approved
Erdős problem 1062 - part ii
Approved in review on 22 Sept 2026
The proof
This proof was approved in review, so the file the kernel accepted is published in full.
Review decision
Approved in reviewREVIEW_APPROVED
Erdős 1062(ii): APPROVED / REVIEW_APPROVED under policy v3. Accepted 2026-09-21 18:55:24.437579 UTC. Production run 38 verified the exact submitted proof and committed task with Lean, Comparator, statement-equality and permitted-axiom checks. The proof establishes both convergence of the native fork-free extremal ratio f(n)/n and irrationality of its limit. No material statement mismatch, earlier eligible submission, or supported disqualification was found. Davis's April 19, 2026 paper establishes existence of the limiting density but explicitly leaves irrationality open; it does not settle this exact target. The full submission-time locked bounty of 3922.148126048 Alpha applies, not a formalization-defect award. Source: https://arxiv.org/html/2604.17613v1 ; original problem: https://www.erdosproblems.com/1062 . The human operator accepted the recorded advisory recommendation. One Codex assessment was completed; multiple independent assessments were not completed and no independent-consensus claim is made. Review relied on recorded production verification, integrity checks, selected proof interfaces and bounded prior-work searches, not a fresh kernel replay or complete line-by-line audit. The submitter may request reconsideration from the review team with contrary evidence, citing submission 8d59a0af-6762-4606-93c9-72dd356a57bc.
Approved · decided 22 Sept 2026
Formal statement
(∃ l, Filter.Tendsto (fun n => ↑(Erdos1062.f n) / ↑n) Filter.atTop (nhds l) ∧ Irrational l) ↔ TrueVerification 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