{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:litvak-gaussian-minima-conjecture", "title": "Litvak's Conjecture on Gaussian Minima", "canonical_statement": "Litvak conjectured in 2018 that for every $p > 0$ the quantity $\\mathbb{E}[\\min_{i \\le n} |g_i|^p]$, for $g \\sim \\mathcal{N}(0,\\Sigma)$, is minimized over $n \\times n$ correlation matrices by the Gram matrix of the regular simplex in $\\mathbb{R}^{n-1}$. False: the matrix $\\Sigma^{\\cos}_{ij} = \\cos(\\pi(i-j)/n)$ already gives a strictly smaller value at $p = 2$, $n = 4$.", "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "resolved", "solution_events": [ { "id": "vibemathed:litvak-gaussian-minima-conjecture:event", "problem_id": "vibemathed:litvak-gaussian-minima-conjecture", "type": "disproved", "status": "resolved", "occurred_at": "2026-05-03T00:00:00.000Z", "title": "arXiv:2605.02023 - A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture", "summary": "the paper proposes that the cosine matrix is the true minimizer for all p and n, and proves a stronger stochastic domination statement conditional on a new volumetric extension of Fejes Toth's zone conjecture", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:litvak-gaussian-minima-conjecture:attempt" ], "method_family_ids": [ "vibemathed:litvak-gaussian-minima-conjecture:method" ], "source_assertion_ids": [ "vibemathed:litvak-gaussian-minima-conjecture:assertion" ] } ], "attempts": [ { "id": "vibemathed:litvak-gaussian-minima-conjecture:attempt", "problem_id": "vibemathed:litvak-gaussian-minima-conjecture", "model": "AlphaEvolve, GPT-5.5 Pro", "provider": null, "attempted_at": "2026-05-03T00:00:00.000Z", "human_collaborators": [ "Dmitriy Kunisky" ], "exposure": "unknown", "prompt_public": null, "outcome": "Two separate model contributions, and the author keeps them apart. The counterexample itself came out of black-box minimization with AlphaEvolve; what made it usable was the author then recognising a continuous function underneath the numbers and identifying it as the cosine, which turned a numerical optimum into a clean matrix and then into a stronger conjecture. He notes conventional optimizers such as differential evolution and BFGS also produced matrices that would disprove the conjecture, but rarely converged to this one. Separately, GPT-5.5 Pro surfaced the connection to Fejes Toth's zone conjecture, which the author had not known about, while repeatedly producing proofs that leaned on an unsupported step as though it were established. Isolating that step is what produced the volumetric conjecture stated in the paper, so the model's mistake was itself informative.", "failed_routes": [], "artifacts": [], "sources": [ { "label": "arXiv:2605.02023 - A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture", "url": "https://arxiv.org/abs/2605.02023", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Litvak's Conjecture on Gaussian Minima", "url": "https://vibemathed.com/problem/litvak-gaussian-minima-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:litvak-gaussian-minima-conjecture:verification", "problem_id": "vibemathed:litvak-gaussian-minima-conjecture", "solution_event_id": "vibemathed:litvak-gaussian-minima-conjecture:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Reproduced here. For $\\Sigma^{\\cos}$ with $n = 4$ the matrix has rank two, so $g_i = R\\cos(\\Theta - \\pi i/4)$ with $R^2 \\sim \\chi^2_2$, giving the closed form $\\mathbb{E}[\\min_i |g_i|^2] = 1 - 2\\sqrt{2}/\\pi = 0.0996836838$. The regular-simplex Gram matrix was evaluated by deterministic quadrature over the sphere, stable at $0.1421833$ across grid refinements and corroborated by a 20-million-sample simulation at $0.142238$. The cosine matrix is smaller by about 30 percent, far outside any numerical doubt. arXiv preprint, not peer-reviewed.", "sources": [ { "label": "VibeMathed: Litvak's Conjecture on Gaussian Minima", "url": "https://vibemathed.com/problem/litvak-gaussian-minima-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "arXiv:2605.02023 - A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture", "url": "https://arxiv.org/abs/2605.02023", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "arXiv:2605.02023 - A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture", "url": "https://arxiv.org/abs/2605.02023", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Litvak's Conjecture on Gaussian Minima", "url": "https://vibemathed.com/problem/litvak-gaussian-minima-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:litvak-gaussian-minima-conjecture:signal:watch_only", "problem_id": "vibemathed:litvak-gaussian-minima-conjecture", "kind": "watch_only", "reason": "Current public evidence does not meet the default replay threshold.", "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" }