Conjectures.io

submission

Erdős problem 859

Lean accepted this proof on 18 Sept 2026. What happens to it next is decided by people, and that decision is recorded below.

Formal statement

∃ c₁ > 0,
  ∃ c₂ > 0,
    ∃ d,
      (∀ t > 0, (Erdos859.DivisorSumSet t).HasDensity (d t)) ∧
        Asymptotics.IsEquivalent Filter.atTop (fun t => d t) fun t => c₁ / Real.log ↑t ^ c₂

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