Classical entanglement of purification is nonadditive with three outcomes per party

We exhibit an explicit rational 3×3 classical probability table whose regularized von Neumann entanglement of purification is strictly smaller than its one-copy value. In nats, we prove E_P ≥ 0.07163132826 and E_P^∞ ≤ 0.071592778856911368. For every n ≥ 10^12 we also prove E_P(ρ^(⊗n))/n < 0.071611 by a direct classical construction. It specifies an exact purification through a terminating finite search; it is not a practical small-copy witness. A support-degree rank reduction handles arbitrary finite ancillas and reduces the one-copy problem to two phases. Rational entropy quadrature and elementary minima of three-cosine remainders certify the lower bound. A two-component covering argument supplies the upper bound without invoking the Chen–Winter theorem. The gap persists for strictly positive tables. We also prove an obstruction to the same ensemble strategy for binary classical–quantum targets; binary additivity remains open. Standard-library verifiers replay the finite arithmetic. The accompanying research record documents the Recursive Reasoning Exploration run and subsequent refinements. An accompanying three-view visualization guides the reader through the discovery route, proof dependencies, and provenance of the mathematical inputs.