Coherent risk from finite models

Coherent estimators at different sample sizes need not describe the same population risk. We prove that dyadic estimators do so exactly when duplicating every scenario leaves the estimate unchanged, and recover the population dual set from the finite models. The risk of a Bernoulli loss determines the exact Wasserstein, Kolmogorov and total-variation moduli on a common bounded support. It also characterises uniform estimability, with an impossibility bound for arbitrary randomised estimators. An intrinsic floor on tail levels gives an explicit optimal Lipschitz constant and a uniform approximation defect; every deterministic vanishing floor preserves pointwise strong consistency. A nonstandard construction supplies compatible estimators at all finite sample sizes, explicit block-sum weights, and a formula for the worst-case mass created by infinitesimal tail levels. Finally, we derive uniform regret bounds for data-driven risk-averse optimisation and specialise them to stop-loss retention. Exact binomial calculations compare regularised and original estimators against the same population target.