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REVIEW 6 major objections 5 minor 4 cited by

DeepTheorem: Advancing LLM Reasoning for Theorem Proving Through Natural Language and Reinforcement Learning

T0 review · 6 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that informal theorem proving can be advanced at scale by mining 121K IMO-level theorem–proof pairs, checking them with LLMs, and training with RL-Zero on guaranteed true/false variants, so a 7B model outscores larger…

desk verdict A genuinely useful dataset and RL reward idea, but the headline SOTA claims and the LLM-only label loop need serious revision before the paper is publishable as-is. read the letter →

arxiv 2505.23754 v2 pith:6O3VAIXJ submitted 2025-05-29 cs.CL cs.AI

Ziyin Zhang , Jiahao Xu , Zhiwei He , Tian Liang , Qiuzhi Liu , Yansi Li , Linfeng Song , Zhenwen Liang
show 5 more authors
Zhuosheng Zhang Rui Wang Zhaopeng Tu Haitao Mi Dong Yu
This is my paper · ORCID
classification cs.CLcs.AI
keywords informaltheoremprovingreinforcementlearningRL-ZeroLLMreasoningvariantsmathematicalbenchmarkGRPOnaturallanguageproofs
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

DeepTheorem claims that large language models can get better at proving hard mathematical theorems without a formal proof assistant, by training on natural-language theorem–proof pairs and by rewarding consistent answers on deliberately modified theorem variants. The paper builds a 121K-sample dataset of IMO-level statements, each with a correctness label, a difficulty score, a topic tag, and a proof written by an advanced LLM, plus entailing and contradictory variants of each theorem whose truth value is guaranteed by the original. On that data it runs RL-Zero with a binary reward: the model must end by saying proved or disproved, and the variant's known truth value decides the reward. Across three olympiad-level benchmarks, RL-Zero on DeepTheorem beats supervised fine-tuning and the prior theorem-proving dataset baseline, with a 7B model reaching 47.22 average outcome score and 34.04 process score. The paper takes this as evidence that scalable, web-mined informal theorem proving is a viable path to stronger mathematical reasoning in LLMs.

What carries the argument

The load-bearing object is the set of entailing and contradictory theorem variants. For each original theorem, a variant is produced by a minimal change—for example, from $x>1$ to $x>0$ (entailing) or to $x<1$ (contradictory)—so that one variant is guaranteed true and the other guaranteed false whenever the original theorem has a definite truth value. This gives the RL training a binary outcome signal: GRPO, a policy-gradient algorithm, rewards the model with 1 when its proved/disproved answer matches the variant's known truth value and 0 otherwise, plus sanity checks against degenerate outputs. The same variant structure defines the outcome evaluation, which counts a test case as passed only when the original theorem and all entailing variants get the same truth value and all contradictory variants get the inverse.

What would settle it

A human audit of a random sample of DeepTheorem theorem–proof–variant triples: expert mathematicians would verify each theorem's truth value, each proof's validity, and each variant's entailment or contradiction relation. If a nontrivial fraction of labels are wrong, the RL reward is noisy and the outcome scores are partly scored against a faulty key; a second check is to generate new variants with a different LLM after training and see whether the model's consistency survives the distribution shift.

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Extended reading notes

Core claim

The central discovery, as the authors state it, is that a theorem statement for training and evaluation does not have to be true: a false statement can be disproved, and both directions can be scored. They therefore transform each theorem into variants by minimal logical edits, some entailed by the original and some contradictory to it, so the correct answer is known from the original theorem alone without checking a proof. This converts informal theorem proving into a binary-reward reinforcement-learning problem, and their outcome metric requires the model to assign consistent truth values across all variants. The paper reports that this scheme, trained from a base 7B model with GRPO, achieves 55.56 outcome on FIMO, 28.81 on HMMT, and 57.29 on PutnamBench, an average of 47.22, above the prior theorem-proving dataset baseline (42.90 outcome) and above every open-source model they compared, while ranking behind only three larger commercial models.

Load-bearing premise

The load-bearing premise is that the automatically generated labels are mathematically correct at every layer: the LLM's True/False validation of each theorem–proof pair, the advanced LLM's proofs, and the 72B model's entailing and contradictory variants all preserve real mathematical truth, so the binary reward and the evaluation scores measure validity rather than agreement with an LLM-generated answer key.

