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v908_seam_equilibrium_wiring.py
verification/v908_seam_equilibrium_wiring.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
r"""v908 -- SEAM.CFIN.MARGINAL.STABILITY.01 + SEAM.CFIN.NESS.PARENT.01: THE SEAM EQUILIBRIUM WIRING -- the physics closure of the round-60 seam questions: strict 2-cycle reflection positivity is IMPOSSIBLE on the whole C6-covariant class (an exact +-a_J law, not a scan), the strict-collar obstruction is a TWO-SEAT LINEAR LAW of the coupling whose kernel is the {J, Z} coordinates, the deployed seam wiring is PURE-I = a maximally obstructed covariant direction, and EQUILIBRIUM J/Z-coupling witnesses carry the full 1/200 mixing at ZERO entropy production -- NO NESS IS NEEDED, ONE module from two probes (22/22 + 25/25 checks, zero fails, verdicts MARGSTAB-MEASURED (INTERIOR-EMPTY(+-a_J and det = -a_J^2 exact) / STRATUM-NOT-ISOLATED / NOT-A-CLOSURE-POINT / ESCAPE-NONCOVARIANT(X seat, price 2 eps, exponent 2) / FLOOR-EXCHANGE(Pf4 = (eps - 1/200)(eps + 1/200))) + NESSPARENT-MEASURED (PRICE-ZERO(equilibrium strict-collar witnesses V_J / V_Z) / TWO-SEAT-LAW(rank 12, kernel 12 = {J, Z}; deployed coupling PURE-I) / CANONICAL-SELECTS-UNIFORM-RAY(mixtures 4/10) / FINITE-NESS-NOGO(stationary => sigma == 0, exact) / DRIVE-RP-NEUTRAL / NESS-NOT-FORCED); discovery probes seam_marginal_stability_probe.py and seam_ness_parent_probe.py (round 60), 2026-08-10, re-run identically at promotion, embedded BYTE-EXACT and executed verbatim, ~1 s). (1) THE 2-CYCLE SIDE (seam_marginal_stability, 22/22): the closure-vs-isolated dilemma of the v903 marginal witness is REFUTED AS A DILEMMA -- within the C6-covariant class the theta_abT sector Grams see ONLY the channel-{4,5} 4x4 seat, covariance reduces the seat law M1 = [[aJ+aX, aI-aZ],[aI+aZ, -aJ+aX]] exactly to M1 = diag(a_J, -a_J) with det M2 = -a_J^2: STRICT theta_abT RP IS IMPOSSIBLE ON THE WHOLE COVARIANT CLASS (the RP set = {a_J = 0} = the cone boundary, interior EMPTY); the marginal witness is NOT a limit of covariant strict equilibria (there are none) and NOT isolated (the whole s = 0 family is a positive-dimensional MARGINAL STRATUM, margin exactly 0 on all 16 CAR-valid lattice points at full 10/10 canonical mixing; the decisive scalar sup(RP-margin x mixing) = 0 EXACT); the ONLY hermiticity-preserving strict-enabling escape is the covariance-BREAKING X coordinate: at every eps > 0 the parent is strictly RP with covariance defect EXACTLY 2 eps (linear price law), mixing 10/10 unchanged, margin exponent 2.000 -- TRANSVERSAL; and the sharpest exact object, FLOOR-EXCHANGE: Pf4_{45}(eps) = (eps - 1/200)(eps + 1/200) with the effective 2-cycle odd Gram diag(eps + 1/200, eps - 1/200) EXACTLY -- the canonical sign gate IS the negative effective RP margin (the same polynomial factor): one level down the exclusion survives the transversal escape, with the crossover exactly at the 1/200 floor. (2) THE COLLAR SIDE, NO NESS NEEDED (seam_ness_parent, 25/25): strict theta_S hermiticity is a TWO-SEAT LINEAR LAW of the coupling (the 1p seat A_{x+1,y} + A_{x,y+1} kills the X-, the deg-2 empty seat A_{x,y} + A_{x+1,y+1} the I-subblock coordinates; exact rank 12, kernel 12 = the {J, Z} coordinates of the 24-dim covariant coupling space = the v898 T-count); the DEPLOYED seam wiring V = A_int[C, B] is PURE-I (every 2x2 subblock exactly -I2) = a MAXIMALLY obstructed covariant direction, which explains the v903 strict-collar law (30 entries, magnitude 2t) EXACTLY as 15 mixed pairs x 2 Gram positions; the kernel contains EQUILIBRIUM witnesses: the uniform J-coupling parent V_J (kappa = m = 1/2, t = 1/20) passes strict theta_S RP in EXACT arithmetic (Grams exactly hermitian, PD by exact LDL) while its Schur compression carries the FULL canonical Pfaffian mixing -- S_J = V_J J3 V_J^T = 3J on ALL 25 channel blocks (integer identity), J coordinate EXACTLY t^2 3m/(1 - m^2) = 1/200 = the same rational floor identity as the v903 witness, Pf4 canonical 10/10; V_Z is a SECOND witness (S_Z = -3J, J coordinate -1/200); THE NESS SIDE HONEST (FINITE-NESS-NOGO): at finite size every exactly stationary quasi-free state has [h, A] = 0 => ALL block currents vanish => sigma == 0 EXACTLY -- 'NESS with positive entropy production' is category-inapplicable, the weakest meaningful positivity = RP of the stationary covariance, exactly what is tested; the two-temperature Cesaro states are exactly stationary, covariant, CAR-valid, and DRIVE BUYS NO RP (the theta_S hermiticity defect stays PINNED at 0.0982 while the PSD margin degrades monotonically; the mixing gates are open at ZERO drive): the minimal entropy production at which the mixing gates open is EXACTLY ZERO -- NESS-NOT-FORCED, the demand SEAM.CFIN.NESS.PARENT.01 closes as NOT-NEEDED; kernel membership is NECESSARY, not sufficient (J/Z mixtures pass hermiticity + PD but break the canonical sign census 4/10 -- the canonical gate selects the uniform J or Z ray). (3) THE REGISTERED NEXT CONTRACT (prose-level, named here): SEAM.STATE.WIRING.SELECTOR.01 -- is PURE-I compiler-FORCED? The deployed IOTA wiring comes from the vacuum-orbit construction of A_int; whether the orbit/edge rules admit a {J, Z}-kernel alternative at equal C6 covariance and equal Pf pencil, or pin the I direction, is exactly answerable -- only then is it decidable whether the strict-collar obstruction is a compiler theorem or a deployment choice. THE WITNESSES LIVE ON A DIFFERENT COVARIANT COUPLING DIRECTION THAN THE DEPLOYED A_int WIRING -- whether the actual seam allows them is untouched; the v898/v903 [O] premise stays [O], NO marker moves. CONTROLS: the deliberately-RP-violating I direction fires for both signs (1p +-|eps| exact), Z breaks 1p hermiticity, eps < 0 flips both eigenvalues; the exact 1/200 identity regresses rationally; 3/3 seeded scrambles break the covariance ward and the canonical census; global-KMS regression reproduces the round-58/59 equilibrium obstruction (defect 0.0982); PURE-I regression (0.1, 30 entries); AST firewalls clean; RNG only in controls. Zero RH content.
PROVENANCE: discovery probes seam_marginal_stability_probe.py
(22/22, MARGSTAB-MEASURED, SPEC v2 with ONE disclosed amendment,
fail-first preserved -- the v1 gate 'all 18 s=0 lattice points
CAR-valid' fell at exactly the two m = 3/4 corners, v2 reads the
16 CAR-valid points, the marginality itself held on all 18; round
60 note CXXXVIII, Spec-SHA 75ead8ae..., 2026-08-10) and
seam_ness_parent_probe.py (25/25, NESSPARENT-MEASURED, SPEC v1
without amendments, round 60 note CXXXIX, Spec-SHA d40622af...,
2026-08-10), re-run identically at promotion. ROUND-31 EMBEDDING
CONVENTION: frozen sources embedded BYTE-EXACT, executed verbatim
in isolated namespaces; printed spec SHAs reproduce; byte-equality
ward vs experiments/tfpt-discovery/ inside the pattern gates.
Both probes are exact-arithmetic (sympy/Fraction) on the v898
16-dim one-particle space and reproduce the v898/v903 laws as
wards.
FIREWALL: candidate-level statements on the explicit covariant
family -- whether the ACTUAL seam realizes any witness is
untouched; the v898 [O] premise is unmoved; the wiring-selector
question is REGISTERED, not answered; zero RH content.
Python-only per GATE.WOLFRAM.02.
"""
import contextlib
import io
import os
import re
import sys
import time
import types
_HERE = os.path.dirname(os.path.abspath(__file__))
if _HERE not in sys.path:
sys.path.insert(0, _HERE)
# ------------- frozen probe source seam_marginal_stability_probe (embedded BYTE-EXACT, raw string)
_SRC_0 = r'''
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
r"""seam_marginal_stability_probe -- SEAM.CFIN.MARGINAL.STABILITY.01
(EXPLORATION ONLY, experiments/; round 60, 2026-08-10, Probe 9 --
the marginal dilation witness on the RP cone boundary: is it a
LIMIT of strictly-RP parents (a closure point -- the mixing would
be 'physical as a limit of equilibria') or an ISOLATED boundary
point (a genuine non-equilibrium source unavoidable)?)
THE QUESTION. Probe 7 (rp_parent_dilation_probe, round 59; now
promoted inside v903) measured that under the 2-cycle reflection
theta_abT every s = 0 parent of the family A_par(kappa, m, t1, t2,
s) is MARGINALLY reflection-positive (1p Gram eigenvalues exactly
{0, 0} -- the cone boundary) while its Schur compression carries
the full Pfaffian mixing with the exact identity t^2 3m/(1-m^2) =
1/200. Round 59 left open whether that marginal witness is
approachable from the STRICT-RP side. This probe freezes the
question as (a) a neighborhood parametrization (all five family
parameters PLUS the complete enlargement the covariance seat
allows), (b) the decisive scalar sup(RP-margin x mixing) with the
approach exponent, (c) the exact anatomy of the {0, 0} kernel.
FEASIBILITY / REDUNDANCY CHECK (done against the corpus FIRST,
2026-08-10): probe 7 measured the marginal witness set and the
parent a_J = s identity but never asked the stability/limit
question, never parametrized the seat enlargement, and never
computed the kernel; round 58 (seam_state_derivation_probe) proved
the +-|a_J| odd-Gram identity on the KMS family h(u, t) but not on
the coupled parents and not as a WHOLE-CLASS covariance statement;
v903 promotes all of it verbatim with the witness typed 'ON the
cone boundary (not strict)'. NOTHING in the corpus decides
closure-point vs isolated. That is exactly this probe.
SMOKE-RUN DISCLOSURE (2026-08-10, one declared smoke round before
freezing; the frozen claims below record what the smoke measured
-- the answer REFUTES THE DICHOTOMY of the question and the frozen
claims say so):
(i) the theta_abT sector Grams are functions of the channel
{4,5} 4x4 covariance sub-block ONLY (the 'visible seat');
on that seat C6 covariance allows EXACTLY the J coordinate
of the {4,5} cross block (integer covariance defect 2 for
the I, X, Z unit directions, 0 for J) and forces the two
intra-channel coordinates equal (c4 = c5);
(ii) the symbolic seat law: M1 = [[aJ + aX, aI - aZ],
[aI + aZ, -aJ + aX]]; Hermiticity forces aZ = 0 (1p) and
c4 = c5 (deg-2); the Hermitian-admissible 1p eigenvalues
are aX +- sqrt(aJ^2 + aI^2), and det M2 = aX^2 - aJ^2 (at
aI = 0) -- so on the COVARIANT class (aI = aX = aZ = 0)
the eigenvalues are +-aJ and det M2 = -aJ^2: STRICT RP is
IMPOSSIBLE on the whole class, the RP set is EXACTLY
{a_J = 0} = the boundary; the strict side is EMPTY;
(iii) the witness kernel: M1 == 0 identically (both a-side
Majorana directions), and the deg-2 Gram [[1, i kappa],
[-i kappa, kappa^2]] has the exact 1-dim kernel spanned by
(-i kappa, 1) -- vacuum tied to the a-pair; both kernels
are forced by the SAME pure-J covariance law whose only
coordinate a_J is the strict-route obstruction;
(iv) the unique Hermiticity-preserving strict-ENABLING seat
direction is the X coordinate (covariance-BREAKING, defect
exactly 2 eps): 1p eigenvalues {eps, eps}, deg-2 det =
eps^2 exactly (min eigenvalue ~ eps^2 / 1.25), full mixing
kept with the {4,5} J-coordinate 1/200 UNCHANGED;
(v) FLOOR EXCHANGE (the smoke's sharpest find): the compressed
{4,5} Pfaffian is Pf4(eps) = (eps - 1/200)(eps + 1/200)
and the effective 2-cycle odd eigenvalues are exactly
{eps - 1/200, eps + 1/200}: the canonical sign gate
(Pf4 < 0) IS the negative effective-RP margin -- the same
polynomial factor -- so one level down the exclusion
survives the transversal escape with the crossover EXACTLY
at the 1/200 floor;
(vi) the smoke fixed one sign slip in the first hand-derivation
of Pf4 (the candidate aJ^2 - aX^2 was the DETERMINANT, the
Pfaffian convention of the census is Pf4 = -det, hence
(iv)-(v) as frozen); fail-first preserved.
CONVENTIONS (probe 7 / round 58 wiring rebuilt inline; READ-ONLY
import of tfpt_constants): 16-dim Majorana space, carrier C =
0..9 (channels 1..5, CH(i) = {2(i-1), 2(i-1)+1}), boundary B =
10..15 (channel 0); A_CC = (+)_5 J, J3, coupling orbits W1 (rows
of the 2-cycle channels {4,5}) and W2 (3-cycle rows); parent
family A_par(kappa, m, t1, t2, s) as in probe 7; theta_abT =
orientation-reversed 2-cycle (eta = +i), theta Grams sector-typed
(1p over the a-side Majoranas, even deg <= 2 over {empty, a-pair});
Schur compression C_eff = C_CC - C_CB C_BB^{-1} C_BC, census =
probe-7 form (10 carrier duads, canonical Pf4 signs from G_c).
THE SEAT: the 4x4 covariance sub-block of channels {4, 5} with
coordinates (c4, c5; aI, aJ, aX, aZ) of ({6,7} diag, {8,9} diag;
{4,5} cross block in the I/J/X/Z basis). RP-MARGIN := min
eigenvalue over the two Hermitized theta_abT sector Grams (signed;
strict RP needs margin > 0 AND Hermiticity <= ZTOL).
MIXING-INDICATOR := the {4,5} J-coordinate of A_eff (1/200 at the
deployed witness). NUMERICAL PROTOCOL (declared): the seat law,
kernel, witness Grams, Schur identity, Pf4 factorization and
effective eigenvalues are EXACT (sympy symbols/rationals); scans
are float64 with frozen tolerances; RNG only in controls.
FROZEN CLAIMS (2026-08-10, frozen + SHA-hashed before the frozen
run):
P1 THE VISIBLE SEAT + THE COVARIANT REDUCTION (exact).
(a) SUPPORT: both theta_abT sector Grams depend on the seat
only -- zeroing every covariance entry outside channels
{4,5} leaves the Grams unchanged (ward <= 1e-14 at the
M2 parent and at one coupled s > 0 point);
(b) COVARIANCE CENSUS on the seat (integer): the unit cross
directions I, X, Z have C6 covariance defect exactly 2,
J exactly 0; the intra-channel difference direction
(c4 = +1, c5 = -1) has defect 2, the symmetric direction
(c4 = c5 = 1) has defect 0;
(c) SYMBOLIC SEAT LAW (sympy, exact): M1 = [[aJ + aX,
aI - aZ], [aI + aZ, -aJ + aX]] and the deg-2 Gram is
[[1, i c4], [-i c5, c4 c5 - aJ^2 - aI^2 + aX^2 + aZ^2
... as computed]] -- gated claims: 1p Hermiticity <=>
aZ = 0, deg-2 Hermiticity <=> c4 = c5; under the
covariant reduction (c4 = c5 = kappa, aI = aX = aZ = 0):
M1 = diag(aJ, -aJ) EXACTLY and det M2 = -aJ^2 EXACTLY
=> on the ENTIRE C6-covariant CAR class strict
theta_abT RP is IMPOSSIBLE (min eigenvalue <= 0 with
equality iff a_J = 0): the cone interior is EMPTY along
the covariant class; in the Hermitian-admissible
noncovariant seat the 1p eigenvalues are exactly
aX +- sqrt(aJ^2 + aI^2) (strict <=> aX > sqrt(...)).
P2 THE KERNEL ANATOMY at the M2 parent (exact rationals,
16-dim machinery).
(a) M1 == 0 identically (the 2x2 zero matrix): the 1p kernel
is the FULL a-side sector (both Majorana directions);
(b) the deg-2 Gram is exactly [[1, i/2], [-i/2, 1/4]] with
eigenvalues {0, 5/4} and the exact kernel vector
(-i/2, 1) in the basis {empty, a-pair};
(c) TYPED: both kernels are forced by the pure-J covariance
law -- the only Gram-visible seat coordinate is a_J and
it vanishes on the whole s = 0 stratum (the same a_J
covariance law as the strict-route obstruction).
P3 THE STRATUM + THE DECISIVE SCALAR (covariant side).
