#!/usr/bin/env python3 """ PROVENANCE: PROOF Computation 121 -- Bridge Premise (B): the all-orders decoupling closed EXACTLY by sufficiency (subsumes the Morse-Bott lemma) ========================================================================= STATUS (v26.29+): closes the one residual rigour task carried by F11 / Comp 118 (the all-orders S_D-equivariant decoupling), and answers the v26.28 referee's M2(i): "extending zero-matrix-element from the quadratic Hessian to the quartic 1PI vertex is a genuinely higher-order statement than F11 as proved." The single open content of (B) left after this -- the SM-side matching -- is quantified by Comp 123: a rigid, D-independent 1.0-sigma test (lambda_SM(M_*) = 0.0927 +- 0.0007 m_t/m_h-dominated vs e^-1/4 = 0.09197). THE POINT ========= F8 (Comp 89) makes the substrate Higgs Hamiltonian EXACTLY the linear collective coordinate, H_Higgs(C) = X_bar(C) = (1/D) sum_a B_a(C), B_a i.i.d. (P1), with NO free parameters. So the interaction F_int = Phi(X_bar) is a function of the EMPIRICAL MEAN of i.i.d. per-site variables -- i.e. a function of a SUFFICIENT STATISTIC for the product measure. By the Fisher-Neyman factorisation, the configurations sharing a given mean X_bar = k/D are counted by the multiplicity (the binomial coefficient C(D,k)), and that multiplicity is INDEPENDENT of Phi. Hence the substrate partition function factorises EXACTLY, at every finite D: Z[Phi] = E_mu[ exp(-Phi(X_bar)) ] = sum_{k=0}^{D} ( C(D,k) / 2^D ) exp(-Phi(k/D)). (*) The transverse (non-collective) modes are EXACTLY the C(D,k) configs at fixed mean; integrating them out returns the Phi-independent factor C(D,k)/2^D. Therefore: - NO transverse mode contributes a Phi-dependent vertex at ANY order (Z_Gamma4 = 1 to all orders, and the same for every n-point) -- EXACTLY, at finite D, not as a Gaussian-order + all-ones-tensor statement. This is the answer to M2(i): the "higher-order" worry dissolves because (*) is an exact all-orders factorisation. - The field-strength piece is the Phi-independent marginal of X_bar, whose D->inf de Moivre-Laplace normalisation gives e^-1 (= the Cramer-Bernoulli value F9 already uses). The only D->inf limit is the VALUE of the 1-D collective integral (*), with the standard O(1/D) correction to the saddle -- the substrate-limit caveat every PST quantity carries. The DECOUPLING (the structure) is exact at all D. - The all-orders S_D-equivariant Morse-Bott lemma (Comp 118) is the Gaussian (de Moivre-Laplace) shadow of (*): the all-ones derivative tensor that annihilates the standard rep W is the leading term of the exact sufficiency factorisation. Comp 118 (perturbative) is subsumed. This computation verifies (*), the exact vertex triviality, the isolation of O(1/D) to the value, and the Morse-Bott reduction, and it runs a CONTROL (an interaction that depends on a transverse mode) to show the vertex test has teeth. ========================================================================= """ import math from itertools import product import numpy as np # --------------------------------------------------------------------------- def exact_partition_by_enumeration(D, weight): """Z = (1/2^D) sum over all 2^D configs of weight(config). Brute force.""" Z = 0.0 for C in product((0, 1), repeat=D): Z += weight(np.array(C)) return Z / 2**D def multiplicity_factorised(D, phi): """RHS of (*): sum_k (C(D,k)/2^D) exp(-phi(k/D)).""" Z = 0.0 for k in range(D + 1): Z += math.comb(D, k) / 2**D * math.exp(-phi(k / D)) return Z def transverse_multiplicity_is_phi_independent(D): """Show that the count of configs at fixed mean k/D is C(D,k), independent of any interaction. Returns max deviation over k.""" counts = [0] * (D + 1) for C in product((0, 1), repeat=D): counts[int(sum(C))] += 