The Constructive Cavity Method for the Ising Perceptron

We apply the Constructive Cavity Method (CCM) to the Ising perceptron. The process starts from positive soft channels admitting an auxiliary representation $$ U(x)=\int_{\mathbb{R}} e^{tx}ρ(\mathrm{d}t),\qquad ρ\ge0, $$ with an auxiliary measure having a finite exponential quadratic moment. This rewrites the perceptron as a dense, bipartite Gaussian model with an Ising layer and a continuous auxiliary layer. The CCM blanket geometry then gives two constructive variance profiles, $Q_\ell$ on the Ising side and $Γ_\ell$ on the auxiliary side. The bipartite shell scale is $$ Q_{\ell+1}Γ_{\ell+1}-Q_\ellΓ_\ell, $$ and Gaussian core comparison removes the core contribution $ΔQ_\ellΔΓ_\ell$ under blanket refinement. The scalar cavity functional has three terms, $$ \mathcal{A}_{\mathrm{perc}}(ν)=\mathbb{E}\left[\log\langle2\cosh x\rangle_ν+α\log\langle U(s)\rangle_ν-\log\langle e^y\rangle_ν\right], $$ where $x$ is the new-spin field, $s$ is the new-constraint field, and $y$ is the cavity correction. CCM identifies the constructive cavity variables with the Aizenman--Sims--Starr random overlap structure (ROSt) variables. Parisi's recursive averaging then gives $$ \mathcal{G}_{\mathrm{perc}}^{\mathrm{CCM}}(ζ,Q,Γ)=X_0(0)+αY_0(0)-\frac{1}{2}\sum_{\ell=0}^{R}ζ_\ell\left(Q_{\ell+1}Γ_{\ell+1}-Q_\ellΓ_\ell\right). $$ The replica-symmetric specialization gives Gardner's formula for the generalized Ising perceptron. Specializing the constraint terminal to the hard channel gives the Krauth--Mézard functional for the binary perceptron.