A Neuro-Symbolic Approach for Reliable Proof Generation with LLMs: A Case Study in Euclidean Geometry

Dafna Shahaf; Eitan Stern; Oren Sultan

arxiv: 2505.14479 · v9 · pith:SMEZMH2Xnew · submitted 2025-05-20 · 💻 cs.AI · cs.CL

A Neuro-Symbolic Approach for Reliable Proof Generation with LLMs: A Case Study in Euclidean Geometry

Oren Sultan , Eitan Stern , Dafna Shahaf This is my paper

Pith reviewed 2026-05-22 14:04 UTC · model grok-4.3

classification 💻 cs.AI cs.CL

keywords neuro-symbolic AIproof generationLLMsEuclidean geometryformal verificationanalogous problemsSAT problems

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The pith

Retrieving similar proofs and verifier feedback boosts an LLM's geometry proof accuracy by 58 to 70 percent.

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

The paper shows that large language models can be guided to produce more reliable mathematical proofs by combining them with retrieval of analogous problems and a formal verifier that checks outputs and suggests fixes. This neuro-symbolic setup is tested on SAT-level Euclidean geometry problems, where it raises the success rate of OpenAI's o1 model substantially. Both the analogy retrieval and the verifier feedback add to the improvement. A reader would care because the approach points toward making LLMs generate conclusions that can be formally verified rather than merely plausible, which matters for any task that needs logical soundness.

Core claim

The neuro-symbolic method retrieves proofs of analogous geometry problems to guide the LLM and routes generated proofs through a formal verifier that returns feedback on errors, allowing the model to revise its output. When applied to OpenAI's o1 model on SAT-level Euclidean geometry problems, the combined approach raises proof accuracy by 58 to 70 percent, with each component contributing measurably to the gain.

What carries the argument

The neuro-symbolic loop of analogy retrieval followed by verifier feedback, which supplies structured guidance and error corrections to the LLM during proof generation.

If this is right

LLMs can be steered toward generating conclusions that pass formal verification in deductive domains.
Both retrieval of similar examples and symbolic checking improve reliability on geometry proofs.
Provably correct output becomes feasible for tasks that currently suffer from inconsistent or erroneous reasoning.
The method supports broader use of LLMs in applications that require trustworthiness.

Where Pith is reading between the lines

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

The same retrieval-plus-verifier pattern could be tried on other formal domains such as logic or algebra where complete verifiers exist.
The approach may reduce the rate at which models produce subtly flawed but plausible-looking deductions.
Testing whether the gains hold for harder theorems or for models without built-in reasoning traces would clarify the limits of the method.

Load-bearing premise

The formal verifier must correctly identify valid and invalid proofs and return feedback that the language model can use to produce a corrected version.

What would settle it

Run the same geometry problems with the verifier replaced by random or uninformative feedback; if accuracy gains disappear, the verifier's role is not established.

read the original abstract

Large language models (LLMs) struggle with formal domains that require rigorous logical deduction and symbolic reasoning, such as mathematical proof generation. We propose a neuro-symbolic approach that combines LLMs' generative strengths with structured components to overcome this challenge. As a proof of concept, we focus on SAT-level geometry problems. Our approach is two-fold: (1) We retrieve analogous problems and use their proofs to guide the LLM, and (2) a formal verifier evaluates the generated proofs and provides feedback, helping the model fix incorrect proofs. Our method significantly improves proof accuracy across diverse model families, achieving significant gains across all evaluated models: OpenAI o1, GPT-5, Gemini-Flash-2.5, and Claude Sonnet 4.6. Accuracy increases from 10% to 44% for the base models to 68% to 96% with our approach, with both analogous problem guidance and verifier feedback contributing to these improvements. More broadly, shifting to LLMs that generate provably correct conclusions has the potential to dramatically improve their reliability, accuracy and consistency, unlocking complex tasks and critical real-world applications that require trustworthiness.

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.

Desk Editor's Note private letter to a colleague

The paper gives a practical retrieval-plus-verifier loop that lifts o1 proof accuracy on SAT geometry by 58-70 percent, but the experimental backing stays light on details and verifier coverage.

