The finite-temperature free energy of the spherical perceptron with bounded continuous potentials (Gyorgyi-Reimann continuous-RSB formula)

Theorem 1.1: for all $\alpha,\beta>0$ and $\phi\in C_b(\mathbb R)$, $\mathbb Ep_N$ and $p_N$ (in probability) converge to $\inf_m\{\alpha V_{\beta\phi}(m)+S(m)\}$ over nondecreasing $m:[0,1]\to[0,1]$, with $V$ a Brownian stochastic-control value and $S$ the spherical entropy; this matches the Gyorgyi-Reimann continuous-RSB functional. Companions: the analogous formula for the Gaussian Ising perceptron with any bounded Borel log-potential at all densities and positive temperatures, and for spherical models with bi-orthogonally invariant disorder without outliers. Not shown: hard constraints or zero temperature (storage capacity), discontinuous potentials in the spherical case, or properties of the optimizer.

Theorem 1.1: for all $\alpha,\beta>0$ and $\phi\in C_b(\mathbb R)$, $\mathbb Ep_N$ and $p_N$ (in probability) converge to $\inf_m\{\alpha V_{\beta\phi}(m)+S(m)\}$ over nondecreasing $m:[0,1]\to[0,1]$, with $V$ a Brownian stochastic-control value and $S$ the spherical entropy; this matches the Gyorgyi-Reimann continuous-RSB functional. Companions: the analogous formula for the Gaussian Ising perceptron with any bounded Borel log-potential at all densities and positive temperatures, and for spherical models with bi-orthogonally invariant disorder without outliers. Not shown: hard constraints or zero temperature (storage capacity), discontinuous potentials in the spherical case, or properties of the optimizer.