Full-RSB in the Sherrington–Kirkpatrick spin glass
resolvedconfidence 70%
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
This work proves full replica symmetry breaking for the zero-field Sherrington–Kirkpatrick model at zero temperature $\beta=\infty$: the Parisi minimizer is absolutely continuous, has a smooth density, and has support $[0,1)$, thereby confirming the prediction by Parisi.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
ChatGPT 5.6
The most explicit authorship disclosure in this catalog. The manuscript's opening note says the proof arguments and prose "were generated by the same model from prompts supplied by Hong-Bin Chen", and then states on its own line:
"ChatGPT 5.6 is the author of the manuscript."
It continues: "Since arXiv's policy on generative AI language tools does not permit such a tool to be listed as an author, Hong-Bin Chen is only formally listed as the author for submission purposes and assumes full responsibility for the submitted text. This formal attribution reflects arXiv's policy rather than the division of labor in producing the manuscript: Hong-Bin Chen's role was limited to prompting, editing, proofreading, and verifying the arguments; in particular, he did not construct the proof arguments. He has read and verified the proofs, although errors or oversights may remain."
The Lean development was also written by the same model. AI-discovered is unambiguous here: a human posed the problem and checked the output, and states outright that he did not construct the argument.
Unreviewed, and the tier was lowered from Lean-checked on inspection of the repository - a labelling correction, not a doubt about the mathematics.
What holds: across 75 files and 5,544 lines of Lean 4 there is no $\texttt{sorry}$, no $\texttt{admit}$ and no $\texttt{native\_decide}$, confirmed on 28 August 2026. What does not: the Lean-checked rung requires no stray axioms, and this project declares seven mathematical axioms in $\texttt{ExternalInputs.lean}$. Three cite prior work (Lopatto, Auffinger-Chen, Chen-Handschy-Lerman), but four stand in for the paper's own unformalized analysis - the minimizer itself, and the analytic data behind Propositions 4.1-4.2, 4.3 and 4.4. The repository's own ledger also marks several items "Open analytic", and notes that four terminal approximation modules are "excluded from the root target and omitted from this GitHub bundle".
What the formalization does establish is real and is credited: given that analytic data, the gap exclusions and the smooth-density conclusion are Lean deductions rather than assumptions, as AXIOMS.md is careful to state. The repository is unusually candid - it ships an axiom ledger and a dependency table, and the paper's own footnote says "This is not an assumption-free verification of the entire paper". The project was not built here, for want of a toolchain. The paper is an unrefereed preprint and no human has independently reviewed the analysis.
FRSB IN THE SK SPIN GLASS: CONVERGENCE TO FULL-INTERVAL SUPPORT AT ZERO TEMPERATURE
For the Sherrington-Kirkpatrick model with no external field, at zero temperature $\beta=\infty$, the paper proves that the zero-temperature Parisi minimizer is absolutely continuous with a smooth density and has support $[0,1)$ - full replica symmetry breaking, confirming the Parisi picture at the ground state. It also shows $q_\beta\to1$ as $\beta\to\infty$.
The positive-temperature input is Lopatto's: for every $\beta>1$ the Parisi measure is supported on the closed interval $[0,q_\beta]$, with a smooth density on $[0,q_\beta)$ and a single atom at the right endpoint $q_\beta$. That endpoint atom is what the paper's own quantitative estimates target, so it is not incidental. This entry is cited as Theorem 1.1 rather than reproved.
Remark 1.4 is worth reading beside the support claim: the half-open interval is essential, because the zero-temperature functional cannot see an endpoint atom at all, so there is no canonical mass there to converge to.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
The source did not supply a subject field. VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
The source did not supply a subject field. VibeMath has not independently audited the mathematical statement, proof, or novelty claim.