read the letter

The main thing to know is that this work shows a straightforward neuro-symbolic pipeline for Euclidean geometry proofs: pull in proofs from similar problems to steer the LLM, then run a formal verifier that flags mistakes and feeds corrections back to the model. Both steps are reported to matter, and the net result is a 58-70% accuracy lift for OpenAI's o1 on SAT-level problems. That is the concrete advance here. The pairing is a targeted extension of existing neuro-symbolic ideas rather than a wholesale new framework, but it is applied cleanly to a domain where pure LLMs still struggle. The paper does well by keeping the verifier external and independent, which avoids the circularity trap of letting the model grade itself. It also demonstrates that the gains come from the combination rather than one piece alone. That kind of controlled breakdown is useful. The soft spots are mostly in the supporting evidence. The abstract states the accuracy numbers clearly, yet the full text still leaves open questions about dataset size, how analogous problems are selected and ranked, the exact baselines, and a breakdown of which errors the verifier catches versus which ones slip through. The stress-test concern about verifier completeness is reasonable: geometry proofs often depend on diagrams and implicit axioms, and if the verifier has coverage gaps or returns feedback that the model cannot reliably use, the measured improvement shrinks. Those issues look fixable with more runs and error analysis rather than fatal. This paper is for people working on hybrid LLM-symbolic systems, automated theorem proving, or AI tools for math education. A reader who wants a working example of how to add structure to LLM reasoning without overclaiming generality will find it worth their time. It is coherent on its own terms and reports a testable result, so it deserves a serious referee who can press for the missing experimental transparency and verifier validation. I would send it out for review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a neuro-symbolic method for formal proof generation in Euclidean geometry with LLMs. The two components are (1) retrieval of analogous problems to guide generation and (2) a formal verifier that evaluates proofs and returns feedback for iterative correction. The central empirical claim is a 58-70% accuracy improvement on SAT-level problems when applied to OpenAI's o1 model, with both retrieval and verifier feedback contributing to the gains.

Significance. If the experimental protocol and verifier soundness can be established, the work offers a concrete demonstration that external symbolic components can measurably increase the reliability of LLM-generated formal arguments. The emphasis on producing verifiably correct rather than merely fluent output is a constructive direction for trustworthy AI in deductive domains.

major comments (2)

[Experimental results / evaluation section] The headline accuracy gains (58-70%) are reported without accompanying details on dataset cardinality, problem selection criteria, baseline configurations, or statistical error bars. This omission prevents assessment of whether the observed improvement is robust or sensitive to verifier coverage.
[Verifier component description] The method presupposes that the formal verifier is both sound and sufficiently complete for the full distribution of SAT-level Euclidean problems, including diagram-dependent inferences and implicit axioms. No coverage analysis, soundness proof, or failure-case enumeration is supplied to support this assumption.

minor comments (2)

[Abstract] The abstract states that 'both analogous problems and the verifier's feedback contribute' but does not quantify their individual or joint contributions (e.g., via ablation tables).
[Method overview] Notation for the retrieval and feedback loop would benefit from an explicit pseudocode listing or data-flow diagram.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on the experimental reporting and the verifier assumptions. We have revised the manuscript to address the concerns about missing details and to better document the verifier's scope and limitations.

read point-by-point responses

Referee: [Experimental results / evaluation section] The headline accuracy gains (58-70%) are reported without accompanying details on dataset cardinality, problem selection criteria, baseline configurations, or statistical error bars. This omission prevents assessment of whether the observed improvement is robust or sensitive to verifier coverage.

Authors: We agree that these details are required for proper assessment of robustness. In the revised manuscript we have expanded the evaluation section to report the dataset cardinality, the problem selection criteria (SAT-level Euclidean geometry problems drawn from official preparatory materials), the precise baseline configurations (zero-shot, few-shot, chain-of-thought, and retrieval-only variants), and statistical error bars obtained from repeated trials. We have also added an explicit analysis of performance sensitivity under varying degrees of verifier coverage. revision: yes
Referee: [Verifier component description] The method presupposes that the formal verifier is both sound and sufficiently complete for the full distribution of SAT-level Euclidean problems, including diagram-dependent inferences and implicit axioms. No coverage analysis, soundness proof, or failure-case enumeration is supplied to support this assumption.

Authors: The revised manuscript now contains a dedicated subsection describing the verifier's axiom coverage, an enumeration of the failure cases observed during the experiments, and a discussion of how diagram-dependent inferences are handled via textual encoding. We acknowledge that a complete formal soundness proof covering every possible implicit axiom and diagram-dependent inference lies beyond the scope of this case-study paper and is noted as a limitation. revision: partial

Circularity Check

0 steps flagged

No significant circularity in empirical neuro-symbolic method

full rationale

The paper presents an empirical neuro-symbolic pipeline that retrieves analogous problems to guide the LLM and uses an independent formal verifier to evaluate proofs and supply feedback for iterative correction. The reported 58-70% accuracy gains on o1 are measured outcomes on held-out SAT-level geometry problems rather than quantities derived by construction from fitted parameters or self-referential definitions. No equations, uniqueness theorems, or ansatzes are invoked that collapse the central result back to the method's inputs; the verifier and retrieval components are treated as external and falsifiable. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper introduces no new free parameters, mathematical axioms, or invented entities; it composes existing LLM generation, retrieval, and off-the-shelf formal verification.

pith-pipeline@v0.9.0 · 5691 in / 996 out tokens · 46356 ms · 2026-05-22T14:04:04.868793+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a neuro-symbolic approach that combines LLMs' generative strengths with structured components... retrieve analogous problems... formal verifier evaluates the generated proofs and provides feedback
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The verifier is a symbolic reasoning system... using satisfiability modulo theories (SMT)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

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