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

PST Computation 140, Robustness of the emergence chain to the form of T(C)
===========================================================================
Advances the T(C)-underdetermination open item (Comp 126) by formalising and
verifying the robustness claim of sec:asymmetric-tension: the downstream
structural derivations read the tension functional T(C) only through

    (i)  the DESCENDING ORDER of the magnitudes |T(C)|  (the instantiation
         sequence), and
    (ii) the single global bias direction  tau_hat = T(D)/|T(D)|,

and nothing else, so the remaining freedom in T(C) is inert.

Formalisation. Let phi be any strictly increasing reparametrisation of the
magnitude scale with phi(0)=0, and let T' have |T'(C)| = phi(|T(C)|) with the
same directions (so tau_hat' = tau_hat). Then at matched threshold
tau' = phi(tau):
    (R1) the instantiation sequence (configs by descending magnitude) is identical;
    (R2) the phase sign(|T(C)| - tau) of every configuration is identical
         (each config is instantiated iff it was before);
    (R3) tau_hat is identical.
A NON-order-preserving reassignment of the magnitudes DOES change the sequence,
confirming the ORDINAL content is the load-bearing part.

Exact invariance (monotone maps preserve order and sign) -> PROVENANCE PROOF.
SCOPE: this proves the reads IDENTIFIED in the body factor through (i)+(ii); a
complete closure would audit every derivation, for which this fixes the exact
invariance criterion, it does not derive a canonical T(C) (still open, Comp 126).
"""
import numpy as np

rng = np.random.default_rng(0)


def sequence(mag):
    return tuple(np.argsort(-mag, kind="stable"))     # descending-magnitude order


def phase(mag, tau):
    return np.sign(mag - tau)


nD = 6
dim = 2 ** nD
all_ok = True
for _ in range(300):
    mag = rng.random(dim)                              # |T(C)| on P(D)
    tau = float(np.quantile(mag, rng.uniform(0.2, 0.8)))
    v = rng.standard_normal(7)
    tau_hat = v / np.linalg.norm(v)                    # global bias direction

    # strictly increasing reparametrisation phi, phi(0)=0 (order-preserving)
    a = rng.uniform(0.5, 3.0)
    c = rng.uniform(0.2, 2.0)
    phi = lambda x: np.expm1(c * x ** a)
    mag2 = phi(mag)
    tau2 = phi(tau)

    r1 = sequence(mag) == sequence(mag2)
    r2 = np.array_equal(phase(mag, tau), phase(mag2, tau2))
    # (R3) is NOT tested here, and the earlier np.allclose(tau_hat, tau_hat) was a
    # value compared with itself dressed up as a check.  phi reparametrises the
    # MAGNITUDES |T(C)|; tau_hat is built from the directions v(a), which phi does
    # not act on at all.  So (R3) holds by construction rather than by measurement,
    # and it is reported as structural below instead of being counted as verified.
    all_ok &= (r1 and r2)

# counter-check: reassigning magnitudes across configs (a permutation, NOT a
# reparametrisation of the same values) changes the sequence for essentially all perms
diffs = 0
for _ in range(300):
    mag = rng.random(dim)
    perm = rng.permutation(dim)
    if sequence(mag) != sequence(mag[perm]):
        diffs += 1
counter_ok = diffs > 295

print(f"300 random (T, phi): (R1) sequence and (R2) phase invariant under monotone "
      f"reparametrisation: {all_ok}")
print("  (R3) tau_hat: structural, not measured.  phi reparametrises the magnitudes")
print("  |T(C)|; tau_hat is built from the directions v(a), which phi does not act on,")
print("  so there is nothing for a reparametrisation to move.  It is stated here rather")
print("  than counted among the verified invariances.")
print(f"counter-check: reassigning magnitudes changes the sequence in {diffs}/300 trials "
      f"(ordinal content is load-bearing): {counter_ok}")
print("\nRESULT:", "PASS  (the reads AUDITED HERE depend only on |T| ordering + tau_hat.  This does "
      "NOT cover the Born-consistency comparison of eq:tau-i-explicit, which tests "
      "tau_0 against Born weights carrying their own normalisation and is therefore "
      "a CARDINAL read; see comp 143.)" if (all_ok and counter_ok) else "FAIL")