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REVIEW 4 major objections 5 minor 44 references

Code-enabled agents can evolve existing math problems into new, solvable ones that are harder for current reasoning models—solve rates drop up to 32 points even for the strongest tested solver.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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Code agents with test-time exploration and dual verification can synthesize solvable math problems that are harder for LLM solvers than their seed problems.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection Worth a serious look; the code-driven evolution framework is new and mostly sound, but the headline difficulty claim has a baseline mismatch that needs fixing. the 4 major comments →

arxiv 2603.03202 v3 pith:VHTJZATC submitted 2026-03-03 cs.CL

Code2Math: Can Your Code Agent Effectively Evolve Math Problems Through Exploration?

Dadi Guo , Yuejin Xie , Qingyu Liu , Weixian Huang , Jiayu Liu , Zhiyuan Fan , Qihan Ren , Shuai Shao
show 6 more authors
Tianyi Zhou Jianjie Feng Wenze Su Yujiu Yang Dongrui Liu Yi R. Fung
This is my paper
classification cs.CL
keywords code agentsmath problem generationtest-time explorationsolvability verificationdifficulty escalationLLM reasoningmulti-agent systemproblem evolution
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether large language models, given a Python sandbox and test-time exploration, can turn existing math problems into new problems that are both solvable and genuinely harder. The authors build a three-agent pipeline: an evolution agent that rewrites a seed problem while aiming to hide the key insight, a solvability verifier that audits proposed solutions for logical flaws, and a difficulty verifier that rejects merely tedious increases. Across five evolver models and six solver models, evolved problems lower solve rates relative to seeds for nearly every pair, with the strongest tested solver falling from 70% to 61% and another dropping 32 points, while an external judge certifies most internally accepted problems as solvable. The paper's point is that executable exploration lets a code agent discover structural variants—new constraints, extremal targets, hidden patterns—rather than only surface paraphrase. A sympathetic reader should care because automatic, scalable generation of hard, valid math problems would relieve a bottleneck for training and evaluating reasoning models.

Core claim

The central claim is that a code-enabled autonomous agent can evolve a mathematical problem into a new problem that is (1) recognized as solvable by a held-out judge and (2) harder for current solvers than the original, where hardness is measured by lower solve rate and greater token usage. The mechanism combines executable exploration with dual verification: the evolution agent writes and runs Python to probe candidate structures, the solvability agent checks the proposed solution chain for logical flaws, and the difficulty agent scores whether the change demands a new insight rather than extra computation. The paper reports consistent Evolution-SR deficits across model pairs, a 49.1% failu

What carries the argument

The load-bearing mechanism is the three-agent validation loop. The Evolution Agent operates in two phases—bottleneck analysis then free exploration—and is instructed to create a 'Burden of Discovery' by hiding the entry point to the solution. The Solvability Verification Agent uses the proposed solution as a proxy: a logically consistent chain implies a solution path exists. The Difficulty Verification Agent scores adaptations from 1 to 5 and rejects scores 1–2 as artificial complexity. A code environment with symbolic computation, constraint solving, and enumeration tools supplies deterministic feedback during exploration and verification.

Load-bearing premise

The claim that evolved problems are solvable rests on the paper's solvability verifier accepting a proposed solution whose logical chain it cannot find a flaw in; if that check is systematically over-permissive for subtle olympiad-level statements, the certified solvability counts and the difficulty comparisons built on them are unsupported, and the human audit covers only 127 validity cases, not the full generated set.

