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.
Why Structure Injection Alone Fails to Support Geometry Reasoning
Recent geometry problem solving (GPS) methods explicitly represent geometric entities, relations, and constraints and provide them to the reasoning model. However, we observe that effective formalization is highly non-trivial. Although a clear structure can guide MLLMs toward relevant entities, relations, and constraints, poorly designed formalizations can also interfere with the reasoning process. On 200 Geometry3K problems, adding explicit geometric entities, relations, and constraints to the input of Qwen3VL-8B corrects 28 previously incorrect answers but turns 13 previously correct answers into errors.
Such interference occurs in two common ways. First, redundant or weakly relevant relations can distract the reasoning from the simplest solution path. A formalized input may introduce additional angle relations that steer the reasoning away from the key segment equality, even when the problem can be solved directly using the fact that the diagonals of a parallelogram bisect each other. Second, ambiguous formalizations can be amplified by downstream reasoning. Angle relations with unclear diagram grounding can cause the model to apply constraints to the wrong 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 addition to extracting geometric facts accurately. This motivates using downstream reasoning feedback to improve the policy that governs these decisions.
Formalization Policy Design: Selecting and Grounding Geometric Information
To make geometric conditions explicit for downstream reasoning, we formulate formalization as an intermediate interface between multimodal problem understanding and reasoning. A formalization policy p defines how geometric information should be extracted, organized, and presented. Given a diagram I and a problem statement T, the structure generator G follows p to produce a formalized representation Sp = G(I,T;p). Inspired by prior GPS systems that use formal geometric structures to support downstream solving, we first define a geometry language space Ω consisting 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 as Triangle(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 initial formalization policy p0 as the starting point for the reflective evolution process. This policy specifies how the structure generator should use the typed literals to describe a geometry problem, and the literals in Sp0 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 policy p, the generated representation Sp is then provided to the reasoning model together with the original problem: â = Solver(I,T,C,Sp), where C denotes the candidate answer choices when available. During evolution, GeoReform updates only the policy. The best policy is fixed and used for all test problems.
Genetic Optimization Loop for Reflective Policy Evolution
GeoReform starts from the initial policy p0 and maintains a candidate pool ℙ of formalization policies. During evolution, we also maintain a best policy candidate according to performance on an evaluation minibatch. GeoReform then enters an optimization loop, repeatedly proposing new candidate policies until the evaluation budget is exhausted. At each iteration, GeoReform proceeds in four steps: (i) it selects a parent policy from the candidate pool, biased toward the current best policy while still allowing random exploration; (ii) 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 feedback; (iii) it sends these failed cases to the reflection model, which produces a child policy by revising 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 minibatch. 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 collection 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. This genetic optimization loop allows GeoReform to search over candidate policies that determine how geometric entities, relations, constraints, and targets are selected, grounded, grouped, and presented to the reasoning model.
Reflective Formalization Mutation: From Failed Rollouts to Policy Updates
Rollout traces reveal how a formalization shapes downstream reasoning, not just whether the final answer is correct. This is crucial because poor formalizations fail in opposite ways: verbose formalizations distract the reasoning model with irrelevant relations, while overly conservative formalizations omit necessary constraints. By pairing rollout traces with final outcomes, GeoReform turns single answer failures into representation diagnoses. The reflection model compares the generated formalization with the reasoning trajectory and evaluation feedback. This comparison exposes the likely policy defect: wrong quantity computation may suggest weak target binding, angle confusion may suggest missing angle mappings, irrelevant derivations may suggest redundant or poorly grouped relations, and incomplete reasoning may suggest missing constraints. Thus, reflection converts final answer errors into actionable feedback on the formalization policy.
Mutation updates only the formalization policy; the structure generator, reasoning model, and reasoning prompt are kept fixed. The update is a general 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 policy is then returned to the genetic loop for gated evaluation and pool update.
