Complete monotonicity as a single-trajectory witness ofanomalous thermal relaxation

Anomalous relaxation is usually identified comparatively, through crossings between distinct relaxation trajectories. Whether a single relaxation curve carries an intrinsic signature of anomalous decay has remained unclear. Here we take complete monotonicity of the relative entropy to equilibrium—its being a positive superposition of pure exponential decays—as such a signature—an intrinsic, all-orders criterion defined for any reversible relaxation, classical or quantum—and settle it exactly in the simplest nontrivial arena, the thermal qubit under a Davies generator: complete monotonicity holds if and only if (r−s)(s− 1 2 ) ≥0, where s and r are the Gibbs and initial ground populations. Heating-like relaxation is completely monotone at every order, while cooling-like relaxation carries a negative cubic Bernstein atom—a genuinely finite-temperature, modular obstruction. Initial coherence cancels it only resonantly: exactly at Γ2/Γ1 = 1 2 (T2 = 2T1), with the closed criterion |c|2 ≥(2s−1)(s−r), and in a certified window at 3 2 ; no other physical commensurate ratio restores it. The mechanism yields falsifiable predictions, including an exact dephasing onset law t∗= ln(q0/q∗ 0)/(2Γφ) for the fitted odd-class amplitude of a single relaxation curve.