Editorial extensions

If this is right

  • If the central claim is correct, a 7B model trained with RL-Zero on DeepTheorem outperforms supervised fine-tuning on the same corpus at every model size tested, so RL-Zero rather than more SFT data is the lever for informal theorem-proving skill.
  • The improvement transfers across FIMO, HMMT, and PutnamBench, so the result is not an artifact of one benchmark's style.
  • Because the reward is binary and derivable from logical relations between statements, the same variant-consistency recipe could be applied to other domains where statements can be generated in entailing and contradictory pairs.
  • Process scores also improve alongside outcome scores, suggesting the trained model produces more complete, logically coherent proof traces, not just better final answers.

Reading between the lines

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

  • A testable extension the paper leaves implicit is whether the variant-consistency signal alone, without the original proof text, accounts for the gain; ablating proofs from training would separate the effect of the RL reward from the effect of seeing high-quality proofs.
  • The outcome metric could be gamed by a model that always outputs the truth value of the original theorem for every variant; scoring on fresh variants generated after training would make the metric more robust and is a natural next experiment.
  • The paper does not report an error-rate analysis of the LLM-generated labels, variants, or judge scores, and its own case study praises a proof with a shaky intermediate claim; a human audit of a random sample would clarify whether the reported outcome score reflects mathematical validity or agreement with the LLM-generated answer key.
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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

6 major / 5 minor

Summary. The paper introduces DeepTheorem, a dataset of 121K informal mathematical theorems with LLM-generated proofs and annotations, together with theorem variants for binary-reward RL. It proposes RL-Zero training using GRPO with rewards based on whether the model's proved/disproved classification matches the variant labels, and an evaluation framework with outcome and process metrics. Experiments on FIMO, HMMT, and Putnam compare DeepTheorem against OpenR1-Math-Proof and several commercial/open-source LLMs, reporting that RL-Zero with DeepTheorem improves over SFT and achieves state-of-the-art results.

Significance. The paper addresses an important problem—scaling informal theorem proving without formal proof assistants—and the release of code and data is a useful community resource. However, the headline claims currently rest on unverified LLM-generated labels, a metric that coincides with the RL reward, and a process judge that demonstrably accepts a proof with a false claim. With independent validation (human audit of variants, a human-verified test subset, multiple seeds, and corrected claims about state-of-the-art), the contributions could be significant; as presented, the evidence is not yet convincing.

major comments (6)
  1. [Section 3.1, Figure 6] The ground-truth labels for the variant construction in Section 3.1 are not verified. The prompt in Appendix A.8 instructs the model to 'negate the original statement' with minimal changes, but for statements of the form ∀x(H(x)→C(x)), negating only the conclusion yields ∀x(H(x)→¬C(x)), which is not the logical negation and can be vacuously true. Table 2's pair x>1 and x<1 is not contradictory, since for x=1 both statements are false. These labels are used both as the RL reward (Section 3.2) and as the outcome criteria (Figure 6, conditions 3–4), so a systematic mislabeling would train models to match wrong targets and make the reported numbers in Tables 4 and 5 measure agreement with an unverified distribution rather than mathematical truth. The paper reports no human audit, no error-rate analysis, and no inter-annotator agreement for the variant labels or the 'completeness annotation' described in Section 3.1.
  2. [Section 3.2 vs. Section 3.3] The outcome metric in Figure 6 is the same signal that RL-Zero optimizes: the reward function in Section 3.2 compares the extracted \boxed{proved}/\boxed{disproved} answer with the variant labels, and the outcome criteria require exactly the same consistency pattern across original, entailing, and contradictory variants. Consequently, the RL-vs-SFT comparisons in Table 4 partly measure how well the model has learned the training objective, not an independent property of theorem-proving ability. To support the claim of improved reasoning, the authors should evaluate on a human-verified test set or on variants whose truth values are formally checked, and report the correlation between the variant-consistency metric and a genuinely independent correctness signal.
  3. [Section 4.3] The case study is direct evidence that the LLM-based process evaluation is unsound. The model's solution asserts that 'the number of integer preperiodic points of P is at most n for each iterate,' which is false; for example, P(x)=x^2-1 has the three integer preperiodic points {-1,0,1} although n=2. The paper's own analysis praises this reasoning as 'clear and correct,' and the GPT-4o judge in Appendix A.10 would assign high sub-scores because it does not check the mathematical validity of this step. This shows that the process scores reported in Tables 4 and 5 cannot be interpreted as proof quality.
  4. [Section 2.1] The data construction pipeline is entirely LLM-driven: theorem justification, rationale summarization, proof generation by o3-mini, logical validation, difficulty annotation, and topic annotation each use LLM outputs with no human audit or reported error rates. The claims that the 121K theorems are 'high-quality' and 'IMO-level' rest on these unverified annotations, yet no sample is shown to have been human-checked. Similarly, the decontamination procedure in Section 2.1 uses an embedding similarity threshold of 0.7 and an LLM 'contamination justification' step, but no precision/recall evaluation is given for this filter, making the 'strict decontamination' claim unquantified.
  5. [Table 4] The experimental comparison is based on single runs with no random seeds, no confidence intervals, and no significance tests. Several reported differences are small (e.g., 1.5B RL outcome: 35.85 vs 35.69; 3B RL outcome: 38.53 vs 38.65), so the assertion in Section 4.2 that 'our RL-Zero training paradigm consistently outperforms SFT' is not statistically supported. The paper should report multiple seeds and variability or refrain from claiming consistent superiority.
  6. [Section 4.2, Table 5] The claim that 'our 7B model ... outperforms SOTA models of much larger sizes' is contradicted by the paper's own Table 5. The DeepTheorem-RL-7B average outcome score is 47.22, whereas o1, Gemini2.5-Pro, and o3-mini score 58.88, 59.78, and 68.28 respectively. The abstract's 'state-of-the-art accuracy' is therefore not supported; the accurate statement is that the 7B model outperforms several larger open-weight models, not the best commercial systems.
minor comments (5)
  1. [Section 3.3] The text says 'we manually expand each question in the three data sources into multiple entailing or contradictory variants,' but Section 3.1 and Appendix A.8 describe an LLM-driven expansion using Qwen2.5-72B-Instruct; please clarify who performs the expansion and whether any human checks the resulting variants.
  2. [Section 2.1 vs. Appendix A.6] The difficulty scale is inconsistent: Section 2 mentions 'level 5~10', Section 2.1 says 'a scale of 1 to 9', and Appendix A.6 instructs annotators to rate 'between 1 and 10, with increments of 0.5'. Please reconcile these statements.
  3. [Appendix A.10] The process evaluation weights (0.4 validity, 0.3 completeness, 0.2 correctness, 0.1 clarity) are introduced without justification or sensitivity analysis; since process scores are reported as primary results in Table 4, the weights should be justified or shown not to affect conclusions.
  4. [Section 4.2] There is a grammatical error in the paragraph beginning 'DeepTheorem achieves strong parameter efficiency': 'We demonstrates' should be 'We demonstrate'.
  5. [References] The citation for the Harvard-MIT Mathematics Tournament uses an inconsistent format with two years and a bare URL; please format it according to the journal's reference style.

