{"data":{"event":{"id":"95a88b67-4c01-4574-b6d2-960cb3c2a1e3","slug":"gpt56-sol-chernoff-strong-log-concavity","title":"GPT-5.6 Sol proves Chernoff’s density is strongly log-concave","short_summary":"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.","full_description":"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.","ledger_type":"benefit","primary_domain_id":"23d935bb-b857-4b2b-9471-d568bb848a66","event_status":"provisional","event_date":"2026-07-21T00:00:00.000Z","discovery_date":"2026-07-21T00:00:00.000Z","first_published_date":"2026-07-21T00:00:00.000Z","last_reviewed_date":"2026-07-21T00:00:00.000Z","geographic_scope":"International","affected_population":"Probability and nonparametric statistics research community","base_impact_tier":2,"base_score":"3.00","attribution_multiplier":"0.8000","evidence_multiplier":"0.3500","realization_multiplier":"0.5000","durability_multiplier":"0.6500","current_event_score":"0.273000","confidence_level":"medium","score_explanation":"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).","methodology_version_id":"d7881163-fb23-4e71-8625-bac1a8662c0f","original_methodology_version_id":"d7881163-fb23-4e71-8625-bac1a8662c0f","published_at":"2026-07-21T00:00:00.000Z","created_at":"2026-09-13T22:56:05.431Z","updated_at":"2026-09-13T22:56:05.431Z","flags":[],"domain_name":"Mathematics","domain_slug":"mathematics","methodology_version":"0.1"},"contributions":[{"id":"50fbc844-88ea-42e3-a70d-b1263100d7fa","event_id":"95a88b67-4c01-4574-b6d2-960cb3c2a1e3","model_id":"8caf038b-a43c-49d6-bd85-c3ff83dbd402","role_description":"Generated the strong log-concavity proof across multiple sessions; humans checked and revised the writeup.","attribution_multiplier":"0.8000","credit_share":"1.0000","contribution_score":"0.273000","attribution_rationale":"The paper states the proof was generated in its entirety by GPT-5.6 Sol.","attribution_confidence":"medium","first_used_date":"2026-07-21T00:00:00.000Z","model_version_if_known":null,"review_status":"approved","created_at":"2026-09-13T22:56:05.431Z","model_slug":"gpt-5-6-sol","model_name":"GPT-5.6 Sol","identity_class":"named_release","is_internal":false,"family_name":"GPT-5.6","family_slug":"openai-gpt-5-6","organization_name":"OpenAI","organization_slug":"openai"}],"sources":[{"id":"7aa219d0-bc25-4d72-91ca-afb7e25979ba","event_id":"95a88b67-4c01-4574-b6d2-960cb3c2a1e3","url":"https://arxiv.org/abs/2607.18619","canonical_url":"https://arxiv.org/abs/2607.18619","source_type":"preprint","publisher":"arXiv","author":null,"publication_date":"2026-07-21T00:00:00.000Z","retrieved_at":"2026-09-13T22:56:05.431Z","title":"Chernoff’s Density Is Strongly Log-Concave","excerpt":null,"content_hash":null,"source_reliability_class":"medium","is_primary_source":true,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-13T22:56:05.431Z"}],"claims":[{"id":"21159d95-617d-45ed-93c4-74b6e5043372","event_id":"95a88b67-4c01-4574-b6d2-960cb3c2a1e3","claim_text":"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.","claim_type":"outcome","claim_status":"supported","confidence_score":null,"created_at":"2026-09-13T22:56:05.431Z","updated_at":"2026-09-13T22:56:05.431Z"}],"revisions":[{"id":"a62ee647-dd30-454f-aad4-45c8f25c6df3","event_id":"95a88b67-4c01-4574-b6d2-960cb3c2a1e3","model_id":null,"previous_score":"0.000000","new_score":"0.273000","previous_factors":{},"new_factors":{"evidence":0.35,"base_score":3,"durability":0.65,"attribution":0.8,"realization":0.5},"change_reason":"Imported events under methodology 0.1.","trigger_type":"editorial_import","trigger_source_ids":null,"reviewer_id":null,"review_status":"published","created_at":"2026-09-13T22:56:05.431Z"}],"secondary":[]},"methodology_version":"0.1"}