Constraint-First Reasoning: A Training-Free Protocol for Exploiting Answer-Space Constraints in Mathematical Problem Solving

CFR operates in two stages: Stage 1 extracts and summarizes constraints from the problem statement, while Stage 2 solves the problem while continuously checking intermediate and final results against those constraints. The approach addresses common failure modes where LLMs generate plausible but invalid outputs, such as omitting modular reductions or returning non-integers. Since it is training-free, it can be applied to any existing LLM immediately, making it a practical enhancement for math-heavy applications like automated theorem proving or quantitative reasoning tasks. The paper validates CFR across mathematical problem sets, showing reduced constraint violations without sacrificing answer accuracy.