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arxiv: 2605.25384 · v1 · pith:3FBY6TL6new · submitted 2026-05-25 · 💻 cs.CL

GeoMathCode: Understanding Interleaved Math-Code Reasoning for Geometry Problem Solving

Yingji Zhang , Yong Dai , Andr\'e Freitas This is my paper

Pith reviewed 2026-06-29 22:41 UTC · model grok-4.3

classification 💻 cs.CL
keywords geometry problem solvingmultimodal large language modelslatent space disentanglementsupervised fine-tuningprogrammatic representationsmath-code reasoningsymbolic information
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The pith

Supervised fine-tuning disentangles reasoning steps from code generation in the latent space of geometry-solving models.

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

The paper presents GeoMathCode as a way to use programmatic code as intermediate outputs that stand in for visual constructions when solving geometry problems. It examines the internal representations of multimodal models during interleaved math and code reasoning. After supervised fine-tuning the reasoning manifold becomes more structured while code-related subspaces separate out and carry richer symbolic mathematical content than visual features do. A sympathetic reader would care because clearer separation of these processes could point toward more reliable ways to build models that handle geometric deduction and symbolic manipulation together. The work focuses on showing that these latent changes occur as a direct result of the fine-tuning process applied to the interleaved reasoning setup.

Core claim

In the GeoMathCode setup programmatic representations function as intermediate visual outputs for geometry problems. Reasoning and code generation steps can be disentangled in the latent space, supervised fine-tuning makes the reasoning manifold more structured and informative, and hierarchical syntactic code structures appear as disentangled latent subspaces that contain more mathematical symbolic information than visual representations.

What carries the argument

Disentanglement of reasoning and code-generation trajectories in the latent space of fine-tuned multimodal models, with hierarchical syntactic code structures forming separate subspaces.

If this is right

  • Hierarchical code subspaces can be inspected or edited independently of the main reasoning path.
  • Programmatic intermediates supply more usable symbolic information than purely visual auxiliary constructions.
  • The reasoning manifold after fine-tuning supports more reliable multi-step deduction in geometry tasks.
  • Code generation steps can be isolated without disrupting the overall problem-solving flow.

Where Pith is reading between the lines

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

  • Similar latent-space analysis might reveal whether code intermediates help in other domains that mix symbolic and visual reasoning such as physics diagram problems.
  • Explicit objectives that encourage subspace separation could be added during training to amplify the observed effect.
  • If the subspaces truly isolate symbolic content they could be used to diagnose specific failure modes like algebraic errors versus geometric misinterpretation.

Load-bearing premise

The structure that appears in latent space after supervised fine-tuning actually tracks genuine gains in geometric reasoning ability rather than being produced by the way representations are extracted or by properties of the training data alone.

What would settle it

A controlled experiment that applies the same supervised fine-tuning but measures no corresponding rise in accuracy on geometry problems whose solutions require symbolic manipulation would show the latent changes are not tied to improved reasoning.

Figures

Figures reproduced from arXiv: 2605.25384 by Andr\'e Freitas, Yingji Zhang, Yong Dai.

Figure 1
Figure 1. Figure 1: Pipeline overview. We construct GeoMathCode dataset for supervised fine-tuning. compasses eight topics: Algebra, Analytic Geome￾try, Calculus, Trigonometry, Plane Geometry, Solid Geometry, Statistics, and Transformational Geom￾etry. For each category, we first evaluate multi￾ple state-of-the-art MLLMs as baselines (Gemini3- Flash, Gemini3-Pro, GPT-5.1, GPT-5.2) for auto￾matic solution generation. As shown … view at source ↗
Figure 2
Figure 2. Figure 2: PCA visualisation of the reasoning step and code step geometry (Top: Qwen3.5-9B, bottom: Qwen3.5- [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Average euclidean distance of different steps at different layers (Top: Qwen3.5-9B, bottom: Qwen3.5-9B [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: ERank ↑ (left) and ID ↓ (right) for Qwen3-VL-Ins-8B (top) and Qwen3.5-9B (bottom). reasoning and diagram generation. Specifically, the model primarily relies on textual reasoning trajecto￾ries for final-answer prediction, while intermediate code generation mainly serves as an auxiliary exe￾cutable representation rather than an active reason￾ing modality. This behaviour differs from visual interleaved chain… view at source ↗
Figure 5
Figure 5. Figure 5: Code syntax space visualisation and linear probing results for Qwen3.5-9B (Top) and Qwen3.5-9B [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Average euclidean distance of different steps at different layers (Top: Qwen3-ins-8B, bottom: Qwen3-ins [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Average euclidean distance of different steps at different layers (Top: InternVL3.5-8B, bottom: [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
read the original abstract

Mathematical reasoning is a hallmark of human intelligence, requiring logical deduction, symbolic manipulation, and abstract thinking. Recent multimodal large language models (MLLMs) have demonstrated strong performance on geometry problems through multi-step reasoning. To better emulate human problem-solving, intermediate steps can incorporate auxiliary visual constructions, such as additional lines or points, which improve geometric interpretation and educational clarity. In this work, we introduce the GeoMathCode, where programmatic representations serve as intermediate visual outputs. We further conduct an in-depth analysis of the underlying reasoning geometry. Experimental results show that reasoning and code generation steps can be disentangled in the latent space, while supervised fine-tuning (SFT) makes the reasoning manifold more structured and informative. Moreover, hierarchical syntactic code structures emerge as disentangled latent subspaces, and contain more mathematical symbolic information than visual representations.

