#!/usr/bin/env python3 """ Computation 72 -- Substrate spectral action: Higgs-quartic Taylor expansion ============================================================================ Attempts to close the Z^2 = e^{-1} conjecture (Computation 71) by directly computing the Higgs-quartic coefficient in the Chamseddine-Connes spectral action of the PST substrate triple at matched scaling Lambda = sqrt(D), under explicit Higgs-perturbation models. CONTEXT ------- Comp 71 proposed: Z^2 := lambda_SM(M_*) / b(M_*) = S_sub / 2^D = e^{-1} EXACTLY, with the structural gap being the spectral-action Higgs-quartic identity (Conjecture in Comp 71 docstring). This computation carries out the explicit symbolic Taylor expansion of the substrate spectral action under a Higgs perturbation, extracts the phi^4 coefficient at matched scaling, and identifies what fixing the substrate-Higgs inner-fluctuation structure would close. SETUP ----- Substrate spectral triple: H = C^{2^D}, eigenvalues of D_sub are +/- sqrt(D), multiplicity 2^(D-1) each. At matched scaling Lambda = sqrt(D): D_sub / Lambda has eigenvalues +/- 1 (at the Gaussian cutoff edge). Higgs perturbation via Yukawa-like coupling y*phi: D_phi = D_sub + y * phi * Sigma_X where Sigma_X is the chirality-flip operator on the substrate spinor. Without loss of generality reduce to a single +/- mode pair: D_phi(2x2) = [[Lambda, y*phi], [y*phi, -Lambda]] Eigenvalues: lambda_+/-(phi) = +/- sqrt(Lambda^2 + y^2 phi^2). Bosonic spectral action with Gaussian cutoff f(x) = exp(-x^2): S(D_phi, Lambda, f) = Tr f(D_phi / Lambda) = 2 * 2^(D-1) * f(sqrt(1 + y^2 phi^2 / Lambda^2)) = 2^D * f(sqrt(1 + alpha)) where alpha := y^2 phi^2 / Lambda^2 The Higgs effective potential at one loop is V_eff(phi) = S(D_phi, Lambda, f). GOAL ---- Expand V_eff(phi) in powers of phi and extract the coefficient of phi^4 (the candidate Higgs-quartic coupling at the matched scaling). Compare against the predicted b(M_*) * (S_sub / 2^D) = (1/4) * e^{-1}. """ from __future__ import annotations import math import sympy as sp def derive_higgs_quartic(): """Symbolically derive the phi^4 coefficient of the substrate spectral action.""" phi, y, Lambda = sp.symbols("phi y Lambda", real=True, positive=True) D = sp.symbols("D", integer=True, positive=True) # Eigenvalue magnitudes of D_phi at the (+/-) mode pair lam_squared = Lambda ** 2 + y ** 2 * phi ** 2 lam_over_Lambda = sp.sqrt(lam_squared) / Lambda # = sqrt(1 + y^2 phi^2 / Lambda^2) # Gaussian cutoff f(x) = exp(-x^2), apply at x = lam_over_Lambda f_val = sp.exp(-lam_over_Lambda ** 2) # f(sqrt(1 + alpha)) = f(1) * exp(-alpha) = e^{-1} exp(-y^2 phi^2 / Lambda^2) # equivalently: f(x)^2 evaluated at lambda_squared/Lambda^2 # Per mode-pair the spectral action contribution is f(lam_+/Lambda) + f(lam_-/Lambda) # = 2 * f(lam_over_Lambda) (since f even) per_pair = 2 * f_val # Total spectral action: 2^(D-1) mode pairs, so S = 2^(D-1) * per_pair = 2^D * f_val S_total = 2 ** D * f_val # Taylor-expand S_total in phi S_taylor = sp.series(S_total, phi, 0, 6).removeO() print("Per-mode spectral-action density f(D_phi/Lambda):") print(f" f(D_phi/Lambda) = exp(-1) * exp(-y^2 phi^2 / Lambda^2)") print(f" = exp(-1) * [1 - (y^2/Lambda^2) phi^2") print(f" + (1/2)(y^4/Lambda^4) phi^4 + O(phi^6)]") print() # Extract coefficient of phi^4 in S_total / 2^D s_density = S_total / 2 ** D # = exp(-1) * exp(-y^2 phi^2 / Lambda^2) s_density_taylor = sp.series(s_density, phi, 0, 6).removeO() print(f"S_total / 2^D Taylor expansion in phi:") print(f" {sp.simplify(s_density_taylor)}") print() # Extract coefficients c_phi0 = s_density_taylor.coeff(phi, 0) c_phi2 = s_density_taylor.coeff(phi, 2) c_phi4 = s_density_taylor.coeff(phi, 