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.

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. 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.