#!/usr/bin/env python3
"""
Computation 85 -- A_F-internal Higgs inner fluctuation for Z^2: A_F-internal Higgs fluctuation
==============================================================
A_F-internal Higgs fluctuation: place the Higgs in the canonical A_F = C (+) H (+)
M_3(C) finite-internal factor (as in the Chamseddine-Connes-Marcolli
SM construction) rather than in the Clifford-volume element ω of
Comps 74, 81.

CANONICAL CC-SM CONSTRUCTION (recap)
------------------------------------
In Chamseddine-Connes-Marcolli 2007 (Ch. 17), the SM finite spectral
triple is (A_F, H_F, D_F, J_F, gamma_F) with:
  A_F = C (+) H (+) M_3(C)
  D_F = off-diagonal block coupling (left-handed leptons, right-handed
        leptons), (left-handed quarks, right-handed quarks)
       containing the Yukawa matrix Y

The Higgs doublet H is built from the INNER FLUCTUATION:
  D_F -> D_F + A_F  with  A_F = sum_i a_i [D_F, b_i],  a_i, b_i in A_F

After inner fluctuation, the projected spectral action gives the SM
Higgs potential:
  V(H) = -mu^2 |H|^2 + lambda |H|^4

The QUARTIC COEFFICIENT lambda is fixed by the spectral-action moments:
  lambda = (a / pi^2) * f_0     (leading-order term)
where:
  a = sum over fermion families of (Y_u^4 + Y_d^4 + Y_e^4) traces
  f_0 = zeroth moment of the cutoff function f

PST APPLICATION
---------------
For PST substrate at matched scaling Lambda = sqrt(D), the substrate
Dirac D_sub has all eigenvalues at +/- sqrt(D), so:
  S/2^D = (1/2^D) * 2^D * f(1) = f(1) = e^{-1}     (Comp 62)

The CCM moment f_0 is the integral
  f_0 = int_0^infty f(x) dx  (for f(x) = exp(-x^2): f_0 = sqrt(pi)/2)
                              numerically ~ 0.886

But the PST substrate spectrum is DISCRETE -- all eigenvalues at +/- sqrt(D).
The "moment" structure of CCM applies to CONTINUOUS spectra
via the Seeley-DeWitt heat-kernel asymptotic expansion.  For the
discrete substrate, f_0 collapses to f(1) = e^{-1} (Comp 62 result).

THE CONJECTURE RESTATED
-----------------------
If the CCM moment-to-coefficient extraction generalises to the discrete
substrate triple WITH f_0 -> S_sub/2^D = e^{-1}, then:
  lambda = (a / pi^2) * (S_sub / 2^D) = (a / pi^2) * e^{-1}

For Z^2 = lambda/b with b = 1/4 (LG quartic):
  Z^2 = (4 a / pi^2) * e^{-1}

For Z^2 = e^{-1} we need (4 a / pi^2) = 1, i.e. a = pi^2 / 4 ≈ 2.467.

The Yukawa-trace "a" depends on which substrate-side Yukawa values are
inserted.  This is essentially the Comp 72 / 74 question restated at
the COEFFICIENT-EXTRACTION level instead of the perturbative-expansion
level.  No new computational test.

STRUCTURAL CONCLUSION
------------------
This approach does not add a new computational test.  It RESTATES the conjecture at the
moment-extraction level.  Closure still requires the substrate-side
Yukawa identification y/Lambda = 2^{-1/4} -- which Comps 74, 81, 84
have shown does not emerge from natural inner fluctuations.

The deeper structural obstacle revealed by Angle B:
  The CCM moment-extraction formalism (f_0 -> spectral-action coefficient)
  is built on the HEAT-KERNEL ASYMPTOTIC EXPANSION.  The expansion is
  defined for CONTINUOUS spectra and breaks down for the discrete
  substrate spectrum (Comp 86 explores this directly).

Comp 85 conclusion: Angle B reduces to the same identification problem
already faced; no new computational closure.
"""

from __future__ import annotations
import math


def main():
    print("=" * 100)
    print("  Computation 85 -- A_F-internal Higgs inner fluctuation for Z^2: A_F-internal Higgs fluctuation")
    print("=" * 100)
    print()

    print("CCM MOMENT-EXTRACTION (continuum SM)")
    print("-" * 100)
    f_0_continuum = math.sqrt(math.pi) / 2
    print(f"  f_0 (continuum, f(x) = exp(-x^2)) = sqrt(pi) / 2 = {f_0_continuum:.6f}")
    print(f"  lambda_continuum ~ (a / pi^2) * f_0 ~ 0.0898 * a")
    print()

    print("PST SUBSTRATE: DISCRETE SPECTRUM")
    print("-" * 100)
    f_at_1 = math.exp(-1.0)
    print(f"  All substrate Dirac eigenvalues at +/- sqrt(D).")
    print(f"  S_sub / 2^D = f(1) = exp(-1) = {f_at_1:.6f}  (Comp 62, exact)")
    print(f"  Replacing f_0 -> S_sub/2^D in the CCM formula:")
    print(f"    lambda_PST ~ (a / pi^2) * e^(-1) = {f_at_1 / math.pi**2:.6f} * a")
    print()

    print("Z^2 = lambda / b   with  b = 1/4 (LG quartic)")
    print("-" * 100)
    a_required = math.pi**2 / 4
    print(f"  For Z^2 = e^(-1): require a = pi^2 / 4 = {a_required:.4f}")
    print(f"  'a' is the sum-of-Yukawa-fourth-powers trace.")
    print(f"  Closure of Z^2 reduces to: do the substrate-emergent Yukawas")
    print(f"  satisfy sum_f Y_f^4 = pi^2 / 4 ?")
    print()
    print("  This is the SAME question as the natural Yukawa identification")
    print("  of Comps 74, 81, 84.  Angle B restates the conjecture; no")
    print("  separate computational test.")
    print()

    print("STRUCTURAL OBSTACLE IDENTIFIED")
    print("-" * 100)
    print()
    print("  The CCM moment-extraction f_0 -> spectral-action coefficient is")
    print("  built on the SEELEY-DEWITT heat-kernel asymptotic expansion.")
    print("  For continuous spectra, this delivers the SM Higgs quartic at")
    print("  one-loop order from the moments f_0, f_2, f_4 of the cutoff.")
    print()
    print("  For the DISCRETE PST substrate, all eigenvalues sit at +/- sqrt(D)")
    print("  exactly (the cutoff edge), so the heat-kernel expansion collapses:")
    print("  there is no Seeley-DeWitt small-eigenvalue regime.  The moment")
    print("  f_0 collapses to f(1) = e^(-1), but the FORMALISM that maps f_0")
    print("  to the Higgs quartic does not survive the collapse.")
    print()
    print("  Comp 86 (Angle C) tackles this directly.")
    print()

    print("Conclusion: This approach restates the identification problem at the moment-extraction level (reduces to the same conjecture, plus")
    print("identifies the structural obstacle for Angle C).")


if __name__ == "__main__":
    main()