Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
r"""v813 -- P1.INDEX.KMS.01: the closed normalization identity c3 = 1/(I_Jones * beta_angle) = 1/(4 * 2pi) = 1/(8pi), its convention audit, and the sharpened P1 contract, one probe (13/13 checks incl. the 2 must-fire controls, ~0.1 s; discovery probe p1_index_kms_probe.py, 2026-08-06, verdict P1-IDENTITY-CLOSED).  THE IDENTITY [E]: I * beta * c3 = 4 * 2pi * 1/(8pi) = 1 EXACT in formal Fraction x pi^k arithmetic (no floats; S2.1-S2.3 are one identity in three rearrangements, counted as one content; the v208 area-entropy factor 1/4 = 1/I; the AX.P1.01 anchor route 1/(2 e1(a) pi) with e1 = 4 aligns anchor-4 with index-4).  THE DEPENDENCY AUDIT (the load-bearing honesty step): the index I = 4 = |mu4| is an INDEPENDENT c3-free theorem (ZERO 'c3' occurrences in all four sources v154/v726/v746/v779, gated by file scan); beta_angle = 2pi is an INDEPENDENT measurement (v526: beta = N clock steps by detailed balance, wrong-beta {N/2, N/4, 2N} excluded; the v519 radian pin is the ONE shared convention and CANCELS in the unit-invariant statement 'response per full modular circle = 1/I'); c3 STAYS the P1 axiom -- the identity EXPLAINS its value, it does not derive it.  THE FINITE EVENNESS PRECURSOR (measured, real): the tracial weight per mu4 deck sector on the l-windows satisfies w_1 = w_3 = 1/4 EXACTLY at every level l = 1..6, with the w_0/w_2 offset the symmetric zero-sum pair +-2^-(l+1) halving per level -- equal quarters asymptotically, NO anomaly offset (= the no-anomaly shape the contract's bridge needs); the index-4 conditional expectation is trace-preserving (dev < 1e-12); and the SEA STATE does NOT equidistribute (deviation 0.188 from 1/4 at the N = 1536 Haar pullback) -- the bridge must normalize the MODULAR RESPONSE, not vacuum weights.  THE CONTRACT (registered [O]): an index-I simple-current extension carries per modular period exactly the normalized response 1/(I * beta_angle) = c3; the ONE open bridge is the even distribution of the modular-response (Quillen/determinant-line) normalization over the I deck sectors; KILL: any asymmetric sector offset (w_1 != w_3 at any level, or a non-vanishing offset as l -> inf).  Controls fire: the wrong index 2 gives 1/(4pi), which the corpus books elsewhere (un-halved Gauss-Bonnet; SdS 1/(4pi) = 2 c3); wrong beta = pi breaks the chain at 1/2.  c3 remains the P1 axiom; no marker move, NO RH claim.  Python-only per GATE.WOLFRAM.02.

PROVENANCE: discovery probe p1_index_kms_probe.py (2026-08-06, 13/13, ~0.1 s, P1-IDENTITY-CLOSED; no spec corrections disclosed); re-run identically at promotion.  Promoted verbatim; the corpus file reads (tfpt_constants/v8/v208/v239/v526/v726/status_ledger + the c3-free scans of v154/v726/v746/v779) resolve against the verification directory (VERI = HERE, the one-line path swap); a module-level _LAST verdict capture inserted (v791 precedent); numbers unchanged; run() encodes the pattern (v757 precedent).

Original p1_index_kms_probe.py docstring (verbatim):
P1.INDEX.KMS.01 -- the closed normalization identity
c3 = 1/(I_Jones * beta_angle) = 1/(4 * 2pi) = 1/(8pi), its convention
audit, and the sharpened P1 contract.

