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REVIEW 4 major objections 4 minor 27 references

INTEGRALBENCH: Benchmarking LLMs with Definite Integral Problems

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper introduces INTEGRALBENCH, a 317-problem benchmark of graduate-level definite integrals with paired symbolic and numerical ground truth, and reports baseline results for nine LLMs, with the best model reaching 56.15% symbolic…

desk verdict Real dataset, broken symbolic-validation rule; headline symbolic scores are self-consistency scores, not correctness scores. read the letter →

arxiv 2507.21130 v1 pith:O7GAAT7U submitted 2025-07-22 cs.AI

Bintao Tang , Xin Yang , Yuhao Wang , Zixuan Qiu , Zimo Ji , Wenyuan Jiang This is my paper · ORCID
classification cs.AI
keywords definiteintegralsLLMevaluationsymboliccomputationmathematicalreasoningbenchmarkdatasetdifficultyannotationnumericalverificationPASS@3metrics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

INTEGRALBENCH is a 317-problem benchmark of graduate-level definite integrals, each with a symbolic answer, a numerical value, a manual difficulty rating, and a source attribution. The paper's central claim is that this dual ground truth plus difficulty stratification makes definite integrals a measurable test bed for LLM mathematical reasoning, and that current models are far from solving it: the best of nine evaluated models, Qwen3-235B-A22B, reaches 56.15% symbolic PASS@3 and 50.16% numerical PASS@3. The paper also establishes that accuracy falls sharply as difficulty rises across all models, validating the difficulty annotations, and that a 32B model (QwQ) outperforms several larger models, arguing that architecture matters as much as scale.

What carries the argument

The load-bearing machinery is the dual-verification evaluation pipeline: a numerical check that compares the model's decimal to ground truth with error below $10^{-6}$, and a symbolic check in which another LLM writes Python/mpmath code to evaluate the model's symbolic answer and the answer is accepted only if that computed value matches the model's own numerical answer within the same tolerance. The benchmark construction pipeline is equally central: bounding-box annotation of problem images, OCR to LaTeX with manual verification, and parameter instantiation for integrals containing free variables, with LLM-assisted random sampling and manual confirmation. Difficulty ratings from 1 to 5 are manually assigned and then validated by the observed difficulty-accuracy correlation. The metrics PASS@3 and ALL@3 convert generation randomness into peak-ability and consistency measurements.

What would settle it

Take a sample of INTEGRALBENCH symbolic failures, have an independent symbolic computation engine check each model's expression for equivalence with the ground-truth symbolic answer, and recompute symbolic PASS@3; if the independently checked accuracy is materially lower than the reported ~56%, the dual-verification scoring overstates symbolic capability. A cheaper probe: feed the verifier a deliberately wrong closed form whose decimal value was chosen to match the ground truth to 10 digits and see whether it passes.

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Extended reading notes

Core claim

On its own terms, the paper's contribution is a focused evaluation framework for definite integrals. It claims that no existing benchmark offers enough challenging integrals, integral-specific metrics, or difficulty gradation, and that INTEGRALBENCH fills that gap with 317 verified problems collected from one graduate textbook and three competition series. Each problem stores the integral body, unsimplified and simplified symbolic answers, a numerical answer, difficulty, and source. The evaluation protocol generates three independent attempts per model and reports PASS@3 and ALL@3; numerical answers are checked against ground truth within $10^{-6}$, and symbolic answers are checked by generating code that evaluates the expression and comparing that value with the model's own numerical answer. The reported baseline is the strongest evidence: Qwen3-235B-A22B at 56.15% symbolic PASS@3, with all models declining sharply on difficulty 4-5 problems and exhibiting a consistent symbolic-over-numerical accuracy gap.

Load-bearing premise

A symbolic answer is judged correct when an automatically generated program evaluates it to the same decimal as the model's own numerical answer; if a wrong expression and a wrong decimal happen to agree, the answer is scored correct even though the symbolic claim is false.