What would settle it

Take a random sample of internally accepted evolved problems, remove the reference solutions, and ask strong independent solvers (or a proof-assistant formalization) to solve them from scratch. If a nontrivial fraction yield no correct solution or no verifiable proof, then 'solvable because the proposed solution had no detected flaw' is too weak. A simpler version: seed the pipeline with deliberately ill-posed problems that have plausible but subtly wrong reference solutions and count how often the solvability verifier passes them.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Automatic evolution could produce large sets of novel olympiad-style problems without hand curation.
  • The observed capability asymmetry—evolvers producing problems that defeat stronger solvers—suggests models can generate training or evaluation data beyond their own current solving ceiling.
  • Because the pipeline filters for structural rather than computational difficulty, generated problems should remain useful for reasoning evaluation rather than merely longer arithmetic.
  • The efficiency numbers (1.56–6.55 failed rollouts per success, with solvability failures dominating) imply that making the solvability check cheaper or stronger would directly lower data-synthesis cost.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper does not show that training on evolved problems improves models; a natural follow-up would test whether the evolved set transfers as training data, which the authors explicitly defer.
  • The solvability proxy is the fragile joint; feeding accepted problem/solution pairs into a formal proof checker would give a stronger guarantee, and the pipeline's code-first format makes this test look feasible.
  • Difficulty as 'Burden of Discovery' invites a falsifiable check: whether human experts solving blind take longer and make more failed attempts on evolved problems than on seeds matched for topic and length.
  • A concrete extension would compare evolved problems against human-written olympiad problems of matched topic to see whether solve-rate drops are due to obscurity or ambiguity rather than mathematical depth.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper introduces Code2Math, a multi-agent framework for autonomously evolving mathematical problems. An Evolution Agent analyzes a seed problem and uses executable Python tools to explore structural variants; a Solvability Verification Agent screens generated problems by checking proposed solution steps for logical flaws; a Difficulty Verification Agent scores adaptations on a 1–5 scale according to 'Burden of Discovery'. Using 100 seed problems, five evolution backbones, and six solver models, the authors report that most internally accepted problems are certified solvable by an external judge (74–98 per evolver), that solve rates generally decrease from originals to evolved problems, that token consumption increases, and that a human audit supports the reliability of the GPT-5.2-High judge. A code ablation and a rollout-efficiency analysis are also presented. The central claim is that, given sufficient test-time exploration, code agents can synthesize solvable problems that are structurally distinct from and more challenging than the originals.

Significance. If the central claim holds, the paper makes a useful contribution to automated problem synthesis and benchmark self-evolution. The strengths are real: the evaluation crosses five evolvers and six solvers, the authors ship code and data, full prompts are provided in the appendix, and the human audit (126/127 validity precision, 92.5% answer-agreement, 95.9% within-one difficulty agreement) is a serious attempt to validate the LLM judge. The failure analysis and code ablation are also informative. However, the main quantitative evidence for 'more challenging' is currently an unmatched comparison between solve rates on different problem sets, and the solvability guarantee is an LLM-based proxy rather than a formal guarantee. Both issues are fixable, but they currently weaken the strength of the paper's headline conclusion.

major comments (4)
  1. [Table 1; §3.4; Appendix A.1] Origin-SR is reported on the full 100-seed dataset, whereas Evolution-SR is computed only on the subset of problems that passed internal verification and external certification (CSC range 74–98 per evolver). The manuscript does not state that Origin-SR is recomputed on the matched subset of seeds whose evolved problems were evaluated. If successful evolution is correlated with seed difficulty, the reported drops (e.g., GPT-5.2-High: 70→64; Gemini-3-Flash-Thinking: 56→24–35) could be overstated or reversed. This is a direct confound for the 'more challenging' claim. Please recompute Origin-SR on the exact matched seed subset, or report paired per-seed deltas restricted to seeds with certified evolved problems.
  2. [§3.2; §4.3] The solvability guarantee rests on the assumption that a logically flaw-less proposed solution implies existence of a solution path. The external judge is itself an LLM, and the human audit covers only 127 GPT-valid problems, not the full certified sets (e.g., 74, 83, 94, 97, 98 per evolver). The audit establishes high precision on a sample of GPT-valid problems, but it cannot rule out false positives among the remaining certified problems or false negatives among rejected ones. Since 'solvable' is a central claim, either extend the human audit to all accepted problems or add independent mechanical/formal verification where feasible.
  3. [Table 1; §4.2] No confidence intervals or significance tests are reported for the solve-rate differences. Several cells are small or even positive (e.g., DeepSeek-Chat row: Origin-SR 16 → Evolution-SR 19 on DeepSeek-Chat evolutions; GPT-5.2-High: 70 → 70 on DeepSeek-Chat). With N≈74–98, differences of 3–9 points are often within sampling noise. Report bootstrap confidence intervals, McNemar tests, or paired tests by seed to support the 'most pairs lower' and 'systematically harder' conclusions.
  4. [§4.4; Table 5] The code ablation shows only a 2.6-point average difference (25.7→23.1) and the direction is inconsistent across pairs (e.g., DS-NT/DS-NT +2, Seed/DS-T +7). No uncertainty estimates are given. Because this is the only direct evidence that code execution, rather than general LLM reasoning, drives the difficulty effect, the 'code-driven' component of the central claim is not yet quantitatively established. Provide paired significance tests and, ideally, a more detailed qualitative coding of the generated problem types.
minor comments (5)
  1. [§3.4] Typo: 'evalaute' should be 'evaluate'.
  2. [Table 1] The cell '70 70±0' is unexplained. What does the ±0 denote — a standard error, a range, or a formatting artifact? Please clarify or remove.
  3. [§A.6, Case 1] Typo: 'riangleABC' should be 'triangle ABC'.
  4. [Figure 3; Appendix A.1] Imputing the maximum token limit for all timeout samples may substantially shift the ATC distribution. At a minimum, report the fraction of timeouts separately and show the distribution without imputation as a sensitivity check.
  5. [Table 1] The header abbreviates 'Gemini-3-Pro-Preview-Thinking' as 'Gemini-3-Pro-Thinking' while the text uses the full name. Use one consistent abbreviation.