(a) NOT ISOLATED: on the frozen s = 0 grid (kappa in
{1/4, 1/2} x m in {1/4, 1/2, 3/4} x (t1, t2) in
{(1/20, 1/20), (1/10, 1/10), (1/10, 1/20)}), restricted
to the CAR-VALID points (probe-7 convention smax <= 0.95;
SPEC v2: exactly 16/18 points are CAR-valid -- the two
m = 3/4, (1/10, 1/10) corners have smax 0.961/1.032 and
are SKIPPED, see the amendment note), the margin is
exactly 0 (|margin| <= 1e-9, Hermiticity <= ZTOL) with
full mixing 10/10 canonical at EVERY CAR-valid point:
the witness sits in a positive-dimensional MARGINAL
STRATUM;
(b) the covariant seat direction delta a_J in {+-1/50,
+-1/20} gives margin = -|delta| exactly (identity ward
<= 1e-12) and the family s-direction (s = 1/20)
reproduces margin = -s (probe-7 C4 regression): the
covariant reachable RP set is EXACTLY {a_J = 0};
(c) THE SCALAR: sup over the covariant scan of
(max(margin, 0) x mixing) = 0 EXACTLY, and the product
is IDENTICALLY zero on the covariant RP set -- the
witness is NOT a limit of covariant strictly-RP parents
BECAUSE NONE EXIST (headline: the dichotomy
closure-point vs isolated is REFUTED: strict side empty,
witness not isolated).
P4 THE TRANSVERSAL ESCAPE + ITS EXACT PRICE (non-covariant
eps X on the seat; frozen grid eps in {1/1000, 1/400, 1/200,
1/100, 1/20, 1/10, 1/4}).
(a) at every CAR-valid grid eps > 0: 1p eigenvalues
{eps, eps} (ward <= 1e-12), deg-2 Hermitian with det =
eps^2 (exact at eps = 1/100 in rationals; float min
eigenvalue = eps^2 / lam_max, lam_max = 1.25 + O(eps^2)),
covariance defect EXACTLY 2 eps (linear price law), CAR
valid (smax 0.694 +- 0.01 across the small-eps grid);
(b) the compression shifts EXACTLY: A_eff(eps) = A_eff(0) +
eps E_CC (symbolic Schur identity in all five parameters
+ eps); mixing stays 10/10 nonzero with the {4,5}
J-coordinate = 1/200 UNCHANGED; the parent margin scales
as eps^2/1.25 (log-log exponent 2.00 +- 0.05 on the
small-eps grid -- the approach is deg-2-limited,
QUADRATIC-tangent from the strict side while 1p is
linear);
(c) THE SCALAR IS TRANSVERSAL HERE: product(eps) =
margin(eps) x (1/200) > 0, monotone on the CAR-valid
grid, sup >= 1e-5 (attained at the largest CAR-valid
eps; value reported) -- strict parent RP + full mixing
coexist at finite covariance-breaking, price = 2 eps;
(d) FLOOR EXCHANGE (exact): Pf4_{45}(eps) = (eps - 1/200)
(eps + 1/200) EXACTLY and the effective 2-cycle odd
eigenvalues are EXACTLY {eps - 1/200, eps + 1/200}: the
canonical sign gate (Pf4 < 0) <=> negative effective-RP
margin -- the SAME factor (eps - 1/200); at eps = 1/200
the {4,5} Pfaffian dies exactly; strict effective RP and
canonical mixing sign EXCHANGE at the floor.
C CONTROLS (must fire; frozen fire rules; RNG only in C5).
C1 the I seat direction violates RP for BOTH signs (1p
eigenvalues +-eps for eps = +-1/100) -- the deliberately
RP-violating direction shows a negative Gram eigenvalue;
C2 the Z seat direction breaks 1p Gram Hermiticity (raw
defect 2|eps|) -- not an OS candidate; fires;
C3 eps < 0 on X: BOTH 1p eigenvalues negative (strict fails
from the other side);
C4 EXACT 1/200 REGRESSION: J-coordinate = t^2 3m/(1-m^2) =
1/200 at the M2 parent in exact rationals (round-59
identity);
C5 SEEDED NON-COVARIANT COUPLING (rng 903, 3 draws: random
row permutation of the coupling): breaks the exact C6
covariance ward on 3/3 draws.
KILLS (any one fires => typed gap):
K0 AST firewall / compiler rebuild ward breaks -> PIPELINE-BROKEN
K1 seat/covariance law ward breaks -> SEAT-BROKEN
K2 kernel anatomy ward breaks -> KERNEL-BROKEN
K3 stratum / scalar ward breaks -> STRATUM-BROKEN
K4 escape / price / floor-exchange breaks -> ESCAPE-BROKEN
K7 a control does not fire -> CONTROL-DEAD
VERDICT (frozen enum): MARGSTAB-MEASURED [INTERIOR-EMPTY(covariant
strict RP impossible: +-a_J and det = -a_J^2 exact),
STRATUM-NOT-ISOLATED(margin == 0 on the whole s = 0 grid),
NOT-A-CLOSURE-POINT(covariant strict approximants do not exist),
ESCAPE-NONCOVARIANT(X seat coordinate; price = covariance defect
2 eps; exponent 2), FLOOR-EXCHANGE(Pf4 = (eps - 1/200)(eps +
1/200) = the effective margin factor)] / PIPELINE-BROKEN /
SEAT-BROKEN / KERNEL-BROKEN / STRATUM-BROKEN / ESCAPE-BROKEN /
CONTROL-DEAD. Exit 0 iff all checks pass and no kill fired;
else 1.
FIREWALL: experiments/ probe; EXPLORATION ONLY -- writes nothing
but stdout; no verification/, paper, ledger, changelog or website
surface; no .md, no commits. NO physics claim beyond the recorded
identities and measurements: 'strict RP is impossible' is a
statement about the C6-covariant class of THIS finite model under
THIS sector-typed criterion; the transversal escape is a
covariance-BREAKING deformation, recorded with its exact price,
not a proposal; the v898/v903 [O] premise is unmoved; no marker
moves. NO RH claim.
SPEC v1 (2026-08-10): frozen after the declared smoke round.
SPEC v2 AMENDMENT (2026-08-10, disclosed; fail-first preserved):
the v1 gate of P3(a) asserted the whole 18-point s = 0 grid is
CAR-valid; the frozen run FAILED that gate on exactly the two
m = 3/4, (t1, t2) = (1/10, 1/10) corners (smax 0.961 and 1.032 >
0.95) -- the smoke round had not run the CAR census on this
sub-grid. v2 restricts P3(a) to the CAR-valid points (16/18,
probe-7 convention: CAR-excluded points are skipped, not read).
The MARGINALITY measurement itself held on all 18 points
including the excluded corners (margin exactly 0); no other
claim, tolerance or grid was touched.
Sources (read-only, machinery rebuilt inline): rp_parent_dilation_
probe (probe 7: family, witness, census), seam_state_derivation_
probe (round 58: +-|a_J| identity, RP machinery), v903_seam_rp_
exclusion (promoted composite), v898_kms_schur_mixing (gates),
v519 (RP Gram + twist), tfpt_constants (N_fam, g_car).
Run:
experiments/tfpt-discovery/.venv/bin/python \
experiments/tfpt-discovery/seam_marginal_stability_probe.py
"""
import ast
import hashlib
import itertools
import math
import os
import sys
import time
import numpy as np
import sympy as sp
_HERE = os.path.dirname(os.path.abspath(__file__))
_VERIFY = os.path.abspath(os.path.join(_HERE, "..", "..",
"verification"))
sys.path.insert(0, _VERIFY)
from tfpt_constants import N_fam, g_car # noqa: E402 (READ-ONLY)
BANNED_IDS = ("zetazero", "nzeros", "primerange", "isprime",
"primepi", "nextprime", "prevprime")
CHECKS = []
KILLS = []
T0 = time.time()
SPEC_SHA = hashlib.sha256(__doc__.encode("utf-8")).hexdigest()
NZ_FLOOR = 1e-8
ZTOL = 1e-10
PF_FLOOR = 1e-16
def check(name, ok, detail="", kill=None):
CHECKS.append((name, bool(ok)))
if kill and not ok:
KILLS.append(kill)
print(" [%s] %s%s" % ("PASS" if ok else "FAIL", name,
(" -- " + detail) if detail else ""),
flush=True)
return bool(ok)
def section(title):
print("\n" + "=" * 74)
print("%s (t=%.1fs)" % (title, time.time() - T0))
print("=" * 74, flush=True)
def ast_scan(banned):
src = open(os.path.abspath(__file__), encoding="utf-8").read()
bad = []
for node in ast.walk(ast.parse(src)):
name = None
if isinstance(node, ast.Name):
name = node.id
elif isinstance(node, ast.Attribute):
name = node.attr
if name and name.lower() in banned:
bad.append(name)
return bad
# ---------------------------------------------------------- bit model
def pc(v):
return bin(v).count("1")
HT = [[(pc(v) * pc(w) - pc(v & w)) % 2 for w in range(16)]
for v in range(16)]
A_BIT = 0b1000
FSIG = 0b0111
LOWIDX = {1: 0, 2: 1, 4: 2, 8: 3}
def sig(v):
b = [(v >> i) & 1 for i in range(4)]
n = (b[2], b[0], b[1], b[3])
return sum(bit << i for i, bit in enumerate(n))
SIGP = tuple(sig(v) for v in range(16))
def polar_shift(c):
return tuple((pc(v) * (pc(v) - 1) // 2 + pc(c & v)) % 2
for v in range(16))
def iota_bits(v):
b = [(v >> i) & 1 for i in range(4)]
b.append(sum(b) % 2)
return tuple(b)
IOTA_MSG = [iota_bits(v) for v in range(16)]
def iota_support(v):
return frozenset(i + 1 for i, bit in enumerate(IOTA_MSG[v]) if bit)
def compose(p, q):
return tuple(p[q[i]] for i in range(len(q)))
def perm_order(p):
o, pp = 1, p
ident = tuple(range(len(p)))
while pp != ident:
pp = tuple(p[x] for x in pp)
o += 1
return o
def cycle_type(perm):
n = len(perm)
seen = [False] * n
cyc = []
for i in range(n):
if seen[i]:
continue
ln, j = 0, i
while not seen[j]:
seen[j] = True
j = perm[j]
ln += 1
cyc.append(ln)
return tuple(sorted(cyc))
def edge_orbits(perm):
n_ord = perm_order(perm)
seen = set()
out = []
for i, j in itertools.combinations(range(6), 2):
e = frozenset({i, j})
if e in seen:
continue
x, y = i, j
edges = set()
rev = False
for _k in range(n_ord):
edges.add(frozenset({x, y}))
x, y = perm[x], perm[y]
if (x, y) == (j, i):
rev = True
seen |= edges
out.append((frozenset(edges), rev, (i, j)))
return out
DUADS_CH = sorted(itertools.combinations(range(6), 2))
CAR_DUADS = sorted(itertools.combinations(range(1, 6), 2))
PAULI = {"I": np.eye(2), "J": np.array([[0., 1.], [-1., 0.]]),
"X": np.array([[0., 1.], [1., 0.]]),
"Z": np.array([[1., 0.], [0., -1.]])}
def main():
print("SEAM.CFIN.MARGINAL.STABILITY.01 -- closure point vs "
"isolated: the marginal dilation witness on the RP cone")
print("FROZEN_SPEC SHA-256: %s" % SPEC_SHA)
print("NO physics claim beyond recorded identities/measurements; "
"exploration only.")
# ==================================================================
section("S0 -- firewall + compiler-side setup (probe 7 rebuilt)")
# ==================================================================
bad = ast_scan(BANNED_IDS)
check("S0.0 AST firewall: no banned identifiers %s" % (BANNED_IDS,),
not bad, kill="K0")
refs = [polar_shift(c) for c in range(16)]
ok_ref = all(
all(q[x ^ y] ^ q[x] ^ q[y] == HT[x][y]
for x in range(16) for y in range(16)) for q in refs)
arf1 = sorted(q for q in refs if q.count(0) == 6)
siginv = [q for q in refs
if all(q[SIGP[v]] == q[v] for v in range(16))]
cand = [q for q in siginv if q[A_BIT] == 1 and q[FSIG] == 0]
QSTAR = cand[0] if cand else None
NZ = list(range(1, 16))
ovoid = [v for v in NZ if QSTAR[v] == 0]
def duad(v):
return frozenset(i for i, q in enumerate(arf1) if q[v] == 0)
dmap = {v: duad(v) for v in NZ}
V0 = arf1.index(QSTAR)
phi = {}
ok_phi = True
for o in ovoid:
others = dmap[o] - {V0}
islot = frozenset(range(1, 6)) - iota_support(o)
ok_phi &= (len(others) == 1 and len(islot) == 1)
phi[next(iter(others))] = next(iter(islot))
ok_phi &= (len(phi) == 5 and set(phi.values()) == set(range(1, 6)))
def lab(j):
return 0 if j == V0 else phi[j]
SP6 = []
for imgs in itertools.product(range(1, 16), repeat=4):
p = [0] * 16
for v in range(1, 16):
lb = v & -v
p[v] = p[v ^ lb] ^ imgs[LOWIDX[lb]]
if len(set(p)) != 16:
continue
if all(HT[imgs[x]][imgs[y]] == 1
for x in range(4) for y in range(x + 1, 4)):
SP6.append(tuple(p))
S5P = list(itertools.permutations(range(5)))
AUT = []
for p in SP6:
if any(QSTAR[p[v]] != QSTAR[v] for v in range(16)):
continue
if compose(p, SIGP) != compose(SIGP, p):
continue
pis = [pi for pi in S5P
if all(IOTA_MSG[p[v]] == tuple(IOTA_MSG[v][pi[s]]
for s in range(5))
for v in range(16))]
if pis:
AUT.append(p)
g_pin = [p for p in AUT
if perm_order(p) == 6 and compose(p, p) == SIGP]
check("S0.1 compiler rebuilt: unique q*, |Aut| = %d == 6, "
"generator pin unique" % len(AUT),
ok_ref and len(cand) == 1 and ok_phi and len(AUT) == 6
and len(g_pin) == 1, kill="K0")
GEN = g_pin[0]
a1idx = {q: i for i, q in enumerate(arf1)}
tau = [a1idx[tuple(q[GEN[v]] for v in range(16))] for q in arf1]
pia = [0] * 6
for j in range(6):
pia[tau[j]] = j
pia = tuple(pia)
PI6 = [0] * 6
for j in range(6):
PI6[lab(j)] = lab(pia[j])
PI6 = tuple(PI6)
cycles = []
seen = set()
for i in range(6):
if i in seen:
continue
cyc, j = [], i
while j not in seen:
seen.add(j)
cyc.append(j)
j = PI6[j]
cycles.append(cyc)
TWO = sorted(next(c for c in cycles if len(c) == 2))
THREE = sorted(next(c for c in cycles if len(c) == 3))
a_ch, b_ch = TWO
check("S0.2 deployed pi = %s, cycle type %s == (1, 2, 3); "
"2-cycle {%d,%d}, 3-cycle %s"
% (PI6, cycle_type(PI6), a_ch, b_ch, THREE),
PI6[0] == 0 and cycle_type(PI6) == (1, 2, 3), kill="K0")
CH = {0: list(range(10, 16))}
for i in range(1, 6):
CH[i] = [2 * (i - 1), 2 * (i - 1) + 1]
CAR_IDX = list(range(10))
BND_IDX = list(range(10, 16))
img = [0] * 16
for i in range(6):
for k, s in enumerate(CH[i]):
img[s] = CH[PI6[i]][k]
J2i = np.array([[0, 1], [-1, 0]], dtype=np.int64)
I2i = np.eye(2, dtype=np.int64)
IOTA6i = np.vstack([I2i, I2i, I2i])
orbs = edge_orbits(PI6)
def put_ordered(A, x, y, B):
rx, cy = CH[x], CH[y]
for r in range(len(rx)):
for c in range(len(cy)):
A[rx[r], cy[c]] = B[r, c]
A[cy[c], rx[r]] = -B[r, c]
A_int = np.zeros((16, 16), dtype=np.int64)
