When can a hotter object never cool faster? An Evidence Press briefing. Put two copies of the same small system, one hotter and one cooler, in contact with the same cold surroundings. You would expect the hotter copy to stay farther from equilibrium all the way down. Sometimes it does not: the two relaxation curves cross, and the hotter copy arrives first. This is called the Markovian Mpemba effect, after the old puzzle about hot water freezing faster. This release asks when such crossings can be ruled out at every moment, without solving the dynamics. It gives a simple test. If the dynamics, and its time reverse, keep a certain family of increasing-on-average quantities increasing on average, then the two copies stay in order at all times, for several standard measures of distance from equilibrium. For a standard model of quantum levels coupled to a bath of oscillators, the test passes for every energy spectrum and every bath temperature exactly when the bath's emission strength bends downwards with frequency. Ohmic baths with a high enough cutoff, Drude-Lorentz baths and sub-Ohmic baths pass. Super-Ohmic baths, such as three-dimensional phonons, fail, and for one such bath the paper certifies a real crossing using exact interval arithmetic. The guarantee has limits: total variation distance can still reverse briefly, and checking small subsystems cannot certify a large system. An earlier paper already excluded the slow-mode version of the effect for similar baths; the new contribution is all-time ordering with an exact boundary. The results have written proofs and exact computer checks. This is an anonymous, AI-assisted, unrefereed candidate, not peer reviewed or independently reproduced. Published on 28 October 2026, its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.