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v526_seam_thermal_kms_nariai_bridge.py
verification/v526_seam_thermal_kms_nariai_bridge.py
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"""v526 -- SEAM.THERMAL.KMS.01: the thermal seam -- KMS structure of the
reconstructed free OS quotient CLOSED onto the Nariai/BH temperature
chain (the THIRD LEG of c3). Verdict SUCCESS per the frozen
preregistration. NO marker moves.
[E] 1. beta MEASURED, NOT ASSUMED: the circle kernel C(d)=(2/N)/sin(pi d/N)
has exactly one finite thermal-mode representation with detailed
balance; N=8: one real frequency omega=arcsinh(2^{-3/4}), weight
ratio kappa=q^{-8} EXACTLY => beta=8=N (Q-polynomial identity
Q^2-(2+sqrt2)Q+1=0); N=16: both mode pairs real, pairing exponent
exactly (N/2)+1=9 => beta=16=N; anchor Pi(u_i-2)=sqrt2 =
Pi sinh(omega_i)*2^{7/4} pin at both levels; wrong beta
{N/2,N/4,2N} excluded (gaps 0.0937/0.3126/0.0108; N/2 witness
C(1)-C(3)=sqrt2*C(3) != 0 exact).
[E] 2. T_seam = 1/N per clock step; in angle units (step = 2pi/N,
v519 pin): beta_angle = 2pi EXACT, T_seam = 1/(2pi) = 4 c3 --
N-independent.
[E] 3. MODULAR HAMILTONIAN CLOSED: rho_OS faithful, spectrum
p={(1+-nu1)(1+-nu2)/4} with nu1=cos(pi/24), nu2=cos(7pi/24) exact;
K = eps1 n1 + eps2 n2 with eps1=2 ln cot(pi/48),
eps2=2 ln cot(7pi/48); full Tomita structure (KMS-beta=1 on all
256 pairs, residual 2.5e-38, J Delta J = Delta^{-1}).
[E] 4. THE c3 CLOSURE (class-1 selection in the frozen anti-numerology
lattice, exact r=1): beta_angle=2pi=1/(4 c3) -- T_seam=4 c3 is the
Bisognano-Wichmann/Hawking normalisation from v208; chain:
T_H=c3/M=1/(8pi M) (the axiom IS the Hawking coefficient, here
from the seam KMS), SdS 1/(4pi)=2 c3, Nariai T_N=4 c3 sqrt(Lambda),
v104 cross-link 2H/T_N=4pi=1/(2 c3) exact. [C]-fence: reading
"seam euclidean circle = thermal circle of the reconstructed
horizon dynamics" -- temperature as THIRD LEG of c3 beside
geometry and anomaly.
[E] 5. ENTROPY: S(8)=0.52187, S(16)=0.63510, S(32)=0.75006 nats;
DeltaS/ln2 -> 1/6 = c_chi/3 monotonically (chiral c=1/2 log law
confirmed); HONESTLY FAILED: v129 horizon fractions {1/3,2/3} and
v190 floor do NOT transfer (S/S_max=0.376/0.229, not N-stable) --
temperature bridge closes, entropy-fraction bridge does not.
[C-5] typed non-gating: naive clock reading fails ({1,sqrt2-1} are NOT
e^{-Delta K} ratios; gaps 0.0496/0.1710), Tomita flow != transfer
flow (0.45-0.67); exact replacement: sqrt2-1 = C(3)/C(1) =
1/(2 cosh 2 omega - 1) -- the silver eigenvalue is a thermal
kernel ratio.
[E] 6. NEGATIVE CONTROLS: site (2,2,4) no state; family A (4,4,0);
anti-chiral (8,8,0); NON-COMPACT: Stieltjes => kappa=0, beta=inf,
T=0 -- the temperature hangs exactly on compactness of the
euclidean circle.
Status: [E] exact sympy/cyclotomic + 40-digit Tomita certificates;
[C] bridge reading typed. Python; Wolfram-mirrored (Q-polynomial /
beta measurement / 2pi=1/(4 c3) chain / nu char-poly -- the 40-digit
Tomita residuals and entropy tower are Python-only), counted per
GATE.WOLFRAM.02. Discovery provenance:
experiments/tfpt-discovery/seam_thermal_kms_nariai_bridge_probe.py
(22/22, 2026-07-23).
=== ORIGINAL PROBE HEADER (kept verbatim) ===
seam_thermal_kms_nariai_bridge_probe.py -- EXPLORATION ONLY (thermal-seam
sandbox). Nothing in verification/, ledger, papers, changelog, website or
scorecard is touched; every status marker stays where it is. No files are
written; standalone; deterministic.
THE ATTACK (KMS -> Nariai/BH bridge): the WOIT-beta2 probe
(woit_os_beta2_os_quotient_probe.py) showed that the OS quotient of the free
16-Majorana NS seam circle is THERMAL: at N = 8 the complete half algebra
reconstructs dim H_phys = 16 = 4^2 (a doubled/two-sided space), there is no
contraction on the compact euclidean circle (silver witnesses), and the
clock T^{N/4} has spectrum exactly {1, sqrt(2)-1}. This probe constructs
the KMS structure of that reconstruction EXACTLY and asks whether its
temperature normalisation closes onto the Nariai/BH temperature chain of
the suite (v208 modular T_H, v104 Nariai clock, v129/v190 entropy ledger).
LEADING HYPOTHESIS (preregistered BEFORE the computation): the TFPT axiom
c3 = 1/(8 pi) is numerically the Hawking coefficient (T_H = 1/(8 pi M) for
Schwarzschild). If the KMS temperature of the reconstructed seam state, in
clock-step units, closes exactly onto the modular normalisation the
Nariai/BH sector of the suite uses (v208: modular beta = 2 pi = 1/(4 c3)),
temperature becomes an independent third leg of the axiom, next to
geometry and anomaly.
=== PREREGISTRATION (frozen BEFORE the probe run; criteria not adjusted
after; scratch pre-derivation produced the EXACT ANCHORS marked
"pre-derived", each to be machine-confirmed here -- no criterion, tolerance
or relation class was changed after seeing probe output) ===
[C-0] system pinned = the v519/beta2 seam data, copied verbatim: chiral NS
vacuum kernel C(d) = (2/N)/sin(pi d/N) (odd d; C(even) = 0), bond-cut OS
reflection (axis k = N-1, NS spin signs, twist eta = +i), half algebra
A_+ on {N/2..N-1}; levels N = 8 (complete 2^4 half algebra) and N = 16
(Gaussian/one-particle level). One clock step = the euclidean rotation
alpha_1 = geometric angle 2 pi / N (v519 Q1.1 pin; re-verified here).
[C-1] rho_OS := the unique element of A_+ (== M_4(C) via a pinned
Jordan-Wigner map) with Tr(rho_OS a) = omega(a) for all a in A_+ -- the
reduced state of the pure circle state on the half algebra, i.e. the
state the OS vector Omega induces on pi(A_+) (pi = left multiplication
on H_phys). SUCCESS-criteria: exact duality on all 16 monomials;
Hermitian, trace 1, POSITIVE DEFINITE (faithful => two-sided 4^2 GNS =
thermal representation). Pre-derived exact anchors: Williamson values
nu_1 = cos(pi/24), nu_2 = cos(7 pi/24) (char-poly identity: nu^2 =
(4 + sqrt2 +- sqrt6)/8); spectrum p = {(1 +- nu_1)(1 +- nu_2)/4};
modular Hamiltonian K = -ln rho_OS with single-particle energies
eps_i = ln((1+nu_i)/(1-nu_i)) = 2 ln cot(pi/48), 2 ln cot(7 pi/48).
[C-2] Tomita/KMS of the reconstruction: on H_phys (exact 16x16 Gram G,
PD per beta2) with Omega = [1] and pi = left multiplication, construct
S[a] = pi(a)^{*G} Omega, Delta = S*S, J = S Delta^{-1/2} explicitly.
SUCCESS-criteria (40-digit certificates, thresholds 1e-30): Delta Omega
= Omega; Delta G-self-adjoint PD; the KMS-beta=1 identity
<Omega, pi(a) Delta pi(b) Omega> = <Omega, pi(b) pi(a) Omega> on ALL
256 monomial pairs; J anti-unitary, J^2 = 1, J Delta J = Delta^{-1},
J pi J in the commutant; modular flow preserves pi(A_+); spec Delta =
{p_i / p_j} (the two-sided ratio spectrum -- the commutant carries the
SAME density: the purification is reflection-symmetric).
[C-3] KMS for the TRANSFER flow -- the beta measurement (beta is NOT an
input): the circle two-point kernel C(d), d = 1..N-1, is tested for a
finite thermal-mode (detailed-balance) representation
C(d) = sum_i w_i (q_i^{-d} + kappa_i q_i^{+d}), q_i > 1 real,
via the palindromic Prony recurrence (the even-d checkerboard zeros
force the +-(momentum 0/pi) mode doubling; the ansatz does NOT input
beta). KMS at inverse temperature beta (clock steps) <=> kappa_i =
q_i^{-beta} (detailed balance). Pre-derived exact anchors, to confirm:
N = 8: recurrence closes with ONE pair, q^2 + q^{-2} = 2 + sqrt2,
i.e. omega := ln q = arcsinh(2^{-3/4}); weights positive;
MEASURED kappa = q^{-8} EXACTLY => beta = 8 = N;
silver chain: C(1)/C(3) = 2 cosh(2 omega) - 1 = 1 + sqrt2.
N = 16: recurrence (2 unknowns, 4 equations) closes EXACTLY
(cyclotomic certificates mod z^16 + 1); both mode pairs REAL
(u_i = q_i^2 + q_i^{-2} > 2); anchor prod_i (u_i - 2) = sqrt2
(both levels: at N = 8, u - 2 = sqrt2); weight pairing =>
beta = 16 = N (j-unit exponent (N/2)+1 = 9, 40-digit).
Fermi-Dirac closure: n = 1/(1 + q^N) with |u|-|v| weight symmetry
(the chiral eta = +i as mode phase). The trivially-true graded
reflection identity (alpha_N = (-1)^F) is recorded as necessary-only.
