The annealed complexity formula for TAP critical points

Let $F_{\mathrm{TAP}}$ be the TAP free energy, and for $f\in\mathbb{R}$ and $\varepsilon>0$ let $\mathcal{N}_\varepsilon(f)$ denote the number of its critical points in the free-energy window $N(f-\varepsilon,f+\varepsilon)$. Define

$$\Lambda(\theta)=\theta\inf_{\zeta:\zeta(\{0\})=\theta}\mathcal{P}\mathrm{arisi}(\zeta).$$

The annealed complexity formula. There exists an open subset $D\subset\mathbb{R}$ containing $[\inf_\zeta\mathcal{P}\mathrm{arisi}(\zeta),+\infty)$ such that, for every $f\in D$,

$$\lim_{\varepsilon\to 0}\lim_{N\to\infty}\frac{1}{N}\log\mathbb{E}\bigl[\mathcal{N}_\varepsilon(f)\bigr]=-\Lambda^*(f)=\inf_\theta\bigl(\Lambda(\theta)-\theta f\bigr).$$

This conjecture asserts that the annealed complexity is governed by the Legendre transform of the Parisi variational formula. The preceding theorem gives a rigorous lower bound under tail-optimality and stationarity assumptions; the equality remains conjectural.