The Exact Worst-Case Tail Probability under Bounded Kurtosis

We determine exactly what a kurtosis bound buys for one-sided tail control. For the class C(kappa) of real random variables with mean 0, variance 1, and fourth moment at most kappa, the skewness left free, we compute the worst-case tail probability V 1(t, kappa)= X in C(kappa) P(X at least t) for every threshold t>0 and every kappa at least 1. The answer is a four-regime map: a Cantelli tongue b(kappa) t c(kappa) on which the two-moment bound 1/(1+t 2) remains tight and the kurtosis constraint is worthless; a tail regime t at least c(kappa) with the closed form V 1=(kappa-1)/((t 2-1) 2+ kappa-1); a plateau regime, present only for kappa 3/2, on which the worst case freezes and the value does not depend on t; and a central regime described exactly by an explicit algebraic system, provably admitting no closed form in nested square roots. Beyond c(kappa) the one-sided and two-sided worst cases coincide: Cantelli's improvement over Chebyshev is annihilated by fourth-moment information. The minimal degree of a sum-of-squares proof of the tight bound is 2 on the closed tongue and 4 everywhere else, an exact phase diagram of proof degree. Every closed-form regime carries an explicit dual certificate and an explicit extremal distribution, re-verified on parameter grids by an independent checker in exact arithmetic. The closed forms invert to exact worst-case quantiles, sharpen a median-of-means constant, and give the exact per-direction tail available to degree-4 reasoning under certifiable kurtosis. We found the map through an AI-guided search around the certifying pipeline, LemmaForge, which is validated on classical benchmarks, independently reproduces the symmetric-slice bound of Zelen (1954), and recovers the 2 sqrt 3-3 constant of He, Zhang, and Zhang (2010) at t=0.