Universality Classes for the Reproducibility of Stochastic Competition

When many agents compete under identical conditions but independent noise,how reproducible is the identity of the winner? We show that this question is governed by the coupling of two distinct limit theorems: extreme-value universality, which fixes the spacings at the top of the initial fitness distribution, and stable-law universality, which fixes the tail of the accumulated noise. Writing the winner heterogeneity as $\mathcal{H}=1-\sum_iP_i^2$, we obtain $\mathcal{H}\sim N^{\theta}$ with an exponent $\theta$ that depends on both universality classes and cannot be deduced from either alone. For initial fitnesses in the Fr\'echet domain with index $\beta$ and noise inthe stable domain with index $\alpha$, $\theta=1-\alpha/\beta$, so $\theta$ \emph{changes sign} at $\alpha=\beta$: when the initial distribution is heavier-tailed than the accumulated noise, enlarging the field of competitors makes the outcome \emph{more} predictable. We verify the full phase diagram numerically without free parameters, and show that the predictable phase is protected by an essential singularity for thin-tailed noise but only by a power law for heavy-tailed noise. A corollary for evolutionary dynamics follows: under the classical Gillespie--Orr assumptions the predictability horizon scales as $(\ln N)^2/N$.