Circularity Check

1 steps flagged · score 4.0 of 10

The outcome metric reuses the RL reward signal on LLM-generated variant labels, so the headline outcome gain is partly a same-signal result; external original theorems and held-out variants keep it from being fully circular.

  1. fitted input called prediction [Section 3.2 (RL reward) and Section 3.3 / Figure 6 (outcome criteria)]
    "In the reward function, we extract this answer and compare it against the ground truth, giving a reward of 1 if the answer matches, and 0 otherwise. ... A test case in a theorem testing set is passed if and only if: ... The predicted truth values for all entailing variants are the same as the original theorem; The predicted truth values for all contradictory variants are the inverse of the original theorem."

    The per-example RL reward is answer-vs-ground-truth matching on entailing/contradictory variants, and the outcome metric is the same matching operation aggregated over the original theorem plus its variants. The test variants are built by the same Section 3.1 / Appendix A.8 variant-generation protocol (Table 3 says 'manually expanded ... following the same protocol as Section 3.1'), and no independent human audit or error-rate analysis is reported for their truth labels. The reported 'state-of-the-art accuracy' therefore measures fidelity to the same LLM-generated label distribution the RL objective was trained to maximize, so it cannot by itself certify theorem-proving ability; the prediction is substantially the training signal renamed as an evaluation.

full rationale

The paper's derivation chain is: web-mined theorems -> GPT-4o validation -> o3-mini proofs -> Qwen2.5-72B variant generation -> binary reward for answer matching -> outcome metric of variant consistency. The last two links are effectively the same comparison: extract the model's \boxed answer and check it against the variant's truth label. Thus the RL-vs-SFT and DeepTheorem-vs-OpenR1 outcome gaps in Table 4 can be read partly as 'the model trained to maximize this exact matching objective matches it better,' which is circular as evidence of mathematical validity. The circularity is moderated because original-theorem truth values come from external benchmarks (FIMO, Putnam, HMMT) and the test variants are held out, so generalization is not vacuous. However, the variant-label correctness is load-bearing and unverified: Appendix A.8's negation prompt can produce non-contradictory variants (the paper's own running example 'x>1' with contradictory variant 'x<1' leaves x=1 unaddressed), and Section 4.3's case study shows the GPT-4o process judge endorsing a disproof whose preperiodic-point bound is unsupported. These facts confirm that the LLM-only validation loop does not independently establish the mathematical truth the outcome and process metrics assume. Score 4 rather than 0 because one reported prediction (outcome accuracy on variants) is evaluated with the training objective and unverified labels; not 6 because the held-out external original theorems and manual variant expansion give the comparison non-tautological content.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper's contribution is empirical, so the ledger records hand-chosen thresholds and assumptions about LLM judgment rather than fitted physical constants. The most consequential items are the assumptions that LLM-produced labels, variant transformations and process scores are correct; a single systematic failure in any of these propagates to both the training signal and the evaluation.

