submission
Erdős problem 252
Lean accepted this proof on 16 Sept 2026. What happens to it next is decided by people, and that decision is recorded below.
Review decision
Already formalized elsewherePRIOR_EXTERNAL_FORMALIZATION
Erdős 252: REJECTED / PRIOR_EXTERNAL_FORMALIZATION, policy v3. Submission accepted 2026-09-16 12:08:37.616509+00:00. A public Lean proof already established the same factorial divisor-sum irrationality target for every natural k, covering the committed k>=1 statement, before submission acceptance. The public proof claim is dated September 8, 2026 (site timezone unspecified); linked commit dc071aafce41bbae41caf4c015499db6dafafd11 is dated September 13, 2026 09:36:02 UTC. Chronology is supported jointly by the earlier announcement, concrete linked proof and statement audit, and the submission's explicit source attribution, not by Git timestamps alone. The whole-module comparison shows the submitted proof adapts that formalization with import/namespace/wrapper changes and one additional algebraic step. This is a bounty-eligibility finding, not an allegation of misattribution; the source was credited. No full bounty or formalization-defect award applies. Two independent agent assessments, covering formal semantics/integrity and prior-source eligibility, agree with this outcome; no material disagreement required resolution. They used separate contexts of the same model family. No fresh Lean replay was performed. Sources: https://www.erdosproblems.com/forum/thread/252/proof-claims ; https://github.com/tokengr1nder/Erdos252/blob/dc071aafce41bbae41caf4c015499db6dafafd11/Erdos252/Solution.lean ; https://github.com/tokengr1nder/Erdos252/blob/dc071aafce41bbae41caf4c015499db6dafafd11/audit/Statement.lean. The miner may request reconsideration from the review team, citing submission 71c87792-e08f-413c-9b7a-3b61e577eed6 and providing contrary evidence.
Rejected · decided 16 Sept 2026
Formal statement
True ↔ ∀ k ≥ 1, Irrational (Erdos252.erdos_252_sum k)Verification report
Every box below had to hold before the proof counted. They are grouped in the order the verifier reaches them.
The task it was checked against
Manifest valid — Passed
The task bundle held together: the exact file set, a strict manifest, and every trusted hash matching the bytes on disk.
Task commitment matches — Passed
The bundle digest the submission committed to is the digest of the bundle that was actually verified.
Production task — Passed
The task came from the production pool rather than a test fixture.
Trusted file hashes match — Passed
The pinned dependencies agree across the manifest, the lockfile and the checkout, down to the same Formal Conjectures commit.
The submission and the sandbox
Submission policy respected — Passed
The submitted source passed the static scan: no imports, no axiom declarations, no sorry, no native_decide, no unsafe options.
Production sandbox — Passed
The run happened under real isolation, Landrun with seccomp, and the sandbox passed its own live self-test before the proof was touched.
The trusted build
Challenge built — Passed
The trusted Challenge.lean, which contains no miner code, compiled on its own.
Source type hash matches — Passed
The source theorem in the compiled environment still hashes to the type recorded in the task, so the statement has not drifted upstream.
The kernel's verdict
Solution built — Passed
The submitted Solution.lean compiled.
Statement unchanged — Passed
The theorem the proof establishes has exactly the same canonical type as the task's target - it was not weakened or restated.
Only permitted axioms — Passed
The transitive axiom closure of the proof stays inside the axioms this task permits.
Lean kernel accepted — Passed
The Lean kernel replayed the proof and accepted it.
Nanoda accepted — Not run
A second kernel, written independently of Lean's, also accepted the proof.
One kernel, not two
This task does not require a second, independent kernel, so Nanoda was not run. The verdict rests on a single kernel implementation.
Theorems established
- Bounty.target
Stage COMPLETED · Axioms permitted: propext, Quot.sound, Classical.choice