arxiv: 2604.16347 · v1 · submitted 2026-03-16 · 💻 cs.HC · cs.AI· cs.LO

Recognition: 1 theorem link

· Lean Theorem

Lean Atlas: An Integrated Proof Environment for Scalable Human-AI Collaborative Formalization

Banri Yanahama , Akiyoshi Sannai

Authors on Pith no claims yet

Pith reviewed 2026-05-15 10:20 UTC · model grok-4.3

classification 💻 cs.HC cs.AIcs.LO

keywords Lean 4dependency graphsemantic verificationhuman-AI collaborationautoformalizationformal mathematicsproof environmentaligned code

0 comments

The pith

Lean Compass extracts the smallest set of nodes whose semantic review guarantees fidelity for selected theorems in a Lean project.

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

AI autoformalization often produces proofs that type-check yet fail to capture the intended mathematical meaning. The paper presents Lean Atlas, an interactive viewer of a Lean 4 project's dependency graph, together with the Lean Compass algorithm that, given any chosen theorem set, automatically identifies the project-specific nodes whose correctness can affect those theorems. Human reviewers then check only this reduced set, after which the code is labeled aligned Lean code. Evaluations on six projects show that proof-heavy formalizations require human review of only 1-6 percent of nodes on average, while definition-heavy ones require more but still see substantial cuts. The approach therefore makes large-scale human-AI collaboration feasible by concentrating limited human attention on the statements that actually matter for semantic fidelity.

Core claim

Lean Atlas visualizes the full dependency graph of a Lean 4 project as an interactive web interface. Its core algorithm, Lean Compass, takes any user-selected theorem set and returns the minimal collection of project-specific nodes whose semantic correctness can influence those targets. By restricting human semantic review to this extracted set, the method produces aligned Lean code—formalization that has passed both type checking and targeted human semantic verification—while sharply reducing the volume of material a human must inspect.

What carries the argument

Lean Compass, the algorithm that traverses the syntactic dependency graph of a Lean 4 project and extracts the nodes whose semantic correctness can affect a given target theorem set.

If this is right

Proof-heavy projects achieve 94-99 percent average reduction in nodes requiring semantic review.
A six-theorem milestone subset of the formalization of Fermat's Last Theorem achieves 59.8 percent reduction.
Mixed formalizations such as PhysLib achieve 69.0 percent reduction.
Definition-heavy projects such as XMSS achieve 27.3 percent reduction.
Aligned Lean code is established as a concrete quality standard that any AI-generated formalization can be measured against.

Where Pith is reading between the lines

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

The same dependency-extraction technique could be adapted to other interactive theorem provers whose kernel produces an explicit dependency graph.
Collections of aligned Lean code could serve as high-quality training corpora that reduce future semantic hallucination rates in AI formalizers.
Integrating Lean Compass with automated theorem provers might allow the system to propose which nodes most deserve human attention based on proof complexity or recent changes.

Load-bearing premise

The syntactic dependency graph contains every node whose semantic error could change the meaning of the target theorems, so human review of only those nodes suffices to catch all semantic mismatches.

What would settle it

A documented case in which a target theorem passes type checking and human review of the Lean Compass nodes yet still expresses mathematics different from the authors' intention because of an undetected semantic error in an unextracted node.

Figures

Figures reproduced from arXiv: 2604.16347 by Akiyoshi Sannai, Banri Yanahama.

**Figure 1.** Figure 1: Lean Atlas web viewer. Visualizing the review cone (227 nodes) of the main theorem IsBrownian_brownian in the Brownian Motion project. The orange nodes (14) are the nodes automatically extracted by Lean Compass as targets for semantic verification (93.8% reduction). ∗ banri.yanahama@nyx.foundation † sannai.akiyoshi.7z@kyoto-u.ac.jp 1 arXiv:2604.16347v1 [cs.HC] 16 Mar 2026 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

**Figure 2.** Figure 2: Architecture of Lean Atlas. The Lean 4 backend extracts and classifies the dependency [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: Dependency graph reduction by Lean Compass (Brownian Motion, main theorem [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

read the original abstract

AI-driven autoformalization of mathematics is advancing rapidly. However, the type checker of a proof assistant guarantees only the logical correctness of proofs; it does not verify whether propositions and definitions faithfully capture their intended mathematical content. Consequently, AI-generated formal proofs can exhibit semantic hallucination-passing the type checker yet failing to express the intended mathematics. We propose a human-in-the-loop approach in which human scientists and AI collaboratively produce formal proofs, with humans responsible for the semantic verification of propositions and definitions. To realize this approach, we develop Lean Atlas, a Lean 4 tool that visualizes the dependency graph of a Lean 4 project as an interactive web viewer, enabling human scientists to grasp the overall structure of a formalization efficiently. Its core feature, Lean Compass, is an algorithm that, given a selected theorem set, automatically extracts the project-specific nodes whose semantic correctness can affect those target statements, thereby reducing the candidate set for semantic review in large-scale formalizations. We further define *aligned Lean code* as formalization code that has undergone human semantic verification, and propose it as a quality standard for AI-generated formalizations. We evaluate the tool on six Lean 4 formalization projects with different structural characteristics; proof-heavy projects (PrimeNumberTheoremAnd, Carleson, Brownian Motion) achieved 94-99% average node reduction, a 6-theorem milestone subset of FLT achieved 59.8%, mixed PhysLib 69.0%, and definition-heavy XMSS 27.3%. Lean Atlas is available as open-source software at https://github.com/NyxFoundation/lean-atlas .

