Diffusion-Proof: Diffusion LLM Theorem Proving

• Diffusion-Proof is a formal Lean4 theorem-proving framework that employs diffusion-based LLMs for whole-proof synthesis and local repair, ensuring long-range tactic consistency.
• It leverages a dual-model architecture—dLLM-Prover-7B for complete proof generation and dLLM-Corrector-7B for local repair—via block-diffusion denoising.
• Empirical results show improvements of 1.61% on ProofNet-Test and 6.14% on MiniF2F-Test, highlighting its efficacy in long-horizon proof planning.

Diffusion-Proof is a Lean4 formal theorem-proving framework that replaces purely auto-regressive proof generation with diffusion LLMs trained for whole-proof synthesis and local proof repair. It is presented as, to the best of the authors’ knowledge, the first framework to train and apply diffusion LLMs for formal theorem proving, and it is organized around two models: dLLM-Prover-7B for complete proof generation and dLLM-Corrector-7B for large-block correction. The framework is motivated by the claim that theorem proving stresses long-range coherence, tactic consistency, and local repair in ways that are poorly matched to left-to-right next-token prediction. On the reported Lean4 benchmarks, it improves over a matched auto-regressive baseline by 1.61% on ProofNet-Test and 6.14% on MiniF2F-Test (Wang et al., 17 Jun 2026).

1. Problem formulation and motivation

Diffusion-Proof is situated in the recent effort to improve formal mathematical reasoning with LLMs. Its starting point is a critique of standard auto-regressive theorem provers: they generate proofs strictly left-to-right, so early mistakes propagate; they tend to lose long-range coherence across multi-step tactic sequences; and they do not naturally support repairing an incorrect middle segment while conditioning on both prefix and suffix (Wang et al., 17 Jun 2026).

The framework argues that diffusion LLMs offer a different generative bias. Rather than predicting one next token at a time, they generate text through iterative denoising of a multi-token block. In the theorem-proving setting, this is used to motivate three expected advantages: bi-directional context awareness, local correction or in-filling, and more coherent long-horizon proof planning. The paper does not present this as a general proof that diffusion models dominate auto-regressive models; rather, it frames Diffusion-Proof as an empirical demonstration that a non-autoregressive generation paradigm can be effective for Lean4 proof construction (Wang et al., 17 Jun 2026).

The target task is Lean4 proof generation from theorem statements. The evaluation protocol explicitly excludes long chain-of-thought reasoning in the main comparison, because the compared systems do not support it. This design choice makes the central empirical claim narrower but clearer: the reported gains are intended to isolate the effect of diffusion-based proof generation and correction relative to a baseline trained on the same data (Wang et al., 17 Jun 2026).

2. Diffusion formulation for formal proof synthesis

Diffusion-Proof formalizes the contrast between auto-regressive and diffusion-based generation in the standard probabilistic way. For a token sequence $\bm{x} = \{x_1, x_2, \cdots, x_L\}$, the auto-regressive model factorizes generation as

$$\mathbb{P}_\theta\left(x_i \mid \bm{x}[0:i - 1]\right).$$

The diffusion formulation instead introduces a time step $t \in [0, 1]$ and a noised sequence $\bm{x}^{(t)}$, where tokens are masked independently with probability $t$, and trains the model to recover the clean sequence:

$$\mathbb{P}_\omega (\bm{x}^{(0)} \mid \bm{x}^{(t)}).$$

The training objective is

$$\mathcal{L}(\theta) = - \mathbb{E}_{t, \bm{x}^{(0)}, \bm{x}^{(t)}} \left[\sum_{i = 1}^L \mathbb{I}[x_i^{(t)} = \text{[MASK]} ] \cdot \log \mathbb{P}_\omega (x_i^{(0)} \mid \bm{x}^{(t)})\right].$$

In the block-diffusion variant used here, the sequence is divided into blocks of size $B$; bi-directional attention is restricted within a block; causal relationships are preserved between blocks; and inference decodes one block at a time until stopping (Wang et al., 17 Jun 2026).

This formulation is central to the framework’s account of proof generation. Whole-proof writing is treated as a block-diffusion completion problem rather than a token-by-token decoding problem, and proof correction is treated as a masked-span denoising problem over a large block. A plausible implication is that the same denoising mechanism can support two distinct proof tasks—global synthesis and local repair—without requiring the causal decoding constraint that structures standard theorem-proving LLMs.

