Thermal ordering under Markovian relaxation: a finite-time criterion and its limits

Unrefereed research candidate. When two copies of a system, one hotter and one cooler, relax in the same bath, the Markovian Mpemba effect is a crossing of their distances from equilibrium. This package gives a sufficient test that rules out such crossings at every time: if the dynamics and its time reverse preserve the cone of observables with nondecreasing arithmetic prefix means, cooling Gibbs preparations stay ordered for relative entropy (both orientations), squared Hellinger and Rényi divergences of order 0<α≤2, including for common schedules of noncommuting stages. For complete-graph bosonic population rates with common coupling, the test holds for every spectrum and bath temperature exactly when the spontaneous jump weight is concave; Ohmic baths with large enough cutoff, Drude–Lorentz and sub-Ohmic weights qualify on explicit ranges, super-Ohmic weights do not. A certified super-Ohmic example (J=d3) shows an actual relative-entropy crossing. Limits: an exact five-state cone-preserving example has a brief total-variation reversal, and for every N≥4 there are N-level systems with a slow-mode inversion although no proper induced subsystem shows one. The archive contains the 27-page manuscript (PDF and LaTeX), exact rational and interval certificates, a reproduction runner, semantic negative controls, CI and SHA-256 sums. Assurance: written proofs, exact certificates, internal AI-assisted review only (not peer review, independent reproduction or formal verification); the closest antecedent (Avitan et al. 2026) already excludes the slow-mode effect for Ohmic and sub-Ohmic spectra. Creator: Anonymous. Text and data CC0-1.0; code MIT.