Circularity Check

0 steps flagged

No significant circularity: solvability and difficulty claims rest on independent external-judge certification, cross-model solve-rate drops, and human audit; self-citations are not load-bearing.

full rationale

The paper contains no formal derivation whose output equals an input by construction. The solvability claim is an explicitly stated proxy ('A flaw-less logical chain implies the existence of at least one solution path, serving as a proxy for solvability', §3.2), and it is checked against a held-out GPT-5.2-High judge and a human audit (126/127 validity, 99.2%; 1134/1226 correctness, 92.5%), so it is not a tautology. The difficulty-escalation claim is not defined into existence: the Difficulty Verification Agent filters candidates, but the evidence for difficulty is the measured Evolution-SR vs Origin-SR across six solvers and the ATC shift, which are external to the filter. The main potential concern is that Origin-SR is computed on the full 100-seed set while Evolution-SR is computed on the accepted subset (e.g., AR cells like 83/94, 98/98), so the baseline may be unmatched; however, that is a validity/confound issue, not a circular reduction, and it does not make the difficulty claim true by definition. Self-citations (TRACE, failure modes, self-evolving risks) appear only as background or prompt-design references and are not load-bearing. No uniqueness theorem, ansatz, or fitted parameter is imported from the authors' prior work to force the conclusion.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 2 invented entities

The framework's success metrics rest on several unproven operational assumptions: LLM-judged solvability, solve-rate drop as difficulty, GPT-5.2-High reliability, and the Burden-of-Discovery construct. These are partially supported by human audit on samples, but they remain assumptions rather than externally verified facts.

free parameters (3)
  • Difficulty acceptance threshold = score >= 3 on the 1-5 scale
    Hand-chosen by the authors; determines which evolved problems count as successes and affects all difficulty-escalation statistics.
  • Rollout budget and trajectory limit = 20 rollouts; 30 agent steps
    Arbitrary budgets that define 'sufficient test-time exploration' and the success/failure counts; results would shift if the budgets changed.
  • Solver attempts, timeout, and ATC imputation = 3 attempts; 30-minute wall-clock; timeout imputed as max token limit
    Evaluation choices that shape solve rates and token-consumption distributions; the imputation rule directly affects the ATC difficulty evidence.
axioms (5)
  • domain assumption A flaw-free candidate solution implies the evolved problem is solvable.
    Stated in §3.2 Solvability Verification Agent: 'a flaw-less logical chain implies the existence of at least one solution path, serving as a proxy for solvability.' This underpins all solvability claims.
  • domain assumption Solve-rate decline and higher token consumption indicate increased mathematical difficulty.
    Used to operationalize difficulty in §4.2 and Appendix A.1; competing explanations such as ambiguity, judge bias, or surface-level rewording are not fully controlled.
  • domain assumption GPT-5.2-High is a reliable external judge for solvability and solver-answer correctness.
    The evaluation and human audit depend on this; human agreement is high but only on sampled subsets, so full-set reliability is assumed.
  • domain assumption Executable Python feedback guides the evolver toward valid, harder constructions.
    The code ablation supports this, but the effect is not deterministic across evolver-solver pairs, and the mechanism is not guaranteed (§4.4).
  • domain assumption The Difficulty Verification Agent's Theory-of-Mind scoring captures 'Burden of Discovery' for expert solvers.
    This is a subjective LLM-judgment construct; human audit shows 79.3% exact agreement, but the construct itself is not externally validated.
invented entities (2)
  • Burden of Discovery no independent evidence
    purpose: Operational definition of difficulty used by the Difficulty Verification Agent to accept or reject evolved problems.
    Defined in §3.2 and scored by LLM and human judges on a 1-5 scale; there is no external falsifiable measurement outside the model/human judgment loop.
  • Aha moment no independent evidence
    purpose: The psychological event the evolver is instructed to inject into evolved problems; used as a construction target in prompts.
    Introduced in §1 and §3.2 as a design heuristic; it is not independently measured or verified outside the LLM's stated intent.

