#!/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 89 -- substrate Higgs Hamiltonian derivation (Z^2 bridge step 1)
=============================================================================
Substrate-side closure for the Z² bridge: derive the substrate-side Higgs Hamiltonian
H_Higgs(C) from substrate primitives, with the goal of closing the
KO-tempered Bernoulli MGF identification:

  Z^2 = lambda_SM(M_*) / b(M_*)  <-?->  E_mu[exp(-(KO_total mod 8) X_bar)]

Comp 88 established the RIGHT-HAND SIDE structurally (asymptotic Bernoulli
MGF at KO-temperature beta = 2 delivers e^{-1}).  Comp 89 attacks the
LEFT-HAND SIDE: identifying H_Higgs(C) such that the substrate Bernoulli
expectation E_mu[exp(-beta H_Higgs)] equals the substrate-side Higgs
partition function.

STRUCTURAL DERIVATION OF H_Higgs(C) = X_bar(C)
------------------------------------------------
The substrate Higgs field on substrate configurations C in P(D) must
satisfy three structural constraints:

  (i) ADDITIVITY (independent Bernoulli bits).  The substrate measure
      mu = otimes Bern(1/2) factors across sites.  An extensive
      Hamiltonian must be additive:
        H(C) = sum_a h_a(B_a)
      where B_a = 1[a in C] is the bit at site a, h_a a per-site contribution.

  (ii) MATCHED SCALING (per-site energy 1/D).  At matched scaling
       Lambda = sqrt(D), the substrate's energy units are rescaled by
       1/Lambda^2 = 1/D.  Per-site energy is therefore 1/D.

  (iii) UNIFORMITY (symmetric Bernoulli sites).  P1 distinguishability
        treats all sites equivalently; no site is privileged in the
        Bernoulli measure.  Per-site energy h_a is uniform across a.

Together, (i)-(iii) force:
  H_Higgs(C) = (1/D) sum_a B_a = X_bar(C)

This is the SIMPLEST possible substrate-Higgs Hamiltonian: empirical
mean of bit-occupation.  No free parameters, no model choice.  All three
ingredients trace to P1 + matched-scaling.

BRIDGE IDENTIFICATION (PARTIAL CLOSURE)
----------------------------------------
With H_Higgs(C) = X_bar(C) derived structurally, the KO-tempered
Bernoulli partition function is:
  Z_H(beta) := E_mu[exp(-beta H_Higgs(C))] = E_mu[exp(-beta X_bar)]

At KO-temperature beta = KO_total mod 8 = 2:
  Z_H(2) -> exp(-1)  asymptotically  (Comp 88)

The SUBSTRATE-SIDE bridge is now closed structurally:
  e^{-1} = Z_H(beta_KO) = E_mu[exp(-(KO_total mod 8) X_bar)]

with all ingredients (H_Higgs, beta_KO, asymptotic limit) derived from
P1 + 10-foundational + matched-scaling.

REMAINING OPEN GAP (SM-SIDE)
----------------------------
Why does the SM Higgs coupling ratio Z^2 = lambda_SM(M_*) / b(M_*)
equal the substrate-side Z_H(beta_KO) at the matched scaling?

The candidate physical correspondence: at the matched scaling M_*, the
SM Higgs quartic equals the LG quartic times the substrate's KO-tempered
partition function:
  lambda_SM(M_*) = b * Z_H(beta_KO) = (1/4) * exp(-1) ≈ 0.0920

NUMERICAL VERIFICATION
----------------------
Observed lambda_SM(M_*) ≈ 0.0927 (Buttazzo et al. 2013)
Predicted from bridge: (1/4) * exp(-1) ≈ 0.0920
Ratio: 0.0927 / 0.0920 = 1.008 -- the same 0.8% near-coincidence as Z^2.

STATUS
------
Comp 89 closes the SUBSTRATE SIDE of the Z^2 bridge: H_Higgs = X_bar
is structurally derived (additivity + matched scaling + uniformity),
and the KO-tempered partition function delivers e^{-1} asymptotically.

What remains is the SM-side identification: deriving lambda_SM(M_*) =
b * Z_H(beta_KO) from a Connes-Chamseddine partition-function
correspondence (rather than the obstructed heat-kernel correspondence
of Comps 85, 86).

This is a research thread opened in the closing-act work.  Comp 89 reduces the
remaining-open content of Z^2 to a single specific question:

  WHY does the SM Higgs effective coupling at the matched scaling
  M_* equal the LG quartic times the substrate's KO-tempered
  Bernoulli partition function?

A candidate answer: a discrete-substrate analogue of the Connes-Chamseddine
spectral-action correspondence operating at the partition-function level
rather than the heat-kernel level.  Future research.
"""

from __future__ import annotations
import math
import numpy as np


def H_Higgs(C: set, D: int) -> float:
    """Substrate Higgs Hamiltonian: H(C) = X_bar(C) = |C|/D."""
    return len(C) / D


def Z_H_exact(D: int, beta: float, n_samples: int = 100_000, rng_seed: int = 42) -> float:
    """Empirical Bernoulli partition function E[exp(-beta H_Higgs)].

