e^-1.") print(" (b) single-mode residue of the order parameter = e^{-2s}.") print() print(f" vacuum amplitude (1/2^D)Tr exp(-Delta/D):") for D in (100, 1000, 10000): print(f" D={D:>6}: {vacuum_amplitude(D):.6f} (-> {e_inv:.6f})") print() print(f" single-mode residue e^{{-2s}} at various heat-times s:") for s in (0.10, 0.25, 0.50, 1.00): r = single_mode_residue(0, s) hit = "= e^-1" if abs(r - e_inv) < 1e-9 else "" print(f" s={s:>4.2f}: {r:.6f} {hit}") print() print(" The single-mode residue equals e^-1 ONLY at the tuned value") print(" s = 1/2; at generic s it does not. So the order-parameter") print(" two-point residue, by itself, does NOT reproduce e^-1 -- the") print(" vacuum amplitude (all modes) is a genuinely different object.") print() # ---- 2. they reconcile iff Pi is the TOTAL reduction ---- print("2. RECONCILIATION: Pi as the TOTAL substrate->Higgs reduction") print("-" * 72) print(" The Higgs is the UNIQUE surviving order parameter: every") print(" substrate degree of freedom is integrated out onto X_bar.") print(" Then the Higgs field-strength is the full fluctuation") print(" normalisation = the vacuum amplitude over ALL D modes:") print() print(" Z_phi^2 = (1/2^D) Tr exp(-Delta/D) = E_mu[exp(-2 X_bar)]") print(" -> e^-1.") print() print(" Verify the two vacuum-amplitude forms coincide exactly") print(" (trace-over-modes == tempered expectation), small D:") print(f" {'D':>4} {'(1/2^D)Tr e^{-D.L/D}':>22} {'E_mu[e^{-2 Xbar}]':>20}") for D in (2, 4, 6, 8): trace = 0.0 for k in range(D + 1): trace += math.comb(D, k) * math.exp(-laplacian_eigenvalue(k) / D) trace /= 2 ** D emu = sum(math.exp(-2.0 * xbar(C)) for C in all_configs(D)) / 2 ** D print(f" {D:>4} {trace:>22.8f} {emu:>20.8f}") print() print(" Exact agreement: the trace over all Walsh shells IS the") print(" tempered expectation of the order parameter (X_bar = |C|/D,") print(" Laplacian eigenvalue 2|S| with binomial multiplicity mirror") print(" the bit-sum). So IF Pi folds in all modes, Z_phi^2 = e^-1") print(" is exact, not approximate.") print() # ---- 3. honest assessment ---- print("=" * 72) print("ASSESSMENT: how far M4(b) carries Bridge Premise (B)") print("=" * 72) print() print(" ESTABLISHED:") print(" - The matched-scaling trace and the tempered order-parameter") print(" expectation are EXACTLY the same number (verified, part 2).") print(" So the only question is WHICH object is the Higgs") print(" field-strength -- and a total-reduction Pi makes it the") print(" vacuum amplitude = e^-1, exactly.") print(" - The naive single-field two-point residue does NOT give e^-1") print(" at generic heat-time (part 1): the identification genuinely") print(" requires the FULL mode content, i.e. a total reduction.") print() print(" NOT ESTABLISHED (the residual M4(b) isolates):") print(" - That Pi IS the total substrate->Higgs reduction -- i.e.") print(" that the single surviving order parameter's field-strength") print(" absorbs the entire substrate vacuum amplitude. This is") print(" plausible (the Higgs is the unique order parameter past the") print(" LG threshold, P3) and consistent with Comp 98 (all |S|>=2") print(" modes are trivial fixed points integrated out), but it is") print(" a STRUCTURAL PROPERTY OF Pi, asserted, not proven on the") print(" lattice.") print() print(" NET RESULT:") print(" M4(b) does NOT close (B) ab initio. It reduces the residual to") print(" ONE precise, defensible structural statement:") print() print(" (Pi-total) the substrate->Higgs projection Pi is the total") print(" reduction: the field-strength of the unique") print(" surviving order parameter X_bar equals the full") print(" substrate vacuum amplitude (1/2^D)Tr exp(-Delta/D).") print() print(" This is strictly cleaner than the original residual ('the") print(" multiplicative partition-function CC correspondence is a") print(" structural hypothesis'): (Pi-total) is a single, concrete,") print(" falsifiable property of one map, and Comp 98 already supplies") print(" the mechanism (all |S|>=2 modes are trivial fixed points, so") print(" only the order parameter survives -- consistent with total") print(" reduction). Proving (Pi-total) is the genuinely-remaining") print(" mathematics; it is no longer a vague correspondence.") if __name__ == "__main__": main()