A Decision-theoretic Categorical Synthesis of Tail Classied Probabilities and Risk Functionals

We organise the interaction between tailed-probability laws and the functionals usedto evaluate them (utilities, coherent and convex risk measures, distortion and spectral riskmeasures, ergodic time-average growth rates) into a unied categorical framework. Objectsare distributionfunctional pairs; morphisms are tail-preserving stochastic maps. Withinthis setting we recast the St. Petersburg paradox and the Supercentenarian paradox [GN22]as obstructions to extending a nite-valued risk functor across opposite ends of the tailspectrum. We prove a categorical strengthening of Menger's theorem [Men34]: the expected-utility functor Eu : D+ → R+ factors through R+ precisely when u is bounded above, andthe obstruction is realized by an explicit law in R−α ⊆ S ⊆ L whenever u is at mostpolynomial. We prove a dual statement for the supercentenarian regime: conditional survivalF(t + Δt)/F(t) is a colimit-vanishing functor exactly when the hazard rate diverges, aphenomenon located outside the long-tailed hierarchy in the super-exponential class. Thetwo paradoxes are therefore structurally dual , one at the Extremistan end of the tailspectrum, the other at the super-Mediocristan end [Tal07] , and a single profunctorialobstruction class unies them. We characterize which risk functionals (coherent, convex,spectral, distortion, entropic, time-average) are robust to each obstruction.We then generalize to a sequential decision-theoretic setting in which the distribution isupdated by Bayesian conditioning at each decision epoch. The Bayesian lter becomes afunctor B: Mod → [N0,HT], the tail-class assignment becomes an Ft-adapted stochasticprocess, and the black-swan obstruction class becomes a martingale-adapted process Bρt .We isolate a phenomenon of epistemic tail amplication (Theorem 8.3): even when everycomponent of the likelihood family lies in R−α for some α > 0, the predictive distributioncan lie in R0 (slowly varying), strictly heavier than any component, with the amplica-tion resolving only asymptotically under posterior consistency. The dynamic Menger anddynamic Bayesian black-swan-resolution theorems characterize the conditions under whichthe obstructions of the static theory either persist or dissolve under accumulating evidence.The entire framework is T -indexed for an arbitrary poset T of distribution classes, withthe heavy-tail hierarchy R ⊆ S ⊆ L as a running example; the wider poset includes sub-Gaussian, exponential-decay, super-exponential, and Weibull-shape classes, and we identifythe unique boundary in T at which each natural risk functional (E, entropic, CVaR, dis-tortion) loses niteness, giving a Galois connection between Risk and the upper sets ofT .