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
$2,372at today's rate
3,156.6776 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. For every primitive set of six distinct positive integer speeds with maximum loneliness p/q<1/6 in lowest terms, every speed is less than 3q. The denominator is that of the loneliness value, not a maximizing time. The arbitrary-modulus classification and finite/unbounded certificate assembly discharge the final hypotheses. Exact proof and committed task passed production run 55; independently recomputed proof, task and report hashes match. Cordella's paper expressly leaves the universal maximum-speed bound as Question 6.6. The proof shares three critical families and nine sporadic examples with that source, but supplies the arbitrary-modulus closure absent from the paper. Shared partial results do not meet the exact-target prior-resolution criterion. No earlier eligible holder, material formalization defect or other evidenced published disqualification was established. Full locked bounty: 3156677607599 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/html/2609.03444v2#S6.SS4. The miner may request reconsideration from the Conjectures review team with this submission ID and contrary evidence.
Approved · decided 12 Oct 2026
Formal statement
Math15.LonelyRunner.Target06Verification 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.
Stage COMPLETED · Axioms permitted: propext, Quot.sound, Classical.choice
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