{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:counting-linear-extensions-below-two-to-the-n", "title": "Counting Linear Extensions Below the $2^n$ Barrier", "canonical_statement": "Koivisto asked at Dagstuhl in 2013 whether the linear extensions of an arbitrary $n$-element poset can be counted exactly in time $O^*(c^n)$ for some $c < 2$. Yes: a deterministic exact algorithm runs in $O^*(1.89^n)$, breaking the $2^n$ barrier for the general problem.", "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "candidate", "solution_events": [ { "id": "vibemathed:counting-linear-extensions-below-two-to-the-n:event", "problem_id": "vibemathed:counting-linear-extensions-below-two-to-the-n", "type": "proved", "status": "candidate", "occurred_at": "2026-08-19T00:00:00.000Z", "title": "Breaking the $2^n$ Barrier for Counting Linear Extensions with a Short Elementary Algorithm", "summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:counting-linear-extensions-below-two-to-the-n:attempt" ], "method_family_ids": [ "vibemathed:counting-linear-extensions-below-two-to-the-n:method" ], "source_assertion_ids": [ "vibemathed:counting-linear-extensions-below-two-to-the-n:assertion" ] } ], "attempts": [ { "id": "vibemathed:counting-linear-extensions-below-two-to-the-n:attempt", "problem_id": "vibemathed:counting-linear-extensions-below-two-to-the-n", "model": "Claude Opus 5, ChatGPT 5.6 Sol", "provider": "Anthropic, OpenAI", "attempted_at": "2026-08-19T00:00:00.000Z", "human_collaborators": [ "Keigo Oka" ], "exposure": "unknown", "prompt_public": null, "outcome": "The paper's disclosure: \"The core mathematical ideas underlying the new part of the algorithm and proof were discovered by Claude Opus 5 (Anthropic) during AI-assisted mathematical exploration\" - naming the first-upper-element pattern representation, multiplicity-profile decoding, the deadline dynamic program and the state-counting strategy of Sections 3 to 5. The chain-partition bound of Section 2 refines Kozma and is not new. The research prompt supplied to Claude Opus 5 was itself generated by ChatGPT 5.6 Sol, modelled on OpenAI's publicly released prompt for their cycle double cover work.", "failed_routes": [], "artifacts": [], "sources": [ { "label": "Breaking the $2^n$ Barrier for Counting Linear Extensions with a Short Elementary Algorithm", "url": "https://arxiv.org/abs/2608.19505", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Counting Linear Extensions Below the $2^n$ Barrier", "url": "https://vibemathed.com/problem/counting-linear-extensions-below-two-to-the-n", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:counting-linear-extensions-below-two-to-the-n:verification", "problem_id": "vibemathed:counting-linear-extensions-below-two-to-the-n", "solution_event_id": "vibemathed:counting-linear-extensions-below-two-to-the-n:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Checked by this site on 21 August 2026 against the paper (arXiv:2608.19505v1): the disclosure is verbatim as quoted and Koivisto's Dagstuhl 2013 question is cited in the abstract. This is an exact deterministic algorithm with a proved worst-case bound, not a heuristic, so it clears the methodology's exclusion. The ancillary Python script cross-checks correctness against brute force on small posets and does not certify the running time; it was not re-run here. Days-old preprint, no independent review.", "sources": [ { "label": "VibeMathed: Counting Linear Extensions Below the $2^n$ Barrier", "url": "https://vibemathed.com/problem/counting-linear-extensions-below-two-to-the-n", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Breaking the $2^n$ Barrier for Counting Linear Extensions with a Short Elementary Algorithm", "url": "https://arxiv.org/abs/2608.19505", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Breaking the $2^n$ Barrier for Counting Linear Extensions with a Short Elementary Algorithm", "url": "https://arxiv.org/abs/2608.19505", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Counting Linear Extensions Below the $2^n$ Barrier", "url": "https://vibemathed.com/problem/counting-linear-extensions-below-two-to-the-n", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:counting-linear-extensions-below-two-to-the-n:signal:verify_now", "problem_id": "vibemathed:counting-linear-extensions-below-two-to-the-n", "kind": "verify_now", "reason": "The source labels this result as a candidate; verification should precede reuse.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "The source status is candidate and must not be represented as solved.", "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }