SAGE combines algebraic sparsification and hyperbolic structural guidance to curb long-horizon reasoning biases, beating baselines across 12 benchmarks and 7 model families with up to 8-fold gains on Andrews-Curtis
The work introduces Symbolic Closure Analysis (SCA) to characterize exploration bias and compounding bias in long-horizon reasoning, and builds SAGE, a framework that injects structural priors during post-training via algebraic sparsification and hyperbolic structural guidance, outperforming SFT, GRPO, EMPO, and GRPO-PRM across 12 benchmarks and 7 model families and achieving up to an 8-fold improvement in Lean-verified proofs on the open Andrews-Curtis task.