#!/usr/bin/env python3
"""
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_*) ≈ 0.0926 (Buttazzo et al. 2013)
Predicted from bridge: (1/4) * exp(-1) ≈ 0.0920
Ratio: 0.0926 / 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 the new v24.22+ research thread.  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)
    samples = rng.binomial(1, 0.5, size=(n_samples, D))
    X_bars = samples.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_*)     ~  0.0926 (Buttazzo et al. 2013)")
    print("    RATIO            0.0926 / 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()