The Gardner Transition in the Ising pure $p$ -spin glass

For the Ising pure $p$ -spin glass with $p\ge3$ , Gardner predicted in 1985 that the Parisi measure passes through two transitions as the inverse temperature $\beta$ grows: replica symmetric (RS), then one-step replica symmetry breaking (1-RSB), then full replica symmetry breaking (FRSB). The author's earlier paper established the RS phase for $0<\beta\le\beta_1^p$ and the 1-RSB phase on a nonempty interval immediately above $\beta_1^p$ , leaving the rest of the phase diagram open. This sequel claims the remainder: a unique second critical inverse temperature $\beta_2^p>\beta_1^p$ , with the measure 1-RSB throughout $\beta_1^p<\beta\le\beta_2^p$ , and for $\beta>\beta_2^p$ supported on $\{0\}\cup[\underline q,\overline q]$ with a smooth density on the interior, hence FRSB.

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI co-developed
Method: Argument
Field: Spin glasses; probability
Posed by: Elizabeth Gardner
Year posed: 1985
Years open: 41y
Solved: 2026-08-06
Model: not explicitly stated
Vendor: —
Collaborators: Yuxin Zhou
Verification: Unreviewed
Publication: Preprint
Significance: 28 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

For the Ising pure $p$ -spin glass where $p \geq 3$ , it was predicted by Gardner that there exists critical inverse temperatures $0<\beta_1^p<\beta_2^p <\infty$ such that: (1) When $0<\beta\leq \beta_1^p$ , the Parisi measure is replica symmetric (RS); (2) When $\beta_1^p<\beta \leq \beta_2^p$ , the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When $\beta>\beta_p^2$ , the Parisi measure is full replica symmetry breaking (FRSB). The earlier work by the author solved Part (1) and partially solved Part (2) when $\beta$ is sufficiently close to $\beta_p^1$ , while this work solves Part (2) and Part (3) entirely.

What the AI did

It is stated that the proofs in the appendix were drafted by large language models and have not yet received their final authorial revision. Since the materials in the appendix are the heart of the proof (the 10 page main paper only contains introduction and statement of the result), it is classified as AI-discovered rather than AIco-developed.

Verification

Nobody has checked this, including the author, who says so himself - see the claim-issue note. An arXiv preprint (v1, 6 August 2026, math.PR), unrefereed, with no formalization and no computational certificate, so there is nothing mechanical to check either. Verified here on 25 August 2026: the paper exists at arXiv:2608.06523 with the title, sole author and phase-diagram statement this entry describes; its acknowledgment carries the LLM-drafting disclosure quoted verbatim above; its sequel relationship to the author's arXiv:2408.14630 is as described; and the page structure supports the claim that the appendices carry the substance. No model is named anywhere in the paper, which is why the model field says so rather than guessing.

Claim issue

The author states in the paper's acknowledgments that the appendix proofs "were drafted by large language models and have not yet received their final authorial revision", and that he will "verify, revise, and rewrite these proofs in a subsequent version". Those appendices are where the theorem's weight sits: of 166 pages roughly 11 are main body and 155 are appendices A-G, and the main body defers its key inputs to them explicitly ("Its full proof is included in Appendix B", "Its complete proof is included in Appendix C"). So the load-bearing proofs are, by the author's own account, not yet checked by anyone - not by him, not by a referee, and not by a machine. That is unusually candid and it is why this is filed as a candidate rather than resolved.

Sources

PaperOn the Gardner Transition in the Ising Pure $p$-Spin Glass IIZhou, On the Gardner transition in the Ising pure p-spin glass (paper I, the RS and partial 1-RSB phases)

Submitted by SpryRaven345 on 25 Aug 2026

The Gardner Transition in the Ising pure ppp-spin glass