3. Dual-model architecture

Diffusion-Proof is explicitly a two-stage, two-model pipeline rather than a single prover. The first stage uses dLLM-Prover-7B, based on Fast-dLLM-V2-7B, to generate a complete proof from theorem statements in natural language and Lean4. The second stage uses dLLM-Corrector-7B to repair a failed proof when Lean verification fails but the top-level subgoal structure is still correct (Wang et al., 17 Jun 2026).

Component	Role	Key settings
dLLM-Prover-7B	Whole-proof writing	Base model Fast-dLLM-V2-7B; diffusion block size 32
dLLM-Corrector-7B	Local proof repair / in-filling	Fine-tuned from prover; diffusion block size 512

The distinction between the two models is functional as well as architectural. dLLM-Prover-7B is intended to generate a coherent proof skeleton and tactic sequence over long contexts. dLLM-Corrector-7B is trained for large-block correction, using both prefix and suffix information to fill a removed proof region. During corrector training, loss is applied to both the subgoal proof block and the placeholder token, and training examples are sampled from proofs with subgoal decomposition such as internal have-style structure (Wang et al., 17 Jun 2026).

At inference time, the corrector operates on a masked region. If the generated proof fails but preserves the right high-level decomposition, the erroneous subgoal proof is removed and replaced with a masked block. The corrector then denoises the region using a 512-length diffusion block. The reported correction-stage settings are temperature = 1.2, token denoising confidence = 0.95, an inference masked span of 256 tokens, the placeholder token <|fim_middle|> during data preparation, and <|MASK|> during correction inference (Wang et al., 17 Jun 2026).

The paper also reports that a unified corrector-only model is worse than the dual-model design. This is presented as evidence that global proof writing and local correction are different competencies, and that separating them yields better behavior than forcing one diffusion model to do both equally well (Wang et al., 17 Jun 2026).

4. Data construction, training regime, and inference workflow

The training corpus is assembled from prior Lean theorem-proving data. The raw source pool contains 5,595,798 theorem-proving records or code-completion proofs. The data are standardized into NL statements, FL (formal language) statements, and FL proofs, using rule-based extraction. STP-style data lacking natural-language statements are retained as syntax-understanding data (Wang et al., 17 Jun 2026).

From this source pool, the prover is trained on a cleaned 300k SFT dataset, with an approximate 1:2 ratio between pure Lean proofs and proofs with natural-language annotations or comments. The training format is code-completion rather than chat templating. The corrector is then trained on a 128k subset sampled from proofs with subgoal decomposition (Wang et al., 17 Jun 2026).

The training schedule incorporates curriculum data sorting by increasing proof complexity or proof length. The reported optimization settings are:

• Prover and baseline: learning rate 1e-5, context length 2048
• Corrector: learning rate 5e-6
• Warm-up: 3%
• Hardware: 4 H100 GPUs
• Global batch size: 64
• Appendix setting: per-GPU batch size 2 with gradient accumulation 8

The reported training and evaluation costs are about 96 H100 GPU-hours for dLLM-Prover-7B, an additional 48 H100 GPU-hours for dLLM-Corrector-7B, about 16 hours for the evaluation pipeline, and about 24 hours for AR baseline evaluation without vLLM optimization (Wang et al., 17 Jun 2026).

Operationally, the inference workflow is staged. First, the prover attempts whole-proof generation under block-diffusion decoding. Second, if Lean rejects the proof but the top-level subgoal structure survives, the corrector is invoked for up to five rounds of correction per theorem in valid correction cases. This gives the framework a repair loop that is structurally different from the one-shot generation pattern of standard auto-regressive theorem provers (Wang et al., 17 Jun 2026).

5. Benchmarks and empirical performance

Diffusion-Proof is evaluated on MiniF2F-Test and ProofNet-Test. MiniF2F-Test contains 244 problems spanning IMO, AIME, AMC, a subset of Math-500, and hand-crafted problems. ProofNet-Test contains 186 theorems derived from college textbooks and covering topics such as real and complex analysis, linear algebra, and topology. The main metric is pass@32 (Wang et al., 17 Jun 2026).

