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

Computation 124 -- displacement covariance is PSD; the Lorentzian signature is
the tau_null sign cut, not isotropy   (the isotropy-forcing relation correction, v28.64)
=============================================================================
Backs the clause appended to the isotropy-forcing relation in the Mosco/A6 section at
v28.64.  Equation the isotropy-forcing relation writes the empirical displacement
covariance
    C^{mu nu} = (1/|D|) sum_a (rho(a)^mu - x^mu)(rho(a)^nu - x^nu)  ->  (d_0^2/4) g^{mu nu}.
But C is a sum of outer products, hence POSITIVE-SEMIDEFINITE, and O(4) isotropy
forces it to the Euclidean form (d_0^2/4) delta^{mu nu}.  A covariance cannot
itself carry the indefinite Lorentzian signature of g^{mu nu}; that signature is
supplied SEPARATELY by the directed modal threshold's tau_null sign cut
(the signed-tension metric, Computation 2 -- hyperbolic well-posedness of the Goldstone wave
equation), and the isotropy-forcing relation is written in that post-signature metric.

This script verifies the three factual pillars of the clause:

  (1) PSD vs INDEFINITE.  The empirical displacement covariance has all
      eigenvalues >= 0 (it is a sum of outer products), whereas the Minkowski
      g = diag(-1,1,1,1) has a negative eigenvalue.  A PSD tensor cannot equal
      (d_0^2/4) g unless g is Euclidean -- the literal the isotropy-forcing relation reads
      in the post-signature metric, isotropy alone delivering only the Euclidean
      magnitude.

  (2) ISOTROPY => delta.  Under O(4)-isotropic displacements the covariance
      converges to (d_0^2/4) delta (eigenvalues all equal, off-diagonal -> 0):
      the only O(4)-covariant rank-2 tensor is delta, a positive Euclidean form,
      with scale d_0^2/4 -- exactly what isotropy fixes, and nothing about the
      signature.

  (3) THE SIGNATURE IS THE tau_null CUT.  The the signed-tension metric sign sgn(T({a,b})-tau_null)
      creates timelike (-) directions; the number of them (the signature) is set
      by tau_null and is INDEPENDENT of the isotropic covariance of (2).  As
      tau_null varies the timelike fraction tracks the cut quantile exactly,
      while the displacement covariance stays PSD throughout.  So the
      indefiniteness is manufactured by the cut, not by isotropy.

A "pass": (1) min eig(C) >= 0 and min eig(Minkowski) < 0; (2) C -> (d_0^2/4)delta
(anisotropy and off-diagonal -> 0 as N grows); (3) timelike fraction == tau_null
quantile across a sweep, with C PSD throughout.
"""

from __future__ import annotations
import numpy as np


def displacement_covariance(N, d0, dim=4, seed=0):
    """Empirical covariance C = (1/N) sum_a xi_a xi_a^T of N O(dim)-isotropic
    displacements with population covariance (d_0^2/4) I_dim."""
    rng = np.random.default_rng(seed)
    xi = rng.normal(0.0, d0 / 2.0, size=(N, dim))     # isotropic, cov (d_0^2/4) I
    return (xi.T @ xi) / N


def main():
    print("=" * 100)
    print("  Computation 124 -- displacement covariance is PSD; signature is the tau_null cut")
    print("=" * 100)
    print()
    d0 = 0.1
    target = d0 ** 2 / 4.0

    # ---- (1) PSD vs indefinite ----------------------------------------------
    print("  (1) PSD (covariance) vs INDEFINITE (Lorentzian g)")
    C = displacement_covariance(2_000_000, d0, dim=4, seed=1)
    eigC = np.sort(np.linalg.eigvalsh(C))
    g = np.diag([-1.0, 1.0, 1.0, 1.0])                  # Minkowski inverse metric
    eigg = np.sort(np.linalg.eigvalsh(g))
    print(f"      eigenvalues of the empirical covariance C (target (d_0^2/4)={target:.4e}):")
    print(f"        {np.array2string(eigC, precision=4, floatmode='fixed')}")
    print(f"        min eig(C) = {eigC[0]:.4e}   (>= 0 => PSD: {eigC[0] >= -1e-12})")
    print(f"      eigenvalues of the Lorentzian g = diag(-1,1,1,1): {eigg.astype(int)}")
    print(f"        min eig(g) = {eigg[0]:+.0f}   (< 0 => INDEFINITE: {eigg[0] < 0})")
    print("      -> a PSD covariance cannot equal (d_0^2/4) g; isotropy gives only the")
    print("         Euclidean magnitude, the isotropy-forcing relation is read post-signature.")
    print()