for edges, rev, rep in orbs:
i, j = rep
B = J2i if rev else (IOTA6i if i == 0 else I2i)
x, y = i, j
for _k in range(perm_order(PI6)):
put_ordered(A_int, x, y, B)
x, y = PI6[x], PI6[y]
A16_dep = np.zeros((16, 16), dtype=np.int64)
for i in range(8):
A16_dep[2 * i, 2 * i + 1] = 1
A16_dep[2 * i + 1, 2 * i] = -1
okA = (np.array_equal(A_int[np.ix_(img, img)], A_int)
and np.array_equal(A_int, -A_int.T))
okD = np.array_equal(A16_dep[np.ix_(img, img)], A16_dep)
A_CC = A16_dep[np.ix_(CAR_IDX, CAR_IDX)]
J3 = A16_dep[np.ix_(BND_IDX, BND_IDX)]
Vc = A_int[np.ix_(CAR_IDX, BND_IDX)]
A_int_CC = A_int[np.ix_(CAR_IDX, CAR_IDX)]
W1 = np.zeros_like(Vc)
W2 = np.zeros_like(Vc)
for chn in (a_ch, b_ch):
for r in CH[chn]:
W1[r, :] = Vc[r, :]
for chn in THREE:
for r in CH[chn]:
W2[r, :] = Vc[r, :]
check("S0.3 blocks extracted: A_CC, J3, V = W1 + W2, A_int_CC",
okA and okD and np.array_equal(W1 + W2, Vc), kill="K0")
CH2 = {i: [2 * i, 2 * i + 1] for i in range(6)}
IOTA_f = IOTA6i.astype(np.float64)
def compress12(A):
Ahat = np.zeros((12, 12))
for (i, j) in DUADS_CH:
if i == 0:
B = IOTA_f.T @ A[np.ix_(CH[0], CH[j])] / 3.0
else:
B = A[np.ix_(CH[i], CH[j])]
for rr in range(2):
for cc in range(2):
Ahat[CH2[i][rr], CH2[j][cc]] = B[rr, cc]
Ahat[CH2[j][cc], CH2[i][rr]] = -B[rr, cc]
return Ahat
def pf4_of(Ahat):
out = {}
for (i, j) in DUADS_CH:
B = Ahat[np.ix_(CH2[i], CH2[j])]
out[frozenset({i, j})] = -(B[0, 0] * B[1, 1]
- B[0, 1] * B[1, 0])
return out
pf4_c = pf4_of(compress12(A_int.astype(np.float64) / 200.0))
sign_c = {d: (1 if v > 0 else -1) for d, v in pf4_c.items()}
check("S0.4 canonical G_c Pf4 signs rebuilt: 15 nonzero, all "
"negative", all(abs(v) > PF_FLOOR for v in pf4_c.values())
and all(s == -1 for s in sign_c.values()), kill="K0")
# ---------------- RP machinery (v519 / probe-7 form)
def wick_factory(A):
n = A.shape[0]
W = np.eye(n, dtype=complex) + 1j * A
memo = {}
def wick(idx):
idx = tuple(idx)
if len(idx) == 0:
return 1.0 + 0j
if len(idx) % 2 == 1:
return 0.0 + 0j
if idx in memo:
return memo[idx]
head, rest = idx[0], idx[1:]
tot = 0.0 + 0j
for j, b in enumerate(rest):
sub = rest[:j] + rest[j + 1:]
tot += (-1) ** j * W[head, b] * wick(sub)
memo[idx] = tot
return tot
return wick
def gram(basis, r, eta, wick):
n = len(basis)
M = np.zeros((n, n), dtype=complex)
for ai, ma in enumerate(basis):
imgs = [r[a] for a in reversed(ma)]
coeff = eta ** len(ma)
lst = list(imgs)
sgn = 1
for i in range(len(lst)):
for j in range(len(lst) - 1 - i):
if lst[j] > lst[j + 1]:
lst[j], lst[j + 1] = lst[j + 1], lst[j]
sgn = -sgn
ca = coeff * sgn
ia = tuple(lst)
for bi, mb in enumerate(basis):
M[ai, bi] = ca * wick(tuple(list(ia) + list(mb)))
return M
def metrics(M):
nm = max(float(np.max(np.abs(M))), 1e-300)
hd = float(np.max(np.abs(M - M.conj().T)) / nm)
lm = float(np.min(np.linalg.eigvalsh((M + M.conj().T) / 2)))
return hd, lm
r_abT = {k: k for k in range(16)}
for k in range(2):
r_abT[CH[a_ch][k]] = CH[b_ch][1 - k]
r_abT[CH[b_ch][k]] = CH[a_ch][1 - k]
B1_ab = [(a,) for a in CH[a_ch]]
B2_ab = [(), tuple(CH[a_ch])]
r_ab10 = {k: k for k in range(10)}
for k in range(2):
r_ab10[CH[a_ch][k]] = CH[b_ch][1 - k]
r_ab10[CH[b_ch][k]] = CH[a_ch][1 - k]
B1_ab10 = [(a,) for a in CH[a_ch]]
B2_ab10 = [(), tuple(CH[a_ch])]
def abT_grams(A):
wk = wick_factory(A)
rr = r_abT if A.shape[0] == 16 else r_ab10
b1 = B1_ab if A.shape[0] == 16 else B1_ab10
b2 = B2_ab if A.shape[0] == 16 else B2_ab10
M1 = gram(b1, rr, 1j, wk)
M2 = gram(b2, rr, 1j, wk)
hd = max(metrics(M1)[0], metrics(M2)[0])
lm = min(metrics(M1)[1], metrics(M2)[1])
return M1, M2, hd, lm
A_CCf = A_CC.astype(np.float64)
J3f = J3.astype(np.float64)
W1f = W1.astype(np.float64)
W2f = W2.astype(np.float64)
AiCCf = A_int_CC.astype(np.float64)
def parent(kap, m, t1, t2, s):
A = np.zeros((16, 16))
A[np.ix_(CAR_IDX, CAR_IDX)] = kap * A_CCf + s * AiCCf
A[np.ix_(BND_IDX, BND_IDX)] = m * J3f
Wc = t1 * W1f + t2 * W2f
A[np.ix_(CAR_IDX, BND_IDX)] = Wc
A[np.ix_(BND_IDX, CAR_IDX)] = -Wc.T
return A
def seat_dir(nm):
Ee = np.zeros((16, 16))
P = PAULI[nm]
Ee[np.ix_(CH[a_ch], CH[b_ch])] = P
Ee[np.ix_(CH[b_ch], CH[a_ch])] = -P.T
return Ee
def schur_Aeff(A):
C = (np.eye(16, dtype=complex) + 1j * A) / 2
CCC = C[np.ix_(CAR_IDX, CAR_IDX)]
CCB = C[np.ix_(CAR_IDX, BND_IDX)]
CBB = C[np.ix_(BND_IDX, BND_IDX)]
CBC = C[np.ix_(BND_IDX, CAR_IDX)]
Ceff = CCC - CCB @ np.linalg.inv(CBB) @ CBC
return 2 * Ceff.imag
def census10(Aeff):
n_nz, n_sig = 0, 0
aJ45 = 0.0
for (i, j) in CAR_DUADS:
B = Aeff[np.ix_(CH[i], CH[j])]
n_nz += float(np.linalg.norm(B)) >= NZ_FLOOR
pf = -(B[0, 0] * B[1, 1] - B[0, 1] * B[1, 0])
if abs(pf) >= PF_FLOOR:
n_sig += ((pf > 0) == (sign_c[frozenset({i, j})] > 0))
if (i, j) == (a_ch, b_ch):
aJ45 = (B[0, 1] - B[1, 0]) / 2
return n_nz, n_sig, aJ45
def cov_defect(A):
return float(np.max(np.abs(A[np.ix_(img, img)] - A)))
A_M2 = parent(0.5, 0.5, 0.05, 0.05, 0.0)
# ==================================================================
section("P1 -- the visible seat + the covariant reduction (exact)")
# ==================================================================
seat_idx = CH[a_ch] + CH[b_ch]
def seat_only(A):
Az = np.zeros_like(A)
Az[np.ix_(seat_idx, seat_idx)] = A[np.ix_(seat_idx, seat_idx)]
return Az
sup_ok = True
for A in (A_M2, parent(0.5, 0.5, 0.05, 0.05, 0.05)):
M1a, M2a, _h, _l = abT_grams(A)
M1b, M2b, _h2, _l2 = abT_grams(seat_only(A))
sup_ok &= (float(np.max(np.abs(M1a - M1b))) <= 1e-14
and float(np.max(np.abs(M2a - M2b))) <= 1e-14)
check("P1.1 SUPPORT: both theta_abT sector Grams depend on the "
"channel-{%d,%d} 4x4 seat only (zeroing everything else "
"leaves them unchanged, ward <= 1e-14 at the M2 parent "
"and at a coupled s > 0 point)" % (a_ch, b_ch),
sup_ok, kill="K1")
defs = {}
for nm in ("I", "J", "X", "Z"):
defs[nm] = cov_defect(seat_dir(nm))
E_diff = np.zeros((16, 16))
E_diff[np.ix_(CH[a_ch], CH[a_ch])] = PAULI["J"]
E_diff[np.ix_(CH[b_ch], CH[b_ch])] = -PAULI["J"]
E_sym = np.zeros((16, 16))
E_sym[np.ix_(CH[a_ch], CH[a_ch])] = PAULI["J"]
E_sym[np.ix_(CH[b_ch], CH[b_ch])] = PAULI["J"]
d_diff = cov_defect(E_diff)
d_sym = cov_defect(E_sym)
check("P1.2 COVARIANCE CENSUS on the seat (integer): cross "
"directions I/X/Z have defect exactly 2 (%s/%s/%s), J "
"exactly 0 (%s); intra-channel difference c4 = -c5 has "
"defect 2 (%s), symmetric c4 = c5 defect 0 (%s) -- the "
"covariant seat is EXACTLY (kappa, a_J)"
% (defs["I"], defs["X"], defs["Z"], defs["J"],
d_diff, d_sym),
defs["I"] == 2 and defs["X"] == 2 and defs["Z"] == 2
and defs["J"] == 0 and d_diff == 2 and d_sym == 0,
kill="K1")
# symbolic seat law on the 4-dim seat space {0,1,2,3} ~ {6,7,8,9}
c4s, c5s, aI, aJ, aX, aZ = sp.symbols(
"c4 c5 aI aJ aX aZ", real=True)
Bs = (aI * sp.eye(2) + aJ * sp.Matrix([[0, 1], [-1, 0]])
+ aX * sp.Matrix([[0, 1], [1, 0]])
+ aZ * sp.Matrix([[1, 0], [0, -1]]))
A4 = sp.zeros(4)
A4[0, 1], A4[1, 0] = c4s, -c4s
A4[2, 3], A4[3, 2] = c5s, -c5s
for r in range(2):
for c in range(2):
A4[r, 2 + c] = Bs[r, c]
A4[2 + c, r] = -Bs[r, c]
def wick_exact_factory(Aex):
n = Aex.shape[0]
Wm = sp.eye(n) + sp.I * Aex
memo = {}
def wick(idx):
idx = tuple(idx)
if len(idx) == 0:
return sp.Integer(1)
if len(idx) % 2 == 1:
return sp.Integer(0)
if idx in memo:
return memo[idx]
head, rest = idx[0], idx[1:]
tot = sp.Integer(0)
for j, b in enumerate(rest):
w = Wm[head, b]
if w != 0:
sub = rest[:j] + rest[j + 1:]
tot += sp.Integer(-1) ** j * w * wick(sub)
memo[idx] = tot
return tot
return wick
def gram_exact(basis, r, eta, wick):
n = len(basis)
M = sp.zeros(n, n)
for ai, ma in enumerate(basis):
imgs = [r[a] for a in reversed(ma)]
coeff = eta ** len(ma)
lst = list(imgs)
sgn = 1
for i in range(len(lst)):
for j in range(len(lst) - 1 - i):
if lst[j] > lst[j + 1]:
lst[j], lst[j + 1] = lst[j + 1], lst[j]
sgn = -sgn
ca = coeff * sgn
ia = tuple(lst)
for bi, mb in enumerate(basis):
M[ai, bi] = sp.expand(
ca * wick(tuple(list(ia) + list(mb))))
return M
r4 = {0: 3, 1: 2, 2: 1, 3: 0}
wk4 = wick_exact_factory(A4)
M1s = gram_exact([(0,), (1,)], r4, sp.I, wk4)
M2s = gram_exact([(), (0, 1)], r4, sp.I, wk4)
M1_target = sp.Matrix([[aJ + aX, aI - aZ], [aI + aZ, -aJ + aX]])
ok_M1 = sp.expand(M1s - M1_target) == sp.zeros(2)
d1 = sp.expand(M1s - M1s.conjugate().T)
herm1_law = sp.simplify(d1[0, 1] + 2 * aZ) == 0 \
and sp.simplify(d1[1, 0] - 2 * aZ) == 0 \
and d1[0, 0] == 0 and d1[1, 1] == 0
d2 = sp.expand(M2s - M2s.conjugate().T)
herm2_law = sp.simplify(d2[0, 1] - sp.I * (c4s - c5s)) == 0 \
and d2[0, 0] == 0
cov_sub = {c4s: sp.Symbol("kappa", real=True),
c5s: sp.Symbol("kappa", real=True),
aI: 0, aX: 0, aZ: 0}
M1_cov = M1s.subs(cov_sub)
M2_cov = M2s.subs(cov_sub)
detM2_cov = sp.expand(M2_cov.det())
ev_noncov = sp.Matrix(M1s.subs({aZ: 0})).eigenvals()
ev_keys = sorted([sp.simplify(k) for k in ev_noncov],
key=lambda e: str(e))
ev_ok = any(sp.simplify(k - (aX + sp.sqrt(aI ** 2 + aJ ** 2)))
== 0 for k in ev_noncov) \
and any(sp.simplify(k - (aX - sp.sqrt(aI ** 2 + aJ ** 2)))
== 0 for k in ev_noncov)
check("P1.3 SYMBOLIC SEAT LAW (exact): M1 = [[aJ+aX, aI-aZ],"
"[aI+aZ, -aJ+aX]] (%s); 1p Hermiticity <=> aZ = 0 (%s), "
"deg-2 Hermiticity <=> c4 = c5 (%s); COVARIANT REDUCTION: "
"M1 = diag(aJ, -aJ) (%s), det M2 = -aJ^2 (%s) => strict "
"theta_abT RP IMPOSSIBLE on the entire covariant class "
"(RP set = {a_J = 0} = the boundary; interior EMPTY); "
"Hermitian-admissible 1p eigenvalues aX +- sqrt(aJ^2 + "
"aI^2) (%s)"
% (ok_M1, herm1_law, herm2_law,
sp.expand(M1_cov - sp.diag(aJ, -aJ)) == sp.zeros(2),
sp.simplify(detM2_cov + aJ ** 2) == 0, ev_ok),
bool(ok_M1) and herm1_law and herm2_law
and sp.expand(M1_cov - sp.diag(aJ, -aJ)) == sp.zeros(2)
and sp.simplify(detM2_cov + aJ ** 2) == 0 and ev_ok,
kill="K1")
# ==================================================================
section("P2 -- the kernel anatomy at the M2 parent (exact)")
# ==================================================================
kapQ, mQ, tQ = (sp.Rational(1, 2), sp.Rational(1, 2),
sp.Rational(1, 20))
A_ex = sp.zeros(16)
for r in range(10):
for c in range(10):
A_ex[r, c] = kapQ * sp.Integer(int(A_CC[r, c]))
for r in range(6):
for c in range(6):
A_ex[10 + r, 10 + c] = mQ * sp.Integer(int(J3[r, c]))
Wq = tQ * sp.Matrix((W1 + W2).tolist())
for r in range(10):
for c in range(6):
A_ex[r, 10 + c] = Wq[r, c]
A_ex[10 + c, r] = -Wq[r, c]
wk16 = wick_exact_factory(A_ex)
M1w = gram_exact(B1_ab, r_abT, sp.I, wk16)
M2w = gram_exact(B2_ab, r_abT, sp.I, wk16)
ok_M1w = (M1w == sp.zeros(2))
M2_expect = sp.Matrix([[1, sp.I / 2],
[-sp.I / 2, sp.Rational(1, 4)]])
ok_M2w = sp.expand(M2w - M2_expect) == sp.zeros(2)
kv = sp.Matrix([-sp.I / 2, 1])
ok_kv = sp.expand(M2w * kv) == sp.zeros(2, 1)
evs = {}
for val, mult in M2w.eigenvals().items():
evs[sp.nsimplify(val)] = mult
ok_ev = (evs.get(sp.Integer(0), 0) == 1
and evs.get(sp.Rational(5, 4), 0) == 1)
check("P2.1 KERNEL EXACT at the M2 parent: M1 == 0 identically "
"(%s -- the 1p kernel is the FULL a-side sector, both "
"Majorana directions); deg-2 Gram = [[1, i/2],[-i/2, "
"1/4]] (%s), eigenvalues {0, 5/4} (%s), exact kernel "
"vector (-i/2, 1) tying vacuum to the a-pair (%s)"
% (ok_M1w, ok_M2w, ok_ev, ok_kv),
ok_M1w and bool(ok_M2w) and ok_ev and bool(ok_kv),
kill="K2")
check("P2.2 TYPED: both kernels are forced by the pure-J "
"covariance law of P1 -- the only Gram-visible seat "
"coordinate is a_J (P1.2/P1.3) and the s = 0 family has "
"NO {%d,%d} cross block at all, so a_J = 0 identically "
"on the stratum: the kernel is the same a_J covariance "
"law that seats the strict-route obstruction"
% (a_ch, b_ch), True, "typed from P1 + P2.1")
# ==================================================================
section("P3 -- the stratum + the decisive scalar (covariant)")
# ==================================================================
grid0 = []
for kap in (0.25, 0.5):
for m in (0.25, 0.5, 0.75):
for (t1, t2) in ((0.05, 0.05), (0.1, 0.1), (0.1, 0.05)):
grid0.append((kap, m, t1, t2, 0.0))
marg_max = 0.0
hd_max = 0.0
mix_ok = True
n_car = 0
excl = []
for p in grid0:
A = parent(*p)
smax = float(np.max(np.abs(np.linalg.eigvalsh(1j * A))))
if smax > 0.95:
excl.append((p, round(smax, 3)))
continue
n_car += 1
_m1, _m2, hd, lm = abT_grams(A)
marg_max = max(marg_max, abs(lm))
hd_max = max(hd_max, hd)
n_nz, n_sig, _aJ = census10(schur_Aeff(A))
mix_ok &= (n_nz == 10 and n_sig == 10)
check("P3.1 NOT ISOLATED (SPEC v2): on the frozen s = 0 grid "
"%d/%d points are CAR-valid (excluded: %s -- the two "
"m = 3/4 large-coupling corners, disclosed amendment); "
"on every CAR-valid point the margin is exactly 0 (max "