[C-4] TEMPERATURE BRIDGE -- preregistered relation classes
(anti-numerology; FROZEN): the only admissible closures are
beta_angle / (2 pi) in {1/4, 1/2, 1, 2, 4} (small rationals),
optionally times c3^k (k in {-1, 0, 1}) or |mu_4|^{+-1} or N^{+-1} --
and the BRIDGE counts as EXACT CLOSURE only for the class-1 value
beta_angle = 2 pi, since 2 pi = 1/(4 c3) EXACTLY (v58/v208) is the
Bisognano-Wichmann/Hawking modular normalisation of the suite:
T_seam = 1/beta_angle = 1/(2 pi) = 4 c3, and the Schwarzschild chain
T_H = kappa_h/(2 pi) = c3/M = 1/(8 pi M) (v208 check 1), SdS
1/(4 pi) = 2 c3 (v208 check 3), Nariai T_N = 4 c3 sqrt(Lambda) with
the v104 rate cross-link 2H/T_N = 1/(2 c3) = 4 pi. beta_angle :=
beta_steps * (2 pi / N) with beta_steps the [C-3] measurement.
KILL if beta_steps measured but beta_angle/(2 pi) lands outside the
class lattice (incommensurable); the wrong-beta candidates beta = N/2,
N/4 (angle pi, pi/2) must FAIL detailed balance with quantified gaps.
[C-5] clock sub-expectations (typed check-questions, non-gating; the task
hypothesis "the clock eigenvalues {1, sqrt2 - 1} should appear as
e^{-Delta K} ratios" is tested honestly). Scratch pre-derivation
indicates the NAIVE reading FAILS (min gap ~0.05 to spec Delta^{1/4},
~0.17 to spec Delta) and that the Tomita flow is NOT the transfer flow
(||tau_k - G Delta^{k/N}|| = O(0.5) on the Klein-Landau domains) -- the
interval/half-circle modular flow of a thermal chiral fermion is
non-geometric; machine-confirm both, and confirm the POSITIVE exact
identity that replaces the naive reading: sqrt2 - 1 = C(3)/C(1) =
1/(2 cosh(2 omega) - 1) (the silver clock eigenvalue is a thermal
KERNEL ratio, not a Boltzmann/modular weight ratio). The beta2
compressed-clock spectrum {1, sqrt2 - 1} (both parity sectors) is
re-verified exactly.
[C-6] ENTROPY: S(rho_OS) exact form S = sum_i [ln 2 - ((1+nu)ln(1+nu) +
(1-nu)ln(1-nu))/2] + 40 digits; N = 16 and N = 32 via the Gaussian
covariance. Preregistered candidate structures (informative,
NON-GATING either way -- no fishing beyond these):
(E1) chiral-CFT ln-law: Delta S / ln 2 -> c_chi/3 = 1/6 monotonically
(c_chi = 1/2), PASS-shape: |DS2/ln2 - 1/6| < |DS1/ln2 - 1/6|;
(E2) v129 fraction transfer: S/S_max vs {1/3, 2/3} at N = 8, 16;
(E3) v190 floor transfer: S/S_max >= 2/3.
Scratch pre-derivation: E1 holds, E2/E3 fail (the horizon entropy
fractions do NOT transfer to seam entanglement entropy) -- confirm and
record honestly.
[C-7] negative controls (all mandatory):
(i) SITE cut (N = 8, half {1,2,3}): the OS Gram is indefinite for
every admissible twist => no state, no purification, no KMS --
the thermal reconstruction exists only at the bond placement;
(ii) FAMILY A (clock-centralising antipode, alternating dressing):
Hermitian only for a sub-torsor eta and indefinite (4,4,0) =>
no quotient to be thermal;
(iii) ANTI-CHIRAL state (chi = -1): odd block negative definite,
total (8,8,0) => no state for the wrong chirality;
(iv) NON-COMPACT control (half-line kernel C_inf(d) = 2/(pi d), the
N -> inf Cauchy-Stieltjes limit): the return symmetry C(N-d) =
C(d) dies (exact witness), the kernel is monotone-decaying
(h4 - h3 < 0 vs the circle's +sqrt2 C(3) return), the palindromic
recurrence consistency FAILS with the exact gap 128/105, and the
Stieltjes identity 2/(pi d) = (2/pi) int_0^1 x^{d-1} dx shows all
growing weights vanish => beta = inf, T = 0: the temperature
hangs exactly on the compact circle; contraction is restored
(ratios < 1) where the circle shows the silver expansion;
(v) WRONG-beta candidates: beta = N/2, N/4 (and 2N) violate detailed
balance exactly -- gaps |q^{-8} - q^{-4}|, |q^{-8} - q^{-2}|,
|q^{-8} - q^{-16}| quantified at 40 digits, plus the N/2-return
witness C(1) - C(3) = sqrt2 C(3) != 0 exact.
[C-8] verdict logic (FROZEN): KILL iff (K1) rho_OS fails PD/duality or
the Tomita KMS identity fails, or (K2) the transfer kernel admits the
real-mode representation but detailed balance forces beta != N at
either level (or no real-mode rep exists at N = 16 while N = 8 has
one -- incommensurable temperatures), or (K3) beta_angle/(2 pi) lands
on a non-class value. SUCCESS iff ALL of: [C-1] exact anchors confirm;
[C-2] certificates pass; [C-3] beta = N exact at BOTH levels; [C-4]
class-1 closure beta_angle = 2 pi = 1/(4 c3); [C-6] computed; [C-7]
all five controls behave as typed. Otherwise UNDECIDED with the gap
named. [C-5]/[C-6] outcomes are recorded, not gating. Exactness
policy: scalar identities exact (sympy / cyclotomic Laurent reduction
mod z^N + 1); operator certificates 40-digit (mp.mp.dps = 40,
threshold 1e-30, inertia threshold 1e-25). Runtime < 15 min.
=== FINDINGS (from the frozen-prereg run, 2026-07-23; 22/22) ===
VERDICT: SUCCESS (typed [C]; sandbox only -- no marker moves).
* rho_OS explicit & faithful; nu = {cos(pi/24), cos(7 pi/24)} EXACT;
K = -ln rho_OS with eps = {2 ln cot(pi/48), 2 ln cot(7 pi/48)}
= {5.4501, 1.4139}; Tomita (S, J, Delta) certified at 1e-33..1e-38;
KMS beta = 1 modular identity exact on all 256 pairs; spec Delta =
{p_i/p_j} (two-sided ratio spectrum, commutant carries the same
density).
* TRANSFER KMS: C(d) = 2w (q^{-d} + q^{d-N}) with omega = ln q =
arcsinh(2^{-3/4}) at N = 8 -- MEASURED beta = 8 = N exactly
(kappa = q^{-8} via Q-polynomial identity); N = 16: recurrence
closes exactly (cyclotomic certificates), both pairs real, anchor
prod(u_i - 2) = sqrt2 at both levels, measured exponent 9 =
(N/2)+1 to 1e-30 => beta = 16 = N. Fermi-Dirac n = 1/(1+q^N).
* BRIDGE CLOSES (class 1): beta_angle = N * (2 pi/N) = 2 pi =
1/(4 c3) exactly = the v208 modular normalisation; T_seam =
1/(2 pi) = 4 c3; Schwarzschild T_H = kappa_h/(2 pi) = c3/M =
1/(8 pi M): the axiom c3 IS the seam KMS temperature coefficient;
SdS 1/(4 pi) = 2 c3; Nariai cross-link 2H/T_N = 1/(2 c3) = 4 pi.
beta = N/2, N/4, 2N all excluded exactly (gaps 0.094, 0.31, 0.011).
* CLOCK (typed [C-5] finding): the naive e^{-Delta K} reading FAILS
(min gaps 0.0496/0.1710) and tau_k != G Delta^{k/N} (residuals
0.45/0.67): the Tomita flow of the half-circle cut is NOT the
transfer flow (non-geometric interval modular flow, as in CFT);
the EXACT replacement holds: sqrt2 - 1 = C(3)/C(1) =
1/(2 cosh 2 omega - 1) -- the silver clock eigenvalue is the
thermal kernel ratio at separations 3:1; beta2 compressed clock
{1, sqrt2 - 1} re-verified exactly in both parity sectors.
* ENTROPY: S(8) = 0.52187, S(16) = 0.63510, S(32) = 0.75006 nats;
E1 chiral-CFT ln-law CONFIRMED (DeltaS/ln2: 0.16335 -> 0.16585 ->
1/6, monotone); E2/E3 FAIL as pre-derived: S/S_max = 0.376 (N=8),
0.229 (N=16) -- the v129/v190 horizon fractions {1/3, 2/3} do NOT
transfer to seam entanglement entropy (different objects).
* CONTROLS: site cut indefinite ((2,2,4), 2 negative directions --
no state); family A (4,4,0); anti-chiral odd block (0,8,0), total
(8,8,0); half-line kernel: return dead, recurrence gap 128/105
exact, Stieltjes => beta = inf/T = 0, contraction restored; wrong
beta N/2, N/4, 2N excluded with the quantified gaps above.
Construction base (READ-ONLY): verification/v519_woit_theta_rp_free.py,
verification/v522_woit_beta1_gso_gauge.py, experiments/tfpt-discovery/
woit_os_beta2_os_quotient_probe.py (kernel, Pfaffian-Wick, dihedral
machinery copied verbatim); verification/v104_nariai_clock.py,
v107_quantum_clock_target.py, v190_nariai_entropy_bound.py,
v208_bh_thermodynamics.py, v129_entropy_power_law.py,
v127_ring_resummation.py, v147_clock_gaussian.py (normalisations only).
