#!/usr/bin/env python3
"""
PROVENANCE: PROOF

Computation 118 -- Bridge Premise (B), milestone M8(c):
                   the all-orders S_D-equivariant Morse-Bott lemma for F11
=========================================================================
UPDATE (v26.29+): the all-orders decoupling proved here is SUBSUMED by an
exact, non-perturbative closure in Comp 121. Because X_bar is the empirical
mean of i.i.d. bits (F8), it is a SUFFICIENT STATISTIC, so the transverse
modes factor out as the interaction-independent multiplicity C(D,k) and
Z_Gamma4 = 1 (and every n-point Phi-vertex correction) vanishes EXACTLY at
finite D by Fisher-Neyman factorisation, not merely as a formal all-orders
lemma. The Morse-Bott lemma below is the de Moivre-Laplace (Gaussian) shadow
of that exact factorisation; the O(1/D) is isolated to the VALUE e^-1, not
the decoupling. See Comp 121.

STATUS (v26.19+): M8(c) of the wave-function attack on Bridge Premise (B).
F11 (Comp 115, M6) proved the unique-soft-mode result at Gaussian order;
Comp 116 (M7) verified the all-orders decoupling numerically with the full
exp(-V). The remaining substrate-side rigour task (sub-caveat (c)) was to
state and prove that decoupling as a formal all-orders S_D-equivariant
Morse-Bott lemma. This computation does that, and verifies the load-bearing
steps of the proof.

THE LEMMA
=========
Setup. Substrate fluctuation space R^D (one coordinate per bit) around the
symmetric vacuum, with:
  - kinetic Hessian = the Boolean Laplacian Delta = sum_a (1 - tau_a),
    whose nonzero spectrum on the single-bit (|S|=1) shell is the binary
    gap 2 = 1-(-1) (Comp 112), and >= 2 on all |S|>=1 modes;
  - interaction F_int(C) = Phi(X_bar(C)), a function of the linear
    S_D-invariant collective coordinate X_bar = (1/D) sum_a C_a alone
    (Comp 89: H_Higgs = X_bar; P1: bits i.i.d., so the action is
    S_D-invariant).

Lemma (S_D-equivariant Morse-Bott decoupling). Decompose the
single-bit-fluctuation space R^D under S_D into the singlet (the symmetric
line R|1>, |1> = sum_a e_a, carrying X_bar) and the standard representation
W (dim D-1, the non-collective modes, {v : sum_a v_a = 0}). Then:
  (i)  every derivative tensor of F_int is the totally symmetric all-ones
       tensor, d^n F_int / dC_{a_1}...dC_{a_n} = Phi^{(n)}(X_bar)/D^n,
       which annihilates W in every slot;
  (ii) hence the full Hessian (kinetic + interaction), to ALL orders in
       Phi, restricts on W to the bare Laplacian and is bounded below by
       the binary gap, lambda_min(H|_W) >= 2;
  (iii) only the singlet direction carries a Phi-dependent eigenvalue and
        can be softened to 0 at the LG threshold (P3); the spectral gap
        between this soft mode and W is >= 2 (- the soft eigenvalue) -> 2
        in the substrate limit, finite-D constraint correction O(1/D)
        (Comp 116).
This is the Morse-Bott structure: the critical/soft locus is the collective
(order-parameter) direction; the normal bundle W is uniformly gapped, and
the gap is S_D-equivariantly protected (W is a non-trivial irrep, so no
S_D-invariant interaction has any matrix element on it).

PROOF (the engine, verified below)
==================================
X_bar is LINEAR, so its second and higher derivatives vanish; the chain
rule gives d^n F_int/dC_{a_1}...dC_{a_n} = Phi^{(n)}(X_bar) * prod_k
(dX_bar/dC_{a_k}) = Phi^{(n)}(X_bar)/D^n, independent of the indices --
the all-ones tensor T (T_{a_1...a_n} = 1). For any v in W (sum_a v_a = 0),
contracting T in one slot gives T(v, .,...,.) = (sum_a v_a)(...) = 0. So
F_int contributes nothing to the Hessian or to any higher vertex on W, at
every order n. The W-Hessian is therefore the bare kinetic Hessian
Delta|_W, with eigenvalues 2|S| >= 2. QED (continuum/substrate limit; the
discrete O(1/D) correction is the sum-constraint correlation, Comp 116).
=========================================================================
"""

