#!/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 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()