THE THREE CORPUS DATA (located; module names are the anchors):
 (1) I = 4 = |mu4|, the carrier-to-E8 transition index:
     GATE.METRIC.11 / v154_simple_current_theorem.py ("Jones index
     [B:A] = |L| = 4 = |mu4|", KLM, isotropic glue line; C[Z4] Longo
     Q-system v125) and the LOCAL measurement chain
     v726_phys_car_pp_index.py ("Pimsner-Popa / Watatani index 4 on
     the CAR ladder"), v746_phys_gnet_local_functor.py,
     v779_gnet_gns_arf.py.
 (2) beta_angle = 2pi and T_seam = 4 c3:
     v526_seam_thermal_kms_nariai_bridge.py ("beta MEASURED, NOT
     ASSUMED": beta = N clock steps exactly, by the detailed-balance
     Q-polynomial identity; wrong beta {N/2, N/4, 2N} excluded
     exactly; angle units via the v519 Q1.1 pin step = 2pi/N =>
     "beta_angle = 2pi EXACT, T_seam = 1/(2pi) = 4 c3");
     v239_kms_thermal_time.py (Tomita-Takesaki KMS at beta = 1
     exact; the geometric normalisation beta_phys = 2pi with
     "2 pi = 1/(4 c3)"; residual = the QGEO.SYM.01 geometric-boost
     premise); v8_horizon.py / HOR.01 ("1/(2 pi) = 4 c3").
 (3) c3 = 1/(8pi): tfpt_constants.py ("c3 = 1 / (8 * PI)", the P1
     axiom); AX.P1.01 ledger row: declared but not a free knob --
     "c3 = 1/(2 e1(a) pi) = 1/(8pi)" with "e1(a)=4=|mu4|"
     (ANCHOR.GEN.01/v23, the parabolic-anchor route);
     GATE.METRIC.12: the |Z2| halving phrasing "c3=1/(2*4pi)".
 (+) area-entropy factor: v208_bh_thermodynamics.py
     ("entropy-area coefficient 1/4 = 1/|mu_4|") -- 1/4 = 1/I.