    Uses Monte Carlo sampling of Bernoulli configurations.
    """
    rng = np.random.default_rng(rng_seed)
    # Drawn a block of rows at a time.  The whole sample matrix is 800 MB at the
    # D = 1000 this file reaches, and the browser runtime this book publishes to
    # cannot be relied on for it.  Only the per-row mean is kept, 0.8 MB in all, and
    # a row sum of 0/1 integers is exact however the rows are grouped, so the mean
    # below is taken over the identical array it was taken over before.
    X_bars = np.empty(n_samples)
    for i in range(0, n_samples, 10_000):
        k = min(10_000, n_samples - i)
        X_bars[i:i + k] = rng.binomial(1, 0.5, size=(k, D)).sum(axis=1) / D
    return float(np.mean(np.exp(-beta * X_bars)))


def Z_H_analytic(D: int, beta: float) -> float:
    """Exact Bernoulli MGF at finite D: ((1 + exp(-beta/D))/2)^D."""
    return ((1 + math.exp(-beta / D)) / 2) ** D


def main():
    print("=" * 100)
    print("  Computation 89 -- substrate Higgs Hamiltonian + Z^2 bridge step 1")
    print("=" * 100)
    print()

    print("STRUCTURAL DERIVATION: H_Higgs(C) = X_bar(C)")
    print("-" * 100)
    print()
    print("  Three constraints on substrate Higgs Hamiltonian:")
    print("    (i)   ADDITIVITY:  Bernoulli sites independent => H = sum_a h_a(B_a)")
    print("    (ii)  MATCHED SCALING:  per-site energy 1/D")
    print("    (iii) UNIFORMITY:  P1 symmetry => h_a uniform across sites")
    print()
    print("  Forces:  H_Higgs(C) = (1/D) sum_a B_a = X_bar(C)")
    print()
    print("  No free parameters.  Derived from P1 + matched scaling.")
    print()

    print("KO-TEMPERED PARTITION FUNCTION Z_H(beta_KO)")
    print("-" * 100)
    print()
    beta_KO = 2  # KO_total mod 8
    print(f"  beta_KO = KO_total mod 8 = {beta_KO}")
    print()
    print(f"  {'D':>8}  {'Z_H analytic':>20}  {'Z_H Monte Carlo':>20}  {'asymptote':>15}")
    for D in [10, 100, 1000]:
        z_analytic = Z_H_analytic(D, beta_KO)
        z_mc = Z_H_exact(D, beta_KO)
        asymptote = math.exp(-1)
        print(f"  {D:>8}  {z_analytic:>20.10f}  {z_mc:>20.10f}  {asymptote:>15.10f}")
    print()
    print(f"  Asymptotic limit (D -> infty):  Z_H(beta_KO) = exp(-1) = {math.exp(-1):.10f}")
    print()

    print("BRIDGE IDENTIFICATION (PARTIAL)")
    print("-" * 100)
    print()
    print("  SUBSTRATE side (now closed structurally):")
    print("    Z_H(beta_KO) = E_mu[exp(-(KO_total mod 8) X_bar)]  ->  exp(-1)")
    print("    Ingredients: H_Higgs = X_bar (Comp 89), beta_KO = KO_total mod 8 = 2")
    print("                 (10-foundational + Bott periodicity, Sec foundational-object),")
    print("                 asymptotic limit (Cramer-Bernoulli machinery)")
    print()
    print("  CANDIDATE SM correspondence:")
    print("    lambda_SM(M_*)  =  b * Z_H(beta_KO)  =  (1/4) * exp(-1)  ~  0.0920")
    print("    OBSERVED         lambda_SM(M_*)     ~  0.0927 (Buttazzo et al. 2013)")
    print("    RATIO            0.0927 / 0.0920    =  1.008  (the same 0.8% near-coincidence)")
    print()

    print("REMAINING OPEN GAP (SM-SIDE)")
    print("-" * 100)
    print()
    print("  Why does the SM Higgs quartic at matched scaling EQUAL the LG quartic times")
    print("  the substrate's KO-tempered partition function?")
    print()
    print("  Candidate answer (research direction): a discrete-substrate analogue of the")
    print("  Connes-Chamseddine spectral-action correspondence operating at the")
    print("  partition-function level rather than the heat-kernel level.  The heat-kernel")
    print("  correspondence is structurally obstructed for discrete substrates (Comps 85,")
    print("  86); the partition-function correspondence might survive.")
    print()
    print("  Comp 89 status: SUBSTRATE SIDE OF BRIDGE CLOSED.  SM SIDE OPEN.")
    print()
    print("  Open content of Z^2 reduced to: derive the partition-function-level")
    print("  substrate-to-SM correspondence at matched scaling.")


if __name__ == "__main__":
    main()