# VERIFY-RL: Verifiable Recursive Decomposition for Reinforcement Learning in Mathematical Reasoning

arXiv: https://arxiv.org/abs/2602.07559 · HTML: https://mlsanity.com/paper/2602.07559/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 25 of its 36 references that are in the map.
- First posted on arXiv 7 Feb 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 25 of 25)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Llemma: An Open Language Model For Mathematics](https://mlsanity.com/paper/2310.10631/) (2023)
- [Beyond Human Data: Scaling Self-Training for Problem-Solving with Language Models](https://mlsanity.com/paper/2312.06585/) (2023)
- [ReST-MCTS*: LLM Self-Training via Process Reward Guided Tree Search](https://mlsanity.com/paper/2406.03816/) (2024)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [DeepSeek-Prover: Advancing Theorem Proving in LLMs through Large-Scale Synthetic Data](https://mlsanity.com/paper/2405.14333/) (2024)
- [V-STaR: Training Verifiers for Self-Taught Reasoners](https://mlsanity.com/paper/2402.06457/) (2024)
- [VinePPO: Refining Credit Assignment in RL Training of LLMs](https://mlsanity.com/paper/2410.01679/) (2024)
- [MathCoder: Seamless Code Integration in LLMs for Enhanced Mathematical Reasoning](https://mlsanity.com/paper/2310.03731/) (2023)
- [Curriculum Reinforcement Learning from Easy to Hard Tasks Improves LLM Reasoning](https://mlsanity.com/paper/2506.06632/) (2025)
- [GRPO-LEAD: A Difficulty-Aware Reinforcement Learning Approach for Concise Mathematical Reasoning in Language Models](https://mlsanity.com/paper/2504.09696/) (2025)
- [Lean Copilot: Large Language Models as Copilots for Theorem Proving in Lean](https://mlsanity.com/paper/2404.12534/) (2024)
- [LADDER: Self-Improving LLMs Through Recursive Problem Decomposition](https://mlsanity.com/paper/2503.00735/) (2025)
- [Recursive Decomposition of Logical Thoughts: Framework for Superior Reasoning and Knowledge Propagation in Large Language Models](https://mlsanity.com/paper/2501.02026/) (2025)
- [Complexity Agnostic Recursive Decomposition of Thoughts](https://mlsanity.com/paper/2601.04210/) (2025)

## Abstract (arXiv)

Training language models to solve complex mathematical problems benefits from curriculum learning progressively training on simpler subproblems. However, existing decomposition methods are often heuristic, offering no guarantees that subproblems are simpler, that solving them aids the parent task, or that their relationships are mathematically grounded. We observe that symbolic differentiation provides a natural structure for verified decomposition: calculus rules explicitly define how expressions reduce to simpler components with provable properties. We introduce Verify-RL, a framework where every parent-child decomposition satisfies three verifiable conditions: strictly decreasing structural complexity, solution containment, and formal rule derivation. Unlike heuristic methods where a significant fraction of decompositions are invalid our properties admit automatic verification through symbolic computation, achieving "verification by construction" Experiments demonstrate that eliminating invalid decompositions yields sizable gains, accuracy on the hardest problems more than doubles from 32% to 68%, with a 40% relative improvement overall.