PREREGISTERED GATES (frozen before running):
 S1 CORPUS ANCHORS: every frozen substring above is found in its
    module (read-only file reads; exact strings in ANCHORS below).
 S2 IDENTITY CHAIN [E]: in exact formal arithmetic (Fraction
    coefficient x pi^k, pi symbolic -- no floats):
      S2.1  I * beta * c3 = 4 * 2pi * 1/(8pi) = 1 exactly;
      S2.2  c3 = 1/(I * beta) exactly;
      S2.3  T_seam := 4 c3 = 1/(2pi) and T_seam * beta = 1 exactly
            (HONEST UNITS NOTE: S2.1-S2.3 are ONE identity in three
            rearrangements once T_seam := 4 c3 = 1/beta_angle; they
            are gated as arithmetic, counted as one content);
      S2.4  1/4 = 1/I (the v208 entropy coefficient);
      S2.5  the anchor route: 1/(2 * e1 * pi) with e1 = 4 equals c3
            (AX.P1.01) -- the identity aligns anchor-4 with index-4.
 S3 CONVENTION AUDIT (the load-bearing honesty step) -- dependency
    graph with typed edges, each gated on measured source facts:
      S3.1  the index derivation is c3-free and beta-free: the
            sources of v154, v726, v746, v779 contain ZERO
            occurrences of "c3" (gated by file scan);
      S3.2  the beta measurement is independent: v526 contains the
            frozen measured-claim strings (detailed balance, wrong-
            beta exclusion) and the unit pin "v519"; c3 enters v526
            only through the closure READING/selection (the frozen
            anti-numerology lattice line "optionally times c3^k" --
            recorded, typed, reported);
      S3.3  typed shared items (typing, no gate): (i) the RADIAN
            convention: the 2pi in beta_angle and the 8pi in c3 use
            the same angle unit (v519 pin) -- it cancels in the
            unit-invariant statement "per full modular circle the
            normalized response is 1/I"; (ii) the two fours: e1(a)=4
            (parabolic anchor, v23) and [B:A]=4 (Jones index, v154)
            are TWO INDEPENDENT THEOREMS about the same mu4 object
            -- alignment, not circularity; (iii) c3 itself is the P1
            AXIOM: the identity EXPLAINS its value, it does not yet
            DERIVE it (that is the open bridge, S5).
 S4 FINITE EVENNESS MEASUREMENT (real, not bookkeeping): the finite
    modular-response object in hand is the TRACIAL weight per mu4
    deck sector on the l-windows (dim law, v779):
      S4.1  exact law (Fractions, l = 1..6): sector trace weights
            w_c = d_c/4^l satisfy w_1 = w_3 = 1/4 EXACTLY at every
            l, and w_0 - 1/4 = +2^-(l+1), w_2 - 1/4 = -2^-(l+1):
            a SYMMETRIC zero-sum offset pair, halving per level --
            equal quarters asymptotically, NO anomaly offset (the
            kill object would be an asymmetric offset);
      S4.2  operator cross-check at l = 2: constructed sector
            projections have traces (6,4,2,4) and the index-4 E is
            TRACE-PRESERVING (tr o E = tr, dev < 1e-12) -- the
            modular(tracial)-response normalization at finite level;
      S4.3  the SEA STATE does not equidistribute: omega(P_q) at the
            Haar-limit level (N = 1536 pullback) has max deviation
            from 1/4 > 0.05 (measured) -- the evenness is a property
            of the TRACIAL/modular normalization, not of the vacuum
            weights: the contract's bridge must normalize the
            modular response functional, not vacuum expectation.
 S5 THE SHARPENED CONTRACT (deliverable text, printed): see the
    CONTRACT block in the output, with the one open bridge and the
    kill criterion.
 CONTROLS (must fire):
      C1  wrong index I = 2 (the Z2 anchor case): the chain gives
          1/(2 * 2pi) = 1/(4pi) != c3; the corpus explicitly
          associates 1/(4pi) with OTHER objects: the un-halved
          Gauss-Bonnet step (GATE.METRIC.12 "the |Z2| halving
          c3=1/(2*4pi)") and SdS "1/(4pi)=2 c3" (v526/v208).
          Measured leg: the Z2-only average on the l = 2 window has
          lambda = 1/2 => measured index 2 (my v726-style machinery)
          => the wrong chain lands on 1/(4pi) exactly.
      C2  wrong beta = pi (half circle): I * beta_wrong * c3 = 1/2
          != 1 (and v526's measured wrong-beta exclusion {N/2, N/4,
          2N} is the corpus-level version of this control).
 VERDICT ENUM (frozen):
      P1-IDENTITY-CLOSED : S1 + S2 exact + S3 independence gates
          pass (index modules c3-free; beta measured with c3 only in
          the reading) + S4 + controls fire.  The chain is exact,
          the inputs audited independent, the contract carries ONE
          open bridge.
      P1-IDENTITY-CONVENTION : S2 exact but an S3 independence gate
          fails (hidden c3 use found in the index/beta derivations)
          -- the identity is then a consistency check only.
      P1-IDENTITY-BROKEN : any S2 identity fails.

HONESTY: c3 remains the P1 axiom; nothing here derives it.  The
identity organizes one axiom + two independently certified
quantities into the sharpened contract P1.INDEX.KMS.01 whose single
open bridge is the modular-response normalization.  No marker moves.
Exploration only (tfpt-experiment firewall): read-only corpus file
reads; NOT wired into run_all.py; no ledger row; no file writes.

Run:
    experiments/tfpt-discovery/.venv/bin/python \
        experiments/tfpt-discovery/p1_index_kms_probe.py
"""


import hashlib
import inspect
import os
import sys
import time
from fractions import Fraction as Fr
from math import comb

import numpy as np

SEED = 20260806
CHECKS = []
_LAST = {}
HERE = os.path.dirname(os.path.abspath(__file__))
VERI = HERE

