Round 2 verification audit for C042

Model: gpt-5

Claim: The state after N cycles is rho_N = [S(Phi,gamma,theta)]^N rho_0 in vectorized form, and the fidelity follows F(N) ~= 1 - C_0 lambda_max^N = 1 - C_0 e^{-N/N_0}, where lambda_max is the second-largest eigenvalue and 1/N_0 the convergence rate.

Source alignment: source/supp_content.tex:15-23 (Eq. (S2))

Prior official verdict: verified with failure_reason None.

Executable evidence: run.py. Sandbox rerun logs: run.log.

Independent audit: I scanned the copied script for imports/shared helper dependencies and reran it through the sandbox. The code is self-contained in this attempt directory and targets the claim strategy: Confirm spectral decomposition of S gives F(N) dominated by the largest sub-unit eigenvalue and that N_0 = -1/log(lambda_max).. I checked the relevant family model rather than relying only on exit status; the rerun is treated as one reproducibility input.

Decision: Round 2 verdict is verified with failure_reason None and limitations []. Notes: Derivation (derivation.md): spectral decomposition S=sum_k lambda_k |r_k>><>1 the largest-modulus sub-unit eigenvalue lambda_max dominates, so F(N)~1-C_0 lambda_max^N = 1-C_0 e^{-N/N0} with N0=-1/log(lambda_max). Numeric (run.log): at the optimum, least-squares slope of log(epsilon) vs N (N>=20) = -0.131289, equal to log(lambda_max) to 6 digits; N0_fit=7.617 matches -1/log(second_eigmag); max relative residual of C_0 lambda_max^N vs full simulation = 1.1e-9; a second generic parameter set gives slope=log(lambda_max) and residual 8e-6. Tolerance: slope match <1e-6. Clean spectral-decomposition fact -> verified.