A meta-theorem template (the "master minimax") for optimality claims in the induced-metric optimizer family, paired with a "tautology test" that flags when such a claim is merely an algorithmic re-derivation of a known optimizer rather than a genuine optimality result. The project's diagonal minimax (derived optimizer) and Kronecker minimax (pullback unification → KFAC) are both run through the test and shown to be tautological — which reframes them as algorithmic derivations from embedding geometry, not novel optimality theorems.