"""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 = 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(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 " " = " "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)