评估大模型推理中各阶段是否值得投入成本，提出可量化的方法。
Constrained Path Reasoning: Measuring When Committed Stages Earn Their Cost
- 用约束路径推理匹配路径假设与阶段开销，判断中间步骤价值。
- 实验显示仅用17.7%尝试量就恢复63.0%的额外可行解产出。
- 适合关注推理效率、成本控制或验证机制的研究者参考。
大模型推理流程中，何时一个已承诺的中间阶段才值得其代价？约束路径推理（CPR）结合源感知路径假设与阶段级成本核算。搜索生成临时状态；可信或验证过的不变量可施加硬约束，其余则保持软性且可修改。CPR预测：任务兼容的承诺能促成状态转移、集中候选集、产生规律性，并在收益超过传播误差与执行成本时暴露反馈信号。该形式化涵盖离散承诺与连续流，度量有效分支、终点集中度及每可用输出的成本。在1,180个生成的QCQPs和40个设计退化的多项式实例（共2,140个终点）上，残差筛选以17.7%的尝试量恢复了63.0%的修复全方案额外可行产出。固定LLM核算（270次唯一调用共享于嵌套分支）显示：直接可用率为41.1%，形式化与确定性执行后升至90.0%，一次凸化后为20.0%，完整路径为21.1%。在120对条件调用中，两动作回滚规则达到90%可用率，远超反馈条件选择器的36.7%。两个终点探针区分来源与验证：72输出跨轨迹移植降低熵与可接受质量；24输出同调用自提案试点显示重复碰撞熵不变，可用率25.0%对比8.3%，且1/8终点检查可确定性确认。
原文摘要 · Abstract (English)
When does a committed intermediate stage in an LLM reasoning pipeline earn its cost? Constrained Path Reasoning (CPR) pairs a source-aware path hypothesis with stage-level accounting. Search generates provisional states; trusted or validated invariants can constrain hard, while other proposals remain soft and revisable. CPR predicts that task-compatible commitments can factor transitions, concentrate candidate mass, induce regularity, and expose feedback when their gains exceed propagated error and execution cost. The formalism covers discrete commitments and continuous flows and measures effective branching, endpoint concentration, and cost per usable output. Across 1,180 generated QCQPs and 40 engineered degenerate polynomial instances (2,140 endpoints), residual triage recovers 63.0% of repair-all's additional feasible yield with 17.7% of its attempts. Fixed-LLM accounting (270 unique calls shared across nested arms) finds usable yield of 41.1% direct, 90.0% after formalization and deterministic execution, 20.0% after one-shot convexification, and 21.1% for the full path. In 120 paired-condition calls, a two-action rollback rule reaches 90% usable yield versus 36.7% for the feedback-conditioned selector. Two endpoint probes separate source from validation: a 72-output cross-trajectory transplant reduces entropy and acceptable mass; a 24-output same-call self-proposal pilot gives unchanged two-repeat collision entropy, 25.0% versus 8.3% usable yield, and 1/8 deterministically confirmed endpoint checks. Model-generated states supply hypotheses; trusted execution earns constraint strength.
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