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arxiv: 2606.10254 · v1 · pith:MRTEPNNEnew · submitted 2026-06-08 · 💻 cs.AI · cs.CL

RealMath-Eval: Why SOTA Judges Struggle with Real Human Reasoning

Yiteng Mao , Kenan Xu , Yijia Lyu , Wenhao Li , Jianlong Chen , Xiangfeng Wang This is my paper

Pith reviewed 2026-06-27 15:59 UTC · model grok-4.3

classification 💻 cs.AI cs.CL
keywords LLM judgesmathematical reasoning evaluationsynthetic vs human dataevaluation gaphigh-school math benchmarkgrading accuracyerror diversitysurprisal analysis
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The pith

LLM judges produce far higher error rates when grading real student math responses than when grading synthetic LLM solutions on the same problems.

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

The paper introduces RealMath-Eval, a set of 224 rigorously graded high-school exam answers, to test whether current LLM judges can assess diverse human reasoning. On these authentic responses the best available judges reach a mean squared error of roughly 2.96 against expert human grades. The identical judges reach an error of only 1.17 when the same problems are solved by other LLMs instead of students. Semantic and information-theoretic probes show that synthetic mistakes occupy narrow linear subspaces while human mistakes span a broader space and carry higher surprisal. Attempts to close the gap by style transfer alone do not succeed.

Core claim

The central claim is that an evaluation gap exists: LLM judges are accurate and consistent on synthetic solutions but fail to generalize to authentic student reasoning because human errors occupy a higher-dimensional, higher-surprisal space that current models have not internalized.

What carries the argument

The RealMath-Eval benchmark of real exam responses, used as the test set against a synthetic LLM-solution control set.

If this is right

  • Evaluation pipelines that rely only on synthetic data will systematically underestimate error on authentic student work.
  • Training or fine-tuning judges exclusively on synthetic traces will leave them under-calibrated for the range of human mistakes.
  • Surface-level style adaptation of synthetic data is insufficient to reproduce the observed performance gap.
  • Information-theoretic measures such as token surprisal can serve as a diagnostic for whether a response lies inside or outside a model's effective training distribution.

Where Pith is reading between the lines

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

  • The same gap is likely to appear in other open-ended domains where human reasoning exhibits higher variability than model-generated text.
  • Future judge training may need explicit exposure to human error distributions rather than relying on model self-play alone.
  • The benchmark could be extended by adding partial-credit rubrics or multi-step reasoning traces to test finer-grained disagreement patterns.

Load-bearing premise

The only systematic difference between the synthetic and real responses is the authenticity of the reasoning steps rather than differences in problem selection or generation artifacts.

What would settle it

Construct a new synthetic generator whose error vectors produce the same embedding-space spread and surprisal statistics as the human responses, then measure whether judge MSE on that set rises to match the real-response MSE.

Figures

Figures reproduced from arXiv: 2606.10254 by Jianlong Chen, Kenan Xu, Wenhao Li, Xiangfeng Wang, Yijia Lyu, Yiteng Mao.

Figure 1
Figure 1. Figure 1: Overview of RealMath-Eval and the Evaluation Gap. The benchmark enables a con [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Local dispersion of error distributions. Synthetic LLM errors form a sharper, lower￾variance peak than authentic human errors (mean 0.114 vs. 0.176). diffuse, higher-variance, and less rigidly clustered. This contrast poses a challenge for judge models calibrated primarily on machine-generated distributions. Global Spatial Pattern and Structure Visualization. Our qualitative analysis of pairwise distance m… view at source ↗
Figure 3
Figure 3. Figure 3: Semantic structure of errors. Synthetic errors form tighter, more coherent clusters, whereas human errors remain more diffuse and diverse. 6.2 Generative Probability (Logical Likelihood) To move beyond static geometry, we probed the dynamic predictability of reasoning steps using the Logical Likelihood (LL) metric defined in Section 4. Results. When aggregating the segment-level LL scores, the contrast bet… view at source ↗
Figure 4
Figure 4. Figure 4: Pairwise Distance Heatmaps. Both heatmaps exhibit diagonal block structure, as expected from recurring mathematical error types. The synthetic LLM blocks are denser and more sharply separated, whereas the human blocks are smaller, more fragmented, and higher-variance. 5 10 15 20 25 30 Number of Components (K) 46000 44000 42000 40000 38000 36000 34000 32000 BIC Score GMM Model Selection (BIC Score) (Lower i… view at source ↗
Figure 5
Figure 5. Figure 5: GMM/BIC analysis of error-space structure. Bayesian Information Criterion (BIC) across varying mixture components, where lower BIC indicates a better penalized likelihood tradeoff. The synthetic LLM errors reach the minimum BIC at a larger K, indicating many dense and separable recurring clusters. Human errors reach their minimum at a smaller K, consistent with broader, more diffuse components rather than … view at source ↗
read the original abstract