1 binom = [math.comb(D, k) for k in range(D + 1)] return max(abs(counts[k] - binom[k]) for k in range(D + 1)) # --------------------------------------------------------------------------- def collective_effective_potential(D, phi): """U(k) = phi(k/D) - log( C(D,k)/2^D ): the effective potential for the collective coordinate after EXACTLY integrating out the transverse modes. The second term is the transverse (multiplicity) contribution.""" U = np.zeros(D + 1) for k in range(D + 1): U[k] = phi(k / D) - math.log(math.comb(D, k) / 2**D) return U def main(): print("=" * 72) print("Computation 121: all-orders decoupling closed EXACTLY by") print("sufficiency (Fisher-Neyman); Morse-Bott (Comp 118) subsumed") print("=" * 72) print() # ---- 1. the exact factorisation (*) and Phi-independent multiplicity ---- print("1. EXACT FACTORISATION Z[Phi] = sum_k (C(D,k)/2^D) e^{-Phi(k/D)}") print("-" * 72) # try several non-linear interactions Phi(x) to stress the identity phis = { "linear -2x": lambda x: -2.0 * x, "quartic 5x^4": lambda x: 5.0 * x**4, "mixed": lambda x: -2.0 * x + 3.0 * x**2 - 4.0 * x**3 + 5.0 * x**4, } for D in (6, 10, 14): print(f" D={D}:") # multiplicity equals binomial, independent of any interaction dev = transverse_multiplicity_is_phi_independent(D) print(f" transverse count at fixed mean == C(D,k)? " f"max|count-binom| = {dev} (Phi-independent multiplicity)") for name, phi in phis.items(): Z_enum = exact_partition_by_enumeration( D, lambda C: math.exp(-phi(C.mean()))) Z_fact = multiplicity_factorised(D, phi) rel = abs(Z_enum - Z_fact) / abs(Z_enum) print(f" Phi={name:>16}: Z_enum={Z_enum:.10f} " f"Z_fact={Z_fact:.10f} rel.diff={rel:.1e}") print() print(" (*) holds to machine precision for every interaction: the") print(" transverse modes enter ONLY through the Phi-independent") print(" multiplicity C(D,k). X_bar is a sufficient statistic.") print() # ---- 2. Z_Gamma4 = 1 EXACTLY: no Phi-dependent transverse vertex -------- print("2. VERTEX TRIVIALITY Z_Gamma4 = 1 EXACT AT FINITE D") print("-" * 72) print(" Effective potential U_Phi(k) = Phi(k/D) - log(C(D,k)/2^D).") print(" Add a quartic probe Phi -> Phi + delta*(x-1/2)^4 and ask") print(" whether the transverse term renormalises the added vertex.") D = 16 base = lambda x: -2.0 * x delta = 0.37 probe = lambda x: delta * (x - 0.5) ** 4 U0 = collective_effective_potential(D, base) U1 = collective_effective_potential(D, lambda x: base(x) + probe(x)) diff = U1 - U0 # the difference must equal the bare probe EXACTLY (transverse term cancels) xs = np.array([k / D for k in range(D + 1)]) bare = delta * (xs - 0.5) ** 4 max_resid = np.max(np.abs(diff - bare)) print(f" max | (U_[Phi+probe] - U_Phi) - bare probe | = {max_resid:.2e}") print(" -> the transverse (multiplicity) term is bit-for-bit identical") print(" under Phi -> Phi+probe, so it cannot renormalise the added") print(" quartic. Z_Gamma4 = 1 exactly, at finite D, ALL orders.") print() # CONTROL: an interaction that depends on a transverse mode DOES get a # vertex correction (the test has teeth). print(" CONTROL (teeth): interaction depending on a TRANSVERSE mode") print(" w = C_0 - C_1 (sum_a w_a = 0, a W-direction). Then the") print(" transverse integration is NOT Phi-independent:") Dc = 12 g = 0.9 # Z with transverse-coupled weight exp(-g*(C0 - C1)^2), enumerated, and the # naive 'multiplicity-only' factorisation that wrongly ignores it: Z_true = exact_partition_by_enumeration( Dc, lambda C: math.exp(-g * (C[0] - C[1]) ** 2)) Z_naive_collective = multiplicity_factorised(Dc, lambda x: 0.0) # = 1 print(f" Z_true (transverse-coupled) = {Z_true:.6f}") print(f" collective-only factorisation = {Z_naive_collective:.6f}") print(f" gap = {abs(Z_true - Z_naive_collective):.6f} " f"-> transverse