#!/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 110 -- Bridge Premise (B) attack, milestones M1 + M2:
                   discrete Wilson-Polchinski flow + wave-function
                   renormalisation identification of the bridge factor
=========================================================================
STATUS (v26.13): IN PROGRESS.  This computation sets up the framework
for closing Bridge Premise (B) from P1-P3 alone, via the wave-function
renormalisation route, and reduces the open problem to a single sharp
statement (milestone M3).  It does NOT close (B).  It is the scaffolding
+ precise target, analogous to how Comp 101 reduced (B) to two sub-
questions before Comp 103 attacked one of them.

THE TARGET (Bridge Premise B)
==============================
The substrate-side factor is closed from P1-P3 (Comp 100):

    Z_H(beta_KO) = E_mu[exp(-beta_KO * X_bar)]
                 = ((1 + e^(-beta_KO/D))/2)^D
                 -> e^(-beta_KO/2) = e^(-1)   at beta_KO = 2, D -> infty.

Bridge Premise (B) is the identification

    lambda_SM(M_*) = b * Z_H(beta_KO) = (1/4) * e^(-1) = 0.0920    (B)

between the SM Higgs quartic at M_* and the substrate bare quartic
b = lambda_PST = 1/4 times the substrate partition function.

WHY THE NAIVE ROUTE FAILED (Comp 103)
======================================
Standard Wilsonian threshold matching of a quartic is ADDITIVE in the
loop corrections (lambda_IR = lambda_UV + finite), not a multiplicative
dressing.  Comp 103 showed the multiplicative form is NOT derivable as a
path-integral (measure change-of-variable) Jacobian:
  - linear-map determinant: configuration-independent (constant);
  - Cramer-Bernoulli LDP: Gaussian in (X_bar - 1/2), not exp(X_bar).

THE OPENING (this computation)
===============================
A coupling flows MULTIPLICATIVELY under the RG -- not additively --
precisely when the flow is dominated by WAVE-FUNCTION (field-strength)
renormalisation Z_phi:

    psi -> Z_phi^(1/2) psi   =>   lambda -> lambda / Z_phi^2 (with Z_Gamma4 = 1).

This is exactly the multiplicative structure (B) needs, and it is the
reinterpretation Comp 101 proposed (Z^2 as wave-function renormalisation
rather than coupling renormalisation).  The sharpened conjecture:

    THE BRIDGE FACTOR IS THE FIELD-STRENGTH RENORMALISATION OF THE
    SUBSTRATE-TO-SM PROJECTION Pi, with Z_phi^2 = Z_H(beta_KO) = e^(-1).

i.e. (B) is NOT a coupling-matching Jacobian (Comp 103 killed that), it
is the wave-function renormalisation of the modal field as it projects
to the SM Higgs.

This computation does:
  M1 -- set up the discrete Wilson-Polchinski flow on the Boolean
        lattice with cutoff f(x) = e^(-x) (Comp 100) and matched scaling
        Lambda^2 = D; show the flow reduces to the low Walsh shells
        V_0 + V_1 where X_bar lives (Comp 98).
  M2 -- define the field-strength renormalisation Z_phi from the
        substrate two-point function of the order-parameter field, and
        show that IF the matched-scaling flow weights the field by the
        substrate Boltzmann factor exp(-beta_KO X_bar), then
        Z_phi^2 = Z_H(beta_KO) = e^(-1) numerically.
  M3 (NOT done here) -- prove the Polchinski flow actually GENERATES the
        weight exp(-beta_KO X_bar) as its field-strength kernel.  This is
        the genuinely new mathematics and the remaining crux.

=========================================================================
"""

import math


# -------------------------------------------------------------------------
# Substrate primitives (P1): Bernoulli measure, Walsh basis, Boolean
# Laplacian.  Conventions match Comp 100: Delta = sum_a (1 - tau_a), each
# (1 - tau_a) has eigenvalues {0, 2}, so Delta chi_S = 2|S| chi_S.
# -------------------------------------------------------------------------

def boolean_laplacian_eigenvalue(s):
    """Delta chi_S = 2|S| chi_S  (|S| = s)."""
    return 2 * s


def cutoff(x):
    """Matched-scaling cutoff f(x) = e^(-x), forced by P1's tensor-product
    independence (Comp 100: f(x+y) = f(x)f(y) => f = exp)."""
    return math.exp(-x)


def propagator_mode(s, D):
    """Substrate propagator in Walsh mode of weight s at matched scaling
    Lambda^2 = D:   G_S = f(2|S|/Lambda^2) / (2|S|)  for |S| >= 1.
    The zero mode (s = 0) is the order parameter / massless direction and
    is handled separately (it is NOT a propagating mode integrated out)."""
    if s == 0:
        return None  # order-parameter zero mode
    lam2 = D
    return cutoff(boolean_laplacian_eigenvalue(s) / lam2) / boolean_laplacian_eigenvalue(s)


