We study the statistical behaviour of reasoning probes in a stylized model of looped reasoning, given by Boolean circuits whose computational graph is a perfect $ν$-ary tree ($ν\ge 2$) and whose output is appended to the input and fed back iteratively for subsequent computation rounds. A reasoning probe has access to a sampled subset of internal computation nodes, possibly without covering the entire graph, and seeks to infer which $ν$-ary Boolean gate is executed at each queried node, representing uncertainty via a probability distribution over a fixed collection of $\mathtt{m}$ admissible $ν$-ary gates. This partial observability induces a generalization problem, which we analyze in a realizable, transductive setting. We show that, when the reasoning probe is parameterized by a graph convolutional network (GCN)-based hypothesis class and queries $N$ nodes, the worst-case generalization error attains the optimal rate $\mathcal{O}(\sqrt{\log(2/δ)}/\sqrt{N})$ with probability at least $1-δ$, for $δ\in (0,1)$. Our analysis combines snowflake metric embedding techniques with tools from statistical optimal transport. A key insight is that this optimal rate is achievable independently of graph size, owing to the existence of a low-distortion one-dimensional snowflake embedding of the induced graph metric. As a consequence, our results provide a sharp characterization of how structural properties of the computational graph govern the statistical efficiency of reasoning under partial access.

翻译：本研究探讨循环推理模型（由布尔电路实现，其计算图为完美ν元树（ν≥2），且输出结果被附加至输入并迭代反馈至后续计算轮次）中推理探针的统计行为。推理探针可访问内部计算节点的抽样子集（可能未覆盖整个图结构），旨在推断每个被查询节点所执行的ν元布尔门类型，并通过在固定集合的m个可容许ν元门上的概率分布来表示不确定性。这种部分可观测性引出了泛化问题，我们在可实现、转导的学习框架下对其进行分析。研究表明：当推理探针由基于图卷积网络（GCN）的假设类参数化并查询N个节点时，最坏情况泛化误差以至少1-δ的概率达到最优速率\(\mathcal{O}(\sqrt{\log(2/δ)}/\sqrt{N})\)（其中δ∈(0,1)）。我们的分析结合了雪花度量嵌入技术与统计最优传输的理论工具。关键发现在于：由于诱导图度量存在低失真的一维雪花嵌入，该最优速率可独立于图规模实现。因此，本研究结果精确刻画了计算图的结构特性如何制约部分可访问条件下推理过程的统计效率。