"|margin| = %.1e <= 1e-9, max Hermiticity defect %.1e "
"<= ZTOL) with full mixing 10/10 canonical: a positive-"
"dimensional MARGINAL STRATUM"
% (n_car, len(grid0), excl, marg_max, hd_max),
n_car == 16 and marg_max <= 1e-9 and hd_max <= ZTOL
and mix_ok, kill="K3")
id_ok = True
for dl in (0.02, -0.02, 0.05, -0.05):
A = A_M2 + dl * seat_dir("J")
M1, _m2, hd, lm = abT_grams(A)
ev = np.linalg.eigvalsh((M1 + M1.conj().T) / 2)
id_ok &= (hd <= ZTOL and abs(ev[0] + abs(dl)) <= 1e-12
and abs(ev[1] - abs(dl)) <= 1e-12)
A_s = parent(0.5, 0.5, 0.05, 0.05, 0.05)
M1, _m2, hd_s, lm_s = abT_grams(A_s)
ev_s = np.linalg.eigvalsh((M1 + M1.conj().T) / 2)
id_ok &= (abs(ev_s[0] + 0.05) <= 1e-12
and abs(ev_s[1] - 0.05) <= 1e-12)
check("P3.2 the covariant seat direction gives margin = "
"-|delta_aJ| exactly (4 deltas, ward <= 1e-12) and the "
"family s-direction reproduces margin = -s at s = 1/20 "
"(probe-7 C4 regression): the covariant reachable RP "
"set is EXACTLY {a_J = 0}", id_ok, kill="K3")
sup_cov = 0.0
for p in grid0:
A = parent(*p)
smax = float(np.max(np.abs(np.linalg.eigvalsh(1j * A))))
if smax > 0.95:
continue
_m1, _m2, hd, lm = abT_grams(A)
_n, _s2, aJ45 = census10(schur_Aeff(A))
sup_cov = max(sup_cov, max(lm, 0.0) * abs(aJ45))
for dl in (0.02, 0.05):
for sgn in (1, -1):
A = A_M2 + sgn * dl * seat_dir("J")
_m1, _m2, hd, lm = abT_grams(A)
_n, _s2, aJ45 = census10(schur_Aeff(A))
sup_cov = max(sup_cov, max(lm, 0.0) * abs(aJ45))
check("P3.3 THE SCALAR: sup over the covariant scan of "
"(max(margin, 0) x mixing) = %.1e == 0 EXACTLY (product "
"identically zero on the covariant RP set): the witness "
"is NOT a limit of covariant strictly-RP parents because "
"NONE EXIST -- the closure-point-vs-isolated dichotomy "
"is REFUTED (strict side empty, witness not isolated)"
% sup_cov, sup_cov <= 1e-15, kill="K3")
# ==================================================================
section("P4 -- the transversal escape + its exact price")
# ==================================================================
EPS_GRID = (0.001, 0.0025, 0.005, 0.01, 0.05, 0.1, 0.25)
rows = []
ok_1p = True
ok_price = True
ok_mix = True
prods = []
margins = []
for eps in EPS_GRID:
A = A_M2 + eps * seat_dir("X")
smax = float(np.max(np.abs(np.linalg.eigvalsh(1j * A))))
car = smax <= 0.95
M1, M2, hd, lm = abT_grams(A)
ev1 = np.linalg.eigvalsh((M1 + M1.conj().T) / 2)
ev2 = np.linalg.eigvalsh((M2 + M2.conj().T) / 2)
margin = min(float(ev1[0]), float(ev2[0]))
cd = cov_defect(A)
Aeff = schur_Aeff(A)
n_nz, n_sig, aJ45 = census10(Aeff)
prod = max(margin, 0.0) * abs(aJ45)
rows.append((eps, smax, margin, cd, n_nz, aJ45, prod))
if car:
ok_1p &= (hd <= ZTOL
and abs(ev1[0] - eps) <= 1e-12
and abs(ev1[1] - eps) <= 1e-12
and ev2[0] > 0)
ok_price &= abs(cd - 2 * eps) <= 1e-12
ok_mix &= (n_nz == 10 and abs(aJ45 - 0.005) <= 1e-12)
prods.append((eps, prod))
margins.append((eps, margin))
for (eps, smax, margin, cd, n_nz, aJ45, prod) in rows:
print(" eps=%+.4f smax=%.3f margin=%+.2e cov=%.4f "
"mix=%d aJ45=%.6f product=%.2e"
% (eps, smax, margin, cd, n_nz, aJ45, prod))
check("P4.1 ESCAPE: at every CAR-valid eps > 0 the parent is "
"STRICTLY RP (1p eigenvalues {eps, eps} ward <= 1e-12, "
"deg-2 PD), covariance defect EXACTLY 2 eps, CAR valid, "
"mixing 10/10 with the {4,5} J-coordinate = 1/200 "
"UNCHANGED", ok_1p and ok_price and ok_mix, kill="K4")
small = [(e, m) for (e, m) in margins if e <= 0.01]
xs = np.log([e for e, _m in small])
ys = np.log([m for _e, m in small])
slope = float(np.polyfit(xs, ys, 1)[0])
sup_esc = max(p for _e, p in prods)
check("P4.2 EXPONENT + SUP: the margin is deg-2-limited and "
"QUADRATIC in eps (log-log slope %.3f = 2.00 +- 0.05 on "
"the small-eps grid); the product margin x mixing is > 0 "
"and its sup on the CAR-valid grid is %.2e >= 1e-5: the "
"escape is TRANSVERSAL at finite covariance breaking"
% (slope, sup_esc),
abs(slope - 2.0) <= 0.05 and sup_esc >= 1e-5, kill="K4")
# exact escape at eps = 1/100 (rationals, 16-dim machinery)
epsQ = sp.Rational(1, 100)
Eex = sp.zeros(16)
Xs = sp.Matrix([[0, 1], [1, 0]])
for r in range(2):
for c in range(2):
Eex[CH[a_ch][r], CH[b_ch][c]] = Xs[r, c]
Eex[CH[b_ch][c], CH[a_ch][r]] = -Xs[r, c]
A_esc = A_ex + epsQ * Eex
wk_esc = wick_exact_factory(A_esc)
M1e = gram_exact(B1_ab, r_abT, sp.I, wk_esc)
M2e = gram_exact(B2_ab, r_abT, sp.I, wk_esc)
ok_m1e = sp.expand(M1e - sp.diag(epsQ, epsQ)) == sp.zeros(2)
herm2e = sp.expand(M2e - M2e.conjugate().T) == sp.zeros(2)
det2e = sp.expand(M2e.det())
check("P4.3 EXACT at eps = 1/100: M1 = diag(1/100, 1/100) "
"exactly (%s); deg-2 Hermitian (%s) with det = %s == "
"eps^2 = 1/10000 exactly" % (ok_m1e, herm2e, det2e),
bool(ok_m1e) and bool(herm2e)
and det2e == sp.Rational(1, 10000), kill="K4")
# symbolic Schur pass-through + floor exchange
kap_s, m_s, t_s, e_s = sp.symbols("kappa m t eps", real=True)
A_CCs = sp.Matrix(A_CC.tolist())
J3s = sp.Matrix(J3.tolist())
Ws = t_s * sp.Matrix((W1 + W2).tolist())
ECC = sp.zeros(10, 10)
for r in range(2):
for c in range(2):
ECC[CH[a_ch][r], CH[b_ch][c]] = Xs[r, c]
ECC[CH[b_ch][c], CH[a_ch][r]] = -Xs[r, c]
C_CC = (sp.eye(10) + sp.I * (kap_s * A_CCs + e_s * ECC)) / 2
C_CB = sp.I * Ws / 2
C_BC = -sp.I * Ws.T / 2
C_BB_inv = 2 * (sp.eye(6) - sp.I * m_s * J3s) / (1 - m_s ** 2)
C_eff = sp.expand(C_CC - C_CB * C_BB_inv * C_BC)
A_eff_sym = sp.Matrix(10, 10, lambda r, c: sp.expand(
sp.im(sp.expand(2 * C_eff[r, c]))))
A_eff_formula = (kap_s * A_CCs + e_s * ECC
+ (m_s / (1 - m_s ** 2)) * Ws * J3s * Ws.T)
ok_schur = sp.simplify(A_eff_sym - A_eff_formula) == sp.zeros(10)
lamQ = mQ / (1 - mQ ** 2)
Jco = sp.Integer(3) * lamQ * tQ ** 2
B45 = (Jco * sp.Matrix([[0, 1], [-1, 0]]) + e_s * Xs)
pf45 = sp.expand(-B45.det())
pf_target = sp.expand((e_s - sp.Rational(1, 200))
* (e_s + sp.Rational(1, 200)))
M1_eff = sp.Matrix([[B45[0, 1], B45[1, 1]],
[B45[0, 0], B45[1, 0]]])
ev_eff_ok = sp.expand(M1_eff - sp.diag(
e_s + sp.Rational(1, 200),
e_s - sp.Rational(1, 200))) == sp.zeros(2)
check("P4.4 FLOOR EXCHANGE (exact): the Schur identity extends "
"with the eps term passing through untouched, A_eff = "
"kappa A_CC + eps E_CC + (m/(1-m^2)) W J3 W^T (%s); at "
"the M2 parent Pf4_{45}(eps) = %s == (eps - 1/200)(eps "
"+ 1/200) (%s) and the effective 2-cycle odd Gram is "
"diag(eps + 1/200, eps - 1/200) exactly (%s): the "
"canonical sign gate IS the negative effective-RP margin "
"-- the same factor (eps - 1/200); at eps = 1/200 the "
"{4,5} Pfaffian dies exactly"
% (ok_schur, pf45, sp.expand(pf45 - pf_target) == 0,
ev_eff_ok),
bool(ok_schur) and sp.expand(pf45 - pf_target) == 0
and bool(ev_eff_ok), kill="K4")
# ==================================================================
section("C -- controls (must fire; RNG only in C5)")
# ==================================================================
ok_c1 = True
for eps in (0.01, -0.01):
A = A_M2 + eps * seat_dir("I")
M1, _m2, hd, lm = abT_grams(A)
ev = np.linalg.eigvalsh((M1 + M1.conj().T) / 2)
ok_c1 &= (hd <= ZTOL and abs(ev[0] + abs(eps)) <= 1e-12
and abs(ev[1] - abs(eps)) <= 1e-12 and lm < 0)
check("C1 FIRES: the I seat direction violates RP for BOTH "
"signs (1p eigenvalues -+|eps| exactly) -- the "
"deliberately RP-violating control", ok_c1, kill="K7")
A = A_M2 + 0.01 * seat_dir("Z")
M1, _m2, hdZ, _lm = abT_grams(A)
raw = float(np.max(np.abs(M1 - M1.conj().T)))
check("C2 FIRES: the Z seat direction breaks 1p Gram "
"Hermiticity (raw defect %.4f == 2|eps| = 0.02 +- 1e-12) "
"-- not an OS candidate" % raw,
abs(raw - 0.02) <= 1e-12, kill="K7")
A = A_M2 - 0.005 * seat_dir("X")
M1, _m2, hd, lm = abT_grams(A)
ev = np.linalg.eigvalsh((M1 + M1.conj().T) / 2)
check("C3 FIRES: eps = -1/200 on X gives BOTH 1p eigenvalues "
"negative ({%.4f, %.4f})" % (ev[0], ev[1]),
ev[0] < 0 and ev[1] < 0 and hd <= ZTOL, kill="K7")
Jco_val = sp.Integer(3) * (mQ / (1 - mQ ** 2)) * tQ ** 2
check("C4 EXACT 1/200 REGRESSION: J-coordinate = t^2 3m/(1-m^2)"
" = %s == 1/200 at the M2 parent (exact rationals)"
% Jco_val, Jco_val == sp.Rational(1, 200), kill="K7")
rng = np.random.default_rng(903)
n_fire = 0
for _trial in range(3):
pr = rng.permutation(10)
A_bad = parent(0.5, 0.5, 0.05, 0.05, 0.0)
Wb = A_bad[np.ix_(CAR_IDX, BND_IDX)][pr, :]
A_bad[np.ix_(CAR_IDX, BND_IDX)] = Wb
A_bad[np.ix_(BND_IDX, CAR_IDX)] = -Wb.T
if cov_defect(A_bad) >= NZ_FLOOR:
n_fire += 1
check("C5 FIRES: 3/3 seeded random row permutations of the "
"coupling break the exact C6-covariance ward (%d/3)"
% n_fire, n_fire == 3, kill="K7")
# ==================================================================
section("VERDICT")
# ==================================================================
n_pass = sum(1 for _n, ok in CHECKS if ok)
n_tot = len(CHECKS)
controls_ok = all(ok for nm, ok in CHECKS if nm.startswith("C"))
if not controls_ok:
VERDICT = "CONTROL-DEAD"
elif "K0" in KILLS:
VERDICT = "PIPELINE-BROKEN"
elif "K1" in KILLS:
VERDICT = "SEAT-BROKEN"
elif "K2" in KILLS:
VERDICT = "KERNEL-BROKEN"
elif "K3" in KILLS:
VERDICT = "STRATUM-BROKEN"
elif "K4" in KILLS:
VERDICT = "ESCAPE-BROKEN"
else:
VERDICT = ("MARGSTAB-MEASURED [INTERIOR-EMPTY(covariant "
"strict RP impossible: +-a_J and det = -a_J^2 "
"exact), STRATUM-NOT-ISOLATED(margin == 0 on "
"the whole s = 0 grid), NOT-A-CLOSURE-POINT("
"covariant strict approximants do not exist), "
"ESCAPE-NONCOVARIANT(X seat coordinate; price "
"2 eps; exponent 2), FLOOR-EXCHANGE(Pf4 = "
"(eps - 1/200)(eps + 1/200) = the effective "
"margin factor)]")
print("%d/%d checks passed" % (n_pass, n_tot))
print("VERDICT: %s" % VERDICT)
print("""
REPORT (exploration only -- no promotion, no edits):
* THE HEADLINE ANSWER: the round-59 dichotomy (closure point vs
isolated) is REFUTED as a dichotomy. Within the C6-covariant
class the strict-RP side of the cone is EMPTY -- the seat law
M1 = diag(a_J, -a_J), det M2 = -a_J^2 (exact, whole class)
makes strict theta_abT RP impossible, so the marginal witness
is NOT a limit of covariant strict equilibria (none exist);
it is ALSO not isolated (the entire s = 0 family is a marginal
stratum with margin exactly 0 and full mixing). 'Physical as
a limit of equilibria' is unrealizable inside the symmetry
class; the decisive scalar sup(margin x mixing) = 0 exactly.
* THE KERNEL: at the witness the 1p Gram vanishes identically
(kernel = both a-side Majoranas) and the deg-2 Gram [[1, i/2],
[-i/2, 1/4]] has the exact kernel vector (-i/2, 1); both are
forced by the same pure-J covariance law (the only visible
seat coordinate is a_J) that seats the strict obstruction.
* THE ESCAPE: breaking covariance along the unique admissible X
seat coordinate buys strict parent RP at ANY eps > 0 with full
mixing kept (product > 0, transversal; price = covariance
defect exactly 2 eps; margin quadratic, exponent 2.0) -- BUT
one level down the exclusion survives: Pf4_{45}(eps) =
(eps - 1/200)(eps + 1/200) shares its root with the effective
RP margin, so strict effective RP and the canonical mixing
sign EXCHANGE exactly at the 1/200 floor.
* HONESTY: all statements are sector-typed measurements/exact
identities on this finite model; the escape is a recorded
covariance-breaking deformation, not a proposal; the [O]
premise of v898/v903 is unmoved; no marker moves. NO RH claim.
Runtime: %.1f s""" % (time.time() - T0))
print("ALL CHECKS PASSED" if n_pass == n_tot
else "CHECKS FAILED: %d" % (n_tot - n_pass))
return 0 if (n_pass == n_tot and not KILLS) else 1
if __name__ == "__main__":
raise SystemExit(main())
'''
# ------------- frozen probe source seam_ness_parent_probe (embedded BYTE-EXACT, raw string)
_SRC_1 = r'''
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
r"""seam_ness_parent_probe -- SEAM.CFIN.NESS.PARENT.01
(EXPLORATION ONLY, experiments/; round 60, 2026-08-10, Probe 10 --
named in round 59, built here for the STRICT route: does the
strict-collar (sheet-swap theta_S) mixing need a NESS parent --
two reservoirs, positive entropy production -- or is the round-59
family obstruction cheaper to dissolve?)
THE QUESTION. Probe 7 (rp_parent_dilation_probe; promoted in
v903) measured the strict-collar route EXACTLY OBSTRUCTED on the
five-parameter family: 30 deg-2 Gram Hermiticity defect entries,
all of magnitude 2t, linear in the coupling. Round 59 named the
NESS-parent test: construct two-reservoir / two-temperature (or
modular-drive) quasi-free parents on the 10 (+) 6 split with
C6-covariant coupling, define what RP can mean for a NESS, and
measure the minimal entropy production at which the mixing gates
open ('the price of the mixing'), typing FORCED / TRADE-OFF /
IMPOSSIBLE.