ANCHORS = [
    ("tfpt_constants.py", "c3 = 1 / (8 * PI)",
     "P1 axiom line (c3 declared)"),
    ("v8_horizon.py", "1/(2 pi) = 4 c3",
     "HOR.01 horizon unit code"),
    ("v208_bh_thermodynamics.py", "entropy-area coefficient 1/4 = 1/|mu_4|",
     "area-entropy factor 1/4 = 1/I"),
    ("v239_kms_thermal_time.py", "2 pi = 1/(4 c3)",
     "seam unit in the KMS module"),
    ("v239_kms_thermal_time.py", "KMS AT beta = 1",
     "Tomita-Takesaki beta = 1 exact"),
    ("v239_kms_thermal_time.py", "QGEO.SYM.01",
     "the geometric-boost residual, typed open"),
    ("v526_seam_thermal_kms_nariai_bridge.py", "beta MEASURED, NOT ASSUMED",
     "beta = N clock steps, detailed balance"),
    ("v526_seam_thermal_kms_nariai_bridge.py", "beta_angle = 2pi EXACT",
     "angle units via the v519 pin"),
    ("v526_seam_thermal_kms_nariai_bridge.py", "{N/2,N/4,2N} excluded",
     "wrong-beta exclusion, measured"),
    ("v526_seam_thermal_kms_nariai_bridge.py", "v519",
     "the angle-unit pin (step = 2pi/N)"),
    ("v526_seam_thermal_kms_nariai_bridge.py", "SdS 1/(4pi)=2 c3",
     "1/(4pi) is a DIFFERENT corpus object (SdS)"),
    ("v726_phys_car_pp_index.py", "Watatani index 4",
     "local index-4 measurement chain"),
    ("status_ledger.csv", "Jones index [B:A] = |L| = 4 = |mu4|",
     "GATE.METRIC.11 / v154 transition index"),
    ("status_ledger.csv", "c3 = 1/(2 e1(a) pi) = 1/(8pi)",
     "AX.P1.01 anchor route for the 8"),
    ("status_ledger.csv", "e1(a)=4=|mu4|",
     "anchor four = |mu4| (v23)"),
    ("status_ledger.csv", "c3=1/(2*4pi)",
     "GATE.METRIC.12 |Z2| halving phrasing"),
]
INDEX_MODULES_C3FREE = [
    "v154_simple_current_theorem.py", "v726_phys_car_pp_index.py",
    "v746_phys_gnet_local_functor.py", "v779_gnet_gns_arf.py"]


def check(name, ok, detail=""):
    CHECKS.append((name, bool(ok)))
    print(("PASS " if ok else "FAIL ") + name
          + ("  -- " + detail if detail else ""))
    return ok


# ---- exact formal arithmetic: value = (Fraction q, int k) = q * pi^k

def pmul(a, b):
    return (a[0] * b[0], a[1] + b[1])


def pinv(a):
    return (1 / a[0], -a[1])


I_JONES = (Fr(4), 0)
BETA = (Fr(2), 1)          # 2 * pi^1
C3 = (Fr(1, 8), -1)        # (1/8) * pi^-1
ONE = (Fr(1), 0)


# ---- minimal ladder machinery (as in the sibling probes) -------------

def spower(N, k):
    P = np.zeros((N, N))
    for j in range(N):
        P[(j + k) % N, j] = (-1.0) ** ((j + k) // N)
    return P


def covariance(N):
    th = 2 * np.pi * (np.arange(N) + 0.5) / N
    th = np.mod(th + np.pi, 2 * np.pi) - np.pi
    th[np.isclose(th, -np.pi)] = np.pi
    F = np.exp(1j * np.outer(np.arange(N), th)) / np.sqrt(N)
    occ = (th < 0).astype(float)
    return (F * occ) @ F.conj().T


def window(N, p, l):
    return [(p + i) % N for i in range(l)] + \
           [(p + N // 2 + i) % N for i in range(l)]


def haar_iso(N):
    V = np.zeros((2 * N, N))
    for j in range(N):
        V[2 * j, j] = V[2 * j + 1, j] = 1.0 / np.sqrt(2.0)
    return V


def jw_ops(n):
    sm = np.array([[0, 1], [0, 0]], dtype=complex)
    sz = np.array([[1, 0], [0, -1]], dtype=complex)
    I2 = np.eye(2, dtype=complex)
    ops = []
    for j in range(n):
        m = np.array([[1.0]], dtype=complex)
        for l in range(n):
            m = np.kron(m, sz if l < j else (sm if l == j else I2))
        ops.append(m)
    return ops


def gamma_partial(Hsub, idx, cops):
    mu, V = np.linalg.eigh(1j * Hsub)
    dim = cops[0].shape[0]
    U = np.eye(dim, dtype=complex)
    for i in range(len(idx)):
        d = sum(np.conj(V[j, i]) * cops[idx[j]] for j in range(len(idx)))
        n_i = d.conj().T @ d
        ev = -1j * mu[i]
        U = U @ (np.eye(dim) + (ev - 1) * n_i)
    return U


