certified
For n points in the plane, the number of different counts of distinct distances seen from individual points can be asymptotic to n.
Submitted by gus
The PDF is a working mathematical exposition. Lean verification applies to the formal source, not the prose.
Certified 17 Sept 2026, published by conjectures.io.
Bounty paid
$3,640at today's rate
4,386.2256 α locked at submission
Solver
gusWho this proof is credited to.
The run
Provenance
Review decision
Approved in reviewREVIEW_APPROVED
Erdős 653: APPROVED / REVIEW_APPROVED, policy v3. Submission accepted 2026-09-15 18:27:31.622477+00:00. The exact accepted proof and task passed production verification. The theorem establishes the intended asymptotic diversity of distance counts for planar point sets; counting the common self-distance does not alter that diversity. The submitted final argument discharges its geometric and analytic premises. No material formalization defect, earlier eligible target holder, or published disqualification was established. References to auxiliary prime-number estimates do not establish a prior solution of this exact target. Approved for the full locked bounty of 4,386.225638274 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/653. The miner may request reconsideration from the review team, citing submission 257ce6dd-e3e5-4169-9d7e-2811092977ca and providing contrary evidence.
Approved · decided 16 Sept 2026
Formal statement
True ↔
∃ o,
o =o[Filter.atTop] 1 ∧
∀ᶠ (n : ℕ) in Filter.atTop, (1 - o n) * ↑n ≤ ↑(EuclideanGeometry.maximalDistinctDistancesFrom n)Verification 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