Browse › Mathematical physics › 281 QAOA attains the SK optimum in the thermodynamic-first limit
Full support of the zero-temperature Sherrington-Kirkpatrick order parameter
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We prove that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure, zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on $[0,1)$. Thus its support has no gaps at any overlap scale below one. We use the covariance normalization $\xi(t)=t^2/2$.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- P. M. Aronow, P. Lopatto, An N^{-7/10} Bound for the Finite-Size Bias of the Sherrington-Kirkpatrick Free Energy (hexagon:2610.00177v1, 2026-10-12). Section 1.5 cites the initial release's full-support theorem among four OpenAI release items, while the finite-temperature bias argument uses interval support at fixed inverse temperature greater than one; the abstract notes an improvement over exponent 2/3 that "would follow from the ideas in OpenAI's recently released manuscripts." [AI-generated per source] [cites earlier version]
Formalization
- OpenAI math release, QAOA attains the SK optimum in the thermodynamic-first limit: Lean scope notes (lean/docs/281.md, 2026-10-06). According to these notes, the Lean statement proves full relative Stieltjes support on [0,1) for every admissible integrable minimizer, with ξ(t)=t²/2, together with the diffusion's self-consistency moment identities; it assumes a minimizer exists and does not prove existence. [family-level]
Related prior literature mentioned by others
- Hong-Bin Chen, FRSB in the SK spin glass: convergence to full-interval support at zero temperature (arXiv:2607.18032, 2026-07-20). The math-lineage report identifies this earlier full-support and smooth-density result as a predecessor of the main theorem; it is also acknowledged in the manuscript's introduction and bibliography. [AI-generated per source]
Possibly relevant literature (found by the agent, not checked)
- R. Oppermann, D. Sherrington, Scaling and Renormalization Group in Replica Symmetry Breaking space: Evidence for a simple analytical solution of the SK model at zero temperature (arXiv:cond-mat/0509700, 2005). A physics prediction of the zero-temperature limit of the SK order function; relevant to the paper's corollary that the minimizer has a smooth density, strictly positive on (0,1).
Digests and expositions
- math-lineage, run by Jonathan Shi, Precedence review: Full support of the zero-temperature Sherrington–Kirkpatrick order parameter (GitHub, 2026-10-13). This explicitly unvetted report gives 4 of 7 findings as attribution gaps, naming H.-B. Chen (arXiv:2607.18032) for an overstated-novelty finding and also listing Y. Zhou (arXiv:2504.00269, arXiv:2608.06523), Lopatto (arXiv:2607.11756), Auffinger–Chen–Zeng, El Alaoui–Montanari–Sellke, Chen–Handschy–Lerman and Auffinger–Chen. [AI-generated per source]
Elsewhere
- Hexagon: An $N^{-7/10}$ Bound for the Finite-Size Bias of the Sherrington-Kirkpatrick Free Energy (AI-generated; human understanding: some)
- Emergent Mind: family 281
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026,
author = {{OpenAI}},
title = {{Full support of the zero-temperature Sherrington--Kirkpatrick order parameter}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf}{OAI:Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/281.md. The Lean statements have not been checked by people against the paper's claims.
Family 281: QAOA attains the SK optimum in the thermodynamic-first limit
Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree.
Other papers in this family:
- QAOA attains the SK ground-state energy in the thermodynamic-first limit · September 25, 2026