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GeoReform: Reflective Formalization Evolution for Multimodal Geometry Problem Solving
Jialu Wang, Ruichen Zhang, Xiaoou Liu, Hua Wei, Tianlong Chen
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 10/11/2026, 2:16:39 AM
Summary
The paper introduces GeoReform, a reflective formalization evolution framework designed to improve multimodal geometry problem solving (GPS) by large language models. It addresses the challenge that naive extraction of geometric entities and relations often introduces redundant or ambiguous constraints that distract reasoning models. GeoReform treats formalization as an optimizable policy, using a genetic search loop with reflective mutation based on failed reasoning rollouts to iteratively refine how geometric information is selected, grounded, and presented. Experiments on the Geometry3K benchmark show significant accuracy improvements (e.g., from 42.0% to 56.0% for Qwen3VL-2B) compared to initial policies and other baselines like InterGPS and AutoGPS.
Entities (10)
Relation Signals (8)
GeoReform → improvesaccuracyon → Geometry3K
confidence 95% · On Geometry3K, GeoReform improves Qwen3VL-2B accuracy from 42.0% to 56.0%.
GeoReform → uses → Reflective Formalization Evolution
confidence 95% · we introduce reflective formalization evolution in Sec. 3.3... GeoReform optimizes the policy that turns a geometry problem into a formal representation.
InternVL3.5 → backbonefor → GeoReform
confidence 90% · Table 2: Performance comparison of different policies on Geometry3K using InternVL3.5.
Qwen3VL-2B → backbonefor → GeoReform
confidence 90% · On Geometry3K, GeoReform improves Qwen3VL-2B accuracy from 42.0% to 56.0%.
GeoReform → evaluatedon → GeoQA
confidence 90% · Experiments were conducted on three specialized GPS benchmarks: Geometry3K, GeoQA, and PGPS9K.
GeoReform → evaluatedon → PGPS9K
confidence 90% · Experiments were conducted on three specialized GPS benchmarks: Geometry3K, GeoQA, and PGPS9K.
GeoReform → outperforms → InterGPS
confidence 90% · Table 2 shows GeoReform achieving higher accuracy than InterGPS across various model scales on Geometry3K.
GeoReform → outperforms → AutoGPS
confidence 90% · Table 2 shows GeoReform achieving higher accuracy than AutoGPS across various model scales on Geometry3K.
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Abstract
Abstract:Multimodal large language models (MLLMs) often struggle to identify and use geometric relations in diagrams. Recent methods address this challenge by converting geometric entities, relations, and constraints into explicit textual representations for the model to reason over. However, effective formalization is highly non-trivial: on Geometry3K, structure injection fixes 28 errors but introduces 13 new ones among 200 examples. Redundant relations can distract the model, while ambiguous references to diagram elements can lead it to apply constraints incorrectly. This suggests that the key challenge is not merely extracting more geometric facts, but organizing them into representations that support downstream reasoning. To fully exploit the power of formalization, we further propose GeoReform, a reflective formalization evolution framework that treats formalization as an optimizable policy rather than a fixed parser output. GeoReform executes the full reasoning pipeline, collects failed rollouts, diagnoses defects in the current representation, and mutates the policy to better select, ground, group, and present geometric entities, relations, constraints, and targets. On Geometry3K, GeoReform improves Qwen3VL-2B accuracy from 42.0\% to 56.0\%. Extensive experiments and analyses across geometry reasoning benchmarks demonstrate that effective formalization is crucial for improving multimodal geometry reasoning.
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- Source: https://arxiv.org/abs/2610.12391v1
- Canonical: https://arxiv.org/abs/2610.12391v1
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GeoReform: Reflective Formalization Evolution for Multimodal Geometry Problem Solving Jialu Wang 1 , Ruichen Zhang 2 , Xiaoou Liu 3 , Hua Wei 3 , Tianlong Chen 2,† 1 Tongji University 2 University of North Carolina at Chapel Hill 3 Arizona State University † Corresponding author: tianlong@cs.unc.edu Abstract Multimodal large language models (MLLMs) often struggle to identify and use geometric relations in diagrams. Recent methods ad- dress this challenge by converting geometric entities, relations, and constraints into explicit textual representations for the model to rea- son over. However, effective formalization is highly non-trivial: on Geometry3K, structure injection fixes 28 errors but introduces 13 new ones among 200 examples. Redundant rela- tions can distract the model, while ambiguous references to diagram elements can lead it to apply constraints incorrectly. This suggests that the key challenge is not merely extract- ing more geometric facts, but organizing them into representations that support downstream reasoning. To fully exploit the power of formal- ization, we further propose GeoReform, a re- flective formalization evolution framework that treats formalization as an optimizable policy rather than a fixed parser output. GeoReform executes the full reasoning pipeline, collects failed rollouts, diagnoses defects in the current representation, and mutates the policy to better select, ground, group, and present geometric entities, relations, constraints, and targets. On Geometry3K, GeoReform improves Qwen3VL- 2B accuracy from 42.0% to 56.0%. Extensive experiments and analyses across geometry rea- soning benchmarks demonstrate that effective formalization is crucial for improving multi- modal geometry reasoning. 1 Introduction Geometry problem solving (GPS) requires both accurate understanding of diagrams and rigorous mathematical reasoning (Seo et al., 2015; Chen et al., 2021; Lu et al., 2021; Chen et al., 2022; Zhang et al., 2024a; Lu et al., 2024; Zhang et al., 2024b; Qiao et al., 2025). Recent multimodal large language models (MLLMs) have shown strong per- formance on a wide range of vision and language tasks, including visual question answering, instruc- tion following, and multimodal reasoning (Alayrac et al., 2022; Li et al., 2023; Dai et al., 2023; Liu et al., 2023; Bai et al., 2023; Zhu et al., 2024). How- ever, their success on general tasks does not directly translate to reliable geometry reasoning. MLLMs often struggle to extract and model detailed geo- metric entities and relations, such as points, seg- ments, angles, perpendicular lines, and equality constraints (Gao et al., 2025; Xia et al., 2025; Ning et al., 2025; Cui and Liu, 2025). To address these difficulties, recent GPS meth- ods explicitly represent geometric entities, rela- tions, and constraints and provide them to the rea- soning model (Gao et al., 2025; Xia et al., 2025; Ning et al., 2025; Cui and Liu, 2025). However, we observe that effective formalization is highly non-trivial. Although a clear structure can guide MLLMs toward relevant entities, relations, and con- straints (Lu et al., 2021; Chen et al., 2021, 2022; Zhang et al., 2023a; Xia et al., 2025), poorly de- signed formalizations can also interfere with the reasoning process. On 200 Geometry3K prob- lems (Lu et al., 2021), adding explicit geometric entities, relations, and constraints to the input of Qwen3VL-8B corrects 28 previously incorrect an- swers but turns 13 previously correct answers into errors. We further observe that such interference occurs in two common ways. First, redundant or weakly relevant relations can distract the reasoning from the simplest solution path. As shown in Figure 1 Case 1, the problem can be solved directly using the fact that the diagonals of a parallelogram bi- sect each other, but the formalized input introduces additional angle relations that steer the reasoning away from the key segment equality. Second, am- biguous formalizations can be amplified by down- stream reasoning. As shown in Figure 1 Case 2, angle relations with unclear diagram grounding can cause the model to apply constraints to the wrong 1 arXiv:2610.12391v1 [cs.AI] 8 Oct 2026 Case 1: Redundant Angle Constraints Distract Reasoning Case 2: Ambiguous Angle Grounding Leads to Wrong Answer InterGPS GeoReform Qwen3-VL-8B response: Qwen3-VL-8B response: InterGPS GeoReform Qwen3-VL-8B response: Qwen3-VL-8B response: Find x so the quadrilateral is a parallelogram. A. 3 B. 5 C. 12 D. 18 Find x to the nearest tenth. A. 37.2 B. 40.6 C. 49.4 D. 52.8 Equals(MeasureOf(Angle(D,B,C)),5y); Equals(LengthOf(Line(E,D)),x+4); Equals(LengthOf(Line(A,E)),3x-6) Quadrilateral(D,C,A,B); |DE| = |EA|; |BE| = |EC| Equals(MeasureOf(Angle(B,C,A)),y); Equals(MeasureOf(Angle(D,B,A)),x) Perpendicular(line(C,D),line(B,D)); Angle x = ∠ABD; m∠ABD = x°; Angle y = ∠BCD; m∠BCD = y° Unrelated angles distract from DE=EA. Focuses on diagonal bisection. Reads x at A, not B; uses the wrong angle. Binds x to vertex B and rays BA / BD. Prediction: DAnswer: BPrediction: DAnswer: A Answer: B. 5Answer: A. 37.2° Redundant angle constraintsSelect relevant constraintsAmbiguous angle groundingExplicit angle grounding Figure 1: Failure modes of naive geometry formalization. Left: redundant angle relations distract the model from the key segment relation. Right: unclear angle grounding causes the model to apply the wrong angle constraint. angles and arrive at an incorrect answer. These failures reveal a gap between describing a diagram and representing it for problem solving. Even valid geometric relations can distract reasoning when they are poorly selected or presented. Effective formalization therefore requires deciding which information to include and how to express it, in ad- dition to extracting geometric facts accurately. This motivates using downstream reasoning feedback to improve the policy that governs these decisions. To better exploit formalization, we propose Geo- Reform, a reflective formalization evolution frame- work that optimizes the formalization policy rather than treating parser outputs as fixed. GeoReform searches over candidate policies that determine how geometric entities, relations, constraints, and tar- gets are selected, grounded, grouped, and presented to the reasoning model. By executing the full reasoning pipeline, collecting failed rollouts, and reflecting on representation-level defects such as missing constraints, ambiguous targets, distracting relations, and unmapped angle labels, GeoReform turns downstream reasoning failures into targeted policy mutations. This process encourages formal- izations that are complete enough to support the solution while remaining concise enough to avoid distracting the reasoning. Our contributions can be summarized as follows: •We show that extracting geometric facts alone is insufficient for reliable geometry reason- ing. Effective formalization must therefore organize these facts into selective and unam- biguous representations. • We introduce GeoReform, a reflective formal- ization evolution framework that optimizes formalization policies through genetic search and reflective formalization mutation. •We show that evolving the formalization pol- icy mitigates these failures and substantially improves GPS performance, raising accuracy from 42.0% to 56.0% on Geometry3K. 2 Related Work Geometry Problem Solving Geometry problem solving (GPS) requires accurate diagram under- standing and rigorous mathematical reasoning. Early systems such as Inter-GPS parse textual and visual inputs into formal representations for sym- bolic reasoning with theorem knowledge (Lu et al., 2021; Zhang et al., 2023a; Peng et al., 2023). Neu- ral approaches explore intermediate representations such as geometric primitives, solution programs, and mathematical expressions (Zhang et al., 2022; Chen et al., 2021, 2022). AlphaGeometry further combines neural search with symbolic deduction for geometry theorem proving (Trinh et al., 2024). Recent MLLM methods improve geometric reason- ing through geometry instruction data and formal- ized vision-language pretraining, as exemplified by G-LLaVA and GeoX (Gao et al., 2025; Xia et al., 2025). AutoGPS, GOLD, and DFE-GPS ex- plicitly incorporate structured diagram information into problem solving (Ping et al., 2025; Zhang and 2 Moshfeghi, 2024; Zhang et al., 2024c). These ap- proaches highlight the value of geometric structure, but errors in grounding, relation selection, or tar- get specification can affect downstream reasoning. GeoReform studies how formalized information should be selected, grounded, and organized to sup- port problem solving. Reflective Prompt Evolution Prompt evolution iteratively searches, evaluates, and refines prompts. CoT demonstrates how reasoning instructions af- fect model behavior (Wei et al., 2022), while APE, ProTeGi, and OPRO optimize instructions through candidate search, textual feedback, or LLM-based optimization (Zhou et al., 2023; Pryzant et al., 2023; Yang et al., 2024). EvoPrompt and Prompt- Breeder evolve prompt populations through muta- tion and selection (Guo et al., 2024; Fernando et al., 2023). Self-Refine, Reflexion, PromptAgent, and GEPA further use model errors or execution feed- back to guide improvement (Madaan et al., 2023; Shinn et al., 2023; Wang et al., 2024; Agrawal et al., 2026). Such feedback can inform both prompt revi- sions and reusable rules for addressing failures. Ex- isting methods generally optimize task instructions, exemplars, or reasoning prompts. GeoReform ap- plies evolution to the intermediate representation within the GPS pipeline: it keeps the downstream reasoning prompt fixed and evolves the formaliza- tion policy governing which geometric entities, re- lations, constraints, and targets are extracted and how they are grounded, organized, and presented. This allows reasoning feedback to improve the ge- ometric information supplied to the model. 3 Method 3.1 Method Overview To address the representation bottleneck in multi- modal geometry reasoning, we introduce GeoRe- form, a reflective formalization evolution frame- work that optimizes how geometric entities, re- lations, constraints, and targets are organized for downstream reasoning. The overall pipeline is il- lustrated in Figure 2. Formalization Policy Design. Prior GPS sys- tems commonly rely on intermediate formal rep- resentations to bridge diagram understanding and downstream solving (Lu et al., 2021; Chen et al., 2021, 2022; Zhang et al., 2023a; Xia et al., 2025). While such representations can expose useful ge- ometric structures, they may also introduce re- dundant relations or ambiguously grounded con- straints that interfere with reasoning. As detailed in Sec. 3.2, GeoReform makes the formalization policy explicit, defining how to select, ground, group, and present geometric entities, relations, constraints, and targets for the reasoning model. This explicit policy view allows GeoReform to treat formalization as an optimizable interface rather than a fixed parser output. Reflective Formalization Evolution. To better exploit formalization, we introduce reflective for- malization evolution in Sec. 3.3. GeoReform starts from an initial formalization policy, evaluates its downstream reasoning behavior, and uses failure cases to identify weaknesses in the current repre- sentation. These weaknesses are then converted into targeted policy updates, allowing GeoReform to iteratively refine how geometric information is selected, grounded, grouped, and presented. During inference, the evolved formalization pol- icy is fixed and applied to each test problem to generate a structured representation of the relevant geometric information. The downstream reasoning model then conditions on both the original prob- lem and this representation to finish reasoning and predict the final answer. 3.2 Formalization Policy Design To make geometric conditions explicit for down- stream reasoning, we formulate formalization as an intermediate interface between multimodal problem understanding and reasoning. A formal- ization policypdefines how geometric information should be extracted, organized, and presented. Given a diagramIand a problem statementT, the structure generatorGfollowspto produce a formalized representationS p = G(I, T ; p). Inspired by prior GPS systems that use formal geometric structures to support downstream solving (Lu et al., 2021; Chen et al., 2021, 2022; Zhang et al., 2023a; Xia et al., 2025), we first define a geometry language spaceΩfollowing Lu et al. (2021). This space consists of typed literals, where each literal applies a predicate to a set of arguments and describes a geometric entity, relation, numerical constraint, or target query. For instance, a formalized representation may contain literals such asTriangle(A,B,C), Parallel(Line(A,B),Line(C,D))and Equals(Angle(A,B,C),54). Based on this language space, we define an ini- 3 Policy Edits: select only useful facts add missing relations clarify symbol links ...... Reasoning Feedback Error signal: irrelevant constraints Diagnosis hint: formalization is redundant or weakly targeted Redundant constraint: Equals(LengthOf(Line(A,B)), LengthOf(Line(A,C))) Reflect and Propose New Formulation P 0 P3 P1P2 P4 Formalize Policy Candidate Pool P Within Budget? Choose parent policy to create new formalization policy P4 MiniBatch Obtain feedback from the reasoning process Apply 푝 4 to generate formalization for the reasoning Add P new to pool P new EvalMiniBatch Performance Improved? Discard P new Formalization Steers Reasoning Formalize Policy Geometry Formalization Equals(LengthOf(Line(A, C)), 10) PointLiesOnLine(D, Line(A, C)) Equals(LengthOf(Line(A, B)), 13) Equals(LengthOf(Line(B, C)), 13) Find(AreaOf(Triangle(A,B,C))) irrelevant constraints Question: Find the area ? ⇒ B is the vertex Equals(MeasureOf(Angle(A,B,C)), 50) Figure 2: GeoReform optimizes the policy that turns a geometry problem into a formal representation. In each iteration, GeoReform reflects on failed reasoning cases from an existing formalization policy and proposes a new policy. The new policy is first tested on a minibatch, and if improved, evaluated on a validation set. Successful policies are added to the candidate pool, guiding formalization toward clearer and more useful geometric representations. tial formalization policyp 0 as the starting point for the reflective evolution process described in Sec. 3.3. This policy specifies how the structure generator should use the typed literals to describe a geometry problem, and the literals inS p 0 provide an explicit structured description of the original problem. The policy also controls the output form. It must not return explanations, derivations, or extra text. This keeps the formalization short and reduces noise for the reasoning. For any formalization pol- icyp, the generated representationS p is then pro- vided to the reasoning model together with the orig- inal problem:ˆa = Solver(I, T, C, S p ),whereC denotes the candidate answer choices when avail- able. During evolution, GeoReform updates only the policy. The best policy is fixed and used for all test problems. 