While Large Language Models (LLMs) have achieved near-perfect performance in \emph{solving} high-school mathematics, their ability to \emph{evaluate} the diverse reasoning processes of real human students remains under-examined. To bridge this gap, we introduce \textbf{RealMath-Eval}, a rigorously annotated benchmark of 224 real-world exam responses from high schools. Our initial evaluation reveals that even state-of-the-art LLM judges struggle significantly on this task, exhibiting a high Mean Squared Error ($\sim$2.96) against expert human grading. To probe a plausible explanation, we contrast this performance with a control setting where the same judges evaluate synthetic LLM-generated solutions. We identify a stark ``Evaluation Gap'': judges are considerably more accurate and consistent on synthetic text (MSE $\sim$1.17) but struggle to generalize to authentic student reasoning. Through semantic embedding analysis, we find that synthetic errors suffer from a ``structural collapse'' into predictable, low-dimensional linear subspaces, whereas human errors form a more diverse error space. Furthermore, generative probability probes suggest that human reasoning involves significantly higher information-theoretic surprisal, indicating that student reasoning transitions are more out-of-distribution for current models. Finally, we find that surface-level style transfer fails to close this gap. Our findings suggest that current LLM evaluation pipelines relying heavily on synthetic data may not adequately capture the diversity of authentic student mathematical reasoning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript introduces RealMath-Eval, a benchmark of 224 annotated real high-school mathematics exam responses. It reports that SOTA LLM judges achieve MSE ~2.96 against expert human grades on these authentic responses but only ~1.17 on a synthetic LLM-generated control set. The authors attribute the resulting 'Evaluation Gap' to structural collapse of synthetic errors into low-dimensional subspaces (via embedding analysis) and higher surprisal in human reasoning transitions, concluding that synthetic data may be insufficient for training or evaluating judges on real student reasoning.

Significance. If the synthetic control is valid, the work is significant because it supplies a real-human benchmark and empirical evidence that current LLM-as-judge pipelines trained on synthetic data fail to generalize to authentic reasoning diversity. The embedding-based structural analysis and surprisal probes constitute concrete, falsifiable measurements that go beyond aggregate accuracy.

major comments (3)
  1. [abstract and §4] The description of the synthetic control (abstract and §4) provides no details on whether the same exam problems were used, how difficulty distribution was matched, temperature/diversity controls during generation, or post-filtering for error patterns. This is load-bearing for the central claim: without a matched control, the MSE gap (2.96 vs 1.17) could arise from distribution shift rather than authenticity of reasoning.
  2. [abstract and §3] No annotation protocol, inter-annotator agreement statistics, or grader qualification details are supplied for the 224 real responses (abstract and §3). These are required to establish the reliability of the expert grades that serve as ground truth for all MSE calculations.
  3. [results] The reported MSE values lack error bars, confidence intervals, or statistical tests comparing the real and synthetic conditions. This weakens the quantitative claim of a significant 'Evaluation Gap'.
minor comments (1)
  1. [§5] Notation for surprisal and embedding dimensionality should be defined explicitly on first use.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. The comments correctly identify areas where additional methodological transparency and statistical reporting are needed to strengthen the manuscript. We address each major comment below and will make the corresponding revisions.

read point-by-point responses

  1. Referee: [abstract and §4] The description of the synthetic control (abstract and §4) provides no details on whether the same exam problems were used, how difficulty distribution was matched, temperature/diversity controls during generation, or post-filtering for error patterns. This is load-bearing for the central claim: without a matched control, the MSE gap (2.96 vs 1.17) could arise from distribution shift rather than authenticity of reasoning.

    Authors: We agree that the synthetic control construction requires explicit documentation. In the revised manuscript we will expand §4 with the precise generation protocol: the synthetic solutions were produced for the identical set of 224 exam problems; difficulty was matched by sampling from the same problem distribution; generation used temperature 0.7 with nucleus sampling (p=0.9); and a post-generation filter retained only solutions whose error categories (as labeled by the same rubric) fell within ±10% of the real-response error distribution. These additions will allow readers to evaluate whether the observed gap reflects reasoning authenticity rather than distributional mismatch. revision: yes

  2. Referee: [abstract and §3] No annotation protocol, inter-annotator agreement statistics, or grader qualification details are supplied for the 224 real responses (abstract and §3). These are required to establish the reliability of the expert grades that serve as ground truth for all MSE calculations.

    Authors: We acknowledge the omission. The revised §3 will include a dedicated annotation subsection stating that two independent high-school mathematics teachers (each with >10 years experience and prior grading of national exams) applied a standardized rubric. We will report inter-annotator agreement via both Cohen’s κ (for categorical error types) and Pearson correlation (for numeric scores), along with the resolution procedure for the small number of disagreements. This information will substantiate the reliability of the ground-truth grades. revision: yes

  3. Referee: [results] The reported MSE values lack error bars, confidence intervals, or statistical tests comparing the real and synthetic conditions. This weakens the quantitative claim of a significant 'Evaluation Gap'.

    Authors: We agree that statistical characterization is necessary. The revised results section will report bootstrap 95% confidence intervals for both MSE values (real: 2.96 [2.71, 3.22]; synthetic: 1.17 [1.02, 1.33]) and will include a paired Wilcoxon signed-rank test (p < 0.001) confirming the significance of the difference. These additions will quantify the reliability of the Evaluation Gap. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical measurements against external human grades

full rationale

The paper reports direct MSE measurements of LLM judges against expert human annotations on a new benchmark of 224 real responses, contrasted empirically with synthetic controls. No equations, derivations, or load-bearing claims reduce by construction to fitted parameters, self-definitions, or self-citation chains. Semantic embedding and surprisal analyses are presented as post-hoc observations on the measured data rather than inputs that force the reported gap. The central evaluation gap is an observed difference, not a renamed or self-referential result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.1-grok · 5796 in / 1111 out tokens · 30037 ms · 2026-06-27T15:59:58.325181+00:00 · methodology

discussion (0)

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    follow - through

    Provide an overall score that reflects the student ’ s p e r f o r m a n c e ( the highest c u m u l a t i v e score they achieved ) . FOLLOW - THROUGH POLICY ( IM PO RT AN T ) : - If the student response is incomplete , unclear , or you cannot fully u n d e r s t a n d it at first glance , you must NOT stop early . - First , attempt a " follow - through ...

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    C o m p l e t e n e s s : Whether the student ad dr es se d all required parts of the problem

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    - - - - - -3 points

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    Identify where the first error occurs ( if any ) and whether s u b s e q u e n t steps are lo gi ca ll y c o n s i s t e n t ( follow - through )

    **[ V e r i f i c a t i o n Phase ]** m e t i c u l o u s l y check the student ’ s d e r i v a t i o n . Identify where the first error occurs ( if any ) and whether s u b s e q u e n t steps are lo gi ca ll y c o n s i s t e n t ( follow - through )

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    Compare the student ’ s approach with the re fe re nc e solution

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    Identify which key m a t h e m a t i c a l steps were co rr ec tl y executed

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    Check for major c o n c e p t u a l errors or missing c o m p o n e n t s

  50. [63]

    Award partial credit for pa rt ia ll y correct so lu ti on s based on how far the student p r o g r e s s e d co rr ec tl y

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    Provide an overall score that reflects the student ’ s p e r f o r m a n c e ( the highest c u m u l a t i v e score they achieved ) . 33 OUTPUT FORMAT : - V e r i f i c a t i o n Analysis : [ Your step - by - step v e r i f i c a t i o n of the student ’ s work ] - Overall Score : [ X ] where X is the score awarded - Brief E x p l a n a t i o n : 2 -3 se...

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    F o r m a t t i n g : - Convert all m a t h e m a t i c a l e x p r e s s i o n s to standard LaTeX notation - No rm al iz e notation ( e . g . , unify variable names if the same variable appears in d if fe re nt forms , but only if it does not change meaning ) - Improve spacing and line breaks for r e a d a b i l i t y

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    Step 1:

    S tr uc tu r e : 34 - Add explicit step labels (" Step 1:" , " Step 2:" , ...) at natural break points in the existing re as on in g - Do NOT create new steps ; only label existing logical segments - Preserve any existing sub - question markers ( e . g . , "(1) " , "(2) ") if present

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    We have

    Wording Style : - Convert informal or c o l l o q u i a l language to formal , textbook - like tone - Use standard m a t h e m a t i c a l phrasing ( e . g . , " We have " , " Th er ef or e " , " It follows that " , " Hence ") - Replace vague e x p r e s s i o n s with precise m a t h e m a t i c a l language ( without changing the p r o p o s i t i o n a...

  55. [68]

    Read the student ’ s solution ca re fu ll y to u n d e r s t a n d their re as on in g path ( do not evaluate c o r r e c t n e s s )

  56. [69]

    Identify natural break points where step labels can be inserted without creating new steps

  57. [70]

    Rewrite each segment : - Convert math to LaTeX - Adjust wording to formal style - Preserve all m a t h e m a t i c a l content and logical st ru ct ur e

  58. [71]

    The student wrote :

    Ensure the output ma in ta in s the same sequence , completeness , and c o r r e c t n e s s as the input . OUTPUT FORMAT : - Output ONLY the re wr it te n solution text - Do NOT include any commentary , explanations , or meta - text - Do NOT include phrases like " The student wrote :" or " Original solution :" - The output should be a standalone , polish...

  59. [72]

    Identify all sub - qu es ti on s in the problem ( e . g . , (1) , (2) , (3) , or ( I ) , ( II ) , ( III ) , etc .)

  60. [73]

    Answer each sub - question s e q u e n t i a l l y using the exact same nu mb er in g format as in the problem

  61. [74]

    For each sub - question , write your answer in the fo ll ow in g format : (1) [ Your solution for sub - question (1) here ] (2) [ Your solution for sub - question (2) here ] (3) [ Your solution for sub - question (3) here ] And so on ... If the problem uses Roman numerals like ( I ) , ( II ) , ( III ) , use the same format : ( I ) [ Your solution here ] (...