{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:improving-randomized-metric-distortion-to-2-3282", "title": "Improving Randomized Metric Distortion to 2.3282", "canonical_statement": "In metric social choice, voters rank candidates by distance in an unknown metric space, while a randomized voting rule must use only these rankings. The paper introduces random-size stable lotteries and proves that, by mixing a suitably chosen random-size stable lottery with Integrated Veto, one obtains a randomized voting rule with metric distortion at most $11641/5000=2.3282$. This improves the previous best upper bound of $2.5$. The proof combines infinite-dimensional conic linear-programming duality, heuristic nonlinear optimization, and exact rational verification using polynomial nonnegativity in the Bernstein basis.", "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "partial", "solution_events": [ { "id": "vibemathed:improving-randomized-metric-distortion-to-2-3282:event", "problem_id": "vibemathed:improving-randomized-metric-distortion-to-2-3282", "type": "proved", "status": "partial", "occurred_at": "2026-08-29T00:00:00.000Z", "title": "arXiv", "summary": "The paper proves that there exists a randomized voting rule using only ordinal rankings with metric distortion at most $11641/5000=2.3282$. This improves the previous best upper bound of $2.5$ and closes about $44\\%$ of the gap to the known asymptotic lower bound of approximately $2.1126$. It does not determine the optimal randomized metric distortion: for $m\\ge4$, the exact optimum and its asymptotic limit remain open.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:improving-randomized-metric-distortion-to-2-3282:attempt" ], "method_family_ids": [ "vibemathed:improving-randomized-metric-distortion-to-2-3282:method" ], "source_assertion_ids": [ "vibemathed:improving-randomized-metric-distortion-to-2-3282:assertion" ] } ], "attempts": [ { "id": "vibemathed:improving-randomized-metric-distortion-to-2-3282:attempt", "problem_id": "vibemathed:improving-randomized-metric-distortion-to-2-3282", "model": "GPT-5.6 Sol; Claude Opus 5.0", "provider": "OpenAI; Anthropic", "attempted_at": "2026-08-29T00:00:00.000Z", "human_collaborators": [ "Nisarg Shah" ], "exposure": "unknown", "prompt_public": null, "outcome": "GPT-5.6 Sol derived all mathematical proofs in the paper from research directions, literature connections, proof and search strategies, and inspiration supplied by Nisarg Shah. It autonomously introduced stable-lottery ingredients, developed progressively stronger bounds, and derived the proofs leading to $2.3282$. Shah then generalized one proposed lottery to random-size stable lotteries, guided the search over distributions, verified all final mathematical details, and rewrote and simplified the exposition with GPT-5.6 Sol and Claude Opus 5.\n\nThe disclosure is in the paper itself, not only in this entry: \"All the proofs in this document were obtained using GPT-5.6-Sol with guidance from the author.\"", "failed_routes": [], "artifacts": [], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.29308", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Improving Randomized Metric Distortion to 2.3282", "url": "https://vibemathed.com/problem/improving-randomized-metric-distortion-to-2-3282", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:improving-randomized-metric-distortion-to-2-3282:verification", "problem_id": "vibemathed:improving-randomized-metric-distortion-to-2-3282", "solution_event_id": "vibemathed:improving-randomized-metric-distortion-to-2-3282:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Unreviewed. The author states that he verified all final mathematical details; by this site's ladder an author's own check does not move the tier, however expert, and Shah is among the leading researchers on metric distortion. The bound $11641/5000$ rests, per the abstract, on an exact rational verification via polynomial nonnegativity in the Bernstein basis, which is checkable in principle but has not been re-run here. No referee and no formalization.", "sources": [ { "label": "VibeMathed: Improving Randomized Metric Distortion to 2.3282", "url": "https://vibemathed.com/problem/improving-randomized-metric-distortion-to-2-3282", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "arXiv", "url": "https://arxiv.org/abs/2608.29308", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.29308", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Improving Randomized Metric Distortion to 2.3282", "url": "https://vibemathed.com/problem/improving-randomized-metric-distortion-to-2-3282", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "expand", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:improving-randomized-metric-distortion-to-2-3282:signal:recent_partial_result", "problem_id": "vibemathed:improving-randomized-metric-distortion-to-2-3282", "kind": "recent_partial_result", "reason": "A source-reported partial result may support bounded expansion.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "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" }