Talagrand's critical Sherrington-Kirkpatrick overlap conjecture

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 9, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-co-developed. Imported under CC BY 4.0.

Canonical aliases: Talagrand's critical Sherrington-Kirkpatrick overlap conjecture · Critical SK overlap

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

Hang Du
human · human collaborator

Brice Huang
human · human collaborator

GPT-5.6 Pro
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Talagrand's critical Sherrington-Kirkpatrick overlap conjecture

specialist registry · passed

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.

Source ↗

Talagrand's critical Sherrington-Kirkpatrick overlap conjecture

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Dey and Kang (arXiv:2603.05636) - quotes Conjecture 11.7.5 as open, five months before this proof

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Du and Huang's companion paper on critical SK free-energy fluctuations, whose ideas this work extends

paper · pending

paper link supplied by the VibeMathed registry.