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

Lean Atlas gives a practical visualization tool and node-pruning algorithm for semantic review in Lean projects, but its central claim rests on an untested assumption about what syntactic dependencies actually capture.

read the letter

The main point is that this paper ships a working Lean 4 tool called Lean Atlas that renders project dependency graphs in a web viewer and includes an algorithm, Lean Compass, that takes a target theorem set and returns a much smaller subset of nodes whose semantic correctness could matter for those targets. They also coin the term aligned Lean code for formally verified material that has passed human semantic checks. The code is open-sourced and they run it on six existing projects, reporting node reductions from 27% up to 99% depending on the project structure. That is the concrete new thing here: a targeted extraction step that aims to make human review of AI-generated formalizations less burdensome. The numbers on proof-heavy libraries like PrimeNumberTheoremAnd and Carleson are the strongest part of the evidence; they show the method can shrink the review set dramatically in practice. The open release is also useful for anyone who wants to try it on their own formalizations. The soft spots are straightforward. The abstract gives reduction percentages but almost no description of how the extraction algorithm is implemented, what baselines it beats, or how edge cases are handled. More critically, the whole approach assumes the static syntactic dependency graph is sufficient to identify every node that could affect the semantic meaning of the targets. Lean’s metaprogramming, custom tactics, attributes, and elaborators can influence proof construction and interpretation without appearing in that graph, so the pruned set might still leave semantic hallucinations undetected. The paper does not discuss or test this possibility. The evaluation also lacks detail on how the authors confirmed that the reduced sets were actually sufficient in the six projects. This work is aimed at people already doing large Lean formalizations who are trying to incorporate AI assistance without losing semantic fidelity. It is a tool paper with empirical results rather than a theoretical advance, so it belongs in a venue that values practical systems. I would send it to peer review. The implementation and the reported reductions are worth a closer look by referees who can check the code and probe the dependency assumption directly.

Referee Report

2 major / 1 minor

Summary. The manuscript presents Lean Atlas, a Lean 4 tool featuring an interactive web viewer for dependency graphs and Lean Compass, an algorithm that extracts project-specific nodes from the syntactic dependency graph for semantic review of selected theorems. It aims to support human-AI collaborative formalization by reducing the scope of human semantic verification to prevent hallucinations in AI-generated proofs. The authors define aligned Lean code as human-semantically-verified formalizations and evaluate the tool on six projects, reporting node reductions ranging from 27.3% in definition-heavy projects to 94-99% in proof-heavy ones.

Significance. Should the syntactic dependency extraction prove sufficient for identifying all semantically relevant nodes, this approach could meaningfully advance scalable formalization efforts by facilitating efficient human oversight in large projects. The open-source availability and the concrete empirical results on established projects such as those involving the Prime Number Theorem and Carleson's theorem are notable strengths that enhance reproducibility and practical applicability.

major comments (2)

The central claim that the syntactic dependency graph identifies precisely the nodes whose semantic correctness can affect the target statements is load-bearing but insufficiently supported. The manuscript does not address potential semantic influences through Lean mechanisms outside the static syntactic graph, such as custom tactics, attributes, elaborators, or metaprogramming code, which could affect proof construction without explicit dependencies.
The reported average node reductions (94-99% for PrimeNumberTheoremAnd, Carleson, Brownian Motion; 59.8% for FLT subset; 69.0% for PhysLib; 27.3% for XMSS) lack supporting details on the algorithm implementation, baseline comparisons, edge case handling, or verification that the extracted sets indeed cover all relevant semantic influences, undermining assessment of the results' reliability.

minor comments (1)

The definition of 'aligned Lean code' as formalization code that has undergone human semantic verification could be clarified with respect to how it is practically applied or measured in the collaborative workflow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment point by point below, providing clarifications and indicating revisions where the manuscript requires strengthening.

read point-by-point responses

Referee: The central claim that the syntactic dependency graph identifies precisely the nodes whose semantic correctness can affect the target statements is load-bearing but insufficiently supported. The manuscript does not address potential semantic influences through Lean mechanisms outside the static syntactic graph, such as custom tactics, attributes, elaborators, or metaprogramming code, which could affect proof construction without explicit dependencies.

