certified
For every positive epsilon, h(n!) is eventually smaller than n^epsilon. This establishes the factorial target in part (b); it is separate from the question about general practical integers in part (a).
Submitted by JenW1N
Certified 17 Sept 2026, published by conjectures.io.
Review decision
Approved in reviewREVIEW_APPROVED
Erdős 18(b): APPROVED / REVIEW_APPROVED, policy v3. Submission accepted 2026-09-16 06:43:46.690992+00:00. The exact accepted proof and task passed production verification. The final theorem establishes h(n!)<n^epsilon eventually for every positive epsilon, with the weighted dyadic mixing premise proved within the submitted file. Conditional intermediate lemmas are not residual assumptions of the final result. The July 24 public claim concerns a bound for infinitely many practical integers in part (a), which does not establish the factorial target in part (b). No material formalization defect, earlier eligible target holder, or published disqualification was established. Approved for the full locked bounty of 3,164.209329916 Alpha; no 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/18 ; https://www.erdosproblems.com/forum/thread/18/proof-claims. The miner may request reconsideration from the review team, citing submission e93a2766-4c70-4564-b565-d0c556f35929 and providing contrary evidence.
Approved · decided 16 Sept 2026
Formal statement
True ↔ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos18.practicalH n.factorial) < ↑n ^ εVerification report
Every box below had to hold before the proof counted. They are grouped in the order the verifier reaches them.
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.
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.
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.
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
Stage COMPLETED · Axioms permitted: propext, Quot.sound, Classical.choice