free parameters (4)
  • Difficulty retention threshold = >= 5 on a 1-10 scale
    Section 2.1 keeps only theorems rated at least 5; this choice determines which of the 1.08M validated pairs survive to form the 121K dataset and underpins the 'IMO-level' claim.
  • Process evaluation weights = 0.4 validity + 0.3 completeness + 0.2 correctness + 0.1 clarity
    Appendix A.10; every 'proc' score in Tables 4 and 5 is this weighted sum, so the reasoning-quality comparisons depend on these hand-chosen weights.
  • Decontamination similarity threshold and recall count = cosine threshold 0.7, top-5 recall
    Section 2.1; defines what counts as contamination, directly affecting all benchmark-integrity claims.
  • RL sanity-check thresholds = whitespace ratio < 0.05 or repetition count > 300
    Section 3.2; collapse guards that shape the reward distribution. Minor but hand-chosen.
assumptions (5)
  • domain assumption GPT-4o and o3-mini annotations of theorem correctness and proof validity are reliable enough for the 1.08M pairs and the final 121K dataset.
    Sections 2.1 and A.2-A.5; no human audit or agreement statistics are reported.
  • domain assumption LLM-constructed entailing and contradictory variants preserve the original theorem's truth value.
    Section 3.1 and A.8; Qwen2.5-72B performs negations and strengthenings; a faulty transformation corrupts both the RL reward and the test ground truth.
  • domain assumption Variant-consistency is a valid proxy for theorem-proving correctness.
    Section 3.3, stated as a proxy by the authors; the metric scores truth-value consistency, not proof validity.
  • domain assumption GPT-4o as process judge scores proof quality accurately.
    Sections 3.3 and A.10; Section 4.3's case study is evidence the judge is lenient with a proof containing a false claim.
  • domain assumption Embedding-based decontamination removes all overlaps with testing benchmarks.
    Section 2.1; the pipeline relies on embeddings, a 0.7 threshold and an LLM judgment; residual overlap would inflate scores.

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Cite this review

Pith. "Pith review of DeepTheorem: Advancing LLM Reasoning for Theorem Proving Through Natural Language and Reinforcement Learning." pith.science (2026). https://pith.science/paper/6O3VAIXJ

@misc{pith2026250523754,
  author       = {Pith},
  title        = {Pith review of: DeepTheorem: Advancing LLM Reasoning for Theorem Proving Through Natural Language and Reinforcement Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6O3VAIXJ}},
  note         = {Machine review of arXiv:2505.23754}
}
read the original abstract

Theorem proving serves as a major testbed for evaluating complex reasoning abilities in large language models (LLMs). However, traditional automated theorem proving (ATP) approaches rely heavily on formal proof systems that poorly align with LLMs' strength derived from informal, natural language knowledge acquired during pre-training. In this work, we propose DeepTheorem, a comprehensive informal theorem-proving framework exploiting natural language to enhance LLM mathematical reasoning. DeepTheorem includes a large-scale benchmark dataset consisting of 121K high-quality IMO-level informal theorems and proofs spanning diverse mathematical domains, rigorously annotated for correctness, difficulty, and topic categories, accompanied by systematically constructed verifiable theorem variants. We devise a novel reinforcement learning strategy (RL-Zero) explicitly tailored to informal theorem proving, leveraging the verified theorem variants to incentivize robust mathematical inference. Additionally, we propose comprehensive outcome and process evaluation metrics examining proof correctness and the quality of reasoning steps. Extensive experimental analyses demonstrate DeepTheorem significantly improves LLM theorem-proving performance compared to existing datasets and supervised fine-tuning protocols, achieving state-of-the-art accuracy and reasoning quality. Our findings highlight DeepTheorem's potential to fundamentally advance automated informal theorem proving and mathematical exploration.

Figures

Figures reproduced from arXiv: 2505.23754 by the authors.

Figure 1
Figure 1. (a): Our dataset surpasses others with extremely challenging theories; (b): RL-Zero training [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. A data sample from the DeepTheorem dataset. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Difficulty density histogram comparison across widely used training dataset (i.e. Lean [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Statistics of DeepTheorem hierarchical topics. Diverse topics We categorize all the data sam￾ples from DeepTheorem following Gao et al. (2025). As shown in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: The process pipeline of constructing DeepTheorem dataset. As shown in [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: Outcome evaluation criteria given a testcase with variants [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Distribution of used proof techniques. DeepTheorem achieves strong parame￾ter efficiency We demonstrates that our DeepTheorem-RL strategy achieves strong parameter efficiency in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Distribution of used proof techniques. Reasoning with theorem proving skills In [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

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Forward citations

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Reference graph

Works this paper leans on

18 extracted references · 15 canonical work pages · cited by 4 Pith papers

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    Identify the key mathematical concepts, ideas, or rationales driving the discussion

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  2. [2]

    Let f:R→C be a smooth function,

    Act as a teacher to formulate a theorem based on the discussion, presented as a formal theorem statement. # Requirements - All mathematical equations must be formatted in LaTeX. - The theorem should be a clear, formal statement (e.g., "Let f:R→C be a smooth function, ..."). - The output must be in JSON format, with the following structure: { 18 DeepTheore...

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    \boxed{proved}

    The original solution ends with " \boxed{proved}". Your new solution should end with "\boxed{disproved}". Output the new question and solution in two blocks: “‘question new question “‘ “‘solution corresponding solution “‘ 29 DeepTheorem A.9 System Prompt for RL Training System Prompt for RL Training A conversation between User and Assistant. The User give...

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    ‘ { "score

    Provide a brief explanation (2-3 sentences) summarizing any errors or issues and justifying the score. Final output format: “‘ { "score": float, "validity": float, "completeness": float, "correctness": float, "clarity": float, 30 DeepTheorem "explanation": str } “‘ where "score" is the total score, and "validity", "completeness", "correctness", "clarity" ...

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    - Check for adherence to mathematical definitions, theorems, or properties cited in the step

    **Analyze Each Step**: - Verify if the step is mathematically correct, logically sound, and relevant to proving or disproving the theorem. - Check for adherence to mathematical definitions, theorems, or properties cited in the step. - Ensure the step follows from previous steps or given assumptions without logical gaps. - If the step involves a disproof, ...

    work page

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    - If invalid, return False and summarize the critical errors and recommend how to fix the proof/disproof

    **Overall Assessment**: - Conclude whether the entire proof or disproof is valid. - If invalid, return False and summarize the critical errors and recommend how to fix the proof/disproof. - If valid, return True and confirm that it fully addresses the theorem. A.6 Difficulty Annotation Meta Prompt for Difficulty Annotation # CONTEXT # I am a teacher, and ...

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    Determine the maximum amount of white balls in the drawer, such that the probability statement is true? 22 DeepTheorem **Problem 2/6: 4.5** Find the lowest possible values from the function f(x) =x 2008 −2x 2007 +3x 2006 −4x 2005 +5x 2004 − · · · −2006x3 +2007x 2 −2008x+2009 for any real numbersx. **Problem 3/7: 5** A pair of integers(m,n)is called good i...

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    Based on the source of the given problem, as well as the difficulty of the problems referenced in these materials and the solution to the current problem, please provide an overall difficulty score for the current problem. The score should be a number between 1 and 10, with increments of 0.5, and should align perfectly with the materials. # STYLE # Data r...

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    ‘json {

    Therefore, the number that replaces the□is 6. [Source]: 2010_Pascal </example math problem> ## Summarization The problem requires finding a value that makes the equation 1 9 + 1 18 = 1 □. This involves adding two fractions and determining the equivalent fraction. ## Difficulty...

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    Summarize the math problem in a brief sentence, describing the concepts involved in the math problem

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    Other" to the corresponding branch. For example: Algebra -> Intermediate Algebra -> Other. Only the LAST NODE is allowed to be

    Categorize the math problem into specific mathematical domains. Please provide a classification chain, for example, Applied Mathematics -> Probability -> Combinations. The following is a basic classification framework in the field of mathematics. <math domains> ... </math doma...

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    prove or disprove

    The original question asks to "prove or disprove" a statement, where the statement can be "proved". Please write a new question by negating the original statement, so that it can now be "disproved". For example, if the original statement is x=y , you may change it to x<y or x̸...

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    since we proved x=y , the statement x<y is disproved

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    Flag any logical errors

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Reviewed August 7, 2026 · model on record in the stance chip above.