def gaussian_rho(CW, cops):
    lam, V = np.linalg.eigh(CW)
    lam = np.clip(lam.real, 0.0, 1.0)
    dim = cops[0].shape[0]
    rho = np.eye(dim, dtype=complex)
    for i in range(len(cops)):
        d = sum(np.conj(V[j, i]) * cops[j] for j in range(len(cops)))
        rho = rho @ ((1 - lam[i]) * np.eye(dim)
                     + (2 * lam[i] - 1) * (d.conj().T @ d))
    return rho


def sector_projs(U):
    return [sum((1j ** (-q * j)) * np.linalg.matrix_power(U, j)
                for j in range(4)) / 4 for q in range(4)]


def lam_of(B, Efun):
    A = Efun(B)
    a, W = np.linalg.eigh((A + A.conj().T) / 2)
    keep = a > 1e-11 * max(a.max(), 1.0)
    Ws = W[:, keep] / np.sqrt(a[keep])
    M = Ws.conj().T @ B @ Ws
    return 1.0 / float(np.linalg.eigvalsh((M + M.conj().T) / 2).max().real)


def onb_of(P):
    w, W = np.linalg.eigh((P + P.conj().T) / 2)
    return W[:, w > 0.5]


def sha_freeze():
    fns = [pmul, pinv, spower, covariance, window, haar_iso, jw_ops,
           gamma_partial, gaussian_rho, sector_projs, lam_of, onb_of]
    blob = "".join(inspect.getsource(f) for f in fns)
    blob += repr((SEED, ANCHORS, INDEX_MODULES_C3FREE, I_JONES, BETA, C3))
    return hashlib.sha256(blob.encode()).hexdigest()


def main():
    t0 = time.time()
    print("p1_index_kms_probe -- c3 = 1/(I_Jones * beta_angle): identity, "
          "audit, contract\n")
    print(f"SHA-freeze: {sha_freeze()[:32]}\n")

    # ============ S1: corpus anchors ==================================
    print("-- S1: located corpus anchors (read-only file scans) --")
    srcs = {}
    ok_anch = True
    for fname, needle, why in ANCHORS:
        path = os.path.join(VERI, fname)
        if fname not in srcs:
            with open(path, encoding="utf-8") as fh:
                srcs[fname] = fh.read()
        hit = needle in srcs[fname]
        ok_anch &= hit
        print(f"   [{'ok' if hit else 'MISSING'}] {fname}: "
              f'"{needle}"  ({why})')
    check("S1 all frozen corpus-anchor strings found in their modules",
          ok_anch)

    # ============ S2: the identity chain (exact, formal pi) ===========
    print("\n-- S2: the identity chain [E] (Fraction x pi^k, no floats) --")
    chain = pmul(pmul(I_JONES, BETA), C3)
    check("S2.1 I * beta * c3 = 4 * 2pi * 1/(8pi) = 1 EXACT",
          chain == ONE, f"= {chain[0]} * pi^{chain[1]}")
    check("S2.2 c3 = 1/(I * beta) EXACT",
          C3 == pinv(pmul(I_JONES, BETA)))
    T_seam = pmul((Fr(4), 0), C3)
    check("S2.3 T_seam := 4 c3 = 1/(2pi) and T_seam * beta = 1 EXACT "
          "(units note: S2.1-S2.3 are ONE identity, three "
          "rearrangements -- counted as one content)",
          T_seam == pinv(BETA) and pmul(T_seam, BETA) == ONE,
          f"T_seam = {T_seam[0]} * pi^{T_seam[1]}")
    check("S2.4 area-entropy factor 1/4 = 1/I (v208 coefficient)",
          Fr(1, 4) == 1 / I_JONES[0])
    anchor_route = pinv(pmul((Fr(2), 1), (Fr(4), 0)))   # 1/(2 e1 pi)
    check("S2.5 anchor route 1/(2 e1(a) pi), e1 = 4, equals c3 "
          "(AX.P1.01): the identity aligns anchor-4 with index-4",
          anchor_route == C3)