    # ---- (2) isotropy => delta ----------------------------------------------
    print("  (2) ISOTROPY => (d_0^2/4) delta  (covariance is Euclidean, isotropic)")
    print(f"      {'N':>10}  {'mean diag':>12}  {'max|off-diag|':>14}  {'eig spread (max/min)':>20}")
    for N in (10_000, 100_000, 1_000_000, 10_000_000):
        C = displacement_covariance(N, d0, dim=4, seed=7)
        diag = np.diag(C).mean()
        offmax = np.abs(C - np.diag(np.diag(C))).max()
        ev = np.linalg.eigvalsh(C)
        print(f"      {N:>10d}  {diag:>12.4e}  {offmax:>14.4e}  {ev.max()/ev.min():>20.4f}")
    print(f"      target diagonal (d_0^2/4) = {target:.4e}; off-diagonal -> 0; eig ratio -> 1.")
    print("      -> the only O(4)-covariant rank-2 tensor is delta: isotropy fixes the")
    print("         positive Euclidean form and its scale, nothing about signature.")
    print()

    # ---- (3) the signature is the tau_null cut ------------------------------
    print("  (3) SIGNATURE is the tau_null SIGN CUT (the signed-tension metric), independent of isotropy")
    rng = np.random.default_rng(20260627)
    Npts = 600
    # a generic pairwise tension field (values irrelevant; only the cut matters)
    T = rng.uniform(0.0, 1.0, size=(Npts, Npts)); T = np.triu(T, 1); T = T + T.T
    pairs = T[np.triu_indices(Npts, 1)]
    print(f"      the signed-tension metric g(a,b) = sgn(T-tau_null) |T-tau_null|^(1/2): timelike <=> T < tau_null")
    print(f"      {'tau_null quantile q':>20}  {'timelike fraction':>18}  {'matches q':>10}")
    sweep_ok = True
    for q in (0.1, 0.25, 0.5, 0.75):
        tau_null = np.quantile(pairs, q)
        timelike_frac = np.mean(pairs < tau_null)
        ok = abs(timelike_frac - q) < 0.01
        sweep_ok &= ok
        print(f"      {q:>20.2f}  {timelike_frac:>18.4f}  {str(ok):>10}")
    # meanwhile the displacement covariance is PSD regardless of any cut
    Cpsd = displacement_covariance(1_000_000, d0, dim=4, seed=3)
    psd_throughout = np.linalg.eigvalsh(Cpsd).min() >= -1e-12
    print(f"      timelike fraction tracks the cut quantile exactly: {sweep_ok}")
    print(f"      the displacement covariance stays PSD throughout  : {psd_throughout}")
    print("      -> the indefinite (Lorentzian) directions are manufactured by tau_null,")
    print("         NOT by the isotropic covariance: signature = Computation 2 / the cut.")
    print()

    # ---- assessment ---------------------------------------------------------
    print("=" * 100)
    print("  ASSESSMENT")
    print("=" * 100)
    C = displacement_covariance(10_000_000, d0, dim=4, seed=7)
    ev = np.linalg.eigvalsh(C)
    p1 = (np.linalg.eigvalsh(displacement_covariance(2_000_000, d0, 4, 1)).min() >= -1e-12) and (eigg[0] < 0)
    p2 = (ev.max() / ev.min() < 1.01) and (abs(np.diag(C).mean() - target) / target < 0.02)
    p3 = sweep_ok and psd_throughout
    print(f"  (1) covariance PSD, Lorentzian g indefinite (cannot be equal): {p1}")
    print(f"  (2) isotropy forces the Euclidean (d_0^2/4) delta form        : {p2}")
    print(f"  (3) signature set by the tau_null cut, not the covariance     : {p3}")
    print()
    ok = p1 and p2 and p3
    print(f"  RESULT: {'CONFIRMS the the isotropy-forcing relation correction (covariance PSD/Euclidean; signature = tau_null cut)' if ok else 'MISMATCH -- revise the clause'}")


if __name__ == "__main__":
    main()