Run: . experiments/tfpt-discovery/.venv/bin/activate
python experiments/tfpt-discovery/seam_thermal_kms_nariai_bridge_probe.py
"""
import time
from itertools import combinations
import mpmath as mp
import sympy as sp
from tfpt_constants import check as _check, summary, reset
mp.mp.dps = 40
I = sp.I
N8 = 8
N16 = 16
ETA = I
RESULTS = []
FLAGS = {}
T0_WALL = time.perf_counter()
TOL = mp.mpf(10) ** -30
TOL_IN = mp.mpf(10) ** -25
def check(name, ok):
RESULTS.append(bool(ok))
_check("[%2d] %s" % (len(RESULTS), name), bool(ok))
def iszero(e):
e2 = sp.expand(e)
if e2 == 0:
return True
e3 = sp.expand(sp.expand_complex(e2))
if e3 == 0:
return True
return sp.simplify(e3) == 0
# ---------------------------------------------------------------------------
# v519/beta2 machinery (verbatim where possible)
# ---------------------------------------------------------------------------
def c_of(d, n):
if d % 2 == 0:
return sp.Integer(0)
return sp.Rational(2, n) / sp.sin(sp.pi * sp.Rational(d, n))
def g2f(a, b, n, chi):
if a == b:
return sp.Integer(1)
return I * chi * c_of(a - b, n)
_WICK = {}
def wick(idx, n, chi=1):
idx = tuple(idx)
if len(idx) == 0:
return sp.Integer(1)
if len(idx) % 2 == 1:
return sp.Integer(0)
key = (idx, n, chi)
if key in _WICK:
return _WICK[key]
head, rest = idx[0], idx[1:]
tot = sp.Integer(0)
for j, b in enumerate(rest):
sub = rest[:j] + rest[j + 1:]
tot += (-1) ** j * g2f(head, b, n, chi) * wick(sub, n, chi)
tot = sp.expand_complex(tot)
_WICK[key] = tot
return tot
def refl_map(k, n):
def r(a):
return (k - a) % n
def s(a):
return -1 if (k - a) % (2 * n) >= n else 1
return r, s
def theta_mono(mono, r, s, eta):
imgs = [r(a) for a in reversed(mono)]
coeff = eta ** len(mono)
for a in mono:
coeff *= s(a)
lst = list(imgs)
sign = 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]
sign = -sign
assert len(set(lst)) == len(lst)
return coeff * sign, tuple(lst)
def mono_mul(m1, m2):
out = list(m1)
sign = 1
for g in m2:
out.append(g)
i = len(out) - 1
while i > 0 and out[i - 1] > out[i]:
out[i - 1], out[i] = out[i], out[i - 1]
sign = -sign
i -= 1
if i > 0 and out[i - 1] == out[i]:
del out[i - 1:i + 1]
return sign, tuple(out)
def tower_maps(n, shift, kmax):
maps = [(tuple(range(n)), (1,) * n)]
for _ in range(kmax):
perm, sign = maps[-1]
np_, ns_ = [], []
for a in range(n):
p, s0 = perm[a], sign[a]
q = p + shift
np_.append(q % n)
ns_.append(s0 * (-1 if (q >= n or q < 0) else 1))
maps.append((tuple(np_), tuple(ns_)))
return maps
def alpha_mono(m, pm):
perm, sign = pm
c = 1
imgs = []
for a in m:
c *= sign[a]
imgs.append(perm[a])
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
assert len(set(lst)) == len(lst)
return c * sgn, tuple(lst)
def spectrum_inertia(M, tol=None):
if tol is None:
tol = TOL_IN
n = M.shape[0]
A = mp.matrix(n, n)
for i in range(n):
for j in range(n):
re_, im_ = M[i, j].evalf(45).as_real_imag()
A[i, j] = mp.mpc(mp.mpf(str(re_)), mp.mpf(str(im_)))
E, _ = mp.eighe(A)
evs = [E[i].real for i in range(n)]
npos = sum(1 for e in evs if e > tol)
nneg = sum(1 for e in evs if e < -tol)
nzero = n - npos - nneg
nz = [abs(e) for e in evs if abs(e) > tol]
return (npos, nneg, nzero), (min(nz) if nz else mp.mpf(0))
def to_mp(M):
n, m = M.shape
A = mp.matrix(n, m)
for i in range(n):
for j in range(m):
re_, im_ = M[i, j].evalf(45).as_real_imag()
A[i, j] = mp.mpc(mp.mpf(str(re_)), mp.mpf(str(im_)))
return A
def ctr(A):
B = mp.matrix(A.cols, A.rows)
for i in range(A.rows):
for j in range(A.cols):
B[j, i] = mp.conj(A[i, j])
return B
def mconj(A):
B = mp.matrix(A.rows, A.cols)
for i in range(A.rows):
for j in range(A.cols):
B[i, j] = mp.conj(A[i, j])
return B
# cyclotomic Laurent certificate: expr(z) == 0 at z = e^{i pi / n}
# (z^n + 1 = 0 is the minimal polynomial for n a power of two)
ZSYM = sp.symbols('z')
def zeta_zero(expr, n, pad=220):
e = sp.cancel(sp.together(sp.expand(expr)))
num, den = sp.fraction(e)
num = sp.expand(num * ZSYM ** pad)
try:
p = sp.Poly(num, ZSYM)
r = sp.rem(p, sp.Poly(ZSYM ** n + 1, ZSYM))
poly_zero = r.is_zero
except sp.PolynomialError:
# fallback: substitute the exact root of unity
poly_zero = sp.simplify(sp.expand(
expr.subs(ZSYM, sp.exp(sp.I * sp.pi / n)))) == 0
if not poly_zero:
return False
zval = mp.e ** (1j * mp.pi / n)
dv = complex(sp.lambdify(ZSYM, den)(complex(zval)))
return abs(dv) > 1e-12
P8 = [4, 5, 6, 7]
R7, S7 = refl_map(7, N8)
BASIS8 = [m for d in range(5) for m in combinations(P8, d)] # 16
BIDX = {m: i for i, m in enumerate(BASIS8)}
TW8 = tower_maps(N8, 1, N8)
C1, C3 = c_of(1, 8), c_of(3, 8)
c3 = sp.Rational(1, 8) / sp.pi
pi_ = sp.pi
# pre-derived exact mode data (to be machine-confirmed in Part B)
QQ_ = sp.Rational(1, 2) * (2 + sp.sqrt(2) + sp.sqrt(2 + 4 * sp.sqrt(2)))
q_ = sp.sqrt(QQ_) # q > 1, omega = ln q
# ---------------------------------------------------------------------------
# Part R -- the reduced state and its modular Hamiltonian (N = 8 exact)
# ---------------------------------------------------------------------------
def r1_kernel():
print(" -- R1: kernel preconditions (compact-circle thermal form)")
ret = all(iszero(c_of(N8 - d, N8) - c_of(d, N8)) for d in (1, 3, 5, 7))
antip = all(iszero(c_of(d + N8, N8) + c_of(d, N8)) for d in (1, 3, 5, 7))
chk = all(c_of(d, N8) == 0 for d in (2, 4, 6))
Cm8 = sp.Matrix(N8, N8, lambda a, b: c_of(a - b, N8))
pure = sp.simplify(Cm8 * Cm8 + sp.eye(N8)) == sp.zeros(N8, N8)
S1 = sp.Matrix(N8, N8, lambda i, j: 0)
for a in range(N8):
b = (a + 1) % N8
S1[b, a] = -1 if a + 1 >= N8 else 1
inv = sp.simplify(S1 * Cm8 * S1.T - Cm8) == sp.zeros(N8, N8)
full_turn = (TW8[8][0] == tuple(range(N8)) and set(TW8[8][1]) == {-1})
FLAGS['r1'] = ret and antip and chk and pure and inv and full_turn
check("R1 KERNEL PRECONDITIONS [exact]: return symmetry C(N-d) = C(d) "
"(%s -- correlations RETURN on the compact circle: the thermal "
"hallmark), NS antiperiodicity C(d+N) = -C(d) (%s -- the "
"fermionic KMS boundary sign), chiral checkerboard C(even) = 0 "
"(%s), C^2 = -1 (%s -- the FULL circle state is PURE: the "
"temperature can only come from the cut), alpha_1 preserves the "
"state (%s) and alpha_1^N = (-1)^F (%s -- the trivially-true "
"graded reflection identity omega(a alpha_t b) = "
"omega(alpha_{t+N}(b) a) is thereby necessary-only: the KMS "
"content sits in detailed balance, Part B)"
% (ret, antip, chk, pure, inv, full_turn), FLAGS['r1'])
def r2_rho():
print(" -- R2: rho_OS explicit (Jordan-Wigner dual basis)")
sx = sp.Matrix([[0, 1], [1, 0]])
sy = sp.Matrix([[0, -I], [I, 0]])
sz = sp.Matrix([[1, 0], [0, -1]])
def kron(A, B):
return sp.Matrix(sp.BlockMatrix(
[[A[i, j] * B for j in range(A.cols)] for i in range(A.rows)]))
GM = {4: kron(sx, sp.eye(2)), 5: kron(sy, sp.eye(2)),
6: kron(sz, sx), 7: kron(sz, sy)}
cliff = all(sp.simplify(GM[a] * GM[b] + GM[b] * GM[a]
- 2 * sp.KroneckerDelta(a, b) * sp.eye(4))
== sp.zeros(4, 4) for a in P8 for b in P8)
def mono_mat(m):
M = sp.eye(4)
for a in m:
M = M * GM[a]
return M
rho = sp.zeros(4, 4)
for m in BASIS8:
om = wick(m, N8)
if om != 0:
rho += om * mono_mat(m).conjugate().T # unitary: inv = dagger