Editorial extensions

If this is right

  • Future LLM work on definite integrals can be compared against a fixed, public 317-problem set with a clear pass threshold, making claims about integration skill reproducible.
  • Because all nine models degrade sharply on difficulty 4-5 problems, the difficulty ratings can be used to target model weaknesses rather than just report averages.
  • The 32B QwQ outperforming larger models means parameter count alone is a poor predictor of symbolic integration ability, so evaluations should report architecture and training details alongside size.
  • The consistent symbolic-over-numerical gap indicates models can derive expressions they cannot evaluate reliably, which motivates tool-augmented pipelines where external evaluators finish the computation.
  • Inference-time scaling plateaus imply that there is a model-specific token budget beyond which additional reasoning yields little accuracy gain, so token efficiency should be reported alongside accuracy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The symbolic verification step implicitly assumes that the LLM-generated evaluator code is faithful to the symbolic expression; an independent check with a computer algebra system would convert the reported symbolic accuracy into a measure of true symbolic equivalence, and would likely yield lower numbers.
  • Because many problems contain free variables, the parameter-instantiation scheme can regenerate fresh variants of the same integral, offering a built-in contamination-resistant re-testing procedure the paper does not demonstrate.
  • The large drop from PASS@3 to ALL@3 (for example Qwen3 symbolic 56.15% vs 38.17%) suggests that peak performance is partly a sampling artifact; comparing PASS@16 would likely reorder models whose success variance differs.
  • The failure modes documented—truncation, circular repetition, and JSON format violations—look like general structured-output failures, so the same evaluation harness could be applied to other computation-heavy math tasks beyond integrals.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. INTEGRALBENCH introduces a new benchmark of 317 graduate-level definite integral problems with symbolic and numerical ground truth, manual difficulty ratings, and source attributions. The paper describes a construction pipeline based on OCR, manual annotation, and LLM-assisted parameter instantiation, then evaluates nine LLMs under PASS@3 and ALL@3 metrics. The main empirical findings are that larger models tend to perform better, that Qwen3-235B-A22B achieves the highest accuracies (50.16% numerical, 56.15% symbolic PASS@3), that performance declines with annotated difficulty, and that inference-time scaling curves show model-specific plateaus. The paper also provides a failure-mode analysis and a cost breakdown for dataset construction.

Significance. If the benchmark and its evaluation protocol were sound, INTEGRALBENCH would be a useful focused resource: it has a moderate number of challenging problems, manual difficulty annotations, source metadata, a publicly released dataset, a transparent cost breakdown, and an interesting failure-mode taxonomy. The manual curation effort and the decision to separate symbolic from numerical accuracy are genuinely valuable design choices. However, the central evaluation protocol for symbolic answers is self-referential, and the numerical ground truth is not independently verified, so the headline symbolic accuracies and the symbolic-versus-numerical comparison are not currently trustworthy. The paper would be significant after the validation chain is reworked and the numbers are recomputed; as it stands, the empirical contribution is not reliable.

major comments (4)
  1. [Section 3.1, Answer validation] The symbolic-answer verification rule compares LLM-generated Python evaluation of the model's symbolic answer against the same model's own numerical answer, rather than against the dataset ground truth. This is a self-consistency check, not a correctness check. A model that emits the same decimal string in both the "answer" and "numerical_answer" JSON fields will pass the check even when both are wrong, since the evaluator will evaluate that constant string. The uniform pattern in Table 2, where PASS@3(Sym) exceeds PASS@3(Num) for all nine models, is exactly the signature one expects from this circular rule: the symbolic score is inflated because it only requires agreement with the model's own numerical output. The headline symbolic baselines, such as Qwen3-235B-A22B at 56.15% PASS@3(Sym), are therefore not measurements of symbolic integration ability. The fix is to validate the model's symbolic answer against the independently computed ground-truth numerical value (or to check symbolic equivalence with a CAS such as SymPy or Mathematica).
  2. [Section 2.2, Ground truth solutions] The numerical ground truth is derived from LLM-generated Python scripts that evaluate the symbolic ground truth, rather than from independent numerical integration or a computer algebra system. As described in Section 2.2, DeepSeek V3 generates scripts that substitute parameter values and evaluate the analytical expressions, and the numerical ground truth is the output of those scripts. No independent cross-check with an established CAS or adaptive quadrature is reported. Since the numerical ground truth is the reference for all numerical-accuracy claims and for the proposed fix to the symbolic-validation protocol, this is a load-bearing gap. The paper must provide independent numerical ground truth, at least for a substantial random sample, before the reported accuracies can be considered verified.
  3. [Table 2, Sections 3.2-3.4] All accuracy comparisons are reported as point estimates with no confidence intervals or significance tests. With 317 problems and N=3 attempts, adjacent rankings can easily be within sampling noise: for example, DeepSeek-R1 and Doubao are both reported at 45.43% PASS@3(Num), and the gap between Qwen3 and DeepSeek-R1 in numerical PASS@3 is only 4.73 percentage points. The claims in Sections 3.2-3.4 about which models are better and about the shape of inference-time scaling curves are therefore stronger than the evidence supports. Bootstrap confidence intervals or per-model standard errors should be reported, and the comparative statements should be softened in their absence.
  4. [Section 5, Related Work] The paper criticizes prior benchmarks for relying on "circular evaluation and multi-solution consistency checks," but the symbolic-validation protocol in Section 3.1 is itself a multi-solution consistency check between two outputs of the same model. Recognizing this internal inconsistency is important not only for presentation but also for the framing of the contribution: the paper should acknowledge that its own symbolic metric is a consistency metric and either replace it with a ground-truth-based check or explicitly restrict the symbolic claims to self-consistency.
minor comments (4)
  1. [Section 3.5, Failure mode analysis] The claim that "correct symbolic solutions" are paired with incorrect numerical evaluations is not supported by the validation protocol, because symbolic correctness was never checked against ground truth. The observed Sym > Num gap should not be interpreted as "strong symbolic reasoning but weak numerical computation" without fixing the validation rule first.
  2. [Figure 6, Section 3.4] The legend of the inference-time scaling figure includes only five of the nine evaluated models; please clarify whether the other four were omitted and, if so, why, or include all models for consistency with the rest of the evaluation.
  3. [Section 2.3, Table 1] The cost table labels the API subtotal as "$5.00" even though the rows sum to $0.83 + $4.17; consider clarifying that this excludes the cost of the DeepSeek calls used for symbolic verification during evaluation, if that is not already included.
  4. [Abstract and Conclusion] The abstract and conclusion describe INTEGRALBENCH as a "rigorous evaluation framework." Given the circular validation chain described in Section 3.1, this descriptor is currently overstated; the rigor claim should be re-earned after the validation protocol is repaired.