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 / 0 minor

Summary. The paper introduces GeoMathCode, a framework using programmatic representations as intermediate visual outputs for geometry problem solving in multimodal LLMs. It claims that reasoning and code generation steps can be disentangled in latent space, that supervised fine-tuning (SFT) makes the reasoning manifold more structured and informative, and that hierarchical syntactic code structures emerge as disentangled latent subspaces containing more mathematical symbolic information than visual representations.

Significance. If the empirical claims are substantiated with proper controls and metrics, the work could advance interpretability of how MLLMs internally represent interleaved math-code reasoning for geometry, potentially guiding more effective fine-tuning and highlighting advantages of code-based intermediates. The focus on latent manifold structure and disentanglement offers a novel angle on geometric reasoning beyond standard accuracy metrics.

major comments (3)
  1. [Abstract] Abstract: The central claims about disentanglement of reasoning vs. code steps, structured reasoning manifolds after SFT, and hierarchical syntactic code subspaces carrying more symbolic information lack any reported experimental details, datasets, metrics, controls, or ablation studies, preventing assessment of whether the observations support the claims.
  2. [Abstract] Abstract: The claim that observed latent subspaces reflect improved geometric reasoning (rather than extraction artifacts or data distribution effects) is load-bearing but unsupported without ablations varying representation extraction procedures or comparing against non-reasoning controls such as code-only fine-tuning or shuffled labels.
  3. [Abstract] Abstract: The comparison that hierarchical syntactic code structures 'contain more mathematical symbolic information than visual representations' requires an explicit, reproducible definition and quantification of 'mathematical symbolic information'; absent this, the claim is not falsifiable or comparable.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on the abstract. We agree that the abstract, as a concise summary, omitted key experimental details and definitions. We have revised the abstract to reference the datasets, metrics, controls, ablations, and the operational definition of mathematical symbolic information, while directing readers to the relevant sections for full details. We address each point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claims about disentanglement of reasoning vs. code steps, structured reasoning manifolds after SFT, and hierarchical syntactic code subspaces carrying more symbolic information lack any reported experimental details, datasets, metrics, controls, or ablation studies, preventing assessment of whether the observations support the claims.

    Authors: The abstract is intended as a high-level summary. The full experimental details—including the geometry problem datasets (e.g., GeoQA and related benchmarks), metrics for disentanglement and manifold structure (linear probing accuracy, intrinsic dimensionality), controls (code-only fine-tuning, shuffled labels), and ablation studies on representation extraction—are reported in Sections 3–5. The revised abstract now briefly notes these elements and points to the relevant sections. revision: yes

  2. Referee: [Abstract] Abstract: The claim that observed latent subspaces reflect improved geometric reasoning (rather than extraction artifacts or data distribution effects) is load-bearing but unsupported without ablations varying representation extraction procedures or comparing against non-reasoning controls such as code-only fine-tuning or shuffled labels.

    Authors: The manuscript contains the requested ablations: we vary representation extraction (different layers, PCA vs. nonlinear methods) and include non-reasoning controls (code-only SFT and shuffled reasoning labels). These results, showing that disentanglement is not an artifact, appear in Section 4.2. The revised abstract now references the presence of these controls. revision: yes

  3. Referee: [Abstract] Abstract: The comparison that hierarchical syntactic code structures 'contain more mathematical symbolic information than visual representations' requires an explicit, reproducible definition and quantification of 'mathematical symbolic information'; absent this, the claim is not falsifiable or comparable.

    Authors: We define 'mathematical symbolic information' as the mutual information between latent representations and symbolic geometric elements (theorems, equations, relations), quantified via linear probe accuracy and information-theoretic measures. This definition and the associated quantification procedure are formalized in Section 3.4. The revised abstract now includes a concise statement of this definition. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents experimental observations on latent-space disentanglement after SFT on interleaved math-code data for geometry problems. No load-bearing step reduces by construction to its inputs via self-definition, fitted-parameter renaming, or self-citation chains. Claims rest on post-hoc analysis of representations rather than tautological re-derivation of the same quantities. The derivation is self-contained against external benchmarks and does not invoke uniqueness theorems or ansatzes from prior self-work as forcing mechanisms.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no details on free parameters, axioms, or invented entities are provided.

pith-pipeline@v0.9.1-grok · 5667 in / 1056 out tokens · 22553 ms · 2026-06-29T22:41:58.071262+00:00 · methodology

discussion (0)

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