    # ============ S3: the convention audit ============================
    print("\n-- S3: convention audit / dependency graph --")
    ok_free = True
    for fname in INDEX_MODULES_C3FREE:
        with open(os.path.join(VERI, fname), encoding="utf-8") as fh:
            n_c3 = fh.read().count("c3")
        ok_free &= n_c3 == 0
        print(f"   {fname}: c3 occurrences = {n_c3}")
    check("S3.1 the index derivation is c3-free and beta-free: ZERO "
          "'c3' occurrences in v154/v726/v746/v779", ok_free)
    v526 = srcs["v526_seam_thermal_kms_nariai_bridge.py"]
    lattice_line = "optionally times c3^k" in v526
    check("S3.2 the beta measurement is independent: measured-claim "
          "strings + wrong-beta exclusion + v519 unit pin present "
          "(S1); c3 enters v526 only via the closure READING/frozen "
          "selection lattice", lattice_line,
          "lattice line found: c3 is a candidate in the class-1 "
          "selection, not an input to the beta measurement")
    print("""   DEPENDENCY GRAPH (edges typed):
     I = 4    <- v154 (KLM, [B:A]=|L|) + v726/v746/v779 (Watatani/PP,
                measured exact)          [INDEPENDENT THEOREM: no c3,
                no beta in any source]
     beta     <- v526 (beta = N steps MEASURED, detailed balance;
                wrong beta excluded)      [INDEPENDENT MEASUREMENT]
                x v519 pin (step = 2pi/N) [SHARED CONVENTION: radian
                unit -- cancels in the unit-invariant statement
                'response per full modular circle = 1/I']
     c3       <- P1 AXIOM (tfpt_constants) [DECLARED]; anchor route
                c3 = 1/(2 e1(a) pi), e1(a)=4 (v23) [ALIGNED: the two
                fours are two independent theorems (parabolic anchor
                vs Jones index) about the SAME mu4 object -- an
                alignment, not a circularity]
     identity I*beta*c3 = 1 [CONSISTENCY, exact]: it EXPLAINS the
                axiom's value; it does not yet DERIVE it (open
                bridge, S5).""")

    # ============ S4: finite evenness measurement =====================
    print("\n-- S4: per-sector modular(tracial) response, finite level --")
    ok_law = True
    print(f"{'l':>3} {'w_0 - 1/4':>12} {'w_1 - 1/4':>12} "
          f"{'w_2 - 1/4':>12} {'w_3 - 1/4':>12}")
    for l in range(1, 7):
        dc = [0, 0, 0, 0]
        for a in range(l + 1):
            for b in range(l + 1):
                dc[(a - b) % 4] += comb(l, a) * comb(l, b)
        w = [Fr(d, 4 ** l) for d in dc]
        dev = [x - Fr(1, 4) for x in w]
        ok_law &= dev[1] == 0 and dev[3] == 0
        ok_law &= dev[0] == Fr(1, 2 ** (l + 1))
        ok_law &= dev[2] == -Fr(1, 2 ** (l + 1))
        ok_law &= sum(dev, Fr(0)) == 0
        print(f"{l:>3} " + " ".join(f"{str(d):>12}" for d in dev))
    check("S4.1 EXACT evenness law (l = 1..6): w_1 = w_3 = 1/4 exactly;"
          " w_0/w_2 offset = +-2^-(l+1), symmetric and zero-sum, "
          "halving per level -- equal quarters asymptotically, NO "
          "anomaly offset", ok_law)
    N0 = 48
    win0 = window(N0, 0, 2)
    HW4 = spower(N0, N0 // 2)[np.ix_(win0, win0)]
    c4 = jw_ops(4)
    U4 = gamma_partial(HW4, list(range(4)), c4)
    Ps = sector_projs(U4)
    trs = [float(np.trace(P).real) for P in Ps]

    def E4(x):
        return sum(P @ x @ P for P in Ps)