Circularity Check

1 steps flagged · score 6.0 of 10

PASS@3(Sym) reduces by construction to self-consistency: §3.1 accepts any symbolic expression that numerically matches the model's own numerical_answer, so the headline symbolic baselines are not measurements of symbolic integral-solving ability.

  1. self definitional [Section 3.1, 'Answer validation' (page 5) and Figure 4 caption]
    "Symbolic answer verification: The model-generated answer is passed to DeepSeek, which generates Python code to numerically evaluate it. If this computed result matches the model’s numerical answer within the10−6 threshold, the symbolic answer is considered valid."

    The predicate 'symbolically correct' is defined as agreement between two outputs of the same model: the candidate expression (evaluated by an LLM-generated program) and the model's own numerical_answer. The evaluator never compares the expression to the provided symbolic ground truth or to an independent CAS. Hence a wrong expression whose numeric value equals a wrong decimal passes, and the degenerate case is putting the decimal string in both JSON fields, which the rule accepts as a constant. Consequently the PASS@3(Sym) numbers in Table 2 measure internal consistency, not whether the model produced the correct antiderivative; the paper's own observation that symbolic accuracy exceeds numerical accuracy for all nine models is exactly the artifact this permissive rule would generate.

full rationale

The numerical track is genuinely benchmarked: numerical answers are compared against numeric ground truth with absolute error below 10^-6, and the dataset construction pipeline (manual annotation, OCR verification, difficulty ratings, source attribution) is independent of the evaluated models. Those parts of the benchmark have real content and are not circular. The circularity is confined to the symbolic-validation rule, which is load-bearing for the paper's central empirical claim of symbolic baselines: the symbolic ground-truth answers (shown in Figure 1 and stored as simplified_answer in the dataset) are never consulted when validating a model's symbolic response. Instead, the reference is the model's own numerical_answer, so the validation is a multi-solution self-consistency check of the kind the paper's Related Work criticizes in prior benchmarks. The consistent PASS@3(Sym) > PASS@3(Num) pattern in Table 2 is the expected signature of this rule, and the paper's failure-mode section interprets it as 'symbolic-numerical inconsistency' rather than as an evaluation artifact. Because one of the two headline metrics reduces by construction to self-consistency while the numerical metrics remain independently anchored, the appropriate finding is partial circularity rather than full collapse of the paper's contribution.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The benchmark's claims rest on the trustworthiness of the source answers, the correctness of LLM-generated evaluation scripts, and the validity of the self-consistency check for symbolic answers. No new physical or conceptual entities are postulated.

free parameters (2)
  • numerical_error_tolerance = 1e-6
    Hand-chosen threshold for marking numerical answers correct; a different tolerance changes all reported accuracies.
  • sampling_trials_N = 3
    Number of independent attempts per problem; PASS@3 and ALL@3 metrics depend on this, and the paper acknowledges the small sample in its limitations.
assumptions (3)
  • domain assumption Ground truth symbolic answers from textbooks and competitions are correct, and the OCR plus manual verification pipeline transcribed them faithfully.
    Dataset validity rests on the fidelity of source answers; Section 2.2 manual annotation and LaTeX conversion.
  • domain assumption The DeepSeek-generated Python scripts that evaluate the analytical answers produce correct numerical ground truth.
    Numerical ground truth is computed by executing parameter-instantiated scripts rather than independent numerical integration; Section 2.2 Ground truth solutions.
  • ad hoc to paper A model's symbolic answer can be validated by numerically evaluating it and comparing with the model's own numerical answer.
    This self-consistency check is introduced for this paper and never compares the symbolic answer with the dataset's symbolic ground truth; Section 3.1 Answer validation.

