A Finite Gap Bound for Local Progression-Free Density

Fix s>=3 and consider increasing integer sequences whose every s consecutive terms contain no nontrivial three-term arithmetic progression. We show that every admissible gap sequence can be decreased coordinatewise to an admissible sequence with all gaps at most B_s=1+2 binomial(s,3). Consequently the unrestricted maximum density is the reciprocal of the minimum cycle mean in an explicitly defined finite graph. In particular, it is rational, is computable for each s, and is attained periodically with integer period at most B_s^(s-1). A nonnegative defect derived from this graph characterizes all extremizers, including those with unbounded gaps, for upper, lower, natural and upper Banach density. A short analytic potential proves the sharp value 4/9 for 5<=s<=8. This is an effective finite characterization, not a short closed formula, sharp large-s asymptotic, optimal gap cap or efficient large-s complexity claim. Existence of a periodic optimum does not mean that all extremizers are periodic. Freiman's source constructions, Konyagin's bounds, the frozen aid's capped graph reduction and the classical minimum-cycle-mean method are credited. No absolute priority is claimed. Source reference: AIM-COMBINATORICS-0209. Unrefereed preprint prepared with AI assistance and originating-researcher self-audit. No independent peer review or formal verification is claimed. Author: Alper Ferudun, Mercury Software GmbH.