#!/usr/bin/env python3 """ Computation 74 -- Chamseddine-Connes inner-fluctuation Yukawa for the PST substrate triple ============================================================================================ First-cut attempt at the Chamseddine-Connes inner-fluctuation calculation identified as the remaining structural content of the Z^2 = e^{-1} conjecture (Computation 72, eq.~\eqref{eq:y-fixed}). CONTEXT ------- Comp 72 reduced the Z^2 conjecture to a single scalar condition: the substrate-Higgs Yukawa coupling at matched scaling Lambda = sqrt(D) must satisfy y_substrate / Lambda = mu_site^{1/4} = (1/2)^{1/4} = 2^{-1/4} ~ 0.8409. Comp 72 left open whether the Chamseddine-Connes inner-fluctuation calculation D -> D + A + epsilon' J A J^{-1} delivers this specific value when A is built from elements of the quaternion factor H subset A_F = C (+) H (+) M_3(C). THIS COMPUTATION ---------------- Performs the inner-fluctuation calculation for an explicit and natural Higgs-perturbation parameterisation, computes the resulting effective Yukawa coupling y_eff, and compares against the Comp 72 target. SETUP ----- Substrate Hilbert space H_sub = C^{2^D} carrying the Clifford algebra Cl(0, D) via Jordan-Wigner generators e_a = sigma_z^{(x)(a-1)} (x) sigma_x (x) I^{(x)(D-a)}, a = 1, ..., D so that {e_a, e_b} = 2 delta_{ab} I. The substrate Dirac is D_sub = sum_{a=1}^D e_a, D_sub^2 = D * I, with eigenvalues +/- sqrt(D) each of multiplicity 2^{D-1}. The quaternion factor H subset A_F embeds naturally into Cl(0, 2) subset Cl(0, D): H = span{I, e_1, e_2, e_1 e_2}, with the volume element omega = e_1 e_2 satisfying omega^2 = -I (the quaternion imaginary unit). NATURAL HIGGS PERTURBATION -------------------------- For a real scalar Higgs field phi, the natural inner-fluctuation parameterisation is A = i phi * [D_sub, omega] / 2 (the factor /2 is the standard Connes-Chamseddine normalisation). Compute [D_sub, omega]: [D_sub, omega] = sum_a [e_a, e_1 e_2] = [e_1, e_1 e_2] + [e_2, e_1 e_2] + sum_{a>=3} [e_a, e_1 e_2] = (anticomm computations below) For a >= 3: e_a anticommutes with both e_1 and e_2, so e_a commutes with the PRODUCT e_1 e_2 (two sign flips); [e_a, e_1 e_2] = 0. For a = 1: [e_1, e_1 e_2] = e_1 e_1 e_2 - e_1 e_2 e_1 = e_2 - (-e_1^2 e_2) = e_2 + e_2 = 2 e_2. For a = 2: [e_2, e_1 e_2] = e_2 e_1 e_2 - e_1 e_2 e_2 = -e_1 - e_1 = -2 e_1. So [D_sub, omega] = 2 e_2 - 2 e_1 = -2 (e_1 - e_2). And A = i phi * (-2)(e_1 - e_2) / 2 = -i phi (e_1 - e_2). PERTURBED DIRAC AND ITS SQUARE ------------------------------ D_phi = D_sub + A = sum_a e_a - i phi (e_1 - e_2) = (1 - i phi) e_1 + (1 + i phi) e_2 + sum_{a>=3} e_a. (The J A J^{-1} term contributes additively under KO-6 reality with epsilon' = +1; this script computes the A contribution alone and notes the doubling factor where relevant.) D_phi^2 = sum_a (coeff_a)^2 + sum_{a != b} coeff_a coeff_b {e_a, e_b}. With {e_a, e_b} = 2 delta_{ab} I for a != b (vanishing), we get D_phi^2 = |1 - i phi|^2 + |1 + i phi|^2 + (D - 2) = (1 + phi^2) + (1 + phi^2) + (D - 2) = D + 2 phi^2. [Note: this is real and proportional to identity, so eigenvalues of D_phi are +/- sqrt(D + 2 phi^2), each with multiplicity 2^{D-1}.] SPECTRAL ACTION --------------- At matched scaling Lambda = sqrt(D), the perturbed eigenvalues are lambda_+/-(phi) / Lambda = +/- sqrt(1 + 2 phi^2 / D). With Gaussian cutoff f(x) = exp(-x^2), f(lambda / Lambda) = exp(-(1 + 2 phi^2 / D)) = exp(-1) * exp(-2 phi^2 / D). Total spectral action / 2^D: S(D_phi, Lambda, f) / 2^D = exp(-1) * exp(-2 phi^2 / D) = exp(-1) * [1 - 2 phi^2 / D + 2 phi^4 / D^2 - ...] The phi^4 coefficient is 2 exp(-1) / D^2. COMPARE TO COMP 72 ------------------ Comp 72 predicted phi^4 coefficient = exp(-1) * y^4 / (2 Lambda^4) = exp(-1) * y^4 / (2 D^2). Equating: y^4 / (2 D^2) = 2 / D^2 => y^4 = 4 => y = sqrt(2). So this Higgs perturbation gives y = sqrt(2) at ALL matched scalings, D-INDEPENDENT (in absolute terms, not as y/Lambda). y / Lambda = sqrt(2) / sqrt(D). For D = 2: y/Lambda = 1. For D = 4: y/Lambda = 0.7071. For D = 6: y/Lambda = 0.5774. The Comp 72 target was y/Lambda = 2^{-1/4} = 0.8409 (D-independent). CONCLUSION ---------- The natural Higgs inner-fluctuation A = i phi [D_sub, omega]/2 does NOT deliver the Comp 72 target. It gives y = sqrt(2) at ALL D (in absolute units), or y/Lambda = sqrt(2/D) (decreasing in D). The Comp 72 target was D-independent y/Lambda = 2^{-1/4}. This is an HONEST partial answer: the specific Higgs perturbation tested here does not close Z^2 = e^{-1}. At D = 4 (the only D at which sqrt(2)/sqrt(D) and 2^{-1/4} agree to 16%), there is approximate agreement; at other D the calibration breaks. Three plausible explanations for the discrepancy: (1) The candidate identification Z^2 = e^{-1} is genuinely a 0.8% numerical near-coincidence, not a structural identity. The substrate-side S_sub/2^D = e^{-1} is exact (Comp 62) independently of any Higgs sector; the Z^2 SM-side value depends on RGE truncations and is naturally near e^{-1} without requiring structural identification. (2) The Higgs perturbation should be parameterised differently, e.g. via a SUM over multiple quaternion generators, or via the full A_F = C (+) H (+) M_3(C) inner-fluctuation (not just H), or with a specific Hermiticity / Reality / order-one constraint that fixes y/Lambda = 2^{-1/4} structurally. This would require working through the full Chamseddine-Connes-Marcolli machinery. (3) The matched scaling Lambda = sqrt(D) is not the right cutoff identification; a different Lambda(D) might deliver the target. REMAINING WORK -------------- The cleanest next step is to compute the y value under the FULL CC inner-fluctuation including the J A J^{-1} term and the order-one constraint on a chosen H_F representation. This is a multi-day calculation that would either: - deliver y = Lambda * 2^{-1/4} exactly (closing Z^2 = e^{-1}), or - confirm the discrepancy structurally (retiring the candidate identification as a genuine 0.8% near-coincidence). """ from __future__ import annotations import math import numpy as np def jordan_wigner(D: int): """Returns the Hermitian Jordan-Wigner generators e_1, ..., e_D on the substrate Hilbert space C^{2^D}. {e_a, e_b} = 2 delta_{ab} I; e_a^dagger = e_a; e_a^2 = I. """ sigma_x = np.array([[0, 1], [1, 0]], dtype=complex) sigma_z = np.array([[1, 0], [0, -1]], dtype=complex) I = np.eye(2, dtype=complex) generators = [] for a in range(1, D + 1): # e_a = sigma_z (x) ... (x) sigma_z (x) sigma_x (x) I (x) ... (x) I op = np.array([[1.0]], dtype=complex) for k in range(1, D + 1): if k < a: op = np.kron(op, sigma_z) elif k == a: op = np.kron(op, sigma_x) else: op = np.kron(op, I) generators.append(op) return generators def main(): print("=" * 100) print(" Computation 74 -- CC inner-fluctuation Yukawa for the PST substrate triple") print("=" * 100) print() print("SETUP: substrate Dirac D_sub = sum_a e_a, Cl(0, D) Jordan-Wigner generators.") print() for D in [2, 4, 6, 8]: gens = jordan_wigner(D) # Verify anticommutation ok = True for a in range(D): for b in range(D): anticomm = gens[a] @ gens[b] + gens[b] @ gens[a] expected = 2 * (1.0 if a == b else 0.0) * np.eye(2 ** D, dtype=complex) if not np.allclose(anticomm, expected): ok = False # Build D_sub and verify D_sub^2 = D * I D_sub = sum(gens) Dsub_sq = D_sub @ D_sub ok2 = np.allclose(Dsub_sq, D * np.eye(2 ** D, dtype=complex)) eigenvalues = np.linalg.eigvalsh(D_sub) eigenvalues_unique = sorted(set(round(float(np.real(v)), 6) for v in eigenvalues)) Lambda = math.sqrt(D) print(f" D = {D}: anticomm OK = {ok}; D_sub^2 = D I OK = {ok2}; " f"eigenvalues = {eigenvalues_unique}; matched Lambda = sqrt({D}) = {Lambda:.4f}") print() print("HIGGS PERTURBATION A = i phi [D_sub, omega] / 2, omega = e_1 e_2.") print("-" * 100) print() print(f" {'D':>4} {'phi^4 coeff':>14} {'y from Comp 72':>14} " f"{'y/Lambda':>10} {'target 2^(-1/4)':>16} {'ratio':>10}") target = 2 ** (-0.25) for D in [2, 4, 6, 8]: # phi^4 coefficient (from analytic derivation in docstring): # exp(-1) * 2 / D^2 phi4_coeff = math.exp(-1) * 2 / D ** 2 # Comp 72: phi^4 coefficient = exp(-1) * y^4 / (2 * Lambda^4) at Lambda = sqrt(D) # So y^4 = 2 * D^2 * phi4_coeff / exp(-1) = 4 y4 = 2 * D ** 2 * phi4_coeff / math.exp(-1) y = y4 ** 0.25 Lambda = math.sqrt(D) y_over_L = y / Lambda print(f" {D:>4d} {phi4_coeff:>14.6e} {y:>14.6f} " f"{y_over_L:>10.4f} {target:>16.4f} {y_over_L / target:>10.4f}") print() print("OBSERVATIONS") print("=" * 100) print() print(" - y = sqrt(2) at every D (D-INDEPENDENT in absolute terms).") print(" - y / Lambda = sqrt(2/D) DECREASES with D.") print(f" - Comp 72 target y / Lambda = 2^(-1/4) ~= {target:.4f} is D-INDEPENDENT.") print(" - The two D-dependences disagree: this specific Higgs perturbation") print(" does NOT deliver the Comp 72 target.") print() print(" At D = 4: y/Lambda = sqrt(2)/2 ~= 0.7071, vs target 0.8409, ratio ~= 0.841.") print(" At D = 8: y/Lambda = sqrt(2)/sqrt(8) = 0.5, vs target 0.8409, ratio ~= 0.595.") print() print("INTERPRETATION") print("=" * 100) print() print(" The naive single-generator Higgs inner fluctuation does not close") print(" Z^2 = e^{-1}. Three possible resolutions:") print() print(" (1) Z^2 ~ e^{-1} is a genuine 0.8% numerical near-coincidence") print(" on the SM side, with no structural identification on the") print(" substrate side. (The substrate-side S_sub/2^D = e^{-1} is") print(" still exact from Comp 62; that exactness does not require") print(" a Z^2 identification.) This is the conservative reading.") print() print(" (2) The Higgs perturbation should be parameterised differently,") print(" e.g. via a SUM over A_F generators or via the FULL") print(" Chamseddine-Connes inner fluctuation including J A J^{-1}") print(" and the order-one constraint. The single-generator A used") print(" here is the SIMPLEST natural choice but is not necessarily") print(" the one CC machinery actually selects.") print() print(" (3) The matched scaling Lambda = sqrt(D) might not be the") print(" right cutoff identification for the Z^2 question; a") print(" different Lambda(D) prescription would deliver y/Lambda") print(" = 2^(-1/4) for the same Higgs perturbation.") print() print(" Honest reading: this computation does NOT close Z^2 = e^{-1}.") print(" It tells us which specific calculation needs to be done next: the") print(" full CC machinery with J reality + order-one constraint, on the") print(" PST substrate triple. Until that calculation is done, Z^2 ~ e^{-1}") print(" remains a numerical near-coincidence with a substrate-side anchor") print(" (S_sub/2^D = e^{-1} exact) but no structural Yukawa-level identity.") if __name__ == "__main__": main()