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Finite tight Blackwell approachability-to-improper phi-regret reduction in Lean

Arthur Freitas Ramos, Ruy Jose Guerra Barretto de Queiroz, David Barros Hulak

Palomar Registrypalomar/PALOMAR-2026-09-20-000001Sep 20, 2026cs.GTscore 0

A Mathlib-only nonempty finite-simplex, algorithmic specialization of Theorem 4 in Dann, Mansour, Mohri, Schneider, and Sivan (COLT 2025). It realizes the tensor action space as a joint simplex, defines the source improper comparator by adding a mixed-constraint tensor of the canonical action marginal, proves that comparator escapes the action set, and proves exact finite-horizon equality between approachability loss and improper phi-regret under explicit causal strategy translations in both directions. It also proves that every joint action is a mixture of at most one rank-one tensor per…

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