rho = sp.expand_complex(rho / 4)
dual = all(iszero(sp.expand_complex((rho * mono_mat(m)).trace())
- wick(m, N8)) for m in BASIS8)
herm = all(iszero(x) for x in (rho - rho.conjugate().T))
tr1 = iszero(rho.trace() - 1)
inert, gap = spectrum_inertia(rho)
FLAGS['rho'] = rho
FLAGS['r2'] = cliff and dual and herm and tr1 and inert == (4, 0, 0)
check("R2 rho_OS EXISTS AND IS FAITHFUL [exact + 40-digit]: pinned "
"Jordan-Wigner map is Clifford (%s); the dual-basis element "
"rho_OS = (1/4) sum_m omega(m) m^{-1} satisfies Tr(rho m) = "
"omega(m) on ALL 16 monomials (%s), is Hermitian (%s), trace 1 "
"(%s), and POSITIVE DEFINITE, inertia %s (min eigenvalue %s) -- "
"the reduced state of the pure circle state on the half algebra "
"is a full-rank (thermal) state: the 4^2 = 16 two-sided "
"dimension of H_phys is its purification"
% (cliff, dual, herm, tr1, inert, mp.nstr(gap, 5)), FLAGS['r2'])
def r3_closed_forms():
print(" -- R3: closed forms -- nu, spectrum, modular Hamiltonian")
Jm = sp.zeros(4, 4)
for i, a in enumerate(P8):
for j, b in enumerate(P8):
if a < b:
v = sp.im(wick((a, b), N8))
Jm[i, j] = v
Jm[j, i] = -v
lam = sp.symbols('lam')
cp = sp.expand((I * Jm - lam * sp.eye(4)).det())
n1s = (4 + sp.sqrt(2) + sp.sqrt(6)) / 8
n2s = (4 + sp.sqrt(2) - sp.sqrt(6)) / 8
target = sp.expand((lam ** 2 - n1s) * (lam ** 2 - n2s))
cp_ok = iszero(cp - target)
cos_ok = (iszero(sp.cos(sp.pi / 24) ** 2 - n1s)
and iszero(sp.cos(7 * sp.pi / 24) ** 2 - n2s))
nu1, nu2 = sp.cos(sp.pi / 24), sp.cos(7 * sp.pi / 24)
ps = [(1 + s1 * nu1) * (1 + s2 * nu2) / 4
for s1 in (1, -1) for s2 in (1, -1)]
sum1 = iszero(sum(ps) - 1)
rho_n = to_mp(FLAGS['rho'])
Er, _ = mp.eighe(rho_n)
got = sorted([Er[i].real for i in range(4)])
want = sorted([mp.mpf(str(sp.N(p, 45))) for p in ps])
spec_ok = max(abs(got[k] - want[k]) for k in range(4)) < TOL
th = sp.symbols('th', positive=True)
cot_id = iszero(sp.simplify((1 + sp.cos(th)) / (1 - sp.cos(th))
- sp.cot(th / 2) ** 2))
eps1 = mp.log((1 + mp.cos(mp.pi / 24)) / (1 - mp.cos(mp.pi / 24)))
eps2 = mp.log((1 + mp.cos(7 * mp.pi / 24))
/ (1 - mp.cos(7 * mp.pi / 24)))
eps_ok = (abs(eps1 - 2 * mp.log(mp.cot(mp.pi / 48))) < TOL
and abs(eps2 - 2 * mp.log(mp.cot(7 * mp.pi / 48))) < TOL)
FLAGS['ps'] = want
FLAGS['eps'] = (eps1, eps2)
FLAGS['r3'] = cp_ok and cos_ok and sum1 and spec_ok and cot_id and eps_ok
check("R3 CLOSED FORMS [exact]: char poly of the half covariance iJ = "
"(lam^2 - nu1^2)(lam^2 - nu2^2) with nu^2 = (4 + sqrt2 +- "
"sqrt6)/8 EXACTLY (%s), and nu_1 = cos(pi/24), nu_2 = "
"cos(7 pi/24) (%s); spectrum p = {(1 +- nu1)(1 +- nu2)/4} "
"(sum 1: %s; 40-digit match: %s) -- the MODULAR HAMILTONIAN "
"K = -ln rho_OS is free with single-particle energies eps_i = "
"ln((1+nu)/(1-nu)) = 2 ln cot(theta/2) (identity: %s), theta = "
"pi/24, 7 pi/24: eps = {%s, %s} (%s) -- K = eps1 n1 + eps2 n2 "
"+ const on the 2-mode Fock space"
% (cp_ok, cos_ok, sum1, spec_ok, cot_id, mp.nstr(eps1, 8),
mp.nstr(eps2, 8), eps_ok), FLAGS['r3'])
def r4_gram():
print(" -- R4: H_phys, Gram, purification identity")
G8 = sp.zeros(16, 16)
for i, ma in enumerate(BASIS8):
ca, mta = theta_mono(ma, R7, S7, ETA)
for j, mb in enumerate(BASIS8):
cs, mm = mono_mul(mta, mb)
G8[i, j] = sp.expand_complex(ca * cs * wick(mm, N8))
herm = all(iszero(x) for x in (G8 - G8.conjugate().T))
inert, gap = spectrum_inertia(G8)
vec = all(iszero(G8[0, j] - wick(BASIS8[j], N8)) for j in range(16))
FLAGS['G8'] = G8
FLAGS['r4'] = herm and inert == (16, 0, 0) and vec
check("R4 PURIFICATION [exact + 40-digit]: the 16x16 bond-cut OS Gram "
"is Hermitian (%s) and PD, inertia %s (min %s -- the beta2 Q1.3 "
"regression 3.3471e-3), so Omega = [1] is CYCLIC and SEPARATING "
"for pi(A_+) = M_4 acting on H_phys = C^4 (x) C^4; the vector "
"state IS the reduced state: <Omega, pi(m) Omega> = omega(m) = "
"Tr(rho_OS m) on all 16 monomials (%s) -- H_phys is the "
"standard-form purification of rho_OS"
% (herm, inert, mp.nstr(gap, 5), vec), FLAGS['r4'])
# ---------------------------------------------------------------------------
# Part T -- Tomita-Takesaki data and the modular KMS condition (40-digit)
# ---------------------------------------------------------------------------
def t1_tomita():
print(" -- T1: S, Delta, J constructed on H_phys")
Gn = to_mp(FLAGS['G8'])
EGv, QG = mp.eighe(Gn)
Gh = QG * mp.diag([mp.sqrt(EGv[i]) for i in range(16)]) * ctr(QG)
Ghi = QG * mp.diag([1 / mp.sqrt(EGv[i]) for i in range(16)]) * ctr(QG)
Gninv = Ghi * Ghi
PIn = {}
for m in BASIS8:
M = mp.matrix(16, 16)
for j, mb in enumerate(BASIS8):
c, mm = mono_mul(m, mb)
M[BIDX[mm], j] = c
PIn[m] = M
e0 = mp.matrix(16, 1)
e0[0, 0] = 1
Gcol0 = mp.matrix(16, 1)
for i in range(16):
Gcol0[i, 0] = Gn[i, 0]
A_S = mp.matrix(16, 16)
for m in BASIS8:
col = Gninv * (ctr(PIn[m]) * Gcol0)
for i in range(16):
A_S[i, BIDX[m]] = col[i, 0]
M_T = ctr(A_S) * Gn * A_S
Delta = Gninv * M_T.T
res_om = mp.mnorm(Delta * e0 - e0, 1)
B = Gh * Delta * Ghi
res_h = mp.mnorm(B - ctr(B), 1)
Bh = (B + ctr(B)) * mp.mpf('0.5')
ED, QD = mp.eighe(Bh)
all_pos = all(ED[i].real > TOL_IN for i in range(16))
Dinvh = Ghi * QD * mp.diag([1 / mp.sqrt(ED[i].real)
for i in range(16)]) * ctr(QD) * Gh
Dinv = Dinvh * Dinvh
A_J = A_S * mconj(Dinvh)
res_j2 = mp.mnorm(A_J * mconj(A_J) - mp.eye(16), 1)
res_au = mp.mnorm(ctr(A_J) * Gn * A_J - Gn.T, 1)
JDJ = A_J * mconj(Delta) * mconj(A_J)
res_jdj = mp.mnorm(JDJ - Dinv, 1) / mp.mnorm(Dinv, 1)
FLAGS.update(Gn=Gn, Gh=Gh, Ghi=Ghi, PIn=PIn, e0=e0, Delta=Delta,
Dinv=Dinv, ED=ED, QD=QD, A_J=A_J)
ok = (res_om < TOL and res_h < TOL and all_pos and res_j2 < TOL
and res_au < TOL and res_jdj < TOL)
FLAGS['t1'] = ok
check("T1 TOMITA DATA EXPLICIT [40-digit certificates]: S[a] = "
"pi(a)^{*G} Omega, Delta = S*S, J = S Delta^{-1/2} -- "
"Delta Omega = Omega (res %s), Delta G-self-adjoint (res %s) "
"and POSITIVE (%s), J^2 = 1 (res %s), J anti-unitary (res %s), "
"J Delta J = Delta^{-1} (res %s): the full modular structure "
"of the thermal reconstruction is explicit"
% (mp.nstr(res_om, 3), mp.nstr(res_h, 3), all_pos,
mp.nstr(res_j2, 3), mp.nstr(res_au, 3), mp.nstr(res_jdj, 3)),
ok)
def t2_kms_modular():
print(" -- T2: modular KMS identity + flow/commutant structure")
Gn, PIn, e0 = FLAGS['Gn'], FLAGS['PIn'], FLAGS['e0']
Delta, Dinv, A_J = FLAGS['Delta'], FLAGS['Dinv'], FLAGS['A_J']
r0 = ctr(e0) * Gn
rowa = {m: r0 * PIn[m] for m in BASIS8}
cola = {m: PIn[m] * e0 for m in BASIS8}
ucol = {m: Delta * cola[m] for m in BASIS8}
kms = mp.mpf(0)
for a in BASIS8:
for b in BASIS8:
lhs = (rowa[a] * ucol[b])[0, 0]
rhs = (rowa[b] * cola[a])[0, 0]
kms = max(kms, abs(lhs - rhs))
worst_c = mp.mpf(0)
for m in BASIS8:
X = A_J * mconj(PIn[m]) * mconj(A_J)
for m2 in [(4,), (5,), (6,), (7,)]:
Y = PIn[m2]
worst_c = max(worst_c, mp.mnorm(X * Y - Y * X, 1))
worst_m = mp.mpf(0)
for m in BASIS8:
X = Delta * PIn[m] * Dinv
R = X.copy()
for m2 in BASIS8:
cc = mp.fsum([mp.conj(PIn[m2][k, i2]) * X[k, i2]
for k in range(16) for i2 in range(16)]) / 16
R = R - cc * PIn[m2]
worst_m = max(worst_m, mp.mnorm(R, 1) / mp.mnorm(X, 1))
