A ⊙_ÿ-Critical Framework for the Perfect Cuboid Problem

We expected the perfect cuboid to yield. The four Diophantine equations sit so neatly together — three Pythagorean triples sharing edges, bound by a space diagonal — that constructing a descent operator felt like it should be a matter of patient algebra. Instead we found: the elementary algebra closes cleanly, the modular constraints tighten predictably, and then the framework stops. What remains is not a missing calculation but a missing construction — a map from any putative cuboid to a strictly smaller one that no one has yet written down. The Lean 4 formalization reviewed here makes this gap precise. Twenty-two lemmas are proved without recourse to sorry. Three axioms mark the exact location where number theory has not yet found its footing. The result is a self-modeling proof framework operating at the ⊙_ÿ critical edge: complete in its structure, honest in its gap.