The Exact Worst-Case Tail Probability under Bounded Kurtosis

Abstract

We determine exactly what a kurtosis bound buys for one-sided tail control. For the class $\mathcal{C}(\kappa)$ 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 $V_1(t,\kappa)=\sup_{X\in\mathcal{C}(\kappa)}\mathbb{P}(X\geq t)$ for every threshold $t>0$ and every $\kappa\geq 1$. The answer is a four-regime map: a Cantelli tongue $b(\kappa)\le t\le c(\kappa)$ on which the two-moment bound $1/(1+t^2)$ remains tight and the kurtosis constraint is worthless; a tail regime $t\geq c(\kappa)$ with the closed form $V_1=(\kappa-1)/((t^2-1)^2+\kappa-1)$; a plateau regime, present only for $\kappa\le 3/2$, 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 $t=0$.