Symmetric binary perceptron ultrametric OGP–parametric RDT equivalence conjecture

Let $\mathbf{\mathcal S}(G,\kappa,\alpha)$ be a statistical symmetric binary perceptron, let $\alpha_{ult_s}(\kappa)$ be its $s$-level ultrametric OGP critical density, let $\bar{\alpha}_{ult_s}(\kappa;\mathbf{k})$ be the corresponding union-bound estimate optimized over configurations $\mathbf{k}$, and let $\alpha_c^{(r)}(\kappa)$ be the $r$-th level parametric fl-RDT estimate. Ultrametric OGP–parametric fl-RDT conjecture in the weak sense. One has

$$\lim_{s\rightarrow\infty}\alpha_{ult_s}(\kappa)=\lim_{s\rightarrow\infty}\min_{\mathbf{k}}\bar{\alpha}_{ult_s}(\kappa;\mathbf{k})=\lim_{r\rightarrow\infty}\alpha_c^{(r)}(\kappa).$$

Moreover, the statistical computational gap satisfies

$$\operatorname{SCG}=\alpha_c(\kappa)-\alpha_a(\kappa)=\alpha_c^{(2)}(\kappa)-\lim_{r\rightarrow\infty}\alpha_c^{(r)}(\kappa)=\alpha_c^{(2)}(\kappa)-\lim_{s\rightarrow\infty}\min_{\mathbf{k}}\bar{\alpha}_{ult_s}(\kappa;\mathbf{k})=\alpha_c^{(2)}(\kappa)-\lim_{s\rightarrow\infty}\alpha_{ult_s}(\kappa).$$

Here $\operatorname{SCG}$ is the statistical computational gap. The conjecture identifies the limiting critical density from ultrametric OGP, its union-bound analysis, and parametric fl-RDT, and further identifies it with the algorithmic threshold. Numerical evidence supports the analogy at the first few levels, but the limiting equalities remain open because higher-level evaluations are computationally difficult.