Odd rings fail, even rings hold. An Evidence Press briefing. Many small systems in biology hop at random around a loop of states, like an enzyme cycling through its shapes. Its characteristic modes say how fast it settles down, and whether it oscillates on the way. Oscillation needs a steady push around the loop, a thermodynamic driving force. In 2025, Xu, Kolchinsky, Delvenne and Ito proposed a neat rule: after a suitable rescaling, these modes always lie inside a classical shape called the Karpelevich region. If true, experimenters could read off the smallest driving force consistent with a measured spectrum. An earlier Evidence Press release showed that the rule fails for general networks, but left open the case of a single ring of states, where the rule had been tested. This paper settles that case. The rule holds for every ring with three states or an even number of states. It fails for every ring with an odd number of states, from five upwards, at every strength of driving. The culprits are pairs of states that are tightly linked to each other but only weakly linked to their neighbours. They create fast modes that oscillate slightly too much. A five-state ring with rates of one, two and three already breaks the rule, and if the rule were used to infer its driving force, it would overstate it by about forty-three per cent. The bounds that have actually been proved remain safe. The proofs are written out in full, and the examples have exact computer certificates. The even-ring half relies on a recent preprint by other authors. This is an anonymous, AI-assisted, unrefereed candidate, not peer reviewed or independently reproduced. Published on the third of November twenty twenty-six, 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.