Full support of the zero-temperature SK Parisi minimizer

For the pure zero-field Sherrington–Kirkpatrick model with covariance ξ(t)=t²/2, does every admissible integrable zero-temperature Parisi minimizer have full relative Stieltjes support on [0,1)?

Why it matters and what counts as progress

Why it matters: Full support excludes gaps, strengthening the earlier conclusion that there are infinitely many support points.

What counts as progress: A proof ruling out every constant interval for the minimizer, an independent method for the same full-support theorem, or a valid gap example; infinitely many support points alone do not suffice.

How the problem developed

20 Mar 2017 · Partial result [1]
Antonio Auffinger, Wei-Kuo Chen, and Qiang Zeng Proved that the zero-temperature mixed p-spin Parisi measure has infinitely many support points, ruling out a step-function order parameter; this did not show full-interval support for the SK model.
16 Jul 2026 · Reported claim [2]
ChatGPT 5.6 The aiXiv preprint claimed full support after closure at 1 for the zero-temperature SK Parisi minimizer and convergence of finite-temperature supports to the full interval.
20 Jul 2026 · Reported claim [3]
ChatGPT 5.6, with Hong-Bin Chen taking responsibility for the arXiv submission The arXiv successor claimed an absolutely continuous zero-field SK minimizer with smooth density and support [0,1); it reports a conditional Lean formalization from seven analytic inputs.
27 Sept 2026 · Reported claim [4]
OpenAI A separate OpenAI manuscript reported full support for the zero-temperature SK order parameter.

AI contribution

Full support of the zero-temperature Sherrington-Kirkpatrick order parameter

OpenAI

The September manuscript proves that every admissible integrable zero-temperature Parisi minimizer for the pure zero-field Sherrington–Kirkpatrick model, with covariance ξ(t)=t²/2, has full relative Stieltjes support on [0,1). It uses a rational coercive identity and strict curvature on constant intervals.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4 (source-reported))

Human understandingFew