3.3 Reflective Formalization Evolution 3.3.1 Genetic Optimization Loop Given a geometry reasoning systemΦconsisting of a structure generator and a downstream reasoning model, GeoReform aims to identify a formalization policy that maximizes downstream solving perfor- mance. GeoReform starts from the initial policy p 0 described in Sec. 3.2 and maintains a candidate poolPof formalization policies. During evolution, we also maintain a best policy candidate according to performance on an evaluation minibatch. GeoRe- form then enters an optimization loop, repeatedly proposing new candidate policies until the evalu- ation budget is exhausted. The overall reflective formalization evolution procedure is summarized in Algorithm 1. At each iteration, GeoReform proceeds in four steps: (i) it selects a parent policy from the can- didate pool, biased toward the current best policy while still allowing random exploration; (i) it runs the full reasoning pipeline on a training minibatch and collects failed cases, including the problem statement, diagram, extracted geometric relations, reasoning process, prediction, and evaluation feed- back; (i) it sends these failed cases to the reflec- tion model, which produces a child policy by re- vising the current formalization policy; and (iv) it evaluates the child policy on the same training minibatch as an efficient filter. If the child policy outperforms its parent, it is added to the candidate pool and further evaluated on an evaluation mini- batch. If its evaluation accuracy is no lower than the best historical evaluation accuracy, it becomes the new best policy candidate. The training minibatch is used for failure col- lection and efficient filtering, while the evaluation minibatch is used to track the best policy candidate. The final test set is held out and is not used during evolution. 3.3.2 Reflective Formalization Mutation Rollout traces reveal how a formalization shapes downstream reasoning, not just whether the final answer is correct. This is crucial because poor for- malizations fail in opposite ways: verbose formal- izations distract the reasoning model with irrele- vant relations, while overly conservative formaliza- tions omit necessary constraints. By pairing rollout traces with final outcomes, GeoReform turns single answer failures into representation diagnoses. The 4 Table 1: Performance comparison of formalization evolution on Geometry3K. All values are accuracy (%). Improvements over the initial policy are shown in parentheses. BackboneScaleMLLM+ Initial Policy+ GeoReform ChoiceCompletionChoiceCompletionChoiceCompletion Qwen3VL 2B33.529.042.037.056.0(+14.0)53.5(+16.5) 4B55.051.569.567.078.0(+8.5)73.0(+6.0) 8B69.054.074.068.579.0(+5.0)74.0(+5.5) 32B76.570.082.074.588.0(+6.0)80.0(+5.5) InternVL3.5 1B41.528.043.027.557.5(+14.5)40.5(+13.0) 2B50.529.554.036.068.0(+14.0)53.5(+17.5) 4B53.037.065.048.573.0(+8.0)57.0(+8.5) 8B61.043.573.560.077.5(+4.0)66.5(+6.5) 14B65.548.574.059.078.5(+4.5)67.5(+8.5) reflection model compares the generated formal- ization with the reasoning trajectory and attributes failures to concrete policy defects. Reflective mu- tation then converts these diagnoses into targeted policy updates, revising the formalization policy to preserve necessary constraints, remove distrac- tions, clarify targets, and better organize geometric information. Given a parent policy selected by the genetic op- timization loop, GeoReform executes it on a mini- batch and collects failed rollouts. For each failure, the reflection model compares the generated for- malization with the reasoning trajectory and evalu- ation feedback. This comparison exposes the likely policy defect: wrong quantity computation may suggest weak target binding, angle confusion may suggest missing angle mappings, irrelevant deriva- tions may suggest redundant or poorly grouped relations, and incomplete reasoning may suggest missing constraints. Thus, reflection converts fi- Algorithm 1: Reflective Formalization Evolution Input: initial policy p 0 , training set D train , evaluation minibatch B eval , selection threshold τ , evolution budget T Output: best policy p ∗ P ←p 0 , p ∗ ← p 0 , s ∗ ← score(p ∗ ,B eval ) for t = 1,...,T do Sample u∼ Uniform(0, 1) and set p t ← p ∗ if u < τ , otherwise p t ∼ Uniform(P) // select parent Sample training minibatch B t ⊂ D train Run Φ(p t ) on B t and collect rolloutsR t Extract failed casesF t ⊂R t // bad cases Construct reflection inputs fromF t and generate child policy p ′ t = R(F t ,p t ) // mutation if score(p ′ t ,B t ) > score(p t ,B t ) then Add p ′ t toP and evaluate p ′ t on B eval // filter if score(p ′ t ,B eval )≥ s ∗ then p ∗ ← p ′ t , s ∗ ← score(p ′ t ,B eval ) // update best return p ∗ nal answer errors into actionable feedback on the formalization policy. Mutation updates only the formalization policy; the structure generator, reasoning model, and rea- soning prompt are kept fixed. The update is a gen- eral instruction revision rather than a correction to a specific example. It may require explicit targets, clearer grouping of entities and constraints, useful properties of named figures, or restraints against unsupported diagram assumptions. The child pol- icy is then returned to the genetic loop for gated evaluation and pool update. 4 Experiment 4.1 Experiment Setup Datasets. Experiments were conducted on three specialized GPS benchmarks: Geometry3K (Lu et al., 2021), GeoQA (Chen et al., 2021), and PGPS9K (Zhang et al., 2023b). Geometry3K con- sists of plane geometry problems with diagrams, textual questions, answer choices, and formal anno- tations, and is designed to support geometry pars- ing and symbolic reasoning. GeoQA focuses on geometry question answering, where many geo- metric conditions are described in the problem text and accompanied by diagrams; we use its English version to maintain linguistic consistency with the other datasets. Since GeoQA lacks official formal geometric annotations, we use GPT-5 to extract structured geometric information from each prob- lem (text and diagram). PGPS9K (Zhang et al., 2023b) contains 9,022 plane geometry problems with fine-grained diagram annotations and inter- pretable solution programs. Metrics.We follow the evaluation protocol used in prior work on Geometry3K (Lu et al., 2021). Specifically, we evaluate model performance using Completion and Choice: In Completion, models directly generate the final answer without being 5 Table 2: Performance comparison of different policies on Geometry3K using InternVL3.5. All values are accuracy (%). Improvements of GeoReform over the best competing method are shown in parentheses. SettingScaleMLLMAutoGPSInterGPSGeoReform GPT4oInternVL3Qwen2.5 Choice InternVL3.5 1B41.544.538.540.541.057.5(+13.0) 2B50.550.549.049.552.568.0(+15.5) 4B53.056.056.562.062.073.0(+11.0) 8B61.060.059.562.063.577.5(+14.0) 14B65.568.566.567.566.078.5(+10.0) Completion InternVL3.5 1B28.032.530.028.025.040.5(+8.0) 2B29.528.528.531.532.553.5(+21.0) 4B 37.043.545.042.544.557.0(+12.0) 8B43.548.550.549.058.566.5(+8.0) 14B48.554.556.054.060.067.5(+7.5) provided candidate options, while in Choice, the model is provided with possible options. These metrics jointly assess the model’s ability to perform direct geometric reasoning, option-based answer selection, and ranked prediction. Baselines. We compare GeoReform with three baselines. (i) Initial Policy uses the same struc- ture generator and downstream reasoning prompt as GeoReform, but keeps the original formalization policy fixed. This baseline isolates the effect of re- flective formalization evolution. (i) InterGPS (Lu et al., 2021) parses text and diagrams into formal geometric relations using an OCR based pipeline. (i) AutoGPS (Ping et al., 2025) produces mul- timodal geometric formalizations and combines them with deductive reasoning. Although these baselines obtain geometric structures from differ- ent sources, we feed all formalization inputs to the same downstream reasoning model with the same Table 3: Effect of explicit geometric structure on GeoQA and PGPS9K. Most geometric information in GeoQA is provided in the problem statement; PGPS9K uses ground-truth geomet- ric annotations. All values are accuracy (%). SettingScaleMetric Baseline+ Geometry Structure GeoQA – Choice Qwen3VL 2B49.056.5(+7.5) 4B 78.581.5(+3.0) 8B81.080.0(-1.0) 32B92.090.0(-2.0) GeoQA – Completion InternVL3.5 1B23.524.0(+0.5) 2B32.034.0(+2.0) 4B51.055.5(+4.5) 8B58.564.0(+5.5) PGPS9K – Choice Qwen3VL 2B32.037.5(+5.5) 4B54.576.0(+21.5) 8B66.576.5(+10.0) 32B76.587.0(+10.5) reasoning prompt, decoding setting, and evaluation protocol, treating them as alternative formalization inputs under a unified evaluation setting. Implementation Details. We use GPT-5 as the structure generation model to provide a strong pool of geometrically valid facts, allowing us to focus on optimizing the formalization policy. Reflective evolution runs for at most 20 update iterations. At each iteration, we randomly sample 32 training problems for failure collection and policy filter- ing, using a fixed random seed of 42 across all experiments. The reflection input contains at most 16 failed cases from the sampled minibatch, in- cluding the original problem, generated formal- ization, downstream reasoning trace, model pre- diction, and gold answer. We use a fixed valida- tion minibatch of 50 problems to select the best evolved policy. The final test examples are held out and never used during evolution. Unless other- wise specified, policy selection is based on choice accuracy. For downstream reasoning, we evalu- ate models from the Qwen3VL (Bai et al., 2025) and InternVL3.5 (Wang et al., 2025) series, using the same reasoning prompt across all policies. We serve models with vLLM (Kwon et al., 2023) and set the maximum generation length to 4096 tokens. All experiments are conducted on NVIDIA A6000 and NVIDIA ProA6000 GPUs. 4.2 Experiment Results Reflective Evolution Improves Formalization Quality. As shown in Table 1, GeoReform im- proves over the initial policy described in Sec. 3.2 across all evaluated backbones, scales, and settings, with gains of 4.0%–17.5%. On Qwen3VL-2B, choice accuracy increases from 42.0% to 56.0%, 6 MLLM only GPT4o InternVL Qwen +InterGPS +GeoReform 30 40 50 60 70 80 Choice Accuracy (%) 2B4B MLLM only GPT4o InternVL Qwen +InterGPS +GeoReform 65 70 75 80 85 90 Choice Accuracy (%) 8B32B MLLM only GPT4o InternVL Qwen +InterGPS +GeoReform 30 40 50 60 70 80 Completion Accuracy (%) 2B4B MLLM only GPT4o InternVL Qwen +InterGPS +GeoReform 55 60 65 70 75 80 85 Completion Accuracy (%) 8B32B Figure 3: Comparison of formalization methods on Geometry3K using Qwen3VL reasoning models. MLLM only uses no additional formalization; GPT4o, InternVL, and Qwen indicate the models providing formalizations for AutoGPS. Circle Triangle Parallelogram Rhombus Trapezoid 50 60 70 80 90 100 Choice Accuracy 52.5 77.577.5 92.5 75.0 82.5 53.8 100.0 87.5 95.8 Other Line Rectangle Square Overall 40 60 80 100 Choice Accuracy 90.9 81.8 60.0 65.0 68.0 72.0 83.3 50.0 70.4 80.8 Circle Triangle Parallelogram Rhombus Trapezoid 50 60 70 80 90 100 Choice Accuracy 75.0 77.5 82.5 92.5 70.0 82.5 76.9 100.0 95.8 95.8 Other Line Rectangle Square Overall 40 60 80 100 Choice Accuracy 68.2 81.8 60.0 65.0 80.0 72.0 33.3 50.0 74.0 80.8 MLLMInterGPSGeoReform Figure 4: Choice accuracy by geometry problem type on Geometry3K on Qwen3VL-8B model. GeoReform achieves the strongest gains on Rhombus and Parallelogram, where solving depends on global shape relations rather