ok = kms < TOL and worst_c < TOL and worst_m < TOL
FLAGS['t2'] = ok
check("T2 MODULAR KMS AT beta = 1 [40-digit]: the KMS identity "
"<Omega, pi(a) Delta pi(b) Omega> = <Omega, pi(b) pi(a) Omega> "
"holds on ALL 256 monomial pairs (max residual %s < 1e-30) -- "
"the OS state is EXACTLY KMS for its modular flow; the flow "
"preserves pi(A_+) (membership res %s), and J pi J lands in "
"the commutant (res %s): (pi(A_+), Omega) is a genuine "
"thermal/two-sided (Tomita standard form) pair"
% (mp.nstr(kms, 3), mp.nstr(worst_m, 3), mp.nstr(worst_c, 3)),
ok)
def t3_spec():
print(" -- T3: spec Delta = two-sided ratio spectrum")
ED = FLAGS['ED']
got = sorted([ED[i].real for i in range(16)])
want = sorted([pi_v / pj_v for pi_v in FLAGS['ps']
for pj_v in FLAGS['ps']])
dev = max(abs(got[k] - want[k]) for k in range(16))
ok = dev < TOL
FLAGS['t3'] = ok
FLAGS['specD'] = got
check("T3 SPEC Delta = {p_i / p_j} [40-digit, max dev %s]: the "
"modular spectrum is the full two-sided RATIO spectrum of "
"rho_OS (16 values incl. 4 exact 1's and the degeneracy "
"p1/p3 = p2/p4) -- the commutant (the reflected half) carries "
"the SAME density: the purification is reflection-symmetric, "
"exactly as the bond-cut theta demands"
% mp.nstr(dev, 3), ok)
def t4_clock_reading():
print(" -- T4: the clock vs the modular data (typed [C-5])")
silver = mp.sqrt(2) - 1
ED = FLAGS['ED']
specD = FLAGS['specD']
gap_quarter = min(abs(x ** mp.mpf('0.25') - silver) for x in specD
if x > 0)
gap_full = min(abs(x - silver) for x in specD if x > 0)
naive_fails = gap_quarter > mp.mpf('0.01') and gap_full > mp.mpf('0.01')
# tau_k vs G Delta^{k/N} on the Klein-Landau domains
Gn, Ghi, Gh, QD = FLAGS['Gn'], FLAGS['Ghi'], FLAGS['Gh'], FLAGS['QD']
def Dpow(s):
return Ghi * QD * mp.diag([ED[i].real ** s for i in range(16)]) \
* ctr(QD) * Gh
def tau_mat(D, k):
M = sp.zeros(len(D), len(D))
for i, ma in enumerate(D):
ca, mta = theta_mono(ma, R7, S7, ETA)
for j, mb in enumerate(D):
cb, mbb = alpha_mono(mb, TW8[k])
cs, mm = mono_mul(mta, mbb)
M[i, j] = sp.expand_complex(ca * cb * cs * wick(mm, N8))
return M
D1 = [m for m in BASIS8 if all(4 <= a <= 6 for a in m)]
D2 = [m for m in BASIS8 if all(4 <= a <= 5 for a in m)]
res = {}
for D, k, s in ((D1, 1, mp.mpf(1) / 8), (D2, 2, mp.mpf(1) / 4)):
tn = to_mp(tau_mat(D, k))
GDs = Gn * Dpow(s)
idx = [BIDX[m] for m in D]
r = max(abs(GDs[a, b] - tn[i, j])
for i, a in enumerate(idx) for j, b in enumerate(idx))
res[k] = r
not_geometric = res[1] > mp.mpf('0.1') and res[2] > mp.mpf('0.1')
# the exact positive replacement + beta2 clock regression
silver_id = iszero(c_of(3, 8) / c_of(1, 8) - (sp.sqrt(2) - 1))
cosh_id = iszero(sp.simplify(
2 * (1 + 2 * (sp.sqrt(2) / 4)) - 1 - (1 + sp.sqrt(2))))
# 2 cosh 2w - 1 with sinh^2 w = sqrt2/4 -> 1 + sqrt2
lamv = sp.symbols('lamv')
G8 = FLAGS['G8']
tau2 = tau_mat(D2, 2)
idx2 = [BIDX[m] for m in D2]
Gs = sp.Matrix(4, 4, lambda i, j: G8[idx2[i], idx2[j]])
io = [k for k, m in enumerate(D2) if len(m) % 2 == 1]
ie = [k for k, m in enumerate(D2) if len(m) % 2 == 0]
ok_clock = True
for sec in (io, ie):
Gb = Gs.extract(sec, sec)
Tb = tau2.extract(sec, sec)
roots = sp.solve(sp.det(Tb - lamv * Gb), lamv)
ok_clock = ok_clock and len(roots) == 2 \
and any(iszero(r - 1) for r in roots) \
and any(iszero(r - (sp.sqrt(2) - 1)) for r in roots)
FLAGS['t4'] = (naive_fails and not_geometric and silver_id and cosh_id
and ok_clock)
check("T4 CLOCK vs MODULAR DATA [typed [C-5] finding; pre-declared]: "
"the NAIVE reading fails exactly as pre-derived -- sqrt2 - 1 is "
"NOT an e^{-Delta K} ratio (min gap to spec Delta^{1/4}: %s, to "
"spec Delta: %s) and the transfer is NOT the modular flow: "
"max|tau_1 - G Delta^{1/8}| = %s, max|tau_2 - G Delta^{1/4}| = "
"%s on the Klein-Landau domains (the half-circle interval "
"modular flow of a thermal chiral fermion is non-geometric, as "
"in CFT); the EXACT replacement holds: sqrt2 - 1 = C(3)/C(1) "
"(%s) = 1/(2 cosh 2 omega - 1) (%s, sinh^2 omega = sqrt2/4) -- "
"the silver clock eigenvalue is a thermal KERNEL ratio; beta2 "
"compressed clock spectrum {1, sqrt2 - 1} re-verified exactly "
"in both parity sectors (%s)"
% (mp.nstr(gap_quarter, 4), mp.nstr(gap_full, 4),
mp.nstr(res[1], 4), mp.nstr(res[2], 4), silver_id, cosh_id,
ok_clock), FLAGS['t4'])
# ---------------------------------------------------------------------------
# Part B -- transfer-flow KMS: the beta measurement (exact)
# ---------------------------------------------------------------------------
def b1_recurrence8():
print(" -- B1: N = 8 palindromic recurrence -- the mode structure")
f = [c_of(d, 8) for d in range(1, 8)] # f_1..f_7
# palindromic ansatz f_{d+4} = a(f_{d+3} + f_{d+1}) + b f_{d+2} - f_d
# d = 2 equation: 0 = a (C5 + C3) with C5 + C3 = 2 C3 != 0 => a = 0
a_forced = (not iszero(f[4] + f[2])) and iszero(f[5]) \
and iszero(f[3]) and iszero(f[1])
b1 = sp.simplify((f[4] + f[0]) / f[2]) # from d = 1
b3 = sp.simplify((f[6] + f[2]) / f[4]) # from d = 3
consist = iszero(b1 - b3)
b_ok = iszero(b1 - (2 + sp.sqrt(2)))
# z^4 - b z^2 + 1 = 0: modes {q, 1/q, -q, -1/q}, q^2 + q^{-2} = b
QQok = iszero(sp.expand(QQ_ ** 2 - (2 + sp.sqrt(2)) * QQ_ + 1))
sinh_ok = iszero(sp.expand(QQ_ + 1 / QQ_ - 2 - sp.sqrt(2)))
om_num = mp.log(mp.mpf(str(sp.N(q_, 45))))
om_ok = abs(om_num - mp.asinh(2 ** mp.mpf('-0.75'))) < TOL
# zeta_8 certificate of the consistency: C1/C3 = 1 + sqrt2 with
# sqrt2 = z^2 + z^{-2} at z = e^{i pi/8}
lhs = (ZSYM ** 3 - ZSYM ** -3) - \
(1 + ZSYM ** 2 + ZSYM ** -2) * (ZSYM - ZSYM ** -1)
zc = zeta_zero(lhs, 8)
FLAGS['b1'] = (a_forced and consist and b_ok and QQok and sinh_ok
and om_ok and zc)
check("B1 N = 8 MODE STRUCTURE [exact]: the palindromic Prony "
"recurrence on C(1..7) forces a = 0 (checkerboard: the +- "
"momentum-0/pi doubling, %s) and CLOSES with ONE consistency "
"equation satisfied exactly: b = (C1+C3)/C3 from d = 1 equals "
"(C7+C3)/C5 from d = 3 (%s), b = 2 + sqrt2 (%s; cyclotomic "
"certificate mod z^8+1: %s) -- modes {+-q, +-1/q} with "
"q^2 + q^{-2} = 2 + sqrt2 (%s), i.e. (q - 1/q)^2 = sqrt2: "
"omega = ln q = arcsinh(2^{-3/4}) = %s (%s) -- ONE real "
"frequency: the seam kernel is a bona fide finite thermal-mode "
"kernel"
% (a_forced, consist, b_ok, zc, QQok, mp.nstr(om_num, 10),
om_ok and sinh_ok), FLAGS['b1'])
def b2_beta8():
print(" -- B2: N = 8 detailed balance -- beta measured = N")
# measured kappa from the d = 1,3 solve (beta NOT input):
# C(d) = v1 q^{-d} + v2 q^{d}; kappa := v2/v1 must equal q^{-8}
v2 = (C3 - q_ ** -2 * C1) / (q_ ** 3 - q_ ** -1)
v1 = (C1 - v2 * q_) * q_
# exact route: kappa = q^{-8} <=> (1+sqrt2) Q = Q^2 - Q + 1
# (the minimal polynomial Q^2 - (2+sqrt2) Q + 1 = 0 again)
kap_id = iszero(sp.expand((1 + sp.sqrt(2)) * QQ_
- (QQ_ ** 2 - QQ_ + 1)))
kap_num = mp.mpf(str(sp.N(v2 / v1, 45)))
q_num = mp.mpf(str(sp.N(q_, 45)))
kap_meas = abs(kap_num - q_num ** -8) < TOL
# full detailed-balance representation, all four odd separations
w_ = C1 / (2 * (q_ ** -1 + q_ ** -7))
rep_ok = all(abs(mp.mpf(str(sp.N(
c_of(d, 8) - 2 * w_ * (q_ ** -d + q_ ** (d - 8)), 45)))) < TOL
for d in (1, 3, 5, 7))
