{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture", "title": "Talagrand's critical Sherrington-Kirkpatrick overlap conjecture", "canonical_statement": "At the critical inverse temperature $\\beta=1$ in the Sherrington-Kirkpatrick spin glass model, Talagrand conjectured that the expected squared overlap of two independent Gibbs replicas has an exact $N^{-2/3}$ scaling: there exists a constant $a>0$ such that\n\n$$\n\\lim_{N\\to\\infty} N^{2/3}\\mathbb{E}\\langle R_{1,2}^2\\rangle=a.\n$$\n\nDu and Huang prove that this limit exists and is positive and finite. More strongly, they determine the full limiting quenched distribution of the rescaled overlap $N^{1/3}R_{1,2}$ in terms of the reflected $\\mathrm{Airy}_1$ point process.", "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:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture:event", "problem_id": "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture", "type": "proved", "status": "resolved", "occurred_at": "2026-08-09T00:00:00.000Z", "title": "Overlap distribution of the critical Sherrington-Kirkpatrick model", "summary": "Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: $N^{2/3}\\mathbb{E}\\langle R_{1,2}^2\\rangle$ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of $N^{1/3}R_{1,2}$ converges to an explicit random probability measure defined from the reflected $\\mathrm{Airy}_1$ point process. The same limiting distribution and second-moment constant are obtained for the spherical SK model. This does not resolve the broader low-temperature overlap structure of the SK model.", "ai_contribution": "ai_co_developed", "attempt_ids": [ "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture:attempt" ], "method_family_ids": [ "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture:method" ], "source_assertion_ids": [ "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture:assertion" ] } ], "attempts": [ { "id": "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture:attempt", "problem_id": "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture", "model": "GPT-5.6 Pro", "provider": "OpenAI", "attempted_at": "2026-08-09T00:00:00.000Z", "human_collaborators": [ "Hang Du", "Brice Huang" ], "exposure": "unknown", "prompt_public": null, "outcome": "The authors state that most of the arguments in the paper were generated using GPT-5.6 Pro, with the goal of exploring further consequences of ideas developed in their companion work on critical SK free-energy fluctuations. The paper does not assign individual lemmas or the central comparison principle specifically to the model, so the contribution is classified conservatively as AI co-developed rather than AI discovered.", "failed_routes": [], "artifacts": [ { "label": "Dey and Kang (arXiv:2603.05636) - quotes Conjecture 11.7.5 as open, five months before this proof", "url": "https://arxiv.org/abs/2603.05636", "kind": "problem-record", "license": null }, { "label": "Du and Huang's companion paper on critical SK free-energy fluctuations, whose ideas this work extends", "url": "https://arxiv.org/abs/2607.02172", "kind": "paper", "license": null } ], "sources": [ { "label": "Overlap distribution of the critical Sherrington-Kirkpatrick model", "url": "https://arxiv.org/abs/2608.08752", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Talagrand's critical Sherrington-Kirkpatrick overlap conjecture", "url": "https://vibemathed.com/problem/talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture:verification", "problem_id": "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture", "solution_event_id": "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture: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 14 August 2026 against the paper's LaTeX source and the prior-art source, both fetched and read. Confirmed verbatim in the paper: \"The following corollary affirmatively resolves [Conjecture 11.7.5]\", attached to the corollary giving $\\lim N^{2/3}\\mathbb{E}\\langle R_{1,2}^2\\rangle$ as the mean of an explicit positive random variable built from the reflected $Airy_1$ process. The prior state is independently pinned: Dey and Kang (arXiv:2603.05636, March 2026) quote Conjecture 11.7.5 with exactly the statement submitted here and treat it as open. The AI disclosure was read in full: most formal arguments were initially generated by GPT-5.6 Pro, the principal human inputs are named (identification of the limiting objects, formulation of the sphere-to-cube comparison theorem, and the overall proof strategy), and the technical body separately credits one step to the model by name - the Hilbert-space kernel argument upgrading the Laplace-transform comparison is introduced as \"an idea due to GPT-5.6 Pro\". What was NOT checked here is the mathematics itself: a 36-page argument through the $Airy_1$ scaling limit at the GOE spectral edge needs a spin-glass or random-matrix specialist. The manuscript is five days old, unrefereed, and the authors taking responsibility for correctness is author-side checking, so the tier stays Unreviewed.", "sources": [ { "label": "VibeMathed: Talagrand's critical Sherrington-Kirkpatrick overlap conjecture", "url": "https://vibemathed.com/problem/talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Overlap distribution of the critical Sherrington-Kirkpatrick model", "url": "https://arxiv.org/abs/2608.08752", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Overlap distribution of the critical Sherrington-Kirkpatrick model", "url": "https://arxiv.org/abs/2608.08752", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Talagrand's critical Sherrington-Kirkpatrick overlap conjecture", "url": "https://vibemathed.com/problem/talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture:signal:watch_only", "problem_id": "vibemathed:talagrand-s-critical-sherrington-kirkpatrick-overlap-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" }