The limiting free energy of orthogonally invariant Ising spin glasses at every temperature

Theorem 1.1: under the stated spectral assumptions, $\lim\mathbb EP_N=\lim P_N=\inf_p\{S(p)+G_1(p)\}$ almost surely, where $p$ ranges over overlap quantile functions, $S$ is a Parisi-type one-site functional and $G_1$ involves the edge-extended $R$-transform of $\mu$; no independence across $N$ is needed. Extensions: deterministic external fields converging in Wasserstein-1, random spectra, the zero-temperature ground-state energy, and Gaussian Hopfield consequences (limiting max and min of $N^{-1}\|G\sigma\|_2$). Not shown: outliers in the spectrum, explicit identification with the published 1RSB formulas, or the structure (full RSB or not) of the optimizer.

Theorem 1.1: under the stated spectral assumptions, $\lim\mathbb EP_N=\lim P_N=\inf_p\{S(p)+G_1(p)\}$ almost surely, where $p$ ranges over overlap quantile functions, $S$ is a Parisi-type one-site functional and $G_1$ involves the edge-extended $R$-transform of $\mu$; no independence across $N$ is needed. Extensions: deterministic external fields converging in Wasserstein-1, random spectra, the zero-temperature ground-state energy, and Gaussian Hopfield consequences (limiting max and min of $N^{-1}\|G\sigma\|_2$). Not shown: outliers in the spectrum, explicit identification with the published 1RSB formulas, or the structure (full RSB or not) of the optimizer.