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Pith. "Pith review of INTEGRALBENCH: Benchmarking LLMs with Definite Integral Problems." pith.science (2026). https://pith.science/paper/O7GAAT7U

@misc{pith2026250721130,
  author       = {Pith},
  title        = {Pith review of: INTEGRALBENCH: Benchmarking LLMs with Definite Integral Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O7GAAT7U}},
  note         = {Machine review of arXiv:2507.21130}
}
read the original abstract

We present INTEGRALBENCH, a focused benchmark designed to evaluate Large Language Model (LLM) performance on definite integral problems. INTEGRALBENCH provides both symbolic and numerical ground truth solutions with manual difficulty annotations. Our evaluation of nine state-of-the-art LLMs reveals significant performance gaps and strong correlations between problem difficulty and model accuracy, establishing baseline metrics for this challenging domain. INTEGRALBENCH aims to advance automated mathematical reasoning by providing a rigorous evaluation framework specifically tailored for definite integral computation.

Figures

Figures reproduced from arXiv: 2507.21130 by the authors.

Figure 1
Figure 1. Example problem from INTEGRALBENCH with symbol￾ic/numerical ground truth solutions, difficulty rating, and source attribution. ticated multi-step reasoning including decomposition of complex expressions, pattern recognition for simplification techniques, and recall of integration methods. These char￾acteristics make integral calculus particularly suitable for evaluating advanced LLM reasoning capabilities. Despite e… view at source ↗
Figure 2
Figure 2. Construction pipeline for INTEGRALBENCH. The work￾flow consists of four steps: (1) problem collection from graduate textbooks and competitions with selection criteria, (2) two-stage manual annotation using bounding boxes, (3) OCR-based con￾version to LaTeX with manual verification, and (4) parameter instantiation for problems with free variables, followed by final dataset validation. We start with the definition of … view at source ↗
Figure 3
Figure 3. Diversity analysis of INTEGRALBENCH using affinity propagation clustering on problem-solution embeddings. Left: PCA visualization of 56 distinct clusters showing well-distributed coverage across the embedding space. Right: Cluster size distribution reveals a balanced representation with most clusters containing 3-13 problems, avoiding over-concentration in any single problem type. of the cost are for the tokens of L… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Evaluation pipeline for INTEGRALBENCH. Nine LLMs generate both numerical and symbolic answers using standardized prompts. Numerical answers are validated by direct comparison with ground truth (error < 10−6 ), while symbolic answers are ver￾ified through LLM-generated …
Figure 5
Figure 5. Figure 5: PASS@3 performance across difficulty levels for symbolic and numerical evaluations. All models show consistent performance decline with increasing difficulty, demonstrating a strong negative correlation between problem complexity and accuracy. and demonstrates that INT…
Figure 6
Figure 6. Figure 6: Inference-time scaling curves showing normalized token consumption versus cumulative PASS@3 accuracy. Most mod￾els exhibit rapid early gains followed by saturation, with varying “sweet spot” locations highlighting differences in inference effi￾ciency across models. 3.5…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 26 canonical work pages

  1. [1]

    Parse the given analytical answer (NOT the integral) and translate it into mpmath code

    work page

  2. [2]

    Generate Python code that directly evaluates this analytical expression to exactly 10 decimal places

    work page

  3. [3]

    No other text or ,→multiple outputs are allowed

    The code must ONLY print a single floating-point number (rounded to 10 decimal places). No other text or ,→multiple outputs are allowed

    work page

  4. [4]

    IMPORTANT: Use mp.nstr(result, n=10) for output formatting, NOT Python string formatting like f’{result:.10f}’ ,→which doesn’t work correctly with mpmath’s mpf objects

    work page

  5. [5]

    Avoid using mpmath.quad or any numerical integration - instead, directly compute the value from the analytical ,→formula

    work page

  6. [6]

    For expressions with parameters, select appropriate parameter values within the given constraints

    work page

  7. [7]

    For expressions with parameters, compute the numerical result by substituting the chosen parameter values into ,→the given analytical answer

    work page

  8. [8]