than only local marks. while completion accuracy rises from 37.0% to 53.5%. Importantly, the initial policy does not al- ways help: it slightly reduces InternVL3.5-1B com- pletion accuracy from 28.0% to 27.5%, whereas GeoReform raises it to 40.5%. These results high- light the value of using reasoning failures to refine formalization. A representation may expose ge- ometric facts yet still leave the model uncertain about which constraints matter or how symbols map to the diagram. Policy evolution addresses this gap by improving how the available facts are selected, grounded, and organized. Sensitivity to Representation Quality. The largest gains generally occur on 1B–2B models, reaching 17.5% on InternVL3.5-2B completion. This pattern suggests that smaller models have more difficulty compensating for unclear or dis- tracting representations. Making relevant con- straints explicit can reduce the need to recover miss- ing connections or resolve ambiguous symbols dur- ing reasoning. The gains are not strictly monotonic with model size, however, and the largest evalu- ated models still improve by 4.5%–8.5%. Thus, stronger model capacity reduces some dependence on representation quality but does not remove the benefit of refining the formalization. Comparison with Explicit Formalization. Ta- ble 2 and Figure 3 compare GeoReform with In- terGPS (Lu et al., 2021) and AutoGPS (Ping et al., 2025). GeoReform performs best across all eval- uated settings. On InternVL3.5, it exceeds the best competing method by 10.0%–15.5% in choice and 7.5%–21.0% in completion. Some compet- ing formalizations nevertheless perform below the original MLLM baseline, showing that an addi- tional structured input can also interfere with rea- soning. The comparison suggests that extracting or aligning geometric relations is insufficient by itself: the resulting representation must make the relevant constraints easy to identify and use. Geo- Reform directly optimizes this downstream useful- ness through feedback from failed solutions. 4.2.1 Ablation Study Effect of Explicit Geometric Structure.Table 3 shows that explicit geometric structure improves performance on both GeoQA and PGPS9K. Even on GeoQA, where most geometric conditions are al- ready stated in the problem text (Figure 5), adding structure improves six of eight settings, with gains of up to 7.5%. On PGPS9K, ground-truth geomet- ric annotations improve choice accuracy across all evaluated Qwen3VL scales by 5.5%–21.5%. These gains suggest that explicit structure helps not only by supplying information from diagrams, but also by making available geometric conditions easier to use in reasoning. The small drops in two GeoQA settings further indicate that the benefit depends on how well the added structure complements the 7 InterGPS: Reads the 70 and 30 marks Fails to combine them into ∠KLM Misses the angle relation Question: In parallelogram JKLM, find the measure of angle JKL. GeoReform: Combines local angles at L ; ∠KLM=70+30=100 Uses supplementary adjacent angles GT : 80 ∘ InterGPS: Confuses MP with a diagonal Applies irrelevant diagonal properties Follows an unnecessary reasoning path Question: In rhombus LMPQ, MP = 10. The opposite angles are labeled 2x² − 10 and 8x. Find the perimeter. GeoReform: Recovers vertex order L−M−P−Q ; Identifies MP as a side Uses equal sides of a rhombus GT : 40 Question: As shown in the figure, AB//CD and EF are intersected by AB, CD at points E, F, ∠1=50°, then the degree of ∠2 is () GT : 140 ∘ Geometry Structure: Parallel(Line(A,B), Line(C,D)) Equals(MeasureOf(Angle(D,F,E)), 50) Determine(MeasureOf(Angle(A,E,F))) Text contains all structure. Question: As shown in the figure, in △ABC, it is known that ∠A=80°, ∠B=60°, DE∥BC, then the size of ∠CED is () GT : 140 ∘ Geometry Structure: PointLiesOnLine(E, Line(A,C)), Parallel(Line(D,E), Line(B,C)) Determine(MeasureOf(Angle(C,E,D))) Text contains all structure. Figure 5: Case studies of geometric formalization. Top: GeoReform improves over InterGPS by linking local marks to global shape relations. Bottom: GeoQA cases show that all required structural information is contained in the question text. Even in this setting, making the structure explicit helps organize the stated facts for more reliable reasoning. original input. 4.3 Case Analysis GeoReform Connects Marks to Shape Relations. GeoReform improves because it produces geomet- ric relations that are more useful for the target ques- tion, capturing local marks such as points, lines, an- gles, and lengths while connecting them to the full shape. In contrast, InterGPS improves less because its output remains more local. Although it can de- tect many useful facts, it often leaves them as sepa- rate pieces. As a result, its representation may show that a rhombus or a parallelogram is present, but still fail to specify how each mark should be used. For example, it may confuse a side with a diago- nal, or miss that two marked angles form one full angle. This pattern is clear across geometry types. As illustrated in Figure 4, GeoReform gains the most on Rhombus and Parallelogram. On Rhom- bus, GeoReform reaches100.0%accuracy, while InterGPS reaches only76.9%. On Parallelogram, GeoReform reaches82.5%, compared with70.0% for InterGPS. These types benefit more because their answers depend strongly on global shape re- lations. Rhombus problems require relations such as equal sides, diagonal bisection, and the link be- tween half diagonals and full diagonals. Parallel- ogram problems often require opposite sides, ad- jacent angles, and angle composition. GeoReform recovers these relations more reliably, leading to more accurate answers. The evolution trace of the selected policy is reported in Appendix A.1. We provide all prompts used by GeoReform, including the initial formalization prompt, reflection prompt, downstream reasoning prompt, and final evolved prompt, in Appendix A.2. 