    rng = np.random.default_rng(SEED)
    dev_tr = 0.0
    for _ in range(6):
        A = rng.normal(size=(16, 16)) + 1j * rng.normal(size=(16, 16))
        dev_tr = max(dev_tr, abs(np.trace(E4(A)) - np.trace(A)))
    check("S4.2 operator cross-check (l = 2): sector traces (6,4,2,4) "
          "and tr o E = tr (E trace-preserving: the finite "
          "modular-response normalization)",
          np.allclose(trs, [6, 4, 2, 4], atol=1e-9) and dev_tr < 1e-12,
          f"traces = {[round(t, 6) for t in trs]}, dev = {dev_tr:.1e}")
    chain_m = np.eye(N0)
    Nk = N0
    for _ in range(5):
        chain_m = haar_iso(Nk) @ chain_m
        Nk *= 2
    CW = (chain_m.T @ covariance(Nk) @ chain_m)[np.ix_(win0, win0)]
    rho = gaussian_rho(CW, c4)
    wq = [float(np.real(np.trace(rho @ P))) for P in Ps]
    dev_sea = max(abs(w - 0.25) for w in wq)
    check("S4.3 the SEA STATE does not equidistribute: omega(P_q) at "
          f"the N = {Nk} Haar pullback deviates from 1/4 by "
          f"{dev_sea:.3f} > 0.05 -- evenness is a TRACIAL/modular "
          "property, not a vacuum property (sharpens the bridge)",
          dev_sea > 0.05,
          "omega(P_q) = " + str([round(w, 4) for w in wq]))

    # ============ S5: the sharpened contract ==========================
    print("""
-- S5: P1.INDEX.KMS.01, the sharpened contract (report only) --
  STATEMENT.  An index-I simple-current extension (here I = 4 =
  |mu4|, GATE.METRIC.11) carries per modular period beta_angle
  exactly the normalized response 1/(I * beta_angle); with the
  measured beta_angle = 2pi (v526, one full seam circle) this VALUE
  is c3 = 1/(8pi) -- the P1 axiom as the index-KMS normalization.
  PROOF NEEDS (the one open bridge): (a) the modular response
  functional's normalization -- the Quillen/determinant-line answer
  per modular period distributing EVENLY over the I deck sectors
  with NO anomaly offset (the finite precursor is measured here:
  S4.1 symmetric zero-sum offset vanishing at 2^-l; S4.2 the
  trace-preserving index-4 E).  ALREADY CORPUS THEOREMS: the index
  (v154/v125 + v726/v746/v779), the angle (v526 measured, v239
  Tomita beta = 1), the tracial evenness precursor (v779 dim law).
  STILL OPEN BESIDES THE BRIDGE: the geometric-boost identification
  (QGEO.SYM.01, typed [O] in v239).
  KILL CRITERION: an anomaly offset making the per-sector
  distribution UNEVEN -- concretely, a sector weight asymmetry that
  is NOT the symmetric zero-sum +-2^-(l+1) pair (e.g. w_1 != w_3 at
  any level, or a non-vanishing offset as l -> inf): any such
  measurement kills the bridge and the contract.""")

    # ============ Controls ============================================
    print("-- Controls (must fire) --")
    v2m = np.zeros(16, dtype=complex)
    for P in Ps:
        o = onb_of(P)
        if o.shape[1]:
            v2m += o[:, 0]
    v2m /= np.linalg.norm(v2m)

    def EZ2(x):
        UU = U4 @ U4
        return (x + UU @ x @ UU.conj().T) / 2

    lam2 = lam_of(np.outer(v2m, v2m.conj()), EZ2)
    I_wrong = (Fr(round(1 / lam2)), 0)
    wrong_c3 = pinv(pmul(I_wrong, BETA))
    check("C1 fires: measured Z2 index = 2 (lambda = 1/2) gives "
          "1/(2*2pi) = 1/(4pi) != c3; corpus books 1/(4pi) elsewhere "
          "(un-halved Gauss-Bonnet, GATE.METRIC.12; SdS 1/(4pi) = "
          "2 c3, v526/v208)",
          abs(lam2 - 0.5) < 1e-8 and wrong_c3 == (Fr(1, 4), -1)
          and wrong_c3 != C3 and wrong_c3 == pmul((Fr(2), 0), C3),
          f"lambda = {lam2:.10f}, wrong chain = "
          f"{wrong_c3[0]} * pi^{wrong_c3[1]}")
    beta_wrong = (Fr(1), 1)
    check("C2 fires: wrong beta = pi (half circle) breaks the chain: "
          "I * beta_wrong * c3 = 1/2 != 1 (corpus-level version: "
          "v526's exact wrong-beta exclusion {N/2, N/4, 2N})",
          pmul(pmul(I_JONES, beta_wrong), C3) == (Fr(1, 2), 0)
          and pmul(T_seam, beta_wrong) != ONE)

