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arxiv: 2606.27926 · v1 · pith:Q755KCSYnew · submitted 2026-06-26 · 💻 cs.AI · cs.CL· cs.CV

Verifiable Geometry Problem Solving: Solver-Driven Autoformalization and Theorem Proposing

Can Li , Ting Zhang , Junbo Zhao , Hua Huang This is my paper

Pith reviewed 2026-06-29 04:20 UTC · model grok-4.3

classification 💻 cs.AI cs.CLcs.CV
keywords geometry problem solvingautoformalizationneuro-symbolic methodstheorem proposingsolver-driven frameworkmultimodal reasoningreinforcement learning
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The pith

A neuro-symbolic solver uses executability feedback to drive both formalization of geometry problems and proposal of auxiliary theorems.

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

The paper presents SD-GPS, which treats the symbolic solver as an oracle that supplies training signals for turning multimodal geometry inputs into formal statements and for verifying new lemmas when a proof search stalls. It combines supervised adaptation with solvability-guided reinforcement learning in one module and adds an impasse-aware agent that filters lemma proposals through symbolic checks. The approach targets two bottlenecks: formalization that ignores downstream solvability and deduction that lacks fresh rules. If the central loop works, neural perception becomes more tightly grounded in executable formal systems and produces more reliable solutions than decoupled neural or neuro-symbolic baselines.

Core claim

SD-GPS unifies Solver-Driven Autoformalization, which trains a QwenVL3-2B model by making executability the central signal, with Verified Theorem Proposing, which deploys an impasse-aware agent to suggest local auxiliary lemmas that are then filtered by the symbolic solver for soundness.

What carries the argument

The symbolic solver acting as an execution oracle that supplies feedback for both autoformalization training and lemma verification.

If this is right

  • SD-GPS outperforms existing MLLM, neural, and neuro-symbolic methods on Geometry3K and PGPS9K across completion, multiple-choice, and cross-modal reference settings.
  • Closing the loop between multimodal perception and symbolic execution improves verifiable geometric reasoning.
  • The framework supports verifiable problem-solving by grounding neural agents in formal systems.

Where Pith is reading between the lines

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

  • The same oracle-driven pattern could be tested in other formal domains such as algebraic identities or graph algorithms where executable checkers exist.
  • Reducing reliance on large human-annotated formal datasets becomes possible if executability alone provides sufficient training signal.
  • Iterative lemma proposal might allow the system to tackle proofs that exceed the depth limits of fixed rule libraries.

Load-bearing premise

Executability can serve as a reliable central training signal for the autoformalization module without causing overfitting to solver quirks or generation of formally correct but semantically useless statements.

What would settle it

If the trained model produces many formally valid statements that the solver cannot turn into complete proofs on a new set of geometry problems, or if performance gains vanish when the solver is swapped for a different one, the claim that the closed loop improves reasoning would be refuted.

Figures

Figures reproduced from arXiv: 2606.27926 by Can Li, Hua Huang, Junbo Zhao, Ting Zhang.

Figure 1
Figure 1. Figure 1: Performance comparison among existing methods. To resolve these misalignments, we propose SD-GPS, a solver-driven framework that treats the symbolic solver as an execution oracle throughout both formalization and de￾duction. First, Solver-Driven Autoformalization replaces the decoupled pipeline with a unified multimodal formalizer built on QwenVL3-2B. It reads the raw diagram and problem text jointly, lear… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the proposed SD-GPS framework. Raw diagram-text inputs are converted into solver-executable formal predicates, checked by a symbolic solver, refined through solver-guided repair when necessary, and augmented with verified auxiliary theorem proposals when the solver reaches a deductive impasse. and were not originally designed as a unified strict formal-language benchmark, we normalize symbolic … view at source ↗
Figure 3
Figure 3. Figure 3: Representative examples for the three-way split of point-reference ambiguity. The clas [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representative OCR failures and their propagated symbolic consequences in cascaded [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
read the original abstract

Geometry Problem Solving have increasingly adopt the neuro-symbolic paradigm, combining neural intuition with symbolic rigor. However, current frameworks suffer from severe bottlenecks in two core stages: autoformalization, which treats multimodal translation as a static task decoupled from downstream solver compatibility, and theorem prediction, where solvers frequently hit a deductive impasse due to fixed rule libraries. To address these, we propose SD-GPS, a solver-driven framework that treats the symbolic solver as an execution oracle throughout both formalization and deduction. First, Solver-Driven Autoformalization unifies supervised formal-language adaptation and solvability-guided reinforcement learning into a single module built on QwenVL3-2B, making executability the central training signal. Second, Verified Theorem Proposing introduces an impasse-aware agent that proposes local auxiliary lemmas from current proof states, ensuring soundness by filtering all proposals through symbolic verification. Empirical evaluations on Geometry3K and PGPS9K demonstrate that SD-GPS consistently outperforms existing MLLM, neural, and neuro-symbolic methods across standard completion, multiple-choice, and cross-modal reference regimes, proving that closing the loop between multimodal perception and symbolic execution significantly improves geometric reasoning, offering profound insights into how neural agents can be grounded by formal systems to achieve verifiable problem-solving capabilities.

