A quantitative replica-symmetric bound of Sherrington--Kirkpatrick model in the entire de Almeida--Thouless region

We consider the Sherrington–Kirkpatrick model with inverse temperature $\beta >0$ and deterministic external field $h>0$. Let $q$ be the replica-symmetric fixed point: $q=\mathbb{E}{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}}^2(h+\beta \sqrt{q}Z)$, where $Z$ is a standard normal. We prove that, uniformly on compact subsets of the strict de Almeida–Thouless region: ${\beta }^2\mathbb{E}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^4(h+\beta \sqrt{q}Z)<1,$ the overlap satisfies the concentration: $$\mathbb{E}⟨(R_{12}-q{)}^2⟩=O(N^{-1})\mathrm{.}$$ The result of the present paper provides an alternative proof of the replica-symmetric free energy formula in de Almeida–Thouless region recently established by Lopatto [arXiv:2604.11921]. Moreover, our approach yields the stronger conclusion of an explicit quantitative bound. As a consequence, we obtain an $O(N^{-1})$ replica-symmetric free-energy correction and identify the finite-volume replicon susceptibility. Our proof is self-contained and does not use the identification of the limiting free energy with the Parisi variational formula. The main result supersedes the corresponding result in our recent preprint arXiv:2607.23427, extending the replica-symmetric bounds to the strict de Almeida-Thouless region. However, we keep the previous preprint, since its argument is different and substantially simpler than the one given here.