Benchmark	Qwen-2.5-Lean-SFT-7B	Diffusion-Proof
ProofNet-Test	5.91%	7.53%
MiniF2F-Test	43.85%	50.00%

The corresponding absolute gains are +1.61% on ProofNet-Test and +6.14% on MiniF2F-Test. On MiniF2F-Test by problem type, the reported changes are IMO: 5.00% → 15.00%, AIME: 33.33% → 33.33%, AMC: 22.22% → 26.67%, Algebra: 55.68% → 67.05%, Number Theory: 58.82% → 63.24%, and Induction: 25.00% → 25.00% (Wang et al., 17 Jun 2026).

The long-proof subset analysis is used to argue that diffusion helps more when proofs require longer-range tactic consistency. On the longest 25% of MiniF2F proofs, Diffusion-Proof achieves 5/54 (9.26%) versus 4/54 (7.41%) for the matched Qwen baseline. On the longest 50%, it achieves 26/108 (24.07%) versus 20/108 (18.52%). The paper interprets this pattern as support for the claim that block diffusion is particularly relevant for long-horizon proof planning (Wang et al., 17 Jun 2026).

A notable single-instance result is the solution of imo_1962_p2. The paper highlights that Diffusion-Proof solves this problem while DeepSeek-Prover-V2-7B fails on it under pass@32, even with Long CoT reasoning. This is presented not as a broad state-of-the-art claim, but as evidence that blockwise denoising and bidirectional local refinement can sometimes succeed where stronger auto-regressive reasoning systems do not (Wang et al., 17 Jun 2026).

At the same time, the broader comparison table in the paper is explicitly described as not a controlled comparison. On MiniF2F-Test, systems such as DeepSeek-Prover-V2 7B report 70.49%, whereas Diffusion-Proof reports 50.0%. The paper attributes the incomparability to differences in data scale, reinforcement learning, search, and Long CoT reasoning. A common misconception is therefore avoided by the paper itself: Diffusion-Proof is not presented as a state-of-the-art universal theorem prover, but as an early diffusion-based alternative that improves over a matched 7B auto-regressive baseline (Wang et al., 17 Jun 2026).

6. Interpretation, limitations, and the broader use of the term

The paper states several limitations. Training scale is constrained by computational resources and remains below state-of-the-art open-source auto-regressive provers. The base model’s Long CoT capability is limited, so the work does not fully explore long-thinking diffusion provers. The framework is confined to Lean4, emphasizes empirical performance rather than a theory of diffusion LLMs, and reports that the corrector is less effective on ProofNet because that benchmark emphasizes advanced knowledge more than decomposable subgoals (Wang et al., 17 Jun 2026).

Within those limits, Diffusion-Proof advances a specific thesis: theorem proving may benefit from a non-autoregressive generation paradigm organized around global proof planning via block diffusion and local proof repair via in-filling. This suggests a research direction rather than a settled replacement for auto-regressive proving. A plausible implication is that future systems may combine diffusion-based proof generation with reinforcement learning, verifier feedback, and stronger reasoning scaffolds, which the paper identifies as natural extensions (Wang et al., 17 Jun 2026).

The term “diffusion-proof” also appears elsewhere in the literature with unrelated meanings, and the theorem-proving framework should be distinguished from them. In PDE analysis, the boundedness-by-entropy principle is described as a “diffusion-proof method” because the proof of existence and boundedness is made robust against severe diffusion coupling by an entropy-based change of variables (Jüngel, 2014). In diffusion-model intellectual-property protection, a watermarking scheme is presented as “diffusion-proof” because the watermark is embedded into the diffusion dynamics and verified from internal reverse-process samples rather than final outputs (Yang et al., 2024). In covert communication with image-based diffusion models, a latent sign-flipping construction is characterized as “diffusion-proof” in the sense of indistinguishability against inversion-aware adversaries under the paper’s threat model (Bauer et al., 13 Mar 2025).

In formal theorem proving, however, Diffusion-Proof denotes the specific 2026 Lean4 framework built around dLLM-Prover-7B and dLLM-Corrector-7B. Its significance lies less in a final performance frontier than in demonstrating that diffusion LLMs can be trained to generate and repair formal proofs in a setting where long-range tactic coherence and structured local correction are central requirements (Wang et al., 17 Jun 2026).