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Benefit Ledger · provisional · Mathematics

GPT-5.6 Sol proves Chernoff’s density is strongly log-concave

A 21 July 2026 arXiv note proves Balabdaoui and Wellner’s 2014 conjecture that Chernoff’s density is strongly log-concave. The authors state the proof was generated in its entirety by GPT-5.6 Sol and then checked and revised by them.

21 Jul 2026Tier 2 NotableMethodology 0.1

Current score

+0.27

3 base · Notable (tier 2 of 5, 3 pts)
× 0.8000 attribution · Primary causal contribution
× 0.3500 evidence · Technical disclosure or preprint
× 0.5000 realization · Experimentally validated
× 0.6500 durability
Event-level product before credit split: 0.27

A complete analytic proof of a 2014 conjecture in probability is notable (tier 2). Sol generated the argument (0.80). Evidence is an arXiv note without Lean or refereeing (0.35). Realization is a public proof the authors say they checked (0.50).

What happened

Chernoff’s density is the law of the argmax of two-sided Brownian motion minus t², and appears as a limit in isotonic regression and related shape-constrained problems. Balabdaoui–Wellner proved log-concavity and conjectured a uniform positive lower bound on −(log f)″. The note (arXiv:2607.18619) gives an analytic proof via an Airy series representation, an exponential peeling lemma, and a Menon–Srinivasan convolution identity. The authors describe three Sol sessions: the first had a fatal algebra error; later sessions produced the argument they checked. No Lean formalization is claimed. This is a named Sol release, not Astra.

Model attribution

+0.27

GPT-5.6

Generated the strong log-concavity proof across multiple sessions; humans checked and revised the writeup.

The paper states the proof was generated in its entirety by GPT-5.6 Sol.

Attribution 0.8000 · Credit share 100% · OpenAI

Claims

  • The arXiv note proves Chernoff’s density is strongly log-concave, settling a Balabdaoui–Wellner conjecture, and attributes the argument to GPT-5.6 Sol.

    outcome · supported

Sources

primary sources

Revision history

  • 13 Sep 2026 · 0.00 → 0.27

    Imported events under methodology 0.1.