4) print(f"Coefficient of phi^0: {sp.simplify(c_phi0)} (= S_sub / 2^D unperturbed)") print(f"Coefficient of phi^2: {sp.simplify(c_phi2)} (Higgs mass squared, negative)") print(f"Coefficient of phi^4: {sp.simplify(c_phi4)} (Higgs quartic, positive)") print() # Substitute matched scaling Lambda = sqrt(D) c_phi4_matched = c_phi4.subs(Lambda, sp.sqrt(D)) print(f"At matched scaling Lambda = sqrt(D):") print(f" Coefficient of phi^4 = {sp.simplify(c_phi4_matched)}") print() # The candidate identification says: # lambda_H(M_*) = b(M_*) * (S_sub / 2^D) * (some normalization) # = (1/4) * exp(-1) * (some normalization) # # From the substrate spectral-action calculation we have: # coefficient of phi^4 in S_total / 2^D at matched scaling = y^4 / (2 D^2) * exp(-1) # # For the identification to hold structurally as # lambda_H(M_*) = b(M_*) * e^{-1} = (1/4) e^{-1}, # we need (after appropriate normalization of phi and Yukawa coupling): # y^4 / (2 D^2) = 1/4 # y^4 = D^2 / 2 # y = D^{1/2} / 2^{1/4} = Lambda / 2^{1/4} # # i.e. the substrate-Higgs Yukawa coupling must be y = Lambda / 2^{1/4} at matched scaling. print("=" * 100) print(" IDENTIFICATION CONDITION") print("=" * 100) print() print(" For the candidate identification") print(" lambda_H(M_*) = b(M_*) * (S_sub / 2^D) = (1/4) * exp(-1)") print(" to hold structurally as an identity between the spectral-action phi^4") print(" coefficient and the SM Higgs quartic at M_*, the substrate-Higgs Yukawa") print(" coupling must satisfy") print() print(" y^4 / (2 Lambda^4) = 1/4 <=> y = Lambda / 2^{1/4} = Lambda * 0.8409") print() print(" In other words: at matched scaling Lambda = sqrt(D) = M_*, the substrate-Higgs") print(" Yukawa coupling is uniquely fixed by the identification at the value") print() print(f" y / Lambda = 2^{{-1/4}} = {2 ** (-0.25):.6f}.") print() print(" If this Yukawa value is itself derived structurally from the substrate's") print(" internal algebra A_F = C + H + M_3(C) (via the Connes-Chamseddine inner") print(" fluctuation of D_sub by elements of A_F), then the identification closes") print(" STRUCTURALLY: Z^2 = e^{-1} would be exact.") print() print(" REMAINING STRUCTURAL CONTENT: derive y = Lambda * 2^{-1/4} as the unique") print(" Connes-Chamseddine inner-fluctuation Yukawa coupling for the substrate") print(" spectral triple at matched scaling. This is a finite calculation in the") print(" spectral-triple framework (Chamseddine-Connes-Marcolli 2007), reducing") print(" the Z^2 conjecture from 'spectral-action identity' to 'inner-fluctuation") print(" Yukawa normalization at matched scaling'.") print() def main(): print("=" * 100) print(" Computation 72 -- Substrate spectral action: Higgs-quartic Taylor expansion") print("=" * 100) print() derive_higgs_quartic() print("=" * 100) print(" STATUS UPDATE ON CONJECTURE Z^2 = e^{-1}") print("=" * 100) print() print(" Before this computation: the Conjecture was 'the spectral-action Higgs-") print(" quartic identity', stated abstractly.") print() print(" After this computation: the Conjecture is reduced to a single scalar") print(" identity:") print() print(" y_substrate(M_*) = M_* * 2^{-1/4} = M_* * 0.8409") print() print(" where y_substrate(M_*) is the substrate-Higgs Yukawa coupling derived") print(" from the Chamseddine-Connes inner fluctuation of D_sub by an element of") print(" the internal algebra A_F = C + H + M_3(C). This is a specific finite") print(" number that the spectral-triple framework either delivers or doesn't.") print() print(" The reduction is real progress: the open question is now a specific") print(" Yukawa-normalization calculation, not an open conjecture about an") print(" abstract identity.") if __name__ == "__main__": main()