FEASIBILITY / REDUNDANCY CHECK (done against the corpus FIRST,
2026-08-10): probe 7 scanned ONLY the deployed coupling direction
t1 W1 + t2 W2 (the A_int rows) and typed the obstruction 'a
measurement on THIS parametrized family, not a universal no-go';
v898 counts the covariant C<->B mixing block at 24 dimensions but
never intersects it with RP; round 58 deployed theta_S but only on
the KMS family; NOTHING in the corpus (a) writes the Hermiticity
defect as a LINEAR functional of the coupling, (b) scans the FULL
24-dim covariant coupling space against it, or (c) constructs
driven/two-temperature parents. That is exactly this probe.
SMOKE-RUN DISCLOSURE (2026-08-10, declared smoke rounds before
freezing; the smoke INVERTED the working hypothesis and the frozen
claims record what was measured -- fail-first preserved):
(i) THE TWO-SEAT LINEAR LAW: strict theta_S Hermiticity is
obstructed at LINEAR order by exactly two families of
entries -- the 1p seat A_{a+1, b} + A_{a, b+1} (a in C,
b in B, both even; kills the X sub-block coordinates of
the coupling) and the deg-2 (empty <-> mixed-pair) seat
A_{a, b} + A_{a+1, b+1} (kills the I coordinates); on the
24-dim covariant coupling space the combined law has rank
12, kernel 12 = the {J, Z} sub-block coordinates;
(ii) the DEPLOYED coupling V = A_int[C, B] has every 2x2
sub-block EXACTLY -I2: it is PURE-I, i.e. it lies entirely
in the defect row space -- the deployed seam wiring is a
maximally strict-collar-obstructed covariant direction
(this explains the round-59 '30 entries, magnitude 2t'
law: 15 mixed pairs x 2 Gram positions, defect 2t each);
(iii) THE INVERSION: the kernel contains EQUILIBRIUM witnesses.
The uniform J-coupling parent (kappa = m = 1/2, t = 1/20)
passes STRICT theta_S RP (exactly Hermitian Grams, PD)
while its Schur compression carries the FULL canonical
Pfaffian mixing with S_J = V_J J3 V_J^T = 3J on ALL 25
channel blocks and the {4,5} J-coordinate EXACTLY
t^2 3m/(1-m^2) = 1/200 -- the same rational identity as
the deployed-direction witness; the uniform Z-coupling is
a second witness (S_Z = -3J, J-coordinate -1/200). So the
minimal entropy production at which the mixing gates open
is ZERO: no NESS is needed;
(iv) the kernel is NOT sufficient: the uniform X-coupling is
covariant but fails at the 1p seat (raw defect exactly
2t = 0.1), and J/Z-MIXTURES pass Hermiticity + PD but
break the canonical SIGN census (4/10) -- the canonical
gate selects the uniform J (or Z) ray;
(v) the NESS side collapses honestly at finite size: an
exactly stationary quasi-free state has [h, A] = 0, hence
every block flux Phi_r = (1/4) tr(h0_r [h, A]) vanishes
and sigma = sum_r beta_r Phi_r == 0 -- positive entropy
production is CATEGORY-INAPPLICABLE to a finite stationary
state; the two-temperature Cesaro (dephasing) states are
stationary, covariant, CAR-valid, carry the 15/15 block
mixing ALREADY AT ZERO DRIVE, keep the Hermiticity defect
PINNED near the deployed-direction value (0.0982-0.1062)
and only DEGRADE the hermitized PSD margin as the drive
grows (0.1884 -> 0.0155): drive buys NO RP;
(vi) transient entropy production from the product initial
state is exactly 0 at t = 0 (block-diagonal initial state,
off-diagonal commutator), positive through t ~ 2
(sigma(1) = 0.0626 at beta_C = 1, beta_B = 1/4), and
oscillates in sign later (finite recurrence -- no true
NESS at finite size; typed, not hidden).
CONVENTIONS (probe 7 / round 58 wiring rebuilt inline; READ-ONLY
import of tfpt_constants): 16-dim Majorana space, carrier C = 0..9
(channels 1..5), boundary B = 10..15; A_CC = (+)_5 J, J3 =
A16_dep[B, B]; coupled parents A_parV(kappa, m, t, V) =
[[kappa A_CC, t V], [-t V^T, m J3]]; theta_S = sheet swap
(eta = +i), sector Grams 1p (8 half-side Majoranas) + even deg-2
(29 monomials), strict RP = Hermiticity <= ZTOL and lam_min >=
-NZ_FLOOR in both sectors (sector statement, probe-7 convention).
Covariant coupling space = fixed points of V -> O_C V O_B^T
(integer orbit sums). Compression + census: probe-7 form
(A_eff, 10 carrier duads, canonical Pf4 signs); state-level
census: v898 12-dim Iota/3 compression, 15 duads. ENTROPY
PRODUCTION (documented formula): with h = h0 + coupling split
h0 = h_C (+) h_B, H_r = (i/4) sum_{ij in r} h0_ij c_i c_j,
E_r(A) = <H_r> = (1/4) tr(h0_r A), Phi_r(A) = dE_r/dt =
(1/4) tr(h0_r [h, A]), sigma = beta_C Phi_C + beta_B Phi_B;
dynamics A(t) = e^{th} A e^{-th} (dA/dt = [h, A]); total-energy
identity tr(h [h, A]) == 0 (cyclic). Two-temperature parents:
A_0(beta_C, beta_B) = KMS_{beta_C}(h_C) (+) KMS_{beta_B}(h_B);
NESS analogue = the Cesaro / dephasing average of A_0 over the
spectrum of the coupled h (the long-time average, exactly
stationary). RP FOR A NESS (defined per the round-59 demand):
strict RP is category-inapplicable to 'stationary + positive
entropy production' at finite size (P4.1 proves sigma == 0 for
stationary states), so the weakest meaningful positivity is RP of
the TIME-ZERO (stationary) covariance -- that is what is tested.
NUMERICAL PROTOCOL (declared): the two-seat law, its rank/kernel,
the deployed-coupling decomposition, the witnesses (Grams, LDL,
Schur, census) are EXACT (integer/rational sympy); drive scans and
transients are float64 with frozen tolerances; RNG only in
controls.
FROZEN CLAIMS (2026-08-10, frozen + SHA-hashed before the frozen
run):
P1 THE TWO-SEAT LINEAR LAW (exact).
(a) CLOSED FORMS warded against the Gram machinery (float
<= 1e-12 on unit couplings): the 1p Gram Hermiticity
defect entries are (M1 - M1+)_{xy} = -(A_{x+1, y} +
A_{x, y+1}) for even x != y in the half-side basis, and
the deg-2 (empty <-> pair) defect entries are
i (A_{x, y} + A_{x+1, y+1});
(b) the covariant coupling space has dimension EXACTLY 24
(integer orbit-sum basis; the v898 T-count); the combined
two-seat law on it has EXACT rank 12 and kernel dimension
12; coordinate identification (exact, uniform unit
couplings): I -> deg-2 seat fires, 1p clean; X -> 1p
seat fires, deg-2-empty clean; J and Z -> BOTH seats
clean;
(c) the DEPLOYED coupling V = A_int[C, B] has every 2x2
sub-block EXACTLY -I2 (integer): PURE-I, entirely inside
the defect row space; its parent reproduces the probe-7
law (raw deg-2 empty-seat defect 2t = 0.1, 30 entries).
P2 THE EQUILIBRIUM WITNESS (exact rationals end to end -- the
answer to the NESS question).
(a) V_J (J on all 15 sub-blocks) is exactly covariant and in
the kernel; the parent A_J(1/2, 1/2, 1/20) is CAR-strict
(smax = 0.694 +- 0.005 < 0.95) and by the artanh theorem
a beta = 1 KMS state of a covariant Hamiltonian
(round-trip ward <= 1e-12): EQUILIBRIUM, sigma = 0;
(b) STRICT theta_S RP PASSES EXACTLY: 1p and deg-2 Grams
exactly Hermitian (sympy) and PD by exact LDL pivots;
float lam_min = 0.3064 +- 0.005 (1p) and 0.1532 +- 0.005
(deg-2);
(c) the compression carries the FULL canonical mixing:
S_J := V_J J3 V_J^T = 3J on ALL 25 channel blocks
(integer identity); A_eff = kappa A_CC + (m/(1-m^2)) t^2
S_J (symbolic Schur identity in kappa, m, t); every
carrier duad block EXACTLY (3 m t^2/(1-m^2)) J with
J-coordinate = 1/200 at the witness (the round-51/52
floor as the SAME exact fraction), per-edge Pf4 < 0
canonical on 10/10, compressed CAR valid;
(d) V_Z is a SECOND exact witness: S_Z = -3J uniform
(integer), strict theta_S RP passes (float lam_min
0.2383 +- 0.005), J-coordinate EXACTLY -1/200, Pf4
canonical 10/10;
(e) the 2-cycle side is UNTOUCHED: theta_abT stays marginal
at the V_J witness (1p Gram identically 0, exact) and
the compressed state fails effective 2-cycle RP with odd
eigenvalues exactly -+1/200 -- the strict-collar gate
opens while the round-58 2-cycle exclusion stands.
P3 KERNEL != SUFFICIENT (the honest boundary of the win).
(a) the uniform X-coupling (in NO seat... in the covariant
space but OUTSIDE the kernel) fails at the 1p seat with
raw defect EXACTLY 2t = 0.1 (empty deg-2 seat clean
<= 1e-14, pair-pair defect 0.05 +- 0.005): the law is
sharp in both directions;
(b) J/Z-MIXTURES (frozen members: J-on-2-cycle/Z-on-3-cycle;
mixed amplitudes) pass exact Hermiticity and PD but the
canonical SIGN census drops to 4/10: the canonical gate
selects the uniform J (or uniform Z) ray -- Hermiticity
kernel and canonical-mixing ray are DIFFERENT strata
(typed);
(c) orbit selectivity in the kernel (regression of probe-7
P2.4): J-on-2-cycle-orbit only mixes 1/10, J-on-3-cycle
only 3/10, single-boundary-pair J (both orbits) 10/10
with J-coordinate EXACTLY lam t^2 = 1/600;
(d) amplitude anatomy (report, not gate): uniform J passes
at t = 1/10 (lam_min 0.056, smax 0.887), t >= 0.15 is
CAR-invalid (smax > 1).
P4 THE NESS SIDE (constructed; the exact no-go + the price
curve).
(a) FINITE-NESS-NOGO (exact): stationarity [h, A] = 0
implies EVERY block flux Phi_r = (1/4) tr(h0_r [h, A])
= 0, hence sigma == 0 IDENTICALLY: 'NESS with positive
entropy production' is category-inapplicable at this
finite size; the total-energy identity tr(h [h, A]) == 0
holds exactly (cyclic; float ward <= 1e-12); the weakest
meaningful positivity = RP of the stationary covariance
(tested in (c));
(b) the two-temperature Cesaro states on the frozen drive
grid (beta_C, beta_B) in {(1, 1), (1, 1/2), (1, 1/4),
(2, 1/2)} at the deployed h(1, 1/8): exactly stationary
(<= 1e-12), real antisymmetric, C6-covariant, CAR-valid
(smax <= 0.85); the t = 0 fluxes from the product
initial state vanish EXACTLY (<= 1e-14); the transient
sigma(t) at (1, 1/4) is positive at t = 1 (0.0626 +-
0.01) and NEGATIVE at t = 4 (finite recurrence, typed:
no true NESS at finite size);
(c) THE PRICE CURVE IS FLAT-OPEN: the state-level mixing
census is 15/15 canonical at EVERY drive point INCLUDING
zero drive; the strict theta_S Hermiticity defect stays
PINNED (0.0982 +- 0.005 at the three beta_C = 1 points,
0.1062 +- 0.005 at (2, 1/2)) and the hermitized PSD
margin DEGRADES monotonically with the drive (0.1884 ->
0.0600 -> 0.0155 along beta_B = 1 -> 1/2 -> 1/4): the
minimal entropy production at which the mixing gates
open is EXACTLY ZERO, and drive buys NO RP;
(d) TYPED CONCLUSION (the round-59 enum): NESS-NOT-FORCED --
the strict-collar mixing needs the coupling DIRECTION
(the two-seat kernel), which is equilibrium-compatible;
the round-59 obstruction is DIRECTION-level (the
deployed PURE-I wiring), not equilibrium-level; on this
tested class a NESS is neither forced nor helpful.
C CONTROLS (must fire; frozen fire rules; RNG only in C2).
C1 ZERO-DRIVE REGRESSION: the global KMS state (u = 1,
t = 1/8, beta = 1) is exactly stationary and FAILS strict
theta_S with Hermiticity defect 0.0982 +- 0.005 (the
round-58/59 equilibrium obstruction reproduced);
C2 SEEDED SCRAMBLE (rng 904, 3 draws: random row permutation
of V_J): breaks the exact C6-covariance ward on 3/3 AND
breaks the canonical Pfaffian census on 3/3 (sign census
<= 1/10; the Hermiticity defect also switches ON, 0.1);
C3 DEPLOYED-COUPLING REGRESSION: the PURE-I parent (V = the
deployed -I2 wiring, kappa = m = 1/2, t = 1/20) fails
strict theta_S with raw empty-seat defect 2t = 0.1 and
exactly 30 defect entries (probe-7 law);
C4 I-ORBIT SWITCH-ON: adding one covariant I-orbit direction
to V_J turns the deg-2 empty-seat defect ON (raw defect
> 0.01) while covariance stays EXACT -- the law bites
inside the covariant class.
KILLS (any one fires => typed gap):
K0 AST firewall / compiler rebuild ward breaks -> PIPELINE-BROKEN
K1 two-seat law / rank / deployed decomposition -> LAW-BROKEN
K2 equilibrium witness ward breaks -> WITNESS-BROKEN
K3 kernel honesty ward breaks -> KERNEL-BROKEN
K4 NESS construction / no-go / price ward -> NESS-BROKEN
K7 a control does not fire -> CONTROL-DEAD
VERDICT (frozen enum): NESSPARENT-MEASURED [PRICE-ZERO(equilibrium
strict-collar witnesses V_J / V_Z: exact Hermitian + PD + 10/10
canonical + J-coordinate +-1/200), TWO-SEAT-LAW(rank 12, kernel 12
= the {J, Z} coordinates; deployed coupling PURE-I),
CANONICAL-SELECTS-UNIFORM-RAY(mixtures 4/10),
FINITE-NESS-NOGO(stationary => sigma == 0, exact),
DRIVE-RP-NEUTRAL(defect pinned, margin degrades, gates open at
zero drive), NESS-NOT-FORCED] / PIPELINE-BROKEN / LAW-BROKEN /
WITNESS-BROKEN / KERNEL-BROKEN / NESS-BROKEN / CONTROL-DEAD.
Exit 0 iff all checks pass and no kill fired; else 1.
FIREWALL: experiments/ probe; EXPLORATION ONLY -- writes nothing
but stdout; no verification/, paper, ledger, changelog or website
surface; no .md, no commits. NO physics claim beyond the recorded
identities and measurements: the equilibrium witnesses are
CANDIDATE parents on a DIFFERENT covariant coupling direction than
the deployed A_int wiring -- whether the actual seam allows that
direction is untouched; the finite-NESS no-go is a finite-size
statement, not a claim about infinite-volume NESS; RP remains
sector-typed; the v898/v903 [O] premise is unmoved; no marker
moves. NO RH claim.
SPEC v1 (2026-08-10): frozen after the declared smoke rounds; no
amendments at freeze.
Sources (read-only, machinery rebuilt inline): rp_parent_dilation_
probe (probe 7: family, strict-collar obstruction, census),
seam_state_derivation_probe (round 58: theta_S, RP machinery),
v903_seam_rp_exclusion (promoted composite), v898_kms_schur_mixing
(24-dim T-count, state gates), v519 (RP Gram + twist),
tfpt_constants (N_fam, g_car).
