certified
A Lean-verified submission was approved for this target. Read the recorded review for its precise mathematical scope.
Submitted by Elias
Certified 14 Oct 2026, published by conjectures.io.
Bounty paid
$2,528at today's rate
3,365.1846 α locked at submission
Solver
EliasWho this proof is credited to.
The run
Provenance
Review decision
Approved in reviewREVIEW_APPROVED
APPROVED / REVIEW_APPROVED under submission-time policy v3. Every proper numerical semigroup with conductor c at most four times its multiplicity satisfies c<=e*L. Conductor, multiplicity, minimal-generator count and the count below c have their standard meanings. The universal depth-four case is closed by the general core/window/strip argument; the special quarter-band branch alone is not the full proof. Exact proof and committed task passed production run 53; independently recomputed proof, task and report hashes match. Eliahou's c<=3m result and Eliahou-Fromentin's special families at c=4m do not establish the universal c<=4m target. No qualifying earlier exact-target resolution with substantial matching use was established. No earlier eligible holder, material formalization defect or other evidenced published disqualification was established. Full locked bounty: 3365184674067 Alpha rao, unchanged. The primary review and a separate fresh-context Astra/xhigh advisory review support approval; its baseline was saved before comparison, with no material outcome disagreement. The team reconciled the specific evidence under v3 rather than treating a model vote as decisive. Review relied on the successful pinned production verification, not a new whole-proof replay; Nanoda was disabled. Critical interfaces and bounded primary-source correspondence were checked, not an exhaustive literature corpus. Sources: https://arxiv.org/abs/1703.01761; https://arxiv.org/abs/1710.03623. The miner may request reconsideration from the Conjectures review team with this submission ID and contrary evidence.
Approved · decided 12 Oct 2026
Formal statement
Math30.Problem23Verification 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
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
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.