Constraint-First Reasoning: A Training-Free Protocol for Exploiting Answer-Space Constraints in Mathematical Problem Solving
Hongbo Ma, Bangji Yang, Yunqian Selina Cheng, Jiajun Fan, Hanwen Zhang, Ge Liu
cs.CL, cs.SC
Submitted: 2026-08-05
Comments: 53 pages, 5 figures, 36 tables
License: http://creativecommons.org/licenses/by/4.0/
The gist: Large language models can derive a plausible mathematical object yet still violate explicit requirements--for example, by omitting a modular reduction, returning a non-integer, or using the wrong
Terminology
Abstract
Large language models can derive a plausible mathematical object yet still violate explicit requirements--for example, by omitting a modular reduction, returning a non-integer, or using the wrong encoded answer form. We introduce Constraint-First Reasoning (CFR), a training-free two-stage prompting protocol: Stage 1 extracts and summarizes constraints entailed by the problem, and Stage 2 solves while checking intermediate and final results against that summary. Routed-CFR activates the two-stage protocol only when a text-only regex router detects restrictive cues; otherwise it uses direct chain-of-thought (CoT). Across AIME, CMIMC, BRUMO, and AIMO AMC, the method improves direct CoT on multiple backbones. We further report convention-controlled routing experiments, matched prompting baselines, problem-level paired tests, decoding robustness, constraint-quality audits, total-token accounting, and an OlympiadBench evaluation. These analyses position CFR as a targeted test-time intervention whose benefit depends on recoverable constraints and reliable Stage 1 extraction, rather than as a general-purpose replacement for mathematical reasoning.
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