5 Conclusion In this paper, we proposed GeoReform, a reflec- tive formalization evolution framework for mul- timodal Geometry Problem Solving (GPS). Geo- Reform shows that effective formalization is not merely about extracting geometric facts, but about organizing them into representations that better sup- port downstream reasoning. By optimizing the formalization policy through feedback from failed rollouts, GeoReform refines how geometric enti- ties, relations, constraints, and targets are selected, grounded, grouped, and presented. Extensive ex- periments and analyses on multiple geometry rea- soning benchmarks demonstrate that GeoReform consistently improves over the initial policy and achieves strong performance across different rea- soning models. Limitations Although GeoReform improves multimodal geom- etry problem solving by evolving the formaliza- tion policy, it still has several limitations. First, our experiments mainly focus on plane geometry 8 benchmarks, which do not fully represent broader mathematical reasoning scenarios, such as solid ge- ometry, algebraic diagrams, physics problems, or proof-oriented geometry tasks. Future work should examine whether reflective formalization evolution can generalize to these more diverse settings. Sec- ond, while GeoReform improves how geometric information is selected, grounded, grouped, and presented, it does not completely eliminate errors from diagram understanding or structure genera- tion. If the underlying model misses a visual mark, misidentifies a geometric entity, or grounds an an- gle or segment incorrectly, the resulting formaliza- tion may still mislead downstream reasoning. Incor- porating stronger visual parsers, symbolic consis- tency checks, or theorem-based verification could further improve the reliability of the generated rep- resentations. 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PathPolicyParentPolicy ChangeTrainVal 0P 0 –Initial conservative parser in the style of InterGPS.68.7575.00 1P 3 P 0 General reflective revision with stronger fitness on the train batch.81.2579.17(+4.17) 2P 12 P 3 Explicit mapping for entities, angles, lines, transversals, and targets.68.7583.33(+8.33) 3P 20 P 12 Named figure policy with restrained expansion of useful properties.75.0087.50(+12.50) Table 4 shows the final policy lineage selected by reflective evolution, where Path denotes lineage depth rather than search order. Starting from the initial policyP 0 , the search first improves general formalization quality withP 3 , and then shifts toward explicit grounding of entities, angles, lines, and targets inP 12 . AlthoughP 12 has lower training accuracy thanP 3 , its validation accuracy is higher, suggesting that better formalization is not simply obtained by fitting the training batch. This highlights the importance of using the validation set to select policies that generalize beyond the sampled failures. The final policyP 20 further adds named figure information while keeping property expansion restrained, achieving the best validation accuracy of 87.50%. A.2 Prompts and Policy Details This appendix reports the prompts and policies used by GeoReform. The initial formalization policy defines the starting representation interface, the reflection prompt specifies how failed rollouts are converted into policy mutations, the reasoning prompt is kept fixed during evolution, and the final evolved policy is used for all test examples. A.2.1 Initial Formalization Policy A.2.2 Reflection Prompt for Policy Mutation A.2.3 Downstream Reasoning Prompt A.2.4 Final Evolved Formalization Policy 12 Initial Prompt SYSTEM_PROMPT_TEMPLATE = """You are a geometry formal parser. Your job is to convert a geometry problem (diagram + text) into a formal relation set for a symbolic geometry solver. Goal: Produce a solver-friendly relation set that resembles the formal language used in InterGPS. Core principles:` 1. If a fact is ambiguous or uncertain, omit it. 2. Keep names canonical and stable: - Points: uppercase labels like A, B, C, O, P, M, ... - Variables: x, y, z, r, d, ... - Lines/segments: Line(A,B) - Angles: Angle(A,B,C) - Circles: Circle(O) if center/label is clear - Arcs: Arc(A,B) or Arc(A,B,C) when needed 3. Prefer literal forms that are closest to InterGPS style: - Triangle(A,B,C) - Quadrilateral(A,B,C,D) - Circle(O) - PointLiesOnLine(D,Line(A,C)) - PointLiesOnCircle(A,Circle(O)) - Parallel(Line(A,B),Line(C,D)) - Perpendicular(Line(A,B),Line(C,D)) - IntersectAt(Line(A,C),Line(B,D),Point(E)) - Equals(Line(A,B),Line(C,D)) - Equals(Angle(A,B,C),54) - Equals(AreaOf(Triangle(A,B,C)),36) - Equals(RadiusOf(Circle(O)),5) - Similar(Triangle(A,B,C),Triangle(D,E,F)) - Congruent(Triangle(A,B,C),Triangle(D,E,F)) - Find(x) - Find(Line(A,B)) - Find(Angle(A,B,C)) - Find(AreaOf(Shape(...))) Output: Return ONLY the final merged relation set. """ USER_PROMPT_TEMPLATE = """Parse the following geometry problem into formal relation set. Problem: problem_block Now output the final formal relation set only. """ Figure 6: Initial formalization policy used by the structure generator. The policy specifies how geometric entities, incidence relations, polygons, angle relations, length constraints, and targets are extracted and organized before downstream reasoning. 13 Reflection Prompt reflection_system_prompt = """ You are improving the prompt of a geometry structure generator. The current prompt of the generator is: <curr_prompt> You are also given several examples. Each example contains: - the original problem - the structure information generated by the current prompt - the downstream solver's output using that structure - the gold answer Your task is to write a new prompt for the structure generator. Goal: Generate structure information that improves downstream geometry reasoning accuracy. Read the examples carefully and infer what kinds of generated structure are helpful or harmful for solving the problems. Examples: <examples> Write an improved prompt for the structure generator only. Return only the new prompt inside triple backticks. """ Figure 7: Reflection prompt used to diagnose failed rollouts and generate a revised formalization policy. Reasoning Prompt reasoning_prompt = """ "problem" "Parsed geometry:" "parsed_result" "Output the thinking process in <think> </think> and final answer (the option letter) " "in <answer> </answer> tags." """ Figure 8: Downstream reasoning prompt used by the reasoning model. 