How far off can the Poisson shortcut be? An Evidence Press briefing. Many everyday counts add up independent yes-or-no events with unequal chances: side effects among patients, defaults in a loan book, faults in a network. The exact law of such a count is awkward, so it is routinely replaced by the Poisson law with the same mean. The useful question for rare outcomes is multiplicative: how many times more likely can any single outcome be than the Poisson calculation says? In twenty twenty, Dümbgen and Wellner showed that the factor is at most one over one minus the largest chance, and conjectured something much better: the mean divided by the variance. They proved this when the mean is at most one. This paper proves the conjecture in general. For every such count, no outcome, and therefore no event, is more likely than under the Poisson law by more than mean over variance. An example shows the difference. Take one event with chance ninety-nine percent and nine thousand nine hundred and ninety-nine events with chance one percent. The old bound allows a factor of one hundred. The new bound gives one point zero two, and the actual worst factor is about one point zero one. The proof moves the count smoothly towards the Poisson law and shows that each probability, weighted by the variance, can only grow on the way. Two limits matter. For many comparable events the true factor seems closer to the square root of mean over variance, which is left open. And the bound is about single outcomes and events; for one-sided tails or total variation distance, classical results are better. The proofs are written out and checked by exact and high-precision computer arithmetic. This is an anonymous, AI-assisted, unrefereed candidate, 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.