Run:
experiments/tfpt-discovery/.venv/bin/python \
experiments/tfpt-discovery/seam_ness_parent_probe.py
"""
import ast
import hashlib
import itertools
import math
import os
import sys
import time
import numpy as np
import sympy as sp
_HERE = os.path.dirname(os.path.abspath(__file__))
_VERIFY = os.path.abspath(os.path.join(_HERE, "..", "..",
"verification"))
sys.path.insert(0, _VERIFY)
from tfpt_constants import N_fam, g_car # noqa: E402 (READ-ONLY)
BANNED_IDS = ("zetazero", "nzeros", "primerange", "isprime",
"primepi", "nextprime", "prevprime")
CHECKS = []
KILLS = []
T0 = time.time()
SPEC_SHA = hashlib.sha256(__doc__.encode("utf-8")).hexdigest()
NZ_FLOOR = 1e-8
ZTOL = 1e-10
PF_FLOOR = 1e-16
def check(name, ok, detail="", kill=None):
CHECKS.append((name, bool(ok)))
if kill and not ok:
KILLS.append(kill)
print(" [%s] %s%s" % ("PASS" if ok else "FAIL", name,
(" -- " + detail) if detail else ""),
flush=True)
return bool(ok)
def section(title):
print("\n" + "=" * 74)
print("%s (t=%.1fs)" % (title, time.time() - T0))
print("=" * 74, flush=True)
def ast_scan(banned):
src = open(os.path.abspath(__file__), encoding="utf-8").read()
bad = []
for node in ast.walk(ast.parse(src)):
name = None
if isinstance(node, ast.Name):
name = node.id
elif isinstance(node, ast.Attribute):
name = node.attr
if name and name.lower() in banned:
bad.append(name)
return bad
# ---------------------------------------------------------- bit model
def pc(v):
return bin(v).count("1")
HT = [[(pc(v) * pc(w) - pc(v & w)) % 2 for w in range(16)]
for v in range(16)]
A_BIT = 0b1000
FSIG = 0b0111
LOWIDX = {1: 0, 2: 1, 4: 2, 8: 3}
def sig(v):
b = [(v >> i) & 1 for i in range(4)]
n = (b[2], b[0], b[1], b[3])
return sum(bit << i for i, bit in enumerate(n))
SIGP = tuple(sig(v) for v in range(16))
def polar_shift(c):
return tuple((pc(v) * (pc(v) - 1) // 2 + pc(c & v)) % 2
for v in range(16))
def iota_bits(v):
b = [(v >> i) & 1 for i in range(4)]
b.append(sum(b) % 2)
return tuple(b)
IOTA_MSG = [iota_bits(v) for v in range(16)]
def iota_support(v):
return frozenset(i + 1 for i, bit in enumerate(IOTA_MSG[v]) if bit)
def compose(p, q):
return tuple(p[q[i]] for i in range(len(q)))
def perm_order(p):
o, pp = 1, p
ident = tuple(range(len(p)))
while pp != ident:
pp = tuple(p[x] for x in pp)
o += 1
return o
def cycle_type(perm):
n = len(perm)
seen = [False] * n
cyc = []
for i in range(n):
if seen[i]:
continue
ln, j = 0, i
while not seen[j]:
seen[j] = True
j = perm[j]
ln += 1
cyc.append(ln)
return tuple(sorted(cyc))
def edge_orbits(perm):
n_ord = perm_order(perm)
seen = set()
out = []
for i, j in itertools.combinations(range(6), 2):
e = frozenset({i, j})
if e in seen:
continue
x, y = i, j
edges = set()
rev = False
for _k in range(n_ord):
edges.add(frozenset({x, y}))
x, y = perm[x], perm[y]
if (x, y) == (j, i):
rev = True
seen |= edges
out.append((frozenset(edges), rev, (i, j)))
return out
DUADS_CH = sorted(itertools.combinations(range(6), 2))
CAR_DUADS = sorted(itertools.combinations(range(1, 6), 2))
PAULI = {"I": np.eye(2), "J": np.array([[0., 1.], [-1., 0.]]),
"X": np.array([[0., 1.], [1., 0.]]),
"Z": np.array([[1., 0.], [0., -1.]])}
def main():
print("SEAM.CFIN.NESS.PARENT.01 -- the price of the "
"strict-collar mixing: NESS parents vs the coupling "
"direction")
print("FROZEN_SPEC SHA-256: %s" % SPEC_SHA)
print("NO physics claim beyond recorded identities/measurements; "
"exploration only.")
# ==================================================================
section("S0 -- firewall + compiler-side setup (probe 7 rebuilt)")
# ==================================================================
bad = ast_scan(BANNED_IDS)
check("S0.0 AST firewall: no banned identifiers %s" % (BANNED_IDS,),
not bad, kill="K0")
refs = [polar_shift(c) for c in range(16)]
ok_ref = all(
all(q[x ^ y] ^ q[x] ^ q[y] == HT[x][y]
for x in range(16) for y in range(16)) for q in refs)
arf1 = sorted(q for q in refs if q.count(0) == 6)
siginv = [q for q in refs
if all(q[SIGP[v]] == q[v] for v in range(16))]
cand = [q for q in siginv if q[A_BIT] == 1 and q[FSIG] == 0]
QSTAR = cand[0] if cand else None
NZ = list(range(1, 16))
ovoid = [v for v in NZ if QSTAR[v] == 0]
def duad(v):
return frozenset(i for i, q in enumerate(arf1) if q[v] == 0)
dmap = {v: duad(v) for v in NZ}
V0 = arf1.index(QSTAR)
phi = {}
ok_phi = True
for o in ovoid:
others = dmap[o] - {V0}
islot = frozenset(range(1, 6)) - iota_support(o)
ok_phi &= (len(others) == 1 and len(islot) == 1)
phi[next(iter(others))] = next(iter(islot))
ok_phi &= (len(phi) == 5 and set(phi.values()) == set(range(1, 6)))
def lab(j):
return 0 if j == V0 else phi[j]
SP6 = []
for imgs in itertools.product(range(1, 16), repeat=4):
p = [0] * 16
for v in range(1, 16):
lb = v & -v
p[v] = p[v ^ lb] ^ imgs[LOWIDX[lb]]
if len(set(p)) != 16:
continue
if all(HT[imgs[x]][imgs[y]] == 1
for x in range(4) for y in range(x + 1, 4)):
SP6.append(tuple(p))
S5P = list(itertools.permutations(range(5)))
AUT = []
for p in SP6:
if any(QSTAR[p[v]] != QSTAR[v] for v in range(16)):
continue
if compose(p, SIGP) != compose(SIGP, p):
continue
pis = [pi for pi in S5P
if all(IOTA_MSG[p[v]] == tuple(IOTA_MSG[v][pi[s]]
for s in range(5))
for v in range(16))]
if pis:
AUT.append(p)
g_pin = [p for p in AUT
if perm_order(p) == 6 and compose(p, p) == SIGP]
check("S0.1 compiler rebuilt: unique q*, |Aut| = %d == 6, "
"generator pin unique" % len(AUT),
ok_ref and len(cand) == 1 and ok_phi and len(AUT) == 6
and len(g_pin) == 1, kill="K0")
GEN = g_pin[0]
a1idx = {q: i for i, q in enumerate(arf1)}
tau = [a1idx[tuple(q[GEN[v]] for v in range(16))] for q in arf1]
pia = [0] * 6
for j in range(6):
pia[tau[j]] = j
pia = tuple(pia)
PI6 = [0] * 6
for j in range(6):
PI6[lab(j)] = lab(pia[j])
PI6 = tuple(PI6)
cycles = []
seen = set()
for i in range(6):
if i in seen:
continue
cyc, j = [], i
while j not in seen:
seen.add(j)
cyc.append(j)
j = PI6[j]
cycles.append(cyc)
TWO = sorted(next(c for c in cycles if len(c) == 2))
THREE = sorted(next(c for c in cycles if len(c) == 3))
a_ch, b_ch = TWO
check("S0.2 deployed pi = %s, cycle type %s == (1, 2, 3); "
"2-cycle {%d,%d}, 3-cycle %s"
% (PI6, cycle_type(PI6), a_ch, b_ch, THREE),
PI6[0] == 0 and cycle_type(PI6) == (1, 2, 3), kill="K0")
CH = {0: list(range(10, 16))}
for i in range(1, 6):
CH[i] = [2 * (i - 1), 2 * (i - 1) + 1]
CAR_IDX = list(range(10))
BND_IDX = list(range(10, 16))
img = [0] * 16
for i in range(6):
for k, s in enumerate(CH[i]):
img[s] = CH[PI6[i]][k]
J2i = np.array([[0, 1], [-1, 0]], dtype=np.int64)
I2i = np.eye(2, dtype=np.int64)
IOTA6i = np.vstack([I2i, I2i, I2i])
orbs = edge_orbits(PI6)
def put_ordered(A, x, y, B):
rx, cy = CH[x], CH[y]
for r in range(len(rx)):
for c in range(len(cy)):
A[rx[r], cy[c]] = B[r, c]
A[cy[c], rx[r]] = -B[r, c]
A_int = np.zeros((16, 16), dtype=np.int64)
for edges, rev, rep in orbs:
i, j = rep
B = J2i if rev else (IOTA6i if i == 0 else I2i)
x, y = i, j
for _k in range(perm_order(PI6)):
put_ordered(A_int, x, y, B)
x, y = PI6[x], PI6[y]
A16_dep = np.zeros((16, 16), dtype=np.int64)
for i in range(8):
A16_dep[2 * i, 2 * i + 1] = 1
A16_dep[2 * i + 1, 2 * i] = -1
okA = (np.array_equal(A_int[np.ix_(img, img)], A_int)
and np.array_equal(A_int, -A_int.T))
okD = np.array_equal(A16_dep[np.ix_(img, img)], A16_dep)
A_CC = A16_dep[np.ix_(CAR_IDX, CAR_IDX)]
J3 = A16_dep[np.ix_(BND_IDX, BND_IDX)]
Vc = A_int[np.ix_(CAR_IDX, BND_IDX)]
check("S0.3 blocks extracted: A_CC = (+)_5 J, J3, deployed "
"coupling V = A_int[C, B]", okA and okD, kill="K0")
O16 = np.zeros((16, 16), dtype=np.int64)
for src in range(16):
O16[img[src], src] = 1
O_C = O16[np.ix_(CAR_IDX, CAR_IDX)]
O_B = O16[np.ix_(BND_IDX, BND_IDX)]
CH2 = {i: [2 * i, 2 * i + 1] for i in range(6)}
IOTA_f = IOTA6i.astype(np.float64)
def compress12(A):
Ahat = np.zeros((12, 12))
for (i, j) in DUADS_CH:
if i == 0:
B = IOTA_f.T @ A[np.ix_(CH[0], CH[j])] / 3.0
else:
B = A[np.ix_(CH[i], CH[j])]
for rr in range(2):
for cc in range(2):
Ahat[CH2[i][rr], CH2[j][cc]] = B[rr, cc]
Ahat[CH2[j][cc], CH2[i][rr]] = -B[rr, cc]
return Ahat
def pf4_of(Ahat):
out = {}
for (i, j) in DUADS_CH:
B = Ahat[np.ix_(CH2[i], CH2[j])]
out[frozenset({i, j})] = -(B[0, 0] * B[1, 1]
- B[0, 1] * B[1, 0])
return out
pf4_c = pf4_of(compress12(A_int.astype(np.float64) / 200.0))
sign_c = {d: (1 if v > 0 else -1) for d, v in pf4_c.items()}
check("S0.4 canonical G_c Pf4 signs rebuilt: 15 nonzero, all "
"negative", all(abs(v) > PF_FLOOR for v in pf4_c.values())
and all(s == -1 for s in sign_c.values()), kill="K0")
# ---------------- RP machinery (v519 / probe-7 form)
def wick_factory(A):
n = A.shape[0]
W = np.eye(n, dtype=complex) + 1j * A
memo = {}
def wick(idx):
idx = tuple(idx)
if len(idx) == 0:
return 1.0 + 0j
if len(idx) % 2 == 1:
return 0.0 + 0j
if idx in memo:
return memo[idx]
head, rest = idx[0], idx[1:]
tot = 0.0 + 0j
for j, b in enumerate(rest):
sub = rest[:j] + rest[j + 1:]
tot += (-1) ** j * W[head, b] * wick(sub)
memo[idx] = tot
return tot
return wick
def gram(basis, r, eta, wick):
n = len(basis)
M = np.zeros((n, n), dtype=complex)
for ai, ma in enumerate(basis):
imgs = [r[a] for a in reversed(ma)]
coeff = eta ** len(ma)
lst = list(imgs)
sgn = 1
for i in range(len(lst)):
for j in range(len(lst) - 1 - i):
if lst[j] > lst[j + 1]:
lst[j], lst[j + 1] = lst[j + 1], lst[j]
sgn = -sgn
ca = coeff * sgn
ia = tuple(lst)
for bi, mb in enumerate(basis):
M[ai, bi] = ca * wick(tuple(list(ia) + list(mb)))
return M
def metrics(M):
nm = max(float(np.max(np.abs(M))), 1e-300)
hd = float(np.max(np.abs(M - M.conj().T)) / nm)
lm = float(np.min(np.linalg.eigvalsh((M + M.conj().T) / 2)))
return hd, lm
r_S = {}
for i in range(8):
r_S[2 * i] = 2 * i + 1
r_S[2 * i + 1] = 2 * i
P_S = [2 * i for i in range(8)]
B1_S = [(a,) for a in P_S]
B2_S = [()] + [tuple(c) for c in itertools.combinations(P_S, 2)]
r_abT = {k: k for k in range(16)}
for k in range(2):
r_abT[CH[a_ch][k]] = CH[b_ch][1 - k]
r_abT[CH[b_ch][k]] = CH[a_ch][1 - k]
B1_ab = [(a,) for a in CH[a_ch]]
B2_ab = [(), tuple(CH[a_ch])]
def strict_S(A):
wk = wick_factory(A)
M1 = gram(B1_S, r_S, 1j, wk)
M2 = gram(B2_S, r_S, 1j, wk)
hd = max(metrics(M1)[0], metrics(M2)[0])
lm = min(metrics(M1)[1], metrics(M2)[1])
ok = (hd <= ZTOL and lm >= -NZ_FLOOR)
return ok, hd, lm, (M1, M2)
A_CCf = A_CC.astype(np.float64)
J3f = J3.astype(np.float64)
def parentV(kap, m, tt, V):
A = np.zeros((16, 16))
A[np.ix_(CAR_IDX, CAR_IDX)] = kap * A_CCf
A[np.ix_(BND_IDX, BND_IDX)] = m * J3f
A[np.ix_(CAR_IDX, BND_IDX)] = tt * V
A[np.ix_(BND_IDX, CAR_IDX)] = -tt * V.T
return A
def schur_Aeff(A):
C = (np.eye(16, dtype=complex) + 1j * A) / 2
CCC = C[np.ix_(CAR_IDX, CAR_IDX)]
CCB = C[np.ix_(CAR_IDX, BND_IDX)]
CBB = C[np.ix_(BND_IDX, BND_IDX)]
CBC = C[np.ix_(BND_IDX, CAR_IDX)]
Ceff = CCC - CCB @ np.linalg.inv(CBB) @ CBC
return 2 * Ceff.imag
def census10(Aeff):
n_nz, n_sig = 0, 0
aJ45 = 0.0
for (i, j) in CAR_DUADS:
B = Aeff[np.ix_(CH[i], CH[j])]
n_nz += float(np.linalg.norm(B)) >= NZ_FLOOR
pf = -(B[0, 0] * B[1, 1] - B[0, 1] * B[1, 0])
if abs(pf) >= PF_FLOOR:
n_sig += ((pf > 0) == (sign_c[frozenset({i, j})] > 0))
if (i, j) == (a_ch, b_ch):
aJ45 = (B[0, 1] - B[1, 0]) / 2
return n_nz, n_sig, aJ45
def cov_defect(A):
return float(np.max(np.abs(A[np.ix_(img, img)] - A)))
def mkV(spec):
V = np.zeros((10, 6))
for (orb, s), (nm, amp) in spec.items():
chans = [a_ch, b_ch] if orb == "two" else THREE
for c in chans:
V[2 * (c - 1):2 * c, 2 * s:2 * s + 2] = \
amp * PAULI[nm]
return V
V_J = mkV({(o, s): ("J", 1.0) for o in ("two", "three")
for s in range(3)})
V_Z = mkV({(o, s): ("Z", 1.0) for o in ("two", "three")
for s in range(3)})
V_X = mkV({(o, s): ("X", 1.0) for o in ("two", "three")
for s in range(3)})
# ==================================================================
section("P1 -- the two-seat linear law (exact)")
# ==================================================================
# (a) closed-form ward on two unit couplings
ok_cf = True
for (r0, b0) in ((0, 0), (7, 3)):
Vu = np.zeros((10, 6))
Vu[r0, b0] = 1.0
A = parentV(0.5, 0.5, 1.0, Vu)
_ok, _hd, _lm, (M1, M2) = strict_S(A)
D1 = M1 - M1.conj().T
for xi, x in enumerate(P_S):
for yi, y in enumerate(P_S):
if x == y:
continue
pred = -(A[x + 1, y] + A[x, y + 1])
ok_cf &= abs(D1[xi, yi] - pred) <= 1e-12
D2 = M2 - M2.conj().T
for mi, mono in enumerate(B2_S):
if len(mono) != 2:
continue
x, y = mono
if (x < 10) == (y < 10):
continue
pred = 1j * (A[x, y] + A[x + 1, y + 1])
ok_cf &= abs(D2[0, mi] - pred) <= 1e-12
check("P1.1 CLOSED FORMS warded against the Gram machinery on "
"unit couplings: 1p seat (M1 - M1+)_{xy} = -(A_{x+1,y} + "
"A_{x,y+1}); deg-2 empty seat i (A_{x,y} + A_{x+1,y+1}) "
"(<= 1e-12)", ok_cf, kill="K1")
# (b) covariant basis + exact rank
seen_o = set()
cov_basis = []
for r in range(10):
for b in range(6):
v0 = np.zeros((10, 6))
v0[r, b] = 1.0
w = np.zeros((10, 6))
cur = v0
for _k in range(6):
w += cur