w_pos = sp.N(w_, 30) > 0
n_fd = 1 / (1 + q_num ** 8)
FLAGS['q_num'] = q_num
FLAGS['b2'] = kap_id and kap_meas and rep_ok and w_pos
check("B2 N = 8 DETAILED BALANCE -- MEASURED beta = 8 = N [exact]: "
"solving the two-mode system on d = 1, 3 WITHOUT inputting "
"beta gives growing/decaying weight ratio kappa = q^{-8} "
"exactly (Q-polynomial identity Q/(Q^2-Q+1) = C3/C1 = "
"1/(1+sqrt2): %s; 40-digit: %s), and the full representation "
"C(d) = 2w (q^{-d} + q^{d-8}) holds at ALL odd separations "
"d = 1, 3, 5, 7 (%s) with weight w = %s > 0 (%s) -- the "
"reconstructed transfer dynamics sees the seam state as a "
"Gibbs/KMS state at inverse temperature beta = N = 8 clock "
"steps; Fermi-Dirac occupation n = 1/(1 + q^8) = %s"
% (kap_id, kap_meas, rep_ok, mp.nstr(mp.mpf(str(sp.N(w_, 45))),
8), bool(w_pos),
mp.nstr(n_fd, 8)), FLAGS['b2'])
def b3_beta16():
print(" -- B3: N = 16 -- two real pairs, beta = N, sqrt2 anchor")
# exact solve of the j-unit recurrence with cyclotomic h-values:
# h_j = 1/(z^{2j-1} - z^{-(2j-1)}) ~ C(2j-1) up to a common factor
hz = [1 / (ZSYM ** (2 * jj - 1) - ZSYM ** -(2 * jj - 1))
for jj in range(1, 9)]
a, b = sp.symbols('a b')
eqs = [hz[j + 3] - a * (hz[j + 2] + hz[j]) - b * hz[j + 1] + hz[j - 1]
for j in (1, 2, 3, 4)]
sol = sp.solve(eqs[:2], [a, b], dict=True)[0]
res3 = eqs[2].subs(sol)
res4 = eqs[3].subs(sol)
z16 = 16
c34 = zeta_zero(res3, z16) and zeta_zero(res4, z16)
# anchor prod(u_i - 2) = 2 - b - 2a = sqrt2 = z^4 + z^{-4}
anchor = zeta_zero(2 - sol[b] - 2 * sol[a] - (ZSYM ** 4 + ZSYM ** -4),
z16)
zeta16 = sp.exp(sp.I * sp.pi / 16)
a_val = sp.N(sol[a].subs(ZSYM, zeta16), 40)
b_val = sp.N(sol[b].subs(ZSYM, zeta16), 40)
a_re, a_im = [mp.mpf(str(x)) for x in a_val.as_real_imag()]
b_re, b_im = [mp.mpf(str(x)) for x in b_val.as_real_imag()]
# u quadratic: u^2 - a u - (b + 2) = 0 -> both roots real > 2
disc = a_re ** 2 + 4 * (b_re + 2)
u1 = (a_re + mp.sqrt(disc)) / 2
u2 = (a_re - mp.sqrt(disc)) / 2
real_pairs = (abs(a_im) < TOL and abs(b_im) < TOL
and disc > 0 and u1 > 2 and u2 > 2)
# weights + measured exponent (40-digit Prony in j-units)
hval = [mp.mpf(str(sp.N(c_of(2 * jj - 1, 16), 45)))
for jj in range(1, 9)]
Mh = mp.matrix(4, 4)
bv = mp.matrix(4, 1)
for i2 in range(4):
for j2 in range(4):
Mh[i2, j2] = hval[i2 + j2]
bv[i2, 0] = hval[i2 + 4]
rl = mp.lu_solve(Mh, bv)
coeffs = [mp.mpf(1), -rl[3, 0], -rl[2, 0], -rl[1, 0], -rl[0, 0]]
zr = mp.polyroots(coeffs, maxsteps=200, extraprec=100)
zr = [r.real for r in zr]
all_real_pos = all(abs(mp.im(r)) < TOL for r in
mp.polyroots(coeffs, maxsteps=200, extraprec=100)) \
and all(r > 0 for r in zr)
V4 = mp.matrix(4, 4)
for i2 in range(4):
for j2 in range(4):
V4[i2, j2] = zr[j2] ** (i2 + 1)
hv4 = mp.matrix(4, 1)
for i2 in range(4):
hv4[i2, 0] = hval[i2]
Wv = mp.lu_solve(V4, hv4)
expo_ok = True
w_pos16 = all(Wv[i2, 0] > 0 for i2 in range(4))
oms = []
for i2 in range(4):
for j2 in range(i2 + 1, 4):
if abs(zr[i2] * zr[j2] - 1) < mp.mpf(10) ** -20:
zd = min(zr[i2], zr[j2])
wd = Wv[i2, 0] if zr[i2] == zd else Wv[j2, 0]
wg = Wv[j2, 0] if zr[i2] == zd else Wv[i2, 0]
expo = mp.log(wg / wd) / mp.log(zd)
# j-units: exponent (N/2)+1 = 9 <=> beta = 16 = N
expo_ok = expo_ok and abs(expo - 9) < TOL
oms.append(mp.log(1 / zd) / 2)
prod_sinh = mp.mpf(1)
for om in oms:
prod_sinh *= mp.sinh(om)
anchor_num = abs(prod_sinh - 2 ** mp.mpf('-1.75')) < TOL
FLAGS['oms16'] = oms
FLAGS['b3'] = (c34 and anchor and real_pairs and all_real_pos
and expo_ok and w_pos16 and anchor_num)
check("B3 N = 16 -- THE SAME KMS CLOSURE, TWO MODES [exact "
"cyclotomic + 40-digit]: the palindromic recurrence (2 "
"unknowns from j = 1, 2) satisfies the remaining equations "
"j = 3, 4 EXACTLY mod z^16 + 1 (%s); both mode pairs are REAL "
"with u_i = q_i^2 + q_i^{-2} > 2 (%s; omega_1 = %s, omega_2 = "
"%s), all four Prony roots real positive (%s), weights "
"positive (%s); MEASURED weight-pairing exponent = 9 = (N/2)+1 "
"in j-units, i.e. beta = 16 = N (%s); EXACT ANCHOR at both "
"levels: prod_i (u_i - 2) = sqrt2 (cyclotomic: %s; equivalent "
"40-digit: prod sinh omega_i = 2^{-7/4}: %s; at N = 8: "
"(q - 1/q)^2 = sqrt2) -- the |mu_4|-th root of 2 pins the "
"frequency content at every level"
% (c34, real_pairs, mp.nstr(oms[0] if oms else mp.mpf(0), 8),
mp.nstr(oms[1] if len(oms) > 1 else mp.mpf(0), 8),
all_real_pos, w_pos16, expo_ok, anchor, anchor_num),
FLAGS['b3'])
def b4_wrong_beta():
print(" -- B4: wrong-beta candidates excluded (quantified)")
q_num = FLAGS['q_num']
kap = q_num ** -8
gaps = {half: abs(kap - q_num ** -half) for half in (4, 2, 16)}
all_pos = all(g > mp.mpf('0.005') for g in gaps.values())
wit = iszero((C1 - C3) - sp.sqrt(2) * C3)
wit_num = mp.mpf(str(sp.N(C1 - C3, 45)))
FLAGS['b4'] = all_pos and wit
check("B4 WRONG-beta CANDIDATES EXCLUDED [exact + 40-digit]: "
"detailed balance pins kappa = q^{-8}; the candidates beta = "
"N/2, N/4, 2N would need q^{-4}, q^{-2}, q^{-16} -- gaps "
"|q^{-8} - q^{-4}| = %s, |q^{-8} - q^{-2}| = %s, |q^{-8} - "
"q^{-16}| = %s, all > 0 (%s); direct N/2-periodicity witness: "
"KMS at beta = 4 would demand C(d) = C(d+4)-matching, but "
"C(1) - C(3) = sqrt2 C(3) EXACTLY (%s; = %s != 0) -- the "
"seam temperature is UNIQUELY beta = N in the preregistered "
"candidate lattice"
% (mp.nstr(gaps[4], 6), mp.nstr(gaps[2], 6),
mp.nstr(gaps[16], 6), all_pos, wit, mp.nstr(wit_num, 8)),
FLAGS['b4'])
# ---------------------------------------------------------------------------
# Part C -- the temperature bridge to the Nariai/BH chain
# ---------------------------------------------------------------------------
def c1_bridge():
print(" -- C1: beta_angle = 2 pi = 1/(4 c3) -- the v208 closure")
# one clock step = euclidean angle 2 pi/N (v519 pin, re-verified R1)
beta_angle = sp.Integer(N8) * (2 * pi_ / N8)
cls1 = iszero(beta_angle - 2 * pi_)
v58 = iszero(2 * pi_ - 1 / (4 * c3))
Tseam = sp.simplify(1 / beta_angle)
Tid = iszero(Tseam - 4 * c3)
beta16 = sp.Integer(N16) * (2 * pi_ / N16)
cls16 = iszero(beta16 - 2 * pi_)
FLAGS['c1'] = cls1 and v58 and Tid and cls16
check("C1 THE BRIDGE CLOSES IN CLASS 1 [exact]: beta_steps = N "
"(measured, B2/B3) x angle-per-step 2 pi/N (v519 geometric "
"pin) = beta_angle = 2 pi at BOTH levels (%s/%s) -- and "
"2 pi = 1/(4 c3) EXACTLY (%s, v58/v208): T_seam = 1/(2 pi) = "
"4 c3 (%s) -- the reconstructed seam state is KMS at exactly "
"the Bisognano-Wichmann/Hawking modular normalisation "
"Delta^{it} = e^{-2 pi t K} that v208 carries into the "
"horizon sector; N-independence: the temperature in angle "
"units does not depend on the lattice size (8 vs 16), exactly "
"as a universal modular temperature must"
% (cls1, cls16, v58, Tid), FLAGS['c1'])
def c2_hawking_chain():
print(" -- C2: the Hawking/Nariai chain in the same normalisation")
M, Lam, rh = sp.symbols('M Lambda r_h', positive=True)
T_H = (sp.Rational(1, 4) / M) / (2 * pi_)
schw = iszero(T_H - c3 / M) and iszero(T_H - 1 / (8 * pi_ * M))
sds = iszero(1 / (4 * pi_) - 2 * c3)
T_N = sp.sqrt(Lam) / (2 * pi_)
nariai = iszero(T_N - 4 * c3 * sp.sqrt(Lam))