coupling leaves a residue; sufficiency BROKEN,") print(" exactly when the interaction is NOT a function of X_bar.") print(" (Confirms: Z_Gamma4 = 1 is a consequence of F8 -- interaction") print(" = function of the sufficient statistic X_bar -- not generic.)") print() # ---- 3. O(1/D) is in the VALUE, not the decoupling ---------------------- print("3. O(1/D) LIVES IN THE VALUE Z -> e^-1, NOT IN THE STRUCTURE") print("-" * 72) print(" Decoupling (*) is exact at every D (parts 1-2: zero residual).") print(" The VALUE of the collective integral approaches e^-1 with the") print(" standard Cramer-Bernoulli / de Moivre-Laplace O(1/D) tail:") e_inv = math.exp(-1.0) print(f" {'D':>6} {'Z=E[e^{-2 Xbar}]':>18} {'Z - e^-1':>14} " f"{'(Z-e^-1)*D':>12}") for D in (10, 50, 200, 1000, 5000): Z = ((1.0 + math.exp(-2.0 / D)) / 2.0) ** D # = E[exp(-2 Xbar)] print(f" {D:>6} {Z:>18.10f} {Z - e_inv:>14.2e} " f"{(Z - e_inv) * D:>12.4f}") print(" (Z - e^-1)*D -> const: the correction is O(1/D), in the VALUE") print(" only. The structural decoupling carries no D-dependence.") print() # ---- 4. Morse-Bott (Comp 118) is the Gaussian shadow of (*) ------------- print("4. COMP 118 (MORSE-BOTT) IS THE de MOIVRE-LAPLACE SHADOW OF (*)") print("-" * 72) print(" Gaussian-approximate the multiplicity: C(D,k)/2^D ~=") print(" N(mean=1/2, var=1/(4D)) in m=k/D. The transverse Gaussian") print(" fluctuation IS the W-sector Gaussian integral of Comp 118;") print(" the all-ones derivative tensor Phi^(n)/D^n annihilating W is") print(" the leading term of the exact sufficiency factorisation (*).") D = 40 var = 1.0 / (4 * D) ks = np.arange(D + 1) exact = np.array([math.comb(D, k) / 2**D for k in ks]) m = ks / D gauss = (1.0 / D) * np.exp(-(m - 0.5) ** 2 / (2 * var)) / math.sqrt( 2 * math.pi * var) # compare on the bulk (within 3 sigma) sig = math.sqrt(var) bulk = np.abs(m - 0.5) < 3 * sig rel = np.max(np.abs(exact[bulk] - gauss[bulk]) / exact[bulk]) print(f" D={D}: max rel.dev (multiplicity vs Gaussian, |m-1/2|<3sigma)") print(f" = {rel:.3f} -> Comp 118's Gaussian/Hessian picture is") print(" the leading approximation; (*) is the exact parent.") print() # ---- 5. assessment ------------------------------------------------------ print("=" * 72) print("ASSESSMENT: the residual all-orders rigour task is CLOSED") print("=" * 72) print() print(" Given F8 (H_Higgs = X_bar exactly, the empirical mean of i.i.d.") print(" bits), the interaction is a function of a SUFFICIENT STATISTIC.") print(" Fisher-Neyman => the transverse modes factor out as the") print(" Phi-independent multiplicity C(D,k), exactly, at every finite D") print(" and every order (eq. *). Therefore:") print(" - Z_Gamma4 = 1 to all orders, EXACTLY (part 2) -- this answers") print(" the v26.28 M2(i) 'higher-order' worry: there is no") print(" higher-order gap, the factorisation is exact at all orders") print(" simultaneously, not Gaussian-order + numerics;") print(" - the field-strength is the Phi-independent marginal of X_bar,") print(" its de Moivre-Laplace normalisation -> e^-1 (the same") print(" Cramer-Bernoulli value F9 uses), O(1/D) in the VALUE only;") print(" - Comp 118's all-orders S_D-equivariant Morse-Bott lemma is the") print(" Gaussian shadow of (*) and is subsumed.") print() print(" NET: F11's 'residual all-orders rigour task' is closed by an") print(" EXACT, non-perturbative, combinatorial identity (sufficiency of") print(" the empirical mean). The substrate side of (B) no longer carries") print(" a perturbative all-orders caveat -- only the O(1/D) substrate-") print(" limit on the VALUE e^-1, shared by every PST result. The single") print(" open content of (B) remains the SM-side matching (F4 + RGE).") if __name__ == "__main__": main()