# -------------------------------------------------------------------------
# The Higgs Hamiltonian / order-parameter field (Comp 89, Comp 98):
#   X_bar(C) = (1/D) sum_a C_a,  with Walsh decomposition
#   X_bar = (1/2) chi_emptyset - (1/(2D)) sum_{|S|=1} chi_S
# supported entirely on the lowest two Walsh shells V_0 (+) V_1.
# -------------------------------------------------------------------------

def xbar_walsh_support():
    """Return the Walsh-shell support of X_bar (Comp 98)."""
    return {0: 0.5,        # coefficient of chi_emptyset
            1: -0.5}       # coefficient of (1/D) sum_{|S|=1} chi_S


def xbar_mean():
    """E_mu[X_bar] = 1/2 (Bernoulli site mean)."""
    return 0.5


def xbar_variance(D):
    """Var_mu(X_bar) = (1/D^2) * D * Var(C_a) = (1/D^2)*D*(1/4) = 1/(4D)."""
    return 1.0 / (4 * D)


# -------------------------------------------------------------------------
# KO-tempered Bernoulli partition function (Comp 87/88/100):
#   Z_H(beta) = E_mu[exp(-beta X_bar)] = ((1 + e^(-beta/D))/2)^D
#            -> e^(-beta/2)   as D -> infty.
# At beta_KO = 2 (one-bit Clifford eigenvalue range, Comp 100/102): e^(-1).
# -------------------------------------------------------------------------

def Z_H(beta, D):
    """Closed-form KO-tempered partition function."""
    return ((1.0 + math.exp(-beta / D)) / 2.0) ** D


def Z_H_limit(beta):
    """D -> infty limit."""
    return math.exp(-beta / 2.0)


# -------------------------------------------------------------------------
# M2: wave-function renormalisation from the substrate two-point function.
#
# The substrate generating functional with a source J coupled to the
# order-parameter field, and the matched-scaling Boltzmann weight
# exp(-beta X_bar) supplied by the substrate fluctuations:
#
#   Z[J] = E_mu[ exp(-beta X_bar + J * field) ].
#
# The NORMALISATION Z[0] = Z_H(beta) is the multiplicative factor that
# dresses the bare quartic.  Identifying it as a FIELD-STRENGTH
# renormalisation: the projected SM field phi = Z_phi^(1/2) psi is
# canonically normalised only after the substrate fluctuations are
# integrated with their matched-scaling weight, which rescales the field
# normalisation by exactly Z[0]:
#
#   Z_phi^2 = Z[0] / Z[0]_{beta=0} = Z_H(beta_KO) / 1 = Z_H(beta_KO).
#
# Then with trivial vertex renormalisation (Z_Gamma4 = 1 at tree level,
# CC inner fluctuations; Comp 101 sub-question 2 satisfied):
#
#   lambda_SM(M_*) = b * Z_phi^2 = (1/4) * Z_H(beta_KO) -> (1/4) e^(-1).
# -------------------------------------------------------------------------

def wavefunction_renormalisation(beta, D):
    """Z_phi^2 candidate = Z_H(beta) (the field-strength rescaling under
    the matched-scaling projection).  CONDITIONAL on M3 (that the flow
    generates the weight exp(-beta X_bar))."""
    return Z_H(beta, D)


def main():
    print("=" * 72)
    print("Computation 110: Bridge Premise (B) attack -- M1 + M2")
    print("Wave-function renormalisation route (sets up the M3 target)")
    print("=" * 72)
    print()

    BETA_KO = 2  # one-bit Clifford Cl(1,0) eigenvalue range (Comp 100/102)
    b = 0.25     # lambda_PST, doublet convention

    # ----- M1: discrete Wilson-Polchinski flow setup -----
    print("M1.  DISCRETE WILSON-POLCHINSKI FLOW")
    print("-" * 72)
    print("  Cutoff f(x) = e^(-x) (Comp 100, forced by P1 independence).")
    print("  Boolean Laplacian Delta chi_S = 2|S| chi_S.")
    print("  Matched scaling Lambda^2 = D.")
    print("  Propagator in mode S (|S| >= 1): G_S = e^(-2|S|/D) / (2|S|).")
    print()
    D = 12
    print(f"  Sample propagators at D = {D}:")
    print(f"    {'|S|':>4} {'eig 2|S|':>9} {'cutoff':>10} {'G_S':>12}")
    for s in range(0, 6):
        if s == 0:
            print(f"    {s:>4} {'0':>9} {'1 (zero mode = order parameter)':>30}")
        else:
            eig = boolean_laplacian_eigenvalue(s)
            c = cutoff(eig / D)
            g = propagator_mode(s, D)
            print(f"    {s:>4} {eig:>9} {c:>10.5f} {g:>12.6f}")
    print()
    print("  X_bar (the Higgs Hamiltonian, Comp 89) is supported ONLY on")
    print("  the lowest two Walsh shells V_0 (+) V_1 (Comp 98):")
    print("    X_bar = (1/2) chi_emptyset - (1/(2D)) sum_{|S|=1} chi_S.")
    print("  => integrating out the |S| >= 2 modes is a TRIVIAL fixed-point")
    print("     step for X_bar (Comp 98): the flow leaves X_bar unchanged")
    print("     until it reaches the low shells.  The relevant field")
    print("     content at M_* is exactly X_bar.")
    print()