rate_link = iszero(2 * sp.sqrt(Lam) / T_N - 4 * pi_) \
and iszero(4 * pi_ - 1 / (2 * c3))
FLAGS['c2'] = schw and sds and nariai and rate_link
check("C2 HAWKING/NARIAI CHAIN [exact, = v208/v104 normalisations]: "
"with the seam-supplied 1/(2 pi) = 4 c3, Schwarzschild "
"T_H = kappa_h/(2 pi) = c3/M = 1/(8 pi M) (%s) -- THE AXIOM "
"c3 = 1/(8 pi) IS the Hawking coefficient, now derived from "
"the seam KMS temperature rather than postulated; SdS "
"1/(4 pi) = 2 c3 (%s, v208 check 3); Nariai T_N = "
"sqrt(Lambda)/(2 pi) = 4 c3 sqrt(Lambda) (%s); v104 "
"cross-link: the Nariai entropy-deviation rate 2H satisfies "
"2H/T_N = 4 pi = 1/(2 c3) exactly (%s) -- temperature joins "
"geometry (v101/v104) and anomaly as the third leg of c3"
% (schw, sds, nariai, rate_link), FLAGS['c2'])
def c3_class_audit():
print(" -- C3: anti-numerology class audit")
# the measurement kills every class value except r = 1
q_num = FLAGS['q_num']
kap = q_num ** -8
excluded = []
for r_ in (sp.Rational(1, 4), sp.Rational(1, 2), 2, 4):
beta_steps = 8 * r_
need = q_num ** (-float(beta_steps))
excluded.append(abs(kap - need) > mp.mpf('0.005'))
# c3-power / mu4 / N dressings cannot rescue: beta_angle/(2pi) is a
# PURE NUMBER = 1 exactly; any c3^k (k != 0) dressing would make it
# transcendental-inhomogeneous (pi-degree mismatch) -- record
pure = iszero((sp.Integer(8) * (2 * pi_ / 8)) / (2 * pi_) - 1)
FLAGS['c3a'] = all(excluded) and pure
check("C3 ANTI-NUMEROLOGY AUDIT [exact + 40-digit]: within the "
"FROZEN relation classes beta_angle/(2 pi) in {1/4, 1/2, 1, "
"2, 4} (x optional c3^k, |mu_4|, N dressings), the "
"measurement selects EXACTLY r = 1 (%s): the four alternative "
"rationals are excluded by the detailed-balance gaps (%s), "
"and beta_angle/(2 pi) = 1 is a pure number (%s) -- no "
"c3-power dressing can reproduce it (pi-degree mismatch): "
"the closure is the class-1 identity, not a tuned fit"
% (pure, all(excluded), pure), FLAGS['c3a'])
# ---------------------------------------------------------------------------
# Part S -- entropy
# ---------------------------------------------------------------------------
def s1_entropy():
print(" -- S1: S(rho_OS) exact form + Gaussian tower N = 8, 16, 32")
nu1, nu2 = mp.cos(mp.pi / 24), mp.cos(7 * mp.pi / 24)
S_p = -mp.fsum([p * mp.log(p) for p in FLAGS['ps']])
def Hnu(nu):
return (mp.log(2)
- ((1 + nu) * mp.log(1 + nu)
+ (1 - nu) * mp.log(1 - nu)) / 2)
S_form = Hnu(nu1) + Hnu(nu2)
form_ok = abs(S_p - S_form) < TOL
Ss = {}
nus_all = {}
for NN in (8, 16, 32):
half = list(range(NN // 2, NN))
nh = len(half)
Jh = mp.matrix(nh, nh)
for i2, a2 in enumerate(half):
for j2, b2 in enumerate(half):
if (a2 - b2) % 2 == 1:
Jh[i2, j2] = 1j * 2 / mp.mpf(NN) \
/ mp.sin(mp.pi * (a2 - b2) / NN)
Eh, _ = mp.eighe(Jh)
nsv = sorted([abs(Eh[i2].real) for i2 in range(nh)])[::2]
nus_all[NN] = nsv
Ss[NN] = mp.fsum([Hnu(nu) for nu in nsv])
faithful16 = all(mp.mpf(0) < nu < 1 for nu in nus_all[16])
match8 = abs(Ss[8] - S_p) < TOL
FLAGS['Ss'] = Ss
FLAGS['s1'] = form_ok and match8 and faithful16
check("S1 ENTROPY EXACT FORM + TOWER [40-digit]: S(rho_OS) = "
"sum_i [ln 2 - ((1+nu)ln(1+nu) + (1-nu)ln(1-nu))/2] matches "
"-sum p ln p (%s); S(8) = %s, S(16) = %s, S(32) = %s nats "
"(Gaussian covariance; N = 8 cross-check %s); all N = 16 "
"Williamson values strictly inside (0,1) (%s) -- rho_OS is "
"faithful/thermal at N = 16 too (the Gaussian certificate of "
"the same statement proved algebraically at N = 8); N = 16 "
"nu's have no closed cos-form in the pi/48 family (recorded "
"honestly, 40-digit values kept)"
% (form_ok, mp.nstr(Ss[8], 8), mp.nstr(Ss[16], 8),
mp.nstr(Ss[32], 8), match8, faithful16), FLAGS['s1'])
def s2_scaling():
print(" -- S2/S3: preregistered entropy structures (non-gating)")
Ss = FLAGS['Ss']
ds1 = (Ss[16] - Ss[8]) / mp.log(2)
ds2 = (Ss[32] - Ss[16]) / mp.log(2)
sixth = mp.mpf(1) / 6
e1 = abs(ds2 - sixth) < abs(ds1 - sixth) and abs(ds2 - sixth) < 0.02
fr8 = Ss[8] / (2 * mp.log(2))
fr16 = Ss[16] / (4 * mp.log(2))
e2_fails = (min(abs(fr8 - mp.mpf(1) / 3), abs(fr8 - mp.mpf(2) / 3))
> mp.mpf('0.04')
and min(abs(fr16 - mp.mpf(1) / 3),
abs(fr16 - mp.mpf(2) / 3)) > mp.mpf('0.04')
and abs(fr8 - fr16) > mp.mpf('0.1'))
e3_fails = fr8 < mp.mpf(2) / 3 and fr16 < mp.mpf(2) / 3
FLAGS['s2'] = e1 and e2_fails and e3_fails
check("S2/S3 ENTROPY STRUCTURES [non-gating, outcomes as "
"pre-derived]: (E1) chiral-CFT ln-law CONFIRMS -- DeltaS/ln2 "
"= %s -> %s approaching c_chi/3 = 1/6 = %s monotonically (%s): "
"the seam entanglement entropy has the c = 1/2 chiral AREA/log "
"law, thermal-but-critical, NOT extensive; (E2) the v129 "
"horizon fractions {1/3, 2/3} do NOT transfer: S/S_max = %s "
"(N=8), %s (N=16), not N-stable (%s); (E3) the v190 floor 2/3 "
"does not transfer either (%s) -- the Nariai entropy ledger "
"lives on horizon ratios, the seam on entanglement: the "
"TEMPERATURE bridge closes, the entropy-fraction bridge "
"honestly does not"
% (mp.nstr(ds1, 6), mp.nstr(ds2, 6), mp.nstr(sixth, 6), e1,
mp.nstr(fr8, 5), mp.nstr(fr16, 5), e2_fails, e3_fails),
FLAGS['s2'])
# ---------------------------------------------------------------------------
# Part N -- negative controls
# ---------------------------------------------------------------------------
def n1_site_cut():
print(" -- N1: site cut -- no state, no KMS")
P8s = [1, 2, 3]
basis8s = [m for d in range(4) for m in combinations(P8s, d)]
r8_ = lambda a: (-a) % N8
PLUS = lambda a: 1
picked = None
for etac in (sp.Integer(1), sp.Integer(-1), I, -I):
M = sp.zeros(8, 8)
for i, ma in enumerate(basis8s):
ca, mta = theta_mono(ma, r8_, PLUS, etac)
for j, mb in enumerate(basis8s):
cs, mm = mono_mul(mta, mb)
M[i, j] = sp.expand_complex(ca * cs * wick(mm, N8))
if all(iszero(x) for x in (M - M.conjugate().T)):
inert, _ = spectrum_inertia(M)
picked = (etac, inert)
break
FLAGS['n1'] = picked is not None and picked[1][1] >= 2
check("N1 SITE CUT -> NO STATE [exact + 40-digit]: the site-placed "
"OS form on the full site-half algebra {1,2,3} is Hermitian "
"for eta = %s but INDEFINITE, inertia %s (%d negative "
"directions) -- no PSD Gram, no H_phys, no purification, no "
"rho_OS, hence NO KMS state: the thermal reconstruction "
"exists only at the bond placement (beta2 C4.1 reproduced at "
"thermal level)"
% (picked[0] if picked else None,
picked[1] if picked else None,
picked[1][1] if picked else -1), FLAGS['n1'])
def n2_family_a():
print(" -- N2: family A -- no quotient to be thermal")
r_anti = lambda a: (a + 8) % N16
s_alt = lambda a: 1 if a % 2 == 0 else -1
basis_a = [(a,) for a in range(8)]
picked = None
for etac in (sp.Integer(1), sp.Integer(-1), I, -I):
M = sp.zeros(8, 8)
for i, ma in enumerate(basis_a):
ca, mta = theta_mono(ma, r_anti, s_alt, etac)
for j, mb in enumerate(basis_a):
cs, mm = mono_mul(mta, mb)
M[i, j] = sp.expand_complex(ca * cs * wick(mm, N16))
if all(iszero(x) for x in (M - M.conjugate().T)):
inert, _ = spectrum_inertia(M)
diag0 = all(iszero(M[i, i]) for i in range(8))
picked = (etac, inert, diag0)
break
FLAGS['n2'] = (picked is not None and picked[1] == (4, 4, 0)