    # ============ verdict =============================================
    res = dict(CHECKS)
    s1 = res.get("S1 all frozen corpus-anchor strings found in their "
                 "modules", False)
    s2 = all(v for n, v in CHECKS if n.startswith("S2"))
    s3 = all(v for n, v in CHECKS if n.startswith("S3"))
    s4 = all(v for n, v in CHECKS if n.startswith("S4"))
    ctrl = all(v for n, v in CHECKS if n.startswith("C"))
    if not s2:
        verdict = "P1-IDENTITY-BROKEN"
    elif s1 and s2 and s3 and s4 and ctrl:
        verdict = "P1-IDENTITY-CLOSED"
    else:
        verdict = "P1-IDENTITY-CONVENTION" + ("" if ctrl
                                              else " (CONTROL-VOID)")
    _LAST["verdict"] = verdict
    n_pass = sum(1 for _, ok in CHECKS if ok)
    print(f"\n{n_pass}/{len(CHECKS)} checks pass -- VERDICT: {verdict}"
          f"   ({time.time() - t0:.1f} s)")
    print("""
TYPED SUMMARY: the chain 4 * 2pi * 1/(8pi) = 1 is exact; the index
is an independent c3-free theorem; beta is an independent
measurement (radian unit = the one shared convention, cancelling in
the unit-invariant form); c3 stays the P1 axiom -- the identity
EXPLAINS its value and the contract P1.INDEX.KMS.01 names the one
open bridge (even modular-response distribution, kill = anomaly
offset).  The finite evenness precursor is measured (symmetric
zero-sum offset, vanishing at 2^-l; E trace-preserving; the sea
state does NOT equidistribute -- the bridge is about the modular
response, not vacuum weights).  No marker moves; exploration only.""")
    return 0 if n_pass == len(CHECKS) else 1


def run():
    """run_all entry point (v757 precedent): expected pattern 13/13 with
    verdict P1-IDENTITY-CLOSED."""
    rc = main()
    n_pass = sum(1 for _n, ok in CHECKS if ok)
    fails = [n for n, ok in CHECKS if not ok]
    v = _LAST.get("verdict", "")
    ok = (rc == 0 and n_pass == len(CHECKS) == 13 and not fails
          and v == "P1-IDENTITY-CLOSED")
    print("\n[%s] PATTERN GATE: expected 13/13 with verdict "
          "P1-IDENTITY-CLOSED; got %d/%d, fails: %s, verdict: %s"
          % ("PASS" if ok else "FAIL", n_pass, len(CHECKS),
             fails or "none", v))
    print("\nCOMBINED ADJUDICATION: %s -- P1-IDENTITY-CLOSED: I * "
          "beta * c3 = 1 exact; the index is an independent c3-free "
          "theorem, beta an independent measurement (the radian pin "
          "cancels), c3 stays the P1 axiom the identity EXPLAINS; the "
          "finite evenness precursor is measured (w_1 = w_3 = 1/4 "
          "exactly, symmetric zero-sum offset +-2^-(l+1), E trace-"
          "preserving) and the sea state does NOT equidistribute -- "
          "the one open bridge is the even modular-response "
          "normalization, kill = any anomaly offset.  Contract "
          "P1.INDEX.KMS.01 registered open.  NO RH claim."
          % ("PASS" if ok else "FAIL"))
    return 0 if ok else 1


if __name__ == "__main__":
    raise SystemExit(run())

a10437548d7ac68313b25bf7edd6f07edc3e270baec4921357de004cfa1bf594

The public registry includes 116 document views, 6,693 section records and 985 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.