A variance bound for Poisson-binomial density ratios

Unrefereed research candidate. In one sentence: no outcome of a sum of independent yes/no trials is more likely than under the Poisson law with the same mean by more than the factor mean/variance. Counts of rare events with unequal chances (adverse events, defaults, faults) have a Poisson-binomial law and are routinely approximated by the Poisson law with the same mean λ. Dümbgen and Wellner (2020) bounded the worst ratio P(X = k)/Poisson(k) by 1/(1 − max pi) and conjectured the bound λ/v, where v is the variance of the count; they proved it for λ ≤ 1. This paper proves the conjecture in general (Theorem 1.1): for every sum X of independent Bernoulli variables, finite or countable, optionally plus an independent Poisson part, with mean λ and variance v > 0, and every k, v·P(X = k) ≤ λ·e−λλk/k!. Hence every event has probability at most λ/v times its Poisson probability, and the Rényi divergence of order infinity is at most log(λ/v). Along two mean-preserving interpolations to the Poisson law, a logistic decay of the odds with Poisson immigration (Theorem 5.1) and the classical thinning semigroup (Theorem 5.2), every variance-weighted atom v(t)P(Xt = k) is nondecreasing. The engine is a family of explicit nonnegative 3×3 Toeplitz-minor certificates for convexity of three ratio sequences (Proposition 2.4) and a discrete reverse moment inequality. Consequences and limits: ordered conditional inclusion means in rejective sampling are concave in the sample size (Corollary 7.1); the best constant at fixed Δ = 1 − v/λ lies between max{eΔ, (1 − Δ)−1/2} and (1 − Δ)−1 (Proposition 8.1); the inequality does not extend to ultra-log-concave laws (Proposition 8.2). Example: one unit at 0.99 and 9,999 at 0.01 give an earlier bound of 100, the new bound 1.02 and an actual worst factor of about 1.0099. The archive contains the paper (PDF and LaTeX source), the bibliography, three standard-library Python checkers with retained outputs in normal and optimized mode (exact rational identities and certificates, 60-digit checks of the main inequality and both flows, internal negative controls, and a double-precision sharpness scan), a rating-free internal review summary, the research-gate record, CI and SHA-256 sums. Assurance: written analytic proofs; the checks are finite consistency tests, not interval certificates; internal AI-assisted review only (not peer review, independent reproduction or formal verification). Creator: Anonymous. Text and data CC0-1.0; code MIT.