Extreme economic outcomes — the highest incomes, the most successful firms — follow extreme value distributions. Standard extreme value theory characterizes the tail behavior assuming homogeneous opportunity access. But access isn't homogeneous. Some people draw from deeper pools.
The framework (arXiv:2603.21407) separates the tail shape (extreme value index) from the access structure (heterogeneity in opportunity intensity). A mixed Poisson search model generates a Laplace mixture representation for normalized maxima. The heterogeneity isn't noise — it has geometry, and that geometry is measurable through optimal transport.
The key result: a canonical coupling converts differences in opportunity distribution into optimal transport distances. You can compute how far one economy's opportunity structure is from another's, and the distance tells you how much of the inequality difference comes from structural access rather than tail behavior.
This matters because the policy implications differ completely. If extreme inequality comes from heavy tails (a few rare events produce enormous outcomes), redistribution is the lever. If it comes from unequal opportunity access (some groups draw more frequently from the same distribution), access equalization is the lever. The geometry tells you which one you're dealing with.
Second-order expansions separate classical extreme value approximation errors from heterogeneity-specific effects. Entropy-regularized optimization provides a computational framework for opportunity reallocation — finding the minimum-cost path between two economies' opportunity structures.
The structural claim: inequality at the top isn't a scalar. It has dimensions — tail heaviness and access breadth — that interact but are independently measurable. Collapsing them into a single number (Gini, top-1% share) destroys the information you need to choose the right intervention.