cur = O_C.astype(float) @ cur @ O_B.astype(float).T
key = tuple(np.flatnonzero(w.flatten() > 0.5).tolist())
if key in seen_o or not key:
continue
seen_o.add(key)
cov_basis.append(np.rint(w).astype(np.int64))
rows = []
mixed_ee = [(x, y) for x in P_S if x < 10
for y in P_S if y >= 10]
for w in cov_basis:
r2 = [int(w[x, y - 10] + w[x + 1, y - 9])
for (x, y) in mixed_ee]
r1 = [int(w[x + 1, y - 10] + w[x, y - 9])
for (x, y) in mixed_ee]
rows.append(r2 + r1)
Lmat = sp.Matrix(rows).T
rkL = Lmat.rank()
seat_id_ok = True
for nm, Vt, fire1, fire2 in (("I", mkV({(o, s): ("I", 1.0)
for o in ("two", "three")
for s in range(3)}),
False, True),
("X", V_X, True, False),
("J", V_J, False, False),
("Z", V_Z, False, False)):
d2v = max(abs(Vt[x, y - 10] + Vt[x + 1, y - 9])
for (x, y) in mixed_ee)
d1v = max(abs(Vt[x + 1, y - 10] + Vt[x, y - 9])
for (x, y) in mixed_ee)
seat_id_ok &= ((d1v > 0.5) == fire1 and (d2v > 0.5) == fire2)
check("P1.2 the covariant coupling space has dim %d == 24 (v898 "
"T-count); the combined two-seat law has EXACT rank %d "
"== 12, kernel dim 12 = the {J, Z} coordinates (I fires "
"deg-2 only, X fires 1p only, J/Z clean: %s)"
% (len(cov_basis), rkL, seat_id_ok),
len(cov_basis) == 24 and rkL == 12 and seat_id_ok,
kill="K1")
ok_pureI = all(np.array_equal(
Vc[2 * i:2 * i + 2, 2 * s:2 * s + 2],
-np.eye(2, dtype=np.int64))
for i in range(5) for s in range(3))
A_dep_par = parentV(0.5, 0.5, 0.05, Vc.astype(float))
okD, hdD, lmD, (M1D, M2D) = strict_S(A_dep_par)
D2D = M2D - M2D.conj().T
n_def = int(np.sum(np.abs(D2D) > 1e-12))
raw_def = float(np.max(np.abs(D2D)))
check("P1.3 the DEPLOYED coupling is PURE-I (every 2x2 "
"sub-block exactly -I2: %s) -- entirely inside the "
"defect row space; its parent fails strict theta_S with "
"raw empty-seat defect %.4f == 2t = 0.1 and %d == 30 "
"defect entries (probe-7 law reproduced)"
% (ok_pureI, raw_def, n_def),
ok_pureI and (not okD) and abs(raw_def - 0.1) <= 1e-12
and n_def == 30, kill="K1")
# ==================================================================
section("P2 -- the equilibrium witness (exact rationals)")
# ==================================================================
A_Jf = parentV(0.5, 0.5, 0.05, V_J)
smaxJ = float(np.max(np.abs(np.linalg.eigvalsh(1j * A_Jf))))
wA, QA = np.linalg.eigh(1j * A_Jf)
w_h = -2.0 * np.arctanh(wA)
H_herm = (QA * w_h) @ QA.conj().T
h_r = H_herm.imag
occ = 1.0 / (1.0 + np.exp(w_h))
A_back = (-1j * (2 * (QA * occ) @ QA.conj().T
- np.eye(16))).real
rt = float(np.max(np.abs(A_back - A_Jf)))
h_cov = float(np.max(np.abs(h_r[np.ix_(img, img)] - h_r)))
check("P2.1 the V_J parent (1/2, 1/2, 1/20) is covariant "
"(defect %.1e), CAR-strict (smax %.4f = 0.694 +- 0.005 "
"< 0.95) and a beta = 1 KMS state of a covariant "
"Hamiltonian (artanh round-trip %.1e, h covariant %.1e "
"<= 1e-12): EQUILIBRIUM, sigma = 0"
% (cov_defect(A_Jf), smaxJ, rt, h_cov),
cov_defect(A_Jf) <= ZTOL and abs(smaxJ - 0.694) <= 5e-3
and rt <= 1e-12 and h_cov <= 1e-12, kill="K2")
def wick_exact_factory(Aex):
n = Aex.shape[0]
Wm = sp.eye(n) + sp.I * Aex
memo = {}
def wick(idx):
idx = tuple(idx)
if len(idx) == 0:
return sp.Integer(1)
if len(idx) % 2 == 1:
return sp.Integer(0)
if idx in memo:
return memo[idx]
head, rest = idx[0], idx[1:]
tot = sp.Integer(0)
for j, b in enumerate(rest):
w = Wm[head, b]
if w != 0:
sub = rest[:j] + rest[j + 1:]
tot += sp.Integer(-1) ** j * w * wick(sub)
memo[idx] = tot
return tot
return wick
def gram_exact(basis, r, eta, wick):
n = len(basis)
M = sp.zeros(n, n)
for ai, ma in enumerate(basis):
imgs = [r[a] for a in reversed(ma)]
coeff = eta ** len(ma)
lst = list(imgs)
sgn = 1
for i in range(len(lst)):
for j in range(len(lst) - 1 - i):
if lst[j] > lst[j + 1]:
lst[j], lst[j + 1] = lst[j + 1], lst[j]
sgn = -sgn
ca = coeff * sgn
ia = tuple(lst)
for bi, mb in enumerate(basis):
M[ai, bi] = sp.expand(
ca * wick(tuple(list(ia) + list(mb))))
return M
def herm_exact(M):
return sp.expand(M - M.conjugate().T) == sp.zeros(*M.shape)
def psd_exact(M):
n = M.shape[0]
M = sp.Matrix(M)
pivots = []
for k in range(n):
d = sp.nsimplify(sp.re(M[k, k]))
pivots.append(d)
if d == 0:
if any(sp.expand(M[k, j]) != 0 for j in range(k, n)):
return False, pivots
continue
if d < 0:
return False, pivots
for i in range(k + 1, n):
if M[i, k] != 0:
f = M[i, k] / d
for j in range(k, n):
M[i, j] = sp.expand(M[i, j] - f * M[k, j])
return True, pivots
def exact_parent(kapQ, mQ, tQ, Vint):
A_ex = sp.zeros(16)
for r in range(10):
for c in range(10):
A_ex[r, c] = kapQ * sp.Integer(int(A_CC[r, c]))
for r in range(6):
for c in range(6):
A_ex[10 + r, 10 + c] = mQ * sp.Integer(int(J3[r, c]))
for r in range(10):
for c in range(6):
val = tQ * sp.Integer(int(round(Vint[r, c])))
A_ex[r, 10 + c] = val
A_ex[10 + c, r] = -val
return A_ex
kapQ, mQ, tQ = (sp.Rational(1, 2), sp.Rational(1, 2),
sp.Rational(1, 20))
res_wit = {}
for nm, Vw, lm_exp in (("J", V_J, (0.3064, 0.1532)),
("Z", V_Z, (None, 0.2383))):
A_ex = exact_parent(kapQ, mQ, tQ, Vw)
wk = wick_exact_factory(A_ex)
M1S = gram_exact(B1_S, r_S, sp.I, wk)
M2S = gram_exact(B2_S, r_S, sp.I, wk)
h1 = herm_exact(M1S)
h2 = herm_exact(M2S)
p1, piv1 = psd_exact(M1S)
p2, piv2 = psd_exact(M2S)
pd1 = p1 and all(p > 0 for p in piv1)
pd2 = p2 and all(p > 0 for p in piv2)
l1 = float(np.min(np.linalg.eigvalsh(np.array(
M1S.evalf(16), dtype=complex))))
l2 = float(np.min(np.linalg.eigvalsh(np.array(
M2S.evalf(16), dtype=complex))))
res_wit[nm] = (h1, h2, pd1, pd2, l1, l2)
print(" V_%s witness: Hermitian (%s, %s), PD by exact "
"LDL (%s, %s), float lam_min %.4f / %.4f"
% (nm, h1, h2, pd1, pd2, l1, l2))
okJ = res_wit["J"]
okZ = res_wit["Z"]
check("P2.2 STRICT theta_S RP PASSES EXACTLY at the V_J "
"witness: Grams exactly Hermitian, PD by exact LDL; "
"float lam_min %.4f = 0.3064 +- 0.005 (1p), %.4f = "
"0.1532 +- 0.005 (deg-2)" % (okJ[4], okJ[5]),
all(okJ[:4]) and abs(okJ[4] - 0.3064) <= 5e-3
and abs(okJ[5] - 0.1532) <= 5e-3, kill="K2")
# S_J identity + symbolic Schur + exact census
S_J = V_J.astype(np.int64) @ J3 @ V_J.astype(np.int64).T
okSJ = all(np.array_equal(S_J[2 * i:2 * i + 2, 2 * j:2 * j + 2],
3 * J2i)
for i in range(5) for j in range(5))
S_Z = V_Z.astype(np.int64) @ J3 @ V_Z.astype(np.int64).T
okSZ = all(np.array_equal(S_Z[2 * i:2 * i + 2, 2 * j:2 * j + 2],
-3 * J2i)
for i in range(5) for j in range(5))
kap_s, m_s, t_s = sp.symbols("kappa m t", real=True)
A_CCs = sp.Matrix(A_CC.tolist())
J3s = sp.Matrix(J3.tolist())
VJs = sp.Matrix(np.rint(V_J).astype(int).tolist())
Ws = t_s * VJs
C_CC = (sp.eye(10) + sp.I * kap_s * A_CCs) / 2
C_CB = sp.I * Ws / 2
C_BC = -sp.I * Ws.T / 2
C_BB_inv = 2 * (sp.eye(6) - sp.I * m_s * J3s) / (1 - m_s ** 2)
C_eff = sp.expand(C_CC - C_CB * C_BB_inv * C_BC)
A_eff_sym = sp.Matrix(10, 10, lambda r, c: sp.expand(
sp.im(sp.expand(2 * C_eff[r, c]))))
SJs = sp.Matrix(S_J.tolist())
A_eff_formula = (kap_s * A_CCs
+ (m_s / (1 - m_s ** 2)) * t_s ** 2 * SJs)
ok_schur = sp.simplify(A_eff_sym - A_eff_formula) == sp.zeros(10)
lamQ = mQ / (1 - mQ ** 2)
Jco = sp.Integer(3) * lamQ * tQ ** 2
A_eff_ex = kapQ * A_CCs + lamQ * tQ ** 2 * SJs
ok_cen = True
for (i, j) in CAR_DUADS:
Bx = A_eff_ex.extract(CH[i], CH[j])
target = Jco * sp.Matrix([[0, 1], [-1, 0]])
ok_cen &= (sp.expand(Bx - target) == sp.zeros(2))
ok_cen &= (sp.sign(-Bx.det()) == sign_c[frozenset({i, j})])
smax_eff = float(max(abs(x) for x in np.linalg.eigvalsh(
1j * np.array(A_eff_ex.evalf(16), dtype=np.float64))))
check("P2.3 THE COMPRESSION CARRIES THE FULL CANONICAL MIXING: "
"S_J = 3J on ALL 25 blocks (integer: %s); A_eff = kappa "
"A_CC + (m/(1-m^2)) t^2 S_J (symbolic: %s); every duad "
"block = (3 m t^2/(1-m^2)) J with J-coordinate %s == "
"1/200 EXACTLY, Pf4 canonical 10/10 (%s), compressed CAR "
"valid (smax_eff %.4f < 1)"
% (okSJ, ok_schur, Jco, ok_cen, smax_eff),
okSJ and bool(ok_schur) and Jco == sp.Rational(1, 200)
and ok_cen and smax_eff < 1, kill="K2")
lamZ = mQ / (1 - mQ ** 2)
JcoZ = -sp.Integer(3) * lamZ * tQ ** 2
check("P2.4 V_Z SECOND WITNESS: S_Z = -3J uniform (integer: "
"%s); strict theta_S passes exactly (Hermitian %s/%s, PD "
"%s/%s, float lam_min %.4f = 0.2383 +- 0.005); "
"J-coordinate %s == -1/200 exactly"
% (okSZ, okZ[0], okZ[1], okZ[2], okZ[3], okZ[5], JcoZ),
okSZ and all(okZ[:4]) and abs(okZ[5] - 0.2383) <= 5e-3
and JcoZ == sp.Rational(-1, 200), kill="K2")
A_exJ = exact_parent(kapQ, mQ, tQ, V_J)
wkJ = wick_exact_factory(A_exJ)
M1T = gram_exact(B1_ab, r_abT, sp.I, wkJ)
ok_marg = (M1T == sp.zeros(2))
B45e = A_eff_ex.extract(CH[a_ch], CH[b_ch])
M1_eff = sp.Matrix([[B45e[0, 1], B45e[1, 1]],
[B45e[0, 0], B45e[1, 0]]])
ev_eff = sorted(M1_eff.eigenvals().keys(), key=lambda z: sp.re(z))
ok_eff = (ev_eff == [sp.Rational(-1, 200), sp.Rational(1, 200)])
check("P2.5 THE 2-CYCLE SIDE IS UNTOUCHED: theta_abT 1p Gram "
"identically 0 at the V_J witness (%s, marginal) and the "
"compressed state fails effective 2-cycle RP with odd "
"eigenvalues exactly {-1/200, +1/200} (%s) -- the "
"strict-collar gate opens, the round-58 exclusion stands"
% (ok_marg, ok_eff),
ok_marg and ok_eff, kill="K2")
# ==================================================================
section("P3 -- kernel != sufficient (the honest boundary)")
# ==================================================================
A_Xf = parentV(0.5, 0.5, 0.05, V_X)
okX, hdX, lmX, (M1X, M2X) = strict_S(A_Xf)
raw1X = float(np.max(np.abs(M1X - M1X.conj().T)))
D2X = M2X - M2X.conj().T
row0X = float(np.max(np.abs(D2X[0, :])))
ppX = float(np.max(np.abs(D2X[1:, 1:])))
check("P3.1 the uniform X-coupling fails at the 1p seat: raw "
"defect %.4f == 2t = 0.1 (empty deg-2 seat clean %.1e "
"<= 1e-14, pair-pair %.4f = 0.05 +- 0.005) -- the law "
"is sharp in both directions"
% (raw1X, row0X, ppX),
(not okX) and abs(raw1X - 0.1) <= 1e-12
and row0X <= 1e-14 and abs(ppX - 0.05) <= 5e-3,
kill="K3")
mix_members = [
("J two / Z three", dict(
[(("two", s), ("J", 1.0)) for s in range(3)]
+ [(("three", s), ("Z", 1.0)) for s in range(3)]),
0.1861),
("mixed amps", {("two", 0): ("J", 1.0),
("two", 1): ("Z", 0.5),
("two", 2): ("J", -0.5),
("three", 0): ("Z", 1.0),
("three", 1): ("J", 0.7),
("three", 2): ("Z", -0.3)}, 0.1848),
]
ok_mix = True
for nm, spec, lm_exp in mix_members:
A = parentV(0.5, 0.5, 0.05, mkV(spec))
okS, hdS, lmS, _g = strict_S(A)
n_nz, n_sig, _a = census10(schur_Aeff(A))
print(" %-16s hd=%.1e lm=%.4f mix nz=%d sig=%d"
% (nm, hdS, lmS, n_nz, n_sig))
ok_mix &= (hdS <= ZTOL and abs(lmS - lm_exp) <= 5e-3
and n_nz == 10 and n_sig == 4)
check("P3.2 J/Z-MIXTURES pass exact Hermiticity + PD but the "
"canonical SIGN census drops to 4/10 on both frozen "
"members: the canonical gate selects the uniform J (or "
"Z) ray -- Hermiticity kernel and canonical-mixing ray "
"are different strata", ok_mix, kill="K3")
sel_ok = True
for nm, spec, exp_nz, exp_a in (
("two-orbit only", {("two", s): ("J", 1.0)
for s in range(3)}, 1, 0.005),
("three-orbit only", {("three", s): ("J", 1.0)
for s in range(3)}, 3, 0.0),
("one pair both orbits", {("two", 0): ("J", 1.0),
("three", 0): ("J", 1.0)},
10, 1.0 / 600.0)):
A = parentV(0.5, 0.5, 0.05, mkV(spec))
n_nz, n_sig, aJ45 = census10(schur_Aeff(A))
sel_ok &= (n_nz == exp_nz and abs(aJ45 - exp_a) <= 1e-12)
check("P3.3 orbit selectivity in the kernel (probe-7 P2.4 "
"regression): 2-cycle-orbit only 1/10, 3-cycle only "
"3/10, single boundary pair (both orbits) 10/10 with "
"J-coordinate exactly lam t^2 = 1/600", sel_ok,
kill="K3")
A10 = parentV(0.5, 0.5, 0.1, V_J)
ok10, hd10, lm10, _g = strict_S(A10)
smax10 = float(np.max(np.abs(np.linalg.eigvalsh(1j * A10))))
A15 = parentV(0.5, 0.5, 0.15, V_J)
smax15 = float(np.max(np.abs(np.linalg.eigvalsh(1j * A15))))
print(" amplitude anatomy: t=0.1 smax %.3f lm %.4f pass=%s"
"; t=0.15 smax %.3f (CAR-invalid, report only)"
% (smax10, lm10, ok10, smax15))
check("P3.4 amplitude anatomy (report-gated loosely): uniform "
"J passes at t = 1/10 (lm %.4f = 0.056 +- 0.01, smax "
"%.3f < 0.95); t = 0.15 is CAR-invalid (smax %.3f > 1)"
% (lm10, smax10, smax15),
ok10 and abs(lm10 - 0.056) <= 0.01 and smax10 <= 0.95
and smax15 > 1.0, kill="K3")
# ==================================================================
section("P4 -- the NESS side: exact no-go + the price curve")
# ==================================================================
h_full = -(A16_dep.astype(float) + 0.125 * A_int.astype(float))
h_C16 = np.zeros((16, 16))
h_C16[np.ix_(CAR_IDX, CAR_IDX)] = h_full[np.ix_(CAR_IDX,
CAR_IDX)]
h_B16 = np.zeros((16, 16))
h_B16[np.ix_(BND_IDX, BND_IDX)] = h_full[np.ix_(BND_IDX,
BND_IDX)]
def kms_of(h, beta):
w, Q = np.linalg.eigh(1j * h)
f = 1.0 / (1.0 + np.exp(np.clip(beta * w, -700, 700)))
return (-1j * (2 * (Q * f) @ Q.conj().T
- np.eye(len(h)))).real
def two_temp_A0(bC, bB):
A0 = np.zeros((16, 16))
A0[np.ix_(CAR_IDX, CAR_IDX)] = kms_of(
h_full[np.ix_(CAR_IDX, CAR_IDX)], bC)
A0[np.ix_(BND_IDX, BND_IDX)] = kms_of(
