In [97,99,100], an fl-RDT framework is introduced to characterize \emph{statistical computational gaps} (SCGs). Studying \emph{symmetric binary perceptrons} (SBPs), [100] obtained an \emph{algorithmic} threshold estimate $α_a\approx α_c^{(7)}\approx 1.6093$ at the 7th lifting level (for $κ=1$ margin), closely approaching $1.58$ local entropy (LE) prediction [18]. In this paper, we further connect parametric RDT to overlap gap properties (OGPs), another key geometric feature of the solution space. Specifically, for any positive integer $s$, we consider $s$-level ultrametric OGPs ($ult_s$-OGPs) and rigorously upper-bound the associated constraint densities $α_{ult_s}$. To achieve this, we develop an analytical union-bounding program consisting of combinatorial and probabilistic components. By casting the combinatorial part as a convex problem and the probabilistic part as a nested integration, we conduct numerical evaluations and obtain that the tightest bounds at the first two levels, $\barα_{ult_1} \approx 1.6578$ and $\barα_{ult_2} \approx 1.6219$, closely approach the 3rd and 4th lifting level parametric RDT estimates, $α_c^{(3)} \approx 1.6576$ and $α_c^{(4)} \approx 1.6218$. We also observe excellent agreement across other key parameters, including overlap values and the relative sizes of ultrametric clusters. Based on these observations, we propose several conjectures linking $ult$-OGP and parametric RDT. Specifically, we conjecture that algorithmic threshold $α_a=\lim_{s\rightarrow\infty} α_{ult_s} = \lim_{s\rightarrow\infty} \barα{ult_s} = \lim_{r\rightarrow\infty} α_{c}^{(r)}$, and $α_{ult_s} \leq α_{c}^{(s+2)}$ (with possible equality for some (maybe even all) $s$). Finally, we discuss the potential existence of a full isomorphism connecting all key parameters of $ult$-OGP and parametric RDT.

翻译：文献[97,99,100]引入fl-RDT框架描述统计计算差距（SCGs）。通过研究对称二进制感知器（SBPs），[100]在第7提升层级（$κ=1$裕度）获得算法阈值估计$α_a≈α_c^{(7)}\approx 1.6093$，接近$1.58$局部熵（LE）预测[18]。本文进一步将参数化RDT与重叠间隙特性（OGPs）——解空间的另一关键几何特征——建立关联。具体而言，对任意正整数$s$，我们考虑$s$级超度量OGPs（$ult_s$-OGPs）并严格界定关联约束密度$α_{ult_s}$的上界。为此，我们开发了由组合与概率两部分组成的解析联合界化方案：将组合部分转化为凸问题，概率部分转化为嵌套积分，并通过数值计算获得前两级最紧界限$\barα_{ult_1} \approx 1.6578$和$\barα_{ult_2} \approx 1.6219$，分别逼近第3、第4提升层级的参数化RDT估计值$α_c^{(3)} \approx 1.6576$和$α_c^{(4)} \approx 1.6218$。我们还观察到其他关键参数（包括重叠值和超度量簇相对大小）的高度吻合。基于这些发现，我们提出若干关于$ult$-OGP与参数化RDT关联的猜想，具体推测算法阈值$α_a=\lim_{s\rightarrow\infty} α_{ult_s} = \lim_{s\rightarrow\infty} \barα{ult_s} = \lim_{r\rightarrow\infty} α_{c}^{(r)}$，且$α_{ult_s} \leq α_{c}^{(s+2)}$（可能对某些甚至所有$s$取等）。最后，我们讨论$ult$-OGP与参数化RDT所有关键参数间可能存在完全同构关系的潜在可能性。