Experiment Setup: Datasets, Metrics, and Baselines
Experiments were conducted on three specialized GPS benchmarks: Geometry3K, GeoQA, and PGPS9K. Geometry3K consists of plane geometry problems with diagrams, textual questions, answer choices, and formal annotations, and is designed to support geometry parsing and symbolic reasoning. GeoQA focuses on geometry question answering, where many geometric 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 problem (text and diagram). PGPS9K contains 9,022 plane geometry problems with fine-grained diagram annotations and interpretable solution programs.
We follow the evaluation protocol used in prior work on Geometry3K. Specifically, we evaluate model performance using Completion and Choice: In Completion, models directly generate the final answer without being 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.
We compare GeoReform with three baselines. (i) Initial Policy uses the same structure generator and downstream reasoning prompt as GeoReform, but keeps the original formalization policy fixed. This baseline isolates the effect of reflective formalization evolution. (ii) InterGPS parses text and diagrams into formal geometric relations using an OCR based pipeline. (iii) AutoGPS produces multimodal geometric formalizations and combines them with deductive reasoning. Although these baselines obtain geometric structures from different sources, we feed all formalization inputs to the same downstream reasoning model with the same reasoning prompt, decoding setting, and evaluation protocol, treating them as alternative formalization inputs under a unified evaluation setting.
Results on Geometry3K and Model Scales
As shown in the performance comparison, GeoReform improves over the initial policy 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%, while completion accuracy rises from 37.0% to 53.5%. Importantly, the initial policy does not always help: it slightly reduces InternVL3.5-1B completion accuracy from 28.0% to 27.5%, whereas GeoReform raises it to 40.5%. These results highlight the value of using reasoning failures to refine formalization. A representation may expose geometric 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.
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 distracting representations. Making relevant constraints explicit can reduce the need to recover missing connections or resolve ambiguous symbols during reasoning. The gains are not strictly monotonic with model size, however, and the largest evaluated 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.
GeoReform performs best across all evaluated 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 competing formalizations nevertheless perform below the original MLLM baseline, showing that an additional structured input can also interfere with reasoning. 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. GeoReform directly optimizes this downstream usefulness through feedback from failed solutions.
Ablation Results: Explicit Structure and Reasoning Quality
Table results show that explicit geometric structure improves performance on both GeoQA and PGPS9K. Even on GeoQA, where most geometric conditions are already stated in the problem text, adding structure improves six of eight settings, with gains of up to 7.5%. On PGPS9K, ground-truth geometric 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 original input.
Case Analysis: How GeoReform Connects Local Marks to Global Shape
GeoReform improves because it produces geometric relations that are more useful for the target question, capturing local marks such as points, lines, angles, and lengths while connecting them to the full shape. In contrast, InterGPS improves less because its output remains more local. Although it can detect many useful facts, it often leaves them as separate 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 diagonal, or miss that two marked angles form one full angle. This pattern is clear across geometry types. As illustrated in the case studies, GeoReform gains the most on Rhombus and Parallelogram. On Rhombus, GeoReform reaches 100.0% accuracy, while InterGPS reaches only 76.9%. On Parallelogram, GeoReform reaches 82.5%, compared with 70.0% for InterGPS. These types benefit more because their answers depend strongly on global shape relations. Rhombus problems require relations such as equal sides, diagonal bisection, and the link between half diagonals and full diagonals. Parallelogram problems often require opposite sides, adjacent 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 the appendix. We provide all prompts used by GeoReform, including the initial formalization prompt, reflection prompt, downstream reasoning prompt, and final evolved prompt, in the appendix as well.
Conclusion
In this paper, we proposed GeoReform, a reflective formalization evolution framework for multimodal Geometry Problem Solving. GeoReform shows that effective formalization is not merely about extracting geometric facts, but about organizing them into representations that better support downstream reasoning. By optimizing the formalization policy through feedback from failed rollouts, GeoReform refines how geometric entities, relations, constraints, and targets are selected, grounded, grouped, and presented. Extensive experiments and analyses on multiple geometry reasoning benchmarks demonstrate that GeoReform consistently improves over the initial policy and achieves strong performance across different reasoning models.