An algebraic framework for extremal stability

We propose an algebraic–probabilistic framework that turns finite-sample extrema into a simple exponential race. For one-parameter families whose cumulative distribution function (CDF) or survival function (SF) has the separable exponential form Hθ(x)=exp⁡{−ϕ(x)g−1(θ)}, the extremum of independent variables stays in the family with the parameter given by the pseudo-sum ⨁i=1nθi=g(∑i=1ng−1(θi)) induced by the generator g. The exponential-race view yields two general consequences: (i) the index of the extremum (arg-extreme) is independent of the extremum itself; (ii) its law is Boltzmann–Gibbs (softmax) with weights proportional to g−1(θi). The same mapping gives a closed-form description of the full ranking, clarifies links with existing constructions and extends immediately to exceedances and shortfalls. Under mild regularity, the separable exponential form is also necessary and sufficient for exact finite-n extremal stability. The framework unifies classical examples (Weibull, Pareto, Gumbel and Fréchet), with aggregation rules emerging as harmonic sums, ℓα norms or log-sum-exp, and it offers a boundary-behaviour perspective on the Fisher–Tippett–Gnedenko (FTG) trichotomy. The resulting theory is simple, closed-form and portable across reliability, insurance and finance, and it suggests natural Bayesian and dependence extensions.