    Use mpmath.dps = 15 for internal precision

    work page

Show all 27 references
  1. [9]

    This helps avoid syntax errors and makes the code more readable and debuggable

    IMPORTANT: Break complex calculations into multiple steps with intermediate variables, instead of doing ,→everything in one line. This helps avoid syntax errors and makes the code more readable and debuggable

  2. [10]

    Verify all denominators and logarithm arguments will be ,→ non-zero and positive respectively

    IMPORTANT: Carefully check your selected parameter values to ensure they do not cause division by zero, ,→undefined logarithms, or other mathematical errors. Verify all denominators and logarithm arguments will be ,→ non-zero and positive respectively

  3. [11]

    EXTREMELY IMPORTANT: Always use the EXACT correct mpmath function names with the ’mp.’ prefix. Common mistakes ,→ to avoid: - Use mp.asin(x), NOT mp.arcsin(x) - arcsin is not a valid mpmath function - Use mp.acos(x), NOT mp.arccos(x) - arccos is not a valid mpmath function - U...

  4. [12]

    code": The Python code (plain text, no markdown fences) that evaluates the analytical expression. -

    Return a JSON object with: - "code": The Python code (plain text, no markdown fences) that evaluates the analytical expression. - "parameters": A dictionary of chosen parameter values (for parameter cases only; null for non-parameter cases ,→). - "substituted_answer": The anal...

  5. [15]

    Instead, translate the analytical answer into mpmath code and evaluate it ,→directly

    Do NOT use numerical integration. Instead, translate the analytical answer into mpmath code and evaluate it ,→directly

  6. [19]

    Verify all denominators and logarithm arguments to ensure no mathematical errors

  7. [21]

    : Python code using mpmath to DIRECTLY evaluate the analytical answer to 10 decimal places (without using ,→numerical integration). The code must only print a single number. -

    For printing the result, use ONLY: print(mp.nstr(result, n=10)) user_prompt_without_param: | Given the integral: ${problem} Analytical answer: ${answer} Generate a JSON object containing: - "code": Python code using mpmath to DIRECTLY evaluate the analytical answer to 10 decim...

  8. [22]

    Instead, translate the analytical answer into mpmath code

    Do NOT use mpmath.quad or any numerical integration. Instead, translate the analytical answer into mpmath code

  9. [23]

    Break complex expressions into smaller, manageable parts using intermediate variables

  10. [24]

    Do not compute everything in a single line - use multiple steps to avoid errors

  11. [25]

    Include comments explaining each significant calculation step

  12. [26]

    Use the EXACT correct mpmath functions with the ’mp.’ prefix: - Use mp.asin(x), NOT mp.arcsin(x) - Use mp.acos(x), NOT mp.arccos(x) - Use mp.atan(x), NOT mp.arctan(x) - Use mp.log(x), NOT mp.ln(x) - Use mp.ellipe(x), NOT mp.ellipd(x) - Use mp.struveh(v, x) for the Hankel Struv...

  13. [27]

    Integral prompt

    For printing the result, use ONLY: print(mp.nstr(result, n=10)) Listing 2. Integral prompt. integral_prompt: | Please calculate the definite integral ${problem}. Follow these steps:

  14. [28]

    Compute the integral analytically to obtain the exact answer

  15. [29]

    Provide all necessary steps to reach the solution

  16. [30]

    Include a numerical approximation of the result, rounded to 10 decimal places

  17. [31]

    answer":

    Output the final answer in the following JSON format on the last line, with no additional text: {"answer": "exact_answer_in_latex", "numerical_answer": "numerical_approximation_of_the_result"} C. Full Failure Cases 13 INTEGRAL BENCH : Benchmarking LLMs with Definite Integral P...

  18. [2023]

    problem":

    Curran Associates Inc. 11 INTEGRAL BENCH : Benchmarking LLMs with Definite Integral Problems A. Data Format The processed data is stored in the same JSON format for convenient use in the subsequent steps. An example is shown below: { "problem": "\\int\\limits_{0}ˆ{1}\\frac{(1-...

  19. [2025]

    Azerbayev, Z., Schoelkopf, H., Paster, K., Dos Santos, M., McAleer, S., Jiang, A

    URL https://www.anthropic.com/ news/claude-3-7-sonnet . Azerbayev, Z., Schoelkopf, H., Paster, K., Dos Santos, M., McAleer, S., Jiang, A. Q., Deng, J., Biderman, S., and Welleck, S. Llemma: An open language model for math- ematics. arXiv preprint arXiv:2310.06786, 2023. Bavare...

    arXiv

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