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Code2Math: Can Your Code Agent Effectively Evolve Math Problems Through Exploration?." pith.science (2026). https://pith.science/paper/VHTJZATC

@misc{pith2026260303202,
  author       = {Pith},
  title        = {Pith review of: Code2Math: Can Your Code Agent Effectively Evolve Math Problems Through Exploration?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VHTJZATC}},
  note         = {Machine review of arXiv:2603.03202}
}
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read the original abstract

As large language models (LLMs) advance their mathematical capabilities toward the IMO and research level, the scarcity of challenging, high-quality problems has become a significant bottleneck for training, evaluation and self-evolution of LLMs. Simultaneously, recent code agents have demonstrated sophisticated skills in agentic coding and reasoning, suggesting that code execution can serve as a scalable environment for mathematical experimentation. In this paper, we investigate the potential of code agents to autonomously evolve existing math problems into more complex variations. We introduce a multi-agent framework designed to perform problem evolution while validating the solvability and increased difficulty of the generated problems. Our experiments demonstrate that, given sufficient test-time exploration, code agents can synthesize new, solvable problems that are structurally distinct from and more challenging than the originals. This work provides empirical evidence that code-driven agents can serve as a viable mechanism for synthesizing high-difficulty mathematical reasoning problems within scalable computational environments. Code and data is available at https://github.com/TarferSoul/Code2Math.

Figures

Figures reproduced from arXiv: 2603.03202 by Dadi Guo, Dongrui Liu, Jianjie Feng, Jiayu Liu, Qihan Ren, Qingyu Liu, Shuai Shao, Tianyi Zhou, Weixian Huang, Wenze Su, Yi R. Fung, Yuejin Xie, Yujiu Yang, Zhiyuan Fan.

Figure 1
Figure 1. Figure 1: Example of code-driven problem evolution. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of our multi-agent system. Our pipeline consists of three components: the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of Average Token Consump￾tion (ATC) across original and agent-evolved problems. For each problem, we compute the average output to￾kens across all solver models. Timeout samples (where solvers failed to produce output) are assigned the maxi￾mum token limit to reflect their high difficulty [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Efficiency Analysis of Agentic Problem Evolution. We visualize the distribution of failure counts [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The prompt template of our Evolution Agent. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The prompt template of our Solvability Verification Agent. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The prompt template of our Difficulty Verification Agent. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The prompt template of the solvability evaluator. [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The prompt template of the problem solver. [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The prompt template of the solution evaluator. [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗

discussion (0)

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Reference graph

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    The **Probabilistic Limit**,B, defined as the almost-everywhere limit of the geometric mean: B= lim n→∞ fn(x) 1/n (You may assume that this limit exists and is constant for almost allx∈[0, π] in the sense of Lebesgue measure.) Calculate the value of the ratio A B . Comment: This is an exemplary adaptation (Score 5) that transforms a standard inequality pr...

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    **Conceptual Depth:** The adaptation elevates the mathematical landscape from elementary calculus (analyzing a recurrence relation’s fixed point) to advanced analysis. It requires two distinct, sophisticated insights: - **Harmonic Analysis (for Limit A):** Recognizing that the integral of the product simplifies due to the orthogonality of the cosine terms...

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    It demonstrates that while the original problem’s bound (0.75n) holds, the typical decay (0.25n) is much faster, providing a deeper understanding of the sequence’s behavior

    **Mathematical Significance:** The problem highlights the subtle difference between the ’Analytic Mean’ (dominated by large outliers) and the ’Probabilistic Limit’ (the typical behavior almost everywhere). It demonstrates that while the original problem’s bound (0.75n) holds, the typical decay (0.25n) is much faster, providing a deeper understanding of th...

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    It perfectly embodies the criteria of ’Anti-Templating’ and ’Mathematical Beauty’

    **Elegance:** Despite the high-level machinery required, the final result is a clean integer ratio, and the solution path is aesthetically satisfying. It perfectly embodies the criteria of ’Anti-Templating’ and ’Mathematical Beauty’. 30 Case 10 Evolver: Gemini-3-Pro-Preview-Thinking Original Problem (From AIME-2024): A list of positive integers has the fo...

    2024

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    **Required Insight (The ’Eureka’ Element):** The problem creates a non-trivial optimization landscape. The solver must discover that the median acts as a ’gate’ for density (choosing M= 3 optimally allows packing 1s and 2s) and that there is a critical trade-off between the mode frequency and the sum: increasing the frequency of the mode (10) allows for m...

    2024

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    **Mathematical Elevation**: The original solution relies on an arithmetic progression derived from a specific ’worst-case’ construction. The new problem’s solution ( f(N, K) = PK j=1 inomN j− (2K −1) ) links the dynamical process to the volume of a Hamming ball in the Boolean lattice (the number of subsets of size ≤K ). This is a profound structural insig...

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    The solver cannot simply ’guess’ the worst case; they must understand the underlying state graph’s properties

    **Burden of Discovery**: The optimal initial configuration in the new problem is far less obvious than in the original. The solver cannot simply ’guess’ the worst case; they must understand the underlying state graph’s properties. My computational verification confirmed that the pattern holds, but establishing it requires a non-trivial ’Aha!’ moment conne...

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    37 Case 18 Evolver: Gemini-3-Pro-Preview-Thinking Original Problem (From CMO-2025): Given an odd integer n >1 such that every prime divisor p of n satisfies gcd(p−1, n) = 1

    **Elegance**: By stripping away the ’30 colors, 70 cards’ clutter and replacing it with parameters N, K, the adapter has created a problem of textbook quality that tests deep combinatorial intuition rather than just careful accounting. 37 Case 18 Evolver: Gemini-3-Pro-Preview-Thinking Original Problem (From CMO-2025): Given an odd integer n >1 such that e...

    2025

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    For all other primes, a standard heuristic (’all lifts work’) sufficed

    **Generalization of Difficulty:** In the original problem, the rigorous analysis of ’lifting’ (deter- mining which coefficient perturbations preserve the permutation property) was only strictly necessary for the prime p= 3 . For all other primes, a standard heuristic (’all lifts work’) sufficed. The new problem forces this rigorous analysis for *every* pr...

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    This requires the solver to not only identify the ’double root 1’ condition (insight from the original) but also to classify and sum over the possible behaviors of the third root

    **Structural Depth:** By increasing the order to 3, the problem introduces a free parameter (the third root y) for the characteristic polynomial. This requires the solver to not only identify the ’double root 1’ condition (insight from the original) but also to classify and sum over the possible behaviors of the third root. This integrates the ’root findi...

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    **Anti-Templating:** The new problem invalidates the ’count roots and multiply by powers of p’ template. The density of valid solutions is non-trivial ( <1 ) for all cases, requiring a genuine understanding of how the recurrence coefficients influence the distribution of values modulop k. 38

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    circular

    Find the least integerb≥2for which there are more than ten b-eautiful integers. Evolved Problem: Let b≥2 be an integer. A positive integer n is called "circular" in base b if it has exactly two digits in base b, say n= (xy) b (where x∈ {1, . . . , b−1}and y∈ {0, . . . , b−1}), and the sum of the squares of these two digits equalsn(i.e.,x 2 +y 2 =n). Find ...

    2025

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    Niklas Muennighoff, Zitong Yang, Weijia Shi, Xi- ang Lisa Li, Li Fei-Fei, Hannaneh Hajishirzi, Luke Zettlemoyer, Percy Liang, Emmanuel Candès, and Tatsunori B Hashimoto

    Sand-math: Using llms to generate novel, difficult and useful mathematics questions and answers.arXiv preprint arXiv:2507.20527. Niklas Muennighoff, Zitong Yang, Weijia Shi, Xi- ang Lisa Li, Li Fei-Fei, Hannaneh Hajishirzi, Luke Zettlemoyer, Percy Liang, Emmanuel Candès, and Tatsunori B Hashimoto. 2025. s1: Simple test-time scaling. InProceedings of the 2...

    arXiv 2025

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.