Authors: We acknowledge that the syntactic dependency graph does not capture every possible semantic influence, including those mediated by custom tactics, attributes, elaborators, or metaprogramming. The Lean Compass algorithm is explicitly designed as a syntactic extraction method that identifies nodes reachable via the static dependency graph; it does not claim to enumerate all conceivable semantic effects. In the projects we evaluated, which rely primarily on standard Lean constructs without heavy metaprogramming, this approximation proved effective for reducing the review scope. We will revise the manuscript to add an explicit limitations subsection in Section 3 and the evaluation discussion, clarifying the scope of the syntactic approach and noting that human reviewers remain responsible for any undetected influences. revision: yes
Referee: The reported average node reductions (94-99% for PrimeNumberTheoremAnd, Carleson, Brownian Motion; 59.8% for FLT subset; 69.0% for PhysLib; 27.3% for XMSS) lack supporting details on the algorithm implementation, baseline comparisons, edge case handling, or verification that the extracted sets indeed cover all relevant semantic influences, undermining assessment of the results' reliability.

Authors: We agree that the current presentation of results would benefit from greater transparency. The Lean Compass implementation relies on traversing the Lean 4 environment's dependency information via the Lean server API, starting from the target theorems and collecting all project-local declarations reachable through syntactic dependencies while excluding imported libraries. We will expand Section 4 with pseudocode for the extraction procedure, explicit baseline comparisons against full project node counts, discussion of edge cases such as mutually recursive definitions and tactic-generated dependencies, and a description of how we manually verified coverage on the smaller evaluated subsets. These additions will be placed in the main text and an appendix to support reproducibility. revision: yes

Circularity Check

0 steps flagged

No circularity: syntactic extraction applied to external projects yields measured reductions

full rationale

The paper defines Lean Compass explicitly as an algorithm operating on the standard syntactic dependency graph of Lean 4 projects. It then applies this algorithm to six independent, pre-existing formalization projects (PrimeNumberTheoremAnd, Carleson, etc.) and reports direct empirical node-reduction percentages. No equation, parameter, or central claim is obtained by fitting to the target data, by self-citation of an unverified uniqueness result, or by renaming a known pattern as a new derivation. The reported reductions are therefore measurements, not predictions forced by construction. The assumption that syntactic dependencies suffice for semantic review is stated as an explicit modeling choice rather than derived from the tool's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

This is an engineering and tool-development paper. No free parameters or mathematical axioms are involved in the central claim. The primary invented entity is the concept of aligned Lean code.

invented entities (1)

aligned Lean code no independent evidence
purpose: A proposed quality standard for formalizations that have received human semantic verification
Defined in the paper as formalization code that has undergone human semantic verification to serve as a benchmark for AI-generated work.

pith-pipeline@v0.9.0 · 5602 in / 1197 out tokens · 51441 ms · 2026-05-15T10:20:29.004139+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

Lean Compass... automatically extracts the project-specific nodes whose semantic correctness can affect those target statements... Pruning edges from a theorem’s proof is safe because the type checker verifies proof terms

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.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 2 internal anchors

[1]

Kaito Baba, Chaoran Liu, Shuhei Kurita, and Akiyoshi Sannai

URLhttps: //xenaproject.wordpress.com/2025/12/05/formalization-of-erdos-problems/. Kaito Baba, Chaoran Liu, Shuhei Kurita, and Akiyoshi Sannai. Prover agent: An agent-based framework for formal mathematical proofs,

work page 2025
[2]

Kevin Buzzard and Richard Taylor

URLhttps://arxiv.org/abs/2506.19923. Kevin Buzzard and Richard Taylor. FLT: An ongoing Lean formalisation of the proof of Fermat’s Last Theorem. GitHub,

work page arXiv
[3]

URLhttps://arxiv.org/abs/2512.17260. L. Chen et al. Seed-Prover: Deep and broad reasoning for automated theorem proving,

work page arXiv
[4]

Leonardo de Moura and Sebastian Ullrich

URL https://arxiv.org/abs/2507.23726. Leonardo de Moura and Sebastian Ullrich. The Lean 4 theorem prover and programming language. InProceedings of the 28th International Conference on Automated Deduction (CADE 28), volume 12699 ofLNCS, pages 625–635. Springer,

work page arXiv
[5]

doi: 10.1007/978-3-030-79876-5_37. R. Degenne et al. Formalizing Brownian motion and stochastic calculus in Lean 4,

work page doi:10.1007/978-3-030-79876-5_37
[6]

URL https://arxiv.org/abs/2511.20118. J. Hu, T. Zhu, and S. Welleck. miniCTX: Neural theorem proving with (long-)contexts. In Proceedings of the International Conference on Learning Representations (ICLR),

work page arXiv
[7]

doi: 10.1038/s41586-025-08857-x. D. Khovratovich, M. Kudinov, and B. Wagner. At the top of the hypercube—better size-time tradeoffs for hash-based signatures. ePrint 2025/889,

work page doi:10.1038/s41586-025-08857-x 2025
[8]

URLhttps://eprint.iacr.org/ 2025/889. 10 A. Kontorovich and T. Tao. Prime number theorem and more (PrimeNumberTheoremAnd). GitHub,

work page 2025
[9]

Lean Community

URLhttps://github.com/teorth/PrimeNumberTheoremAnd. Lean Community. Validating a Lean proof. Lean documentation, n.d. URