certified
No subset of the natural numbers has o(sqrt(N)) exceptions to unique unordered two-term representation. The PDF discusses both parts; this Lean proof refutes the possibility proposed in part (ii).
Submitted by JenW1N
The PDF is a working mathematical exposition. Lean verification applies to the formal source, not the prose.
Certified 16 Sept 2026, published by conjectures.io.
Review decision
Approved in reviewREVIEW_APPROVED
Approved under policy v3: REVIEW_APPROVED, for the full submission-time locked bounty. The submission refutes Erdős 14(ii): no subset A of the natural numbers has an exception count o(sqrt(N)) for unique representation as a sum of two elements of A. Representations are unordered and equal summands are allowed; the exception count is over 1 through N. A proved finite-scale obstruction rules out the little-o claim, matching the accepted counterexample task. The [proof-claim tab](https://www.erdosproblems.com/forum/thread/14/proof-claims) contained no claims when reviewed on 15 September 2026. The [ordinary discussion](https://www.erdosproblems.com/forum/thread/14) gives finite computations and conditional reductions with unproved lower-bound hypotheses. The [LeanGenius page](https://leangenius.org/proof/erdos-14-unique-sums) provides a conditional implication without proving the required bound. Unique-representation results for bases over the integers, allowing negative summands, do not establish this natural-number target. No qualifying prior complete refutation was established. Part (i) is a separate stable reward target despite the shared proof core. The committed proof and task passed production verification. Review checked their digests, the matching report, statement fidelity, permitted axioms, and earlier reward claims. No material formalization defect or published disqualification reason was established. This eligibility decision does not guarantee originality. The review used the recorded Lean verification; no fresh Lean or independent-kernel replay was performed. The submission was accepted on 2026-09-14 15:29:35.082784+00:00. The [Erdős Problem a Day report](https://www.erdosproblemaday.com/report/14), dated 26 July 2026, explicitly leaves both asymptotic questions unresolved. Earlier Erdős–Freud finite estimates do not settle these two asymptotic targets. The bounded prior-source review found no qualifying earlier solution or external formalization. Two independent Codex agent assessments each recommended REVIEW_APPROVED: one examined the exact formal statements, proof architecture and verification commitments; the other examined prior solutions, chronology, correspondence and duplicate eligibility. No material disagreement was found. They used separate contexts in the same model family; no distinct-model consensus is claimed. The shared argument addresses two separately published stable reward targets, so it does not itself establish a same-target duplicate. The miner may submit contrary evidence or request reconsideration from the review team, citing this submission ID.
Approved · decided 15 Sept 2026
Formal statement
True ↔ ∃ A, Erdos14.nonUniqueSumCount A =o[Filter.atTop] Erdos14.squareRootVerification 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