and picked[2])
check("N2 FAMILY A -> NO THERMAL STATE [exact + 40-digit]: the "
"clock-centralising antipode (alternating dressing) is "
"Hermitian only for eta = %s with ZERO diagonal (%s) and "
"inertia %s -- indefinite at the one-particle level already: "
"no PSD form, no purification, nothing to be KMS (beta2 C4.2 "
"reproduced)"
% (picked[0] if picked else None,
picked[2] if picked else None,
picked[1] if picked else None), FLAGS['n2'])
def n3_antichiral():
print(" -- N3: anti-chiral state -- wrong chirality has no state")
G8a = sp.zeros(16, 16)
for i, ma in enumerate(BASIS8):
ca, mta = theta_mono(ma, R7, S7, ETA)
for j, mb in enumerate(BASIS8):
cs, mm = mono_mul(mta, mb)
G8a[i, j] = sp.expand_complex(ca * cs * wick(mm, N8, chi=-1))
herm = all(iszero(x) for x in (G8a - G8a.conjugate().T))
ev_idx = [i for i, m in enumerate(BASIS8) if len(m) % 2 == 0]
od_idx = [i for i, m in enumerate(BASIS8) if len(m) % 2 == 1]
in_ev, _ = spectrum_inertia(G8a.extract(ev_idx, ev_idx))
in_od, _ = spectrum_inertia(G8a.extract(od_idx, od_idx))
in_all, _ = spectrum_inertia(G8a)
FLAGS['n3'] = (herm and in_od == (0, 8, 0) and in_ev == (8, 0, 0)
and in_all == (8, 8, 0))
check("N3 ANTI-CHIRAL -> NO STATE [exact + 40-digit]: with C -> -C "
"and the same twist the Gram stays Hermitian (%s) but the odd "
"block flips NEGATIVE definite %s (even %s, total %s) -- the "
"OS quotient and hence the thermal representation exist ONLY "
"for the chirality matching the twist (beta2 C4.3 reproduced)"
% (herm, in_od, in_ev, in_all), FLAGS['n3'])
def n4_noncompact():
print(" -- N4: non-compact control -- the temperature needs the "
"compact circle")
# half-line kernel: h_j = 1/(2j-1) (x 2/pi, cancels in all tests)
hL = [sp.Rational(1, 2 * jj - 1) for jj in range(1, 5)]
hC = [c_of(2 * jj - 1, 8) for jj in range(1, 5)]
ret_dead = not iszero(sp.Rational(2, 1) / pi_ / 7
- sp.Rational(2, 1) / pi_) # C(7) != C(1)
ret_alive = iszero(hC[3] - hC[0]) and iszero(hC[2] - hC[1])
mono_line = all(hL[k + 1] < hL[k] for k in range(3))
return_wit = iszero((hC[3] - hC[2]) - sp.sqrt(2) * c_of(3, 8))
# recurrence consistency: b from j=1 vs j=2 (degree-2, j-units)
bL1 = (hL[2] + hL[0]) / hL[1]
bL2 = (hL[3] + hL[1]) / hL[2]
gapL = sp.simplify(bL1 - bL2)
gap_exact = sp.Rational(128, 105)
line_fails = iszero(gapL - gap_exact) and gap_exact != 0
bC1 = sp.simplify((hC[2] + hC[0]) / hC[1])
bC2 = sp.simplify((hC[3] + hC[1]) / hC[2])
circle_closes = iszero(bC1 - bC2)
# Stieltjes: 2/(pi d) = (2/pi) int_0^1 x^{d-1} dx -> no growing part
d_ = sp.symbols('d_', positive=True)
x_ = sp.symbols('x_', positive=True)
stj = iszero(sp.integrate(x_ ** (d_ - 1), (x_, 0, 1)) - 1 / d_)
# contraction restored on the line vs silver expansion on the circle
contr = all(sp.Rational(2 * jj - 1, 2 * jj + 1) < 1
for jj in range(1, 4))
expand_circle = iszero(hC[0] / hC[1] - (1 + sp.sqrt(2)))
FLAGS['n4'] = (ret_dead and ret_alive and mono_line and return_wit
and circle_closes and line_fails and stj and contr
and expand_circle)
check("N4 NON-COMPACT CONTROL -> T = 0 [exact]: on the half-line "
"kernel 2/(pi d) the return symmetry is DEAD (2/(7 pi) != "
"2/pi: %s; circle: C(7) = C(1), C(5) = C(3): %s) and the "
"kernel decays monotonically (%s) where the circle RETURNS "
"(h4 - h3 = +sqrt2 C(3): %s); the palindromic recurrence "
"closes exactly on the circle (%s) but FAILS on the line with "
"exact gap b(j=1) - b(j=2) = %s != 0 (%s); the line kernel is "
"Stieltjes: 1/d = int_0^1 x^{d-1} dx (%s) -- ONLY decaying "
"modes, growing weights identically zero: kappa = 0, beta = "
"inf, T = 0 (ground state, contraction semigroup restored: "
"ratios < 1: %s) while the compact circle carries the silver "
"expansion C(1)/C(3) = 1 + sqrt2 (%s) -- the seam temperature "
"hangs EXACTLY on the compactness of the euclidean circle"
% (ret_dead, ret_alive, mono_line, return_wit, circle_closes,
gap_exact, line_fails, stj, contr, expand_circle),
FLAGS['n4'])
# ---------------------------------------------------------------------------
# Part V -- verdict
# ---------------------------------------------------------------------------
def v1_verdict():
print(" -- V1: verdict per the frozen preregistration")
k1 = not (FLAGS['r2'] and FLAGS['r4'] and FLAGS['t1'] and FLAGS['t2'])
k2 = not (FLAGS['b1'] and FLAGS['b2'] and FLAGS['b3'] and FLAGS['b4'])
k3 = not FLAGS['c1']
kill = k1 or k2 or k3
s_ok = (FLAGS['r1'] and FLAGS['r3'] and FLAGS['t3'] and FLAGS['c2']
and FLAGS['c3a'] and FLAGS['s1'] and FLAGS['n1']
and FLAGS['n2'] and FLAGS['n3'] and FLAGS['n4'])
if kill:
verdict = "KILL"
elif s_ok:
verdict = "SUCCESS"
else:
verdict = "UNDECIDED"
FLAGS['verdict'] = verdict
print("VERDICT: %s" % verdict)
check("V1 VERDICT = %s [frozen C-8 logic]: no kill -- rho_OS "
"faithful + Tomita KMS exact (K1 clear), transfer detailed "
"balance gives beta = N at BOTH levels (K2 clear), bridge "
"lands in class 1 (K3 clear); success block: kernel "
"preconditions, closed forms, ratio spectrum, Hawking/Nariai "
"chain, class audit, entropy computed, all five controls "
"behave. [C-5] recorded findings (non-gating): naive "
"e^{-Delta K} clock reading FAILS; Tomita flow != transfer "
"flow; the exact silver identity sqrt2 - 1 = C(3)/C(1) = "
"1/(2 cosh 2 omega - 1) replaces it. Scope fence: free seam "
"system only; the bridge statement is [C]-typed (seam "
"euclidean circle = thermal circle of the reconstructed "
"horizon dynamics); NOTHING integrated -- all markers stay; "
"sandbox only" % verdict, verdict == "SUCCESS")
def run():
reset()
print("v526 SEAM.THERMAL.KMS.01 -- the thermal seam: KMS -> Nariai/BH bridge"
"KMS -> Nariai/BH bridge (EXPLORATION ONLY)")
r1_kernel()
r2_rho()
r3_closed_forms()
r4_gram()
t1_tomita()
t2_kms_modular()
t3_spec()
t4_clock_reading()
b1_recurrence8()
b2_beta8()
b3_beta16()
b4_wrong_beta()
c1_bridge()
c2_hawking_chain()
c3_class_audit()
s1_entropy()
s2_scaling()
n1_site_cut()
n2_family_a()
n3_antichiral()
n4_noncompact()
v1_verdict()
npass = sum(RESULTS)
print("\n%d/%d checks passed (runtime %.1f s)"
% (npass, len(RESULTS), time.perf_counter() - T0_WALL))
print("FINAL VERDICT: %s" % FLAGS.get('verdict', 'UNDECIDED'))
return summary("v526 SEAM.THERMAL.KMS.01: beta MEASURED (not assumed) = N at both levels via detailed balance (N=8: omega=arcsinh(2^{-3/4}), kappa=q^{-8}; N=16: pairing exponent (N/2)+1=9); T_seam=1/N per clock step, beta_angle=2pi EXACT = 1/(4 c3), T_seam=4 c3 = Bisognano-Wichmann/Hawking normalisation (v208); modular Hamiltonian closed (nu1=cos(pi/24), nu2=cos(7pi/24); eps=2 ln cot(theta/2); Tomita KMS beta=1 on all 256 pairs); c3 third leg (geometry+anomaly+temperature) class-1 selected; entropy S(8/16/32)=0.52187/0.63510/0.75006, DeltaS/ln2 -> 1/6, horizon fractions do NOT transfer; [C-5] silver = C(3)/C(1) thermal kernel ratio; non-compact => T=0; verdict SUCCESS; no marker moves")
if __name__ == "__main__":
raise SystemExit(1 if run() else 0)
Recorded source fingerprint
250a6c344c028159ac3965abb67ca5717c9e1ad59d5fb6a8304a3d2065820347
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