14 Best Prompt best_prompt = """ "problem" You are a geometry structure generator. Given a problem statement (possibly referring to an unseen diagram), produce a concise, unambiguous set of entities, mappings, a nd relations that capture exactly the information needed for downstream geometric reasoning. Do not solve the problem or choose an answer; only encode the structure. Key principles: - Encode only what is justified by the text and standard definitions of named figures (e.g., rhombus, parallelogram). Ignore any diagram features not mentioned. - Prefer figure-level properties (e.g., “adjacent angles in a parallelogram are supplementary”) when a special figure is named, rather than trying to derive them via transversals. Guidelines: 1) Entities, incidence, and polygons - List all named points. - If a polygon/special quadrilateral is named with vertex order (e.g., rhombus LMPQ), record it with that cyclic order and declare its type: - quadrilateral(L, M, P, Q); - rhombus(L, M, P, Q) / parallelogram(L, M, P, Q) / rectangle(...), etc., exactly as stated. - Record sides as segments and the corresponding supporting lines: - Sides: LM, MP, PQ, QL (for quadrilateral(L, M, P, Q)); - Lines: line(L, M), line(M, P), line(P, Q), line(Q, L). - Do NOT add diagonals or extra points unless explicitly referenced in the text. - Record all explicit collinearities only if stated or unavoidable from polygon sides (e.g., collinear(L, M, P) is NOT assumed). For every named/numbered angle ∠PQR, assert noncollinear(P, Q, R). 2) Special-figure properties (encode only those implied by the named figure) - Parallelogram(A, B, C, D): - parallel(line(A, B), line(C, D)); parallel(line(B, C), line(D, A)); - |AB| = |CD|; |BC| = |DA|; - Opposite angles equal: m∠ABC = m∠CDA; m∠BCD = m∠DAB; - Adjacent angles supplementary: m∠ABC + m∠BCD = 180°; m∠BCD + m∠CDA = 180°; m∠CDA + m∠DAB = 180°; m∠DAB + m∠ABC = 180°. - Rhombus(A, B, C, D): include all parallelogram properties plus - All sides equal: |AB| = |BC| = |CD| = |DA|; - Diagonals perpendicular and angle-bisecting are standard properties, BUT DO NOT add diagonal lines or use these properties unless the diagonals or those relations are explicitly referenced in the text. - Rectangle/Square/Kite/Isosceles triangle: encode only standard defining properties relevant to given entities, without inventing extra points/lines. 3) Angles and mappings - Map any numbered angles to concrete named angles (e.g., Angle 4 = ∠CED). - For every named/numbered angle with a measure (numeric or algebraic), create a measure variable and definition (e.g., m∠QLM = 2x^2 − 10°). - Keep a single consistent variable symbol per unknown and state what it represents if not obvious (e.g., x is a free variable; y = m∠ABD). 4) Parallel/perpendicular/transversal relations - Record parallelism/perpendicularity exactly as stated or implied by the named figure’s standard properties (see 2). - Identify a transversal only when a set of explicitly collinear points lies across two known parallel lines. - For angles created by parallel lines and a defined transversal, record the angle-type relation: - corresponding(∠..., ∠...) - alternate_interior(∠..., ∠...) - alternate_exterior(∠..., ∠...) - same_side_interior(∠..., ∠...) [supplementary] Also add the implied equation only when the type determines equality or supplementarity. 5) General angle relations - Record vertical angle equalities where two known lines intersect at a named point (vertical(∠..., ∠...)). - Record linear pairs/supplementary relations arising from stated collinearities (e.g., linear_pair(∠PQA, ∠PQB); m∠PQA + m∠PQB = 180°). - If two angles differ only by swapping an endpoint along the same known straight line, add the explicit identity (e.g., since collinear(B, E, C), m∠DCE = m∠DCB and m∠EBD = m∠CBD). 6) Triangles and polygons - When three noncollinear points form a triangle that is directly referenced, declare it (triangle(P, Q, R)). - For named polygons, do not add diagonals or interior intersection points unless the problem mentions them. Prefer interior-angle relations from the figure type (e.g., adjacent supplementary in a parallelogram/rhombus) over constructing transversals. 7) Lengths and equalities - Record all given segment lengths or equalities. - When a special figure is named, add only those side equalities implied by its definition (e.g., in a rhombus: all sides equal). - Use segment notation for lengths (e.g., |MP| = 10). If this, together with figure properties, determines other side equalities (e.g., |QL| = |MP|), record those equalities explicitly. 8) Targets - Clearly state the target(s) to solve for (e.g., Target: |QL|; or Target: y where y = m∠ABD). - Include all algebraic links between given expressions and the target that are explicitly provided or follow from encoded figure properties and angle-type relations. 9) Restraints - Do not invent points, lines, diagonals, or intersections not supported by the text. - Do not assume collinearity unless explicitly given or inherent from declared sides of a polygon. - Do not choose acute vs. obtuse interpretations arbitrarily. Where a relation depends on orientation, encode the relation type (e.g., same_side_interior) and the implied equality/supplement statement rather than a numeric measure. - Do not perform numeric calculations or algebraic simplifications. Output format: - A single bracketed list containing one structured block with semicolon-separated statements, for example: [ "Points: ...; quadrilateral(A, B, C, D); rhombus(A, B, C, D); Sides: AB, BC, CD, DA; Lines: line(A, B), line(B, C), line(C, D), line(D, A); parallel(line(A, B), line(C, D)); parallel(line(B, C), line(D, A)); |AB| = |BC| = |CD| = |DA|; noncollinear(P, Q, R); Angle n = ∠PQR; m∠ABC = ...; m∠DAB + m∠ABC = 180°; corresponding(∠..., ∠...); m∠... = m∠...; linear_pair(∠..., ∠...); m∠... + m∠... = 180°; triangle(P, Q, R); Target: ...;" ] Ensure consistency, completeness, and minimality needed to support correct downstream reasoning. """ Figure 9: Final evolved formalization policy selected by GeoReform. 15