    # ----- M2: wave-function renormalisation -----
    print("M2.  WAVE-FUNCTION RENORMALISATION FROM THE TWO-POINT FUNCTION")
    print("-" * 72)
    print(f"  Order-parameter field statistics under mu:")
    print(f"    E_mu[X_bar]   = {xbar_mean()}")
    for Dv in (10, 100, 1000):
        print(f"    Var_mu(X_bar) = 1/(4D) = {xbar_variance(Dv):.6e}   (D = {Dv})")
    print()
    print("  Field-strength renormalisation candidate (CONDITIONAL on M3):")
    print("    Z_phi^2 = Z[0] = E_mu[exp(-beta_KO X_bar)] = Z_H(beta_KO)")
    print()
    print(f"  Numerical Z_phi^2 = Z_H(beta_KO = {BETA_KO}):")
    print(f"    {'D':>6} {'Z_phi^2':>12} {'dev from e^-1':>16}")
    e_inv = math.exp(-1.0)
    for Dv in (6, 10, 100, 1000, 10000):
        z = wavefunction_renormalisation(BETA_KO, Dv)
        dev = 100.0 * (z - e_inv) / e_inv
        print(f"    {Dv:>6} {z:>12.6f} {dev:>15.3f}%")
    print(f"    {'limit':>6} {e_inv:>12.6f} {'0.000':>15}%")
    print()
    print("  => Z_phi^2 -> e^(-1) (D -> infty), reproducing the substrate")
    print("     factor of Comp 100 in the wave-function-renormalisation")
    print("     reading.")
    print()

    # ----- the resulting bridge value -----
    print("RESULTING BRIDGE VALUE (conditional on M3)")
    print("-" * 72)
    print("  With Z_Gamma4 = 1 (trivial vertex, tree-level CC inner")
    print("  fluctuations, Comp 101 sub-question 2):")
    print(f"    lambda_SM(M_*) = b * Z_phi^2 = {b} * Z_H({BETA_KO})")
    for Dv in (100, 10000):
        lam = b * Z_H(BETA_KO, Dv)
        print(f"      D = {Dv:>5}: {lam:.6f}")
    lam_lim = b * e_inv
    print(f"      limit:     {lam_lim:.6f}   (= e^(-1)/4)")
    print(f"    observed lambda_SM(M_*) ~ 0.0927 (Buttazzo)")
    print(f"    ratio: {lam_lim/0.0927:.4f}")
    print()

    # ----- honest status: what M1+M2 achieve, what M3 must prove -----
    print("=" * 72)
    print("STATUS: what M1 + M2 achieve, and the remaining crux (M3)")
    print("=" * 72)
    print()
    print("  ACHIEVED (M1, M2):")
    print("   - The discrete Wilson-Polchinski flow is set up with the")
    print("     P1-forced cutoff f = exp and matched scaling Lambda^2 = D.")
    print("   - The flow reduces to the low Walsh shells where X_bar lives")
    print("     (Comp 98), so the field content at M_* is exactly X_bar.")
    print("   - The bridge factor is RE-CAST as a field-strength")
    print("     renormalisation Z_phi^2 (multiplicative by construction),")
    print("     side-stepping the additive-matching obstruction that")
    print("     Comp 103 established for the coupling-Jacobian reading.")
    print("   - Numerically, Z_phi^2 = Z_H(beta_KO) -> e^(-1), reproducing")
    print("     the Comp 100 value in the new reading.")
    print()
    print("  NOT ACHIEVED -- the remaining crux (M3):")
    print("   The identification Z_phi^2 = E_mu[exp(-beta_KO X_bar)] is so")
    print("   far IMPOSED (the field is weighted by the substrate Boltzmann")
    print("   factor).  What must be PROVEN is that the Wilson-Polchinski")
    print("   flow GENERATES this weight as its field-strength kernel --")
    print("   i.e. that integrating the substrate fluctuations from")
    print("   Lambda = sqrt(D) down to M_* produces exactly the multiplier")
    print("   exp(-beta_KO X_bar) on the order-parameter field, with")
    print("   beta_KO = 2 inherited from the one-bit Clifford range.")
    print()
    print("   Until M3 is proven, this is the SAME gap as (B) in a sharper")
    print("   form: 'derive the multiplicative bridge factor' has become")
    print("   'show the Polchinski field-strength kernel is the KO-tempered")
    print("   Boltzmann weight'.  That is a concrete, well-posed RG-flow")
    print("   computation -- the natural next milestone -- and it either")
    print("   closes (B) or yields a Comp-103-style sharp obstruction.")
    print()
    print("  This computation does NOT close Bridge Premise (B).")


if __name__ == "__main__":
    main()