import numpy as np


def all_ones_hessian_of_collective(D, Phi2):
    """The interaction Hessian of F_int = Phi(X_bar): (Phi''/D^2) * J,
    J the all-ones D x D matrix."""
    return (Phi2 / D**2) * np.ones((D, D))


def standard_rep_basis(D):
    """Orthonormal basis of W = {v : sum v = 0} (the S_D standard rep)."""
    # Helmert-style contrasts, orthonormalised.
    M = np.zeros((D - 1, D))
    for k in range(1, D):
        M[k - 1, :k] = 1.0
        M[k - 1, k] = -k
        M[k - 1] /= np.linalg.norm(M[k - 1])
    return M  # rows span W


def fd_tensor_entry(Phi, D, idxs, h=1e-3):
    """Central finite-difference of the n-th mixed partial of
    F_int(C) = Phi(mean(C)) at the symmetric point C = 1/2, for the
    multi-index idxs (length n). Used to check the all-ones structure for
    n = 2, 3, 4 without assuming the analytic form."""
    n = len(idxs)
    base = np.full(D, 0.5)

    def F(shift):
        c = base + shift
        return Phi(c.mean())

    # n-th mixed derivative via 2^n-point central difference
    total = 0.0
    for signs in range(2**n):
        shift = np.zeros(D)
        parity = 1
        for k in range(n):
            s = 1 if (signs >> k) & 1 else -1
            shift[idxs[k]] += s * h
            parity *= s
        total += parity * F(shift)
    return total / (2 * h) ** n


def main():
    print("=" * 72)
    print("Computation 118: M8(c) -- all-orders S_D-equivariant Morse-Bott")
    print("=" * 72)
    print()

    # ---- 1. derivative tensors of F_int are all-ones (n = 2, 3, 4) ----
    print("1.  DERIVATIVE TENSORS OF F_int = Phi(X_bar) ARE ALL-ONES")
    print("-" * 72)
    print("    Test Phi(x) = exp(3x) (generic, all derivatives nonzero).")
    print("    Check d^n F_int / dC...dC is index-independent = Phi^(n)/D^n.")
    D = 6
    Phi = lambda x: np.exp(3.0 * x)
    for n, label in ((2, "Hessian"), (3, "3rd"), (4, "4th")):
        Phi_n = (3.0**n) * np.exp(3.0 * 0.5)        # Phi^{(n)}(1/2)
        predicted = Phi_n / D**n
        # sample a few distinct multi-indices
        samples = [tuple([0]*n), tuple(range(n)) if n <= D else None,
                   tuple([0,1]+[2]*(n-2))]
        vals = []
        for idx in samples:
            if idx is None:
                continue
            vals.append(fd_tensor_entry(Phi, D, idx))
        spread = max(vals) - min(vals)
        ok = "yes" if abs(spread) < 1e-4 and abs(vals[0] - predicted) < 1e-3 else "NO"
        print(f"    n={n} ({label:>7}): entries ~ {vals[0]:.5f} "
              f"(predicted {predicted:.5f}), index-spread {spread:.1e}  "
              f"all-ones? {ok}")
    print()
    print("    Every order n: the tensor entry is the same regardless of")
    print("    which bits are differentiated -> the all-ones tensor.")
    print()

    # ---- 2. the all-ones tensor annihilates the standard rep W ----
    print("2.  THE ALL-ONES TENSOR ANNIHILATES W (sum_a v_a = 0)")
    print("-" * 72)
    for D in (4, 8, 16):
        W = standard_rep_basis(D)
        Hint = all_ones_hessian_of_collective(D, Phi2=1.0)   # (1/D^2) J
        # contract the V-Hessian with each W basis vector
        max_resid = max(np.linalg.norm(Hint @ W[i]) for i in range(D - 1))
        print(f"    D={D:>2}: max |H_int . v| over v in W  = {max_resid:.2e}"
              f"   (J v = (sum v) 1 = 0)")
    print()
    print("    F_int has zero matrix element on W at every order (the")
    print("    contraction is (sum_a v_a) x ... = 0). The non-collective")
    print("    modes never see the interaction.")
    print()

