certified
Erdős problem 15
Certified 6 Aug 2026, published by conjectures.io.
Review decision
Defect in the published taskFORMALIZATION_DEFECT_AWARD
Approved under FORMALIZATION_DEFECT_AWARD. The production verifier accepted the exact frozen Erdős 15 counterexample task, but the published Lean statement asks for summability over the rational numbers rather than convergence of the intended real-valued series. The miner validly refuted the task we published; this does not settle the informal Erdős problem. The validator therefore authorized the fixed $750 USD-equivalent formalization-defect award. It was paid as 1,044.286814577 Subnet 66 Alpha to hotkey 5FqLp5QmNRiHGyj3xbLVnDHfCx25qxJX5CUhpndF9GFfZZiK and finalized at chain event 8785036-19-218 on 2026-08-06 11:51:12 UTC.
Partial award · decided 25 Aug 2026
Formal statement
True ↔ Summable fun k => (-1) ^ (k + 1) * (↑k + 1) / ↑(Nat.nth Nat.Prime k)The proof
This proof was approved in review, so the file the kernel accepted is published in full.
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