h_full[np.ix_(BND_IDX, BND_IDX)], bB)
return A0
wH, QH = np.linalg.eigh(1j * h_full)
def cesaro(A0):
Ac = QH.conj().T @ A0 @ QH
mask = np.abs(wH[:, None] - wH[None, :]) < 1e-9
return (QH @ np.where(mask, Ac, 0) @ QH.conj().T).real
def evolve(A0, tt):
R = (QH @ np.diag(np.exp(-1j * tt * wH)) @ QH.conj().T)
ri = float(np.max(np.abs(R.imag)))
R = R.real
return R @ A0 @ R.T, ri
def flux(hr, A):
comm = h_full @ A - A @ h_full
return 0.25 * float(np.sum(hr * comm))
# (a) exact no-go, stated + total-energy ward
A0_test = two_temp_A0(1.0, 0.25)
At1, ri1 = evolve(A0_test, 1.0)
e_tot = abs(0.25 * float(np.sum(h_full * (h_full @ At1
- At1 @ h_full))))
check("P4.1 FINITE-NESS-NOGO (exact): [h, A] = 0 => Phi_r = "
"(1/4) tr(h0_r [h, A]) = 0 for every block => sigma == "
"0 identically -- positive entropy production is "
"category-inapplicable to a finite stationary state "
"(algebraic implication; total-energy identity "
"tr(h [h, A]) = %.1e <= 1e-12 warded; evolution real "
"%.1e); the weakest meaningful positivity = RP of the "
"stationary covariance (tested in P4.3)"
% (e_tot, ri1), e_tot <= 1e-12 and ri1 <= 1e-12,
kill="K4")
DRIVES = ((1.0, 1.0), (1.0, 0.5), (1.0, 0.25), (2.0, 0.5))
hd_seq = []
lm_seq = []
ok_ces = True
ok_gates = True
for (bC, bB) in DRIVES:
A0 = two_temp_A0(bC, bB)
Abar = cesaro(A0)
stat = float(np.max(np.abs(h_full @ Abar - Abar @ h_full)))
anti = float(np.max(np.abs(Abar + Abar.T)))
smax = float(np.max(np.abs(np.linalg.eigvalsh(1j * Abar))))
cd = cov_defect(Abar)
f0C = abs(flux(h_C16, A0))
f0B = abs(flux(h_B16, A0))
okS, hdS, lmS, _g = strict_S(Abar)
pf4_d = pf4_of(compress12(Abar))
n_blk = sum(1 for dd in pf4_c
if abs(pf4_d[dd]) > PF_FLOOR
and (pf4_d[dd] > 0) == (pf4_c[dd] > 0))
hd_seq.append(hdS)
lm_seq.append(lmS)
ok_ces &= (stat <= 1e-12 and anti <= 1e-12 and cd <= 1e-12
and smax <= 0.85 and max(f0C, f0B) <= 1e-14)
ok_gates &= (n_blk == 15)
print(" drive (%.1f, %.2f): stat %.1e smax %.3f | "
"thetaS hd %.4f lm %.4f | blocks %d/15"
% (bC, bB, stat, smax, hdS, lmS, n_blk))
check("P4.2 the two-temperature Cesaro states: exactly "
"stationary, real antisym, covariant, CAR-valid, t = 0 "
"product fluxes EXACTLY zero on the whole frozen drive "
"grid", ok_ces, kill="K4")
sig1 = None
sig4 = None
for tt, tgt in ((1.0, None), (4.0, None)):
At, _ri = evolve(two_temp_A0(1.0, 0.25), tt)
s_val = 1.0 * flux(h_C16, At) + 0.25 * flux(h_B16, At)
if tt == 1.0:
sig1 = s_val
else:
sig4 = s_val
check("P4.3 THE PRICE CURVE IS FLAT-OPEN: mixing census 15/15 "
"canonical at EVERY drive point including zero drive "
"(%s); Hermiticity defect PINNED (%.4f/%.4f/%.4f = "
"0.0982 +- 0.005; %.4f = 0.1062 +- 0.005) while the PSD "
"margin degrades (%.4f -> %.4f -> %.4f); transient "
"sigma(1) = %.4f = 0.0626 +- 0.01 > 0, sigma(4) = %.4f "
"< 0 (finite recurrence, typed): the minimal entropy "
"production at which the mixing gates open is EXACTLY "
"ZERO -- drive buys NO RP"
% (ok_gates, hd_seq[0], hd_seq[1], hd_seq[2], hd_seq[3],
lm_seq[0], lm_seq[1], lm_seq[2], sig1, sig4),
ok_gates
and all(abs(h - 0.0982) <= 5e-3 for h in hd_seq[:3])
and abs(hd_seq[3] - 0.1062) <= 5e-3
and lm_seq[0] > lm_seq[1] > lm_seq[2] > 0
and abs(sig1 - 0.0626) <= 0.01 and sig4 < 0, kill="K4")
check("P4.4 TYPED CONCLUSION: NESS-NOT-FORCED -- the "
"strict-collar mixing needs the coupling DIRECTION (the "
"two-seat kernel, equilibrium-compatible: P2), not a "
"non-equilibrium source; the round-59 obstruction is "
"DIRECTION-level (deployed PURE-I wiring, P1.3), not "
"equilibrium-level; on this tested class a NESS is "
"neither forced nor helpful (P4.3)", True,
"typed by measurement")
# ==================================================================
section("C -- controls (must fire; RNG only in C2)")
# ==================================================================
A_kms = kms_of(h_full, 1.0)
statK = float(np.max(np.abs(h_full @ A_kms - A_kms @ h_full)))
okK, hdK, lmK, _g = strict_S(A_kms)
check("C1 FIRES (zero-drive regression): the global KMS state "
"(1, 1/8, 1) is exactly stationary (%.1e) and FAILS "
"strict theta_S with defect %.4f = 0.0982 +- 0.005"
% (statK, hdK),
statK <= 1e-12 and (not okK)
and abs(hdK - 0.0982) <= 5e-3, kill="K7")
rng = np.random.default_rng(904)
n_cov = 0
n_gates = 0
for _trial in range(3):
pr = rng.permutation(10)
Vb = V_J[pr, :]
A_bad = parentV(0.5, 0.5, 0.05, Vb)
if cov_defect(A_bad) >= NZ_FLOOR:
n_cov += 1
_n_nz, n_sig, _a = census10(schur_Aeff(A_bad))
if n_sig <= 1:
n_gates += 1
check("C2 FIRES: 3/3 seeded row permutations of V_J break the "
"covariance ward (%d/3) AND the canonical Pfaffian "
"census (%d/3 with sign census <= 1/10)"
% (n_cov, n_gates), n_cov == 3 and n_gates == 3,
kill="K7")
check("C3 FIRES (deployed-coupling regression): the PURE-I "
"parent fails strict theta_S with raw defect 0.1 and 30 "
"entries (P1.3 doubles as the control; fire rule "
"re-read)", (not okD) and n_def == 30, kill="K7")
v0 = np.zeros((10, 6))
v0[2 * (a_ch - 1):2 * a_ch, 0:2] = PAULI["I"]
wI = np.zeros((10, 6))
cur = v0
for _k in range(2):
wI += cur
cur = O_C.astype(float) @ cur @ O_B.astype(float).T
A_c4 = parentV(0.5, 0.5, 0.05, V_J + 0.5 * wI)
okC4, hdC4, lmC4, (M1C4, M2C4) = strict_S(A_c4)
rawC4 = float(np.max(np.abs(M2C4[0, :]
- M2C4[:, 0].conj().T)))
check("C4 FIRES (I-orbit switch-on): adding one covariant "
"I-orbit to V_J turns the deg-2 empty-seat defect ON "
"(raw %.4f > 0.01) while covariance stays exact (%.1e)"
% (rawC4, cov_defect(A_c4)),
(not okC4) and rawC4 > 0.01
and cov_defect(A_c4) <= ZTOL, kill="K7")
# ==================================================================
section("VERDICT")
# ==================================================================
n_pass = sum(1 for _n, ok in CHECKS if ok)
n_tot = len(CHECKS)
controls_ok = all(ok for nm, ok in CHECKS if nm.startswith("C"))
if not controls_ok:
VERDICT = "CONTROL-DEAD"
elif "K0" in KILLS:
VERDICT = "PIPELINE-BROKEN"
elif "K1" in KILLS:
VERDICT = "LAW-BROKEN"
elif "K2" in KILLS:
VERDICT = "WITNESS-BROKEN"
elif "K3" in KILLS:
VERDICT = "KERNEL-BROKEN"
elif "K4" in KILLS:
VERDICT = "NESS-BROKEN"
else:
VERDICT = ("NESSPARENT-MEASURED [PRICE-ZERO(equilibrium "
"strict-collar witnesses V_J / V_Z: exact "
"Hermitian + PD + 10/10 canonical + "
"J-coordinate +-1/200), TWO-SEAT-LAW(rank 12, "
"kernel 12 = the {J, Z} coordinates; deployed "
"coupling PURE-I), CANONICAL-SELECTS-UNIFORM-"
"RAY(mixtures 4/10), FINITE-NESS-NOGO("
"stationary => sigma == 0, exact), "
"DRIVE-RP-NEUTRAL(defect pinned, margin "
"degrades, gates open at zero drive), "
"NESS-NOT-FORCED]")
print("%d/%d checks passed" % (n_pass, n_tot))
print("VERDICT: %s" % VERDICT)
print("""
REPORT (exploration only -- no promotion, no edits):
* THE HEADLINE ANSWER: the strict-collar mixing does NOT need a
NESS. The round-59 obstruction is a property of the coupling
DIRECTION: strict theta_S Hermiticity is obstructed at linear
order by exactly two seat families (1p kills the X sub-block
coordinates, deg-2 kills the I coordinates; rank 12 on the
24-dim covariant coupling space), and the DEPLOYED seam wiring
is PURE-I -- maximally obstructed. The 12-dim {J, Z} kernel
contains EQUILIBRIUM witnesses: the uniform J- and Z-coupling
parents pass strict theta_S RP in exact arithmetic while their
Schur compressions carry the full canonical Pfaffian mixing
with the J-coordinate EXACTLY +-1/200 (the same rational floor
identity). The minimal entropy production at which the mixing
gates open is ZERO; typing: NESS-NOT-FORCED.
* THE NESS SIDE, honestly: at this finite size an exactly
stationary quasi-free state has zero currents and sigma == 0
(exact) -- 'NESS with positive entropy production' is
category-inapplicable; the two-temperature Cesaro states are
stationary, covariant, mixing-open at ZERO drive, keep the
deployed-direction Hermiticity defect PINNED (~0.098) and only
lose PSD margin as the drive grows: drive buys NO RP.
* THE BOUNDARY OF THE WIN: the kernel is necessary, not
sufficient -- X-couplings fail the 1p seat (raw defect exactly
2t), and J/Z mixtures pass Hermiticity but break the canonical
sign census (4/10): the canonical gate selects the uniform J
(or Z) ray. The 2-cycle exclusion is untouched: the V_J
witness stays theta_abT-MARGINAL and its compression fails
effective 2-cycle RP at exactly -+1/200.
* HONESTY: the witnesses live on a DIFFERENT covariant coupling
direction than the deployed A_int wiring -- whether the actual
seam allows that direction is untouched; finite-size no-go
only; RP sector-typed; the [O] premise of v898/v903 is
unmoved; no marker moves. NO RH claim.
Runtime: %.1f s""" % (time.time() - T0))
print("ALL CHECKS PASSED" if n_pass == n_tot
else "CHECKS FAILED: %d" % (n_tot - n_pass))
return 0 if (n_pass == n_tot and not KILLS) else 1
if __name__ == "__main__":
raise SystemExit(main())
'''
# --------------------------------------------------------------- harness
_PF_RE = re.compile(r"^\s*\[(PASS|FAIL)\]\s+(\S+)", re.M)
_VD_RE = re.compile(r"VERDICT[:\]]")
def _probe_file(name):
cand = os.path.abspath(os.path.join(
_HERE, os.pardir, "experiments", "tfpt-discovery", name + ".py"))
return cand if os.path.isfile(cand) else None
def _census(out):
marks = _PF_RE.findall(out)
fails = sorted({tok for st, tok in marks if st == "FAIL"})
verdict = ""
for line in out.splitlines():
if _VD_RE.search(line):
verdict = line.strip()
return len(marks), fails, verdict
def _exec_probe(name, src, run_entry=True):
"""Execute one embedded frozen probe source BYTE-EXACT in its own
module namespace (round-31 convention); capture and re-emit
stdout; return (stdout, exit_code, byte_equal_or_None)."""
if src[:1] == "\n":
src = src[1:]
path = _probe_file(name)
same = None
if path is not None:
with open(path, encoding="utf-8") as fh:
same = (fh.read() == src)
fname = path or os.path.abspath(__file__)
mod = types.ModuleType(name)
mod.__file__ = fname
sys.modules[name] = mod
buf = io.StringIO()
code = 0
with contextlib.redirect_stdout(buf):
try:
exec(compile(src, fname, "exec"), mod.__dict__)
entry = mod.__dict__.get("main") or mod.__dict__.get("run")
if run_entry and callable(entry):
rc = entry()
code = 0 if rc is None else int(rc)
except SystemExit as exc:
code = 0 if exc.code is None else int(exc.code)
except Exception: # regression guard
import traceback
traceback.print_exc(file=sys.stdout)
code = 99
out = buf.getvalue()
sys.stdout.write(out)
sys.stdout.flush()
return out, code, same
def _gate(name, out, code, same, exp_n, exp_fails, exp_verdict,
exp_code, gates):
n, fails, verdict = _census(out)
ok = (n == exp_n and fails == list(exp_fails)
and exp_verdict in verdict and code == exp_code
and same is not False)
gates.append(ok)
prov = ("byte-exact vs experiments source" if same is True else
"embedded copy (source file not present)" if same is None
else "SOURCE MISMATCH")
print("\n[%s] PATTERN GATE %s: %d checks (exp %d) | FAILs %s "
"(exp %s) | exit %d (exp %d) | %s\n verdict line: %s"
% ("PASS" if ok else "FAIL", name, n, exp_n,
",".join(fails) if fails else "none",
",".join(exp_fails) if exp_fails else "none",
code, exp_code, prov, verdict), flush=True)
return ok
_PLAN = (
('seam_marginal_stability_probe', _SRC_0, 22, (), 'MARGSTAB-MEASURED', 0),
('seam_ness_parent_probe', _SRC_1, 25, (), 'NESSPARENT-MEASURED', 0),
)
def run():
t0 = time.time()
print("=" * 74)
print('v908 -- SEAM.CFIN.MARGINAL.STABILITY.01 + SEAM.CFIN.NESS.PARENT.01: the seam equilibrium wiring -- strict 2-cycle RP impossible on the whole covariant class (exact +-a_J law, floor-exchange Pf4 = (eps - 1/200)(eps + 1/200)); the strict-collar obstruction is a two-seat linear law with kernel {J, Z}, the deployed wiring is PURE-I, and equilibrium J/Z witnesses carry the full 1/200 mixing at zero entropy production -- NO NESS NEEDED')
print("(frozen probes embedded byte-exact and executed verbatim; NO RH claim)")
print("=" * 74, flush=True)
gates = []
for name, src, exp_n, exp_fails, exp_verdict, exp_code in _PLAN:
print("\n" + "-" * 74)
print("EMBEDDED FROZEN PROBE: %s" % name)
print("-" * 74, flush=True)
out, code, same = _exec_probe(name, src)
_gate(name, out, code, same, exp_n, exp_fails,
exp_verdict, exp_code, gates)
ok = all(gates)
print("\n" + "=" * 74)
print("v908: %d/%d pattern gates passed | runtime %.1f s"
% (sum(gates), len(gates), time.time() - t0))
print('the mixing needs no drive: its price is exactly zero entropy production on the {J, Z} kernel; whether PURE-I is compiler-forced is the registered next contract (SEAM.STATE.WIRING.SELECTOR.01); the v898/v903 [O] premise is unmoved')
print("[%s] v908 VERDICT GATE" % ("PASS" if ok else "FAIL"))
return 0 if ok else 1
if __name__ == "__main__":
raise SystemExit(run())
Recorded source fingerprint
8dab7fead65fc50f25eb10ab1365f005c09cfae66ca20e25513550c2ad6aa950
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