The thermal seam and horizon temperature bridge

SEAM.THERMAL.KMS.01

Original ledger classification: SUCCESS ([E] exact sympy/cyclotomic + 40-digit Tomita certificates; [C] bridge reading typed; [C-5] non-gating clock-vs-modular findings); NO marker moves

Original claim text

THE THERMAL SEAM: KMS -> NARIAI/BH BRIDGE (2026-07-23; one module, 22 checks, ~8.4 s; verdict SUCCESS): the reconstructed free OS quotient of the 16-Majorana NS seam circle is KMS, and its temperature normalisation CLOSES onto the Nariai/BH chain as the THIRD LEG of c3. (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). (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. (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}). (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. (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. (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. Discovery provenance: seam_thermal_kms_nariai_bridge_probe.py (22/22). NO marker moves.

Original classification detail

SUCCESS: beta MEASURED = N at both levels; T_seam = 4 c3 = Bisognano-Wichmann/Hawking normalisation; modular Hamiltonian closed; c3 third leg class-1 selected; entropy log-law confirms, horizon fractions do NOT transfer; non-compact => T=0; no marker moves

Dependencies as recorded

WOIT.BETA2.OS.01; WOIT.OS.TWISTOR.01; v58; v104; v208; v519; v524