SAGE: Mitigating Long-Horizon Reasoning Biases via Topological Guidance

Long-horizon reasoning remains a central challenge for large language models (LLMs) under sparse-reward regimes. We argue that this brittleness arises from two biases induced by complex reasoning spaces: an exploration bias, where models are drawn toward locally plausible but structurally unstable branches, and a compounding bias, where small local deviations accumulate across depth and suppress rare rewards. We introduce Symbolic Closure Analysis (SCA) as a theoretical lens characterizing how branching structures and sparse rewards induce these biases in long-horizon reasoning with local admissibility, and as a design principle for structural priors in less formal reasoning tasks. Motivated by this analysis, we propose \model{} (\textbf{S}tructural \textbf{A}dmissibility-\textbf{G}uided \textbf{E}xploration), a unified framework that injects structural guidance to alleviate exploration bias and compounding bias in long-horizon reasoning. \model{} combines two complementary structural dimensions: algebraic sparsification, which projects locally admissible candidates onto operator-indexed algebraic subspaces to suppress spurious branching and mitigate exploration bias, and hyperbolic structural guidance, which embeds reasoning states into a negatively curved space to provide dense depth-wise signals and mitigate compounding bias. Across 12 benchmarks and 7 model families, \model{} outperforms competitive baselines. In particular, \model{} achieves up to an 8-fold improvement on the Andrews-Curtis problem, an open real-world long-horizon task. Code is available at: \url{https://anonymous.4open.science/r/SAGE-Long-Horizon-Reasoning-AD70}.