#!/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())