    # ---- 3. Morse-Bott spectrum: 1 soft singlet + (D-1) gapped normals ----
    print("3.  MORSE-BOTT SPECTRUM ON THE |S|=1 SHELL")
    print("-" * 72)
    print("    Full Hessian = 2 I (binary gap) + (Phi''/D) projector_singlet.")
    print("    Sweep the singlet curvature; W stays pinned at the gap.")
    print(f"    {'Phi-curv':>9} {'singlet eig':>12} {'W eigs (min..max)':>22}")
    D = 8
    for c in (2.0, 0.0, -1.0, -2.0):          # singlet mass shift
        # H = 2 I + (c/D) J  (rank-1 interaction on the singlet)
        H = 2.0 * np.eye(D) + (c / D) * np.ones((D, D))
        eigs = np.sort(np.linalg.eigvalsh(H))
        soft = eigs[0] if c < 0 else eigs[-1]
        # the singlet eigenvalue is 2 + c; the rest are 2
        singlet = 2.0 + c
        Weigs = [e for e in eigs if abs(e - singlet) > 1e-9] or [2.0]
        print(f"    {c:>9.1f} {singlet:>12.4f} "
              f"{min(Weigs):>10.4f}..{max(Weigs):<10.4f}")
    print()
    print("    The singlet (X_bar) eigenvalue tracks 2 + Phi-curv and softens")
    print("    to 0 at the LG threshold; the D-1 normal (W) eigenvalues stay")
    print("    at the binary gap 2 for every curvature. Morse-Bott: a soft")
    print("    critical direction, a uniformly gapped normal bundle.")
    print()

    # ---- 4. equivariance / Schur: no S_D-invariant form mixes singlet & W --
    print("4.  S_D-EQUIVARIANCE PROTECTS THE GAP (Schur)")
    print("-" * 72)
    print("    Any S_D-invariant quadratic form on R^D commutes with the S_D")
    print("    action, so by Schur it is block-scalar on the isotypic pieces:")
    print("    a I on the standard rep W (dim D-1), b on the singlet. An")
    print("    interaction factoring through X_bar can only move b; a stays")
    print("    the bare-kinetic value. Verify: a random S_D-invariant")
    print("    Hessian (alpha I + beta J) has W-eigenvalue alpha independent")
    print("    of beta.")
    rng = np.random.default_rng(0)
    D = 10
    for _ in range(3):
        alpha, beta = rng.normal(), rng.normal()
        H = alpha * np.eye(D) + beta * np.ones((D, D))
        W = standard_rep_basis(D)
        wvals = [float(W[i] @ H @ W[i]) for i in range(D - 1)]
        print(f"    alpha={alpha:+.3f} beta={beta:+.3f}: "
              f"W-eigenvalues all = {np.mean(wvals):+.4f} "
              f"(spread {max(wvals)-min(wvals):.1e}) -- independent of beta")
    print()

    # ---- 5. assessment ----
    print("=" * 72)
    print("ASSESSMENT: does M8(c) close the rigour residual of F11?")
    print("=" * 72)
    print()
    print("  ESTABLISHED (formal, all orders):")
    print("   The decoupling is now a lemma, not a Gaussian-order +")
    print("   numerics statement. Because F_int factors through the LINEAR")
    print("   functional X_bar, its derivative tensor at every order n is")
    print("   the all-ones tensor Phi^(n)/D^n (part 1), which annihilates")
    print("   the standard rep W in every slot (part 2). So the W-Hessian is")
    print("   the bare Boolean Laplacian to all orders, bounded below by the")
    print("   binary gap (part 3). S_D-equivariance (Schur) guarantees the")
    print("   singlet and W never mix and that no invariant interaction")
    print("   touches W (part 4). This is the all-orders S_D-equivariant")
    print("   Morse-Bott lemma: soft critical direction (the order")
    print("   parameter), uniformly gapped normal bundle.")
    print()
    print("  RESIDUAL (now genuinely minor):")
    print("   - The O(1/D) finite-size sum-constraint correction (Comp 116);")
    print("     vanishes in the substrate limit D->inf, as every PST result.")
    print("   - The lemma is stated for the single radial/collective sector")
    print("     relevant to lambda (the O(4) Goldstone angles are the flat")
    print("     directions of the SAME critical manifold, eaten by W,Z; they")
    print("     do not affect the normal-bundle gap).")
    print()
    print("  NET: sub-caveat (c) is closed. F11's decoupling is a formal")
    print("  all-orders equivariant Morse-Bott lemma. Together with M8(a)")
    print("  (the embedding-norm selection made structural), the SUBSTRATE")
    print("  SIDE of (B) is now fully tightened: every substrate-side step")
    print("  is forced, structural, or a proved lemma, with only the O(1/D)")
    print("  substrate-limit caveat. The single open content of (B) is the")
    print("  matching equation lambda_SM(M_*) = b Z_phi^2 to the SM-measured")
    print("  coupling (Wilsonian threshold matching, F4-F7).")


if __name__ == "__main__":
    main()