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

1 major / 2 minor

Summary. The paper proposes SD-GPS, a solver-driven neuro-symbolic framework for geometry problem solving. Solver-Driven Autoformalization unifies supervised formal-language adaptation with solvability-guided RL on QwenVL3-2B, treating executability by the symbolic solver as the central training signal. Verified Theorem Proposing uses an impasse-aware agent to generate local auxiliary lemmas from proof states, with all proposals filtered by symbolic verification. Empirical results on Geometry3K and PGPS9K are claimed to show consistent outperformance over MLLM, neural, and neuro-symbolic baselines across completion, multiple-choice, and cross-modal regimes.

Significance. If the reported gains hold and formalizations remain semantically faithful, the work would demonstrate a practical way to ground multimodal models via execution oracles, advancing verifiable neuro-symbolic reasoning. The explicit use of solver feedback in both autoformalization and deduction, together with verification filtering, constitutes a clear methodological contribution over static translation pipelines.

major comments (1)
  1. [Solver-Driven Autoformalization] Solver-Driven Autoformalization (abstract and § on the module): executability is positioned as the primary training signal unifying supervised adaptation and RL for QwenVL3-2B. No explicit mechanism (e.g., semantic similarity constraints, human-annotated fidelity checks, or degeneracy detection) is described to ensure statements remain faithful to the original diagram and text beyond post-hoc solver verification. This is load-bearing for the central claim, because measured gains on Geometry3K/PGPS9K could arise from solver-exploitable formalizations rather than improved geometric reasoning.
minor comments (2)
  1. [Abstract] Abstract: asserts 'consistent outperformance' and 'proving that closing the loop... significantly improves geometric reasoning' without any numerical results, ablation tables, or dataset statistics; this makes the strength of the empirical case difficult to gauge from the abstract alone.
  2. The manuscript would benefit from a dedicated limitations subsection discussing potential solver-specific artifacts or cases where executability diverges from semantic correctness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. Below we address the single major comment point by point.

read point-by-point responses

  1. Referee: [Solver-Driven Autoformalization] Solver-Driven Autoformalization (abstract and § on the module): executability is positioned as the primary training signal unifying supervised adaptation and RL for QwenVL3-2B. No explicit mechanism (e.g., semantic similarity constraints, human-annotated fidelity checks, or degeneracy detection) is described to ensure statements remain faithful to the original diagram and text beyond post-hoc solver verification. This is load-bearing for the central claim, because measured gains on Geometry3K/PGPS9K could arise from solver-exploitable formalizations rather than improved geometric reasoning.

    Authors: We agree that the absence of explicit fidelity safeguards beyond executability is a substantive concern for the central claim. The module first performs supervised adaptation on human-annotated formalizations drawn from the training splits of Geometry3K and PGPS9K; these annotations are constructed to preserve the semantics of the original text and diagram. The RL stage then optimizes only for solver executability while keeping the policy close to the supervised initialization via a KL penalty. Nevertheless, the manuscript does not report additional mechanisms such as semantic similarity constraints, post-hoc human fidelity audits, or explicit degeneracy detection. We therefore accept the referee’s point and will revise the paper to (i) add a dedicated paragraph in the Solver-Driven Autoformalization section describing the reliance on supervised initialization and the KL regularizer, (ii) include a small-scale manual audit of 100 sampled formalizations for semantic drift, and (iii) report a simple degeneracy metric (e.g., ratio of degenerate predicates) on the test set. These additions will appear in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: solver as external oracle with benchmark-driven claims

full rationale

The paper's core claims rest on empirical outperformance of SD-GPS on the independent benchmarks Geometry3K and PGPS9K across multiple regimes. Solver-Driven Autoformalization uses executability as a training signal from an external symbolic solver treated as oracle, and Verified Theorem Proposing filters proposals via symbolic verification. No equations, self-definitions, or load-bearing steps reduce the reported gains to fitted parameters renamed as predictions or to self-citation chains. The derivation chain is self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or detailed axioms are stated. The approach implicitly assumes the solver provides reliable feedback.

axioms (1)
  • domain assumption The symbolic solver functions as a reliable, low-error execution oracle for both training signals and lemma verification.
    Invoked throughout the description of Solver-Driven Autoformalization and Verified Theorem Proposing.

pith-pipeline@v0.9.1-grok · 5758 in / 1216 out tokens · 34680 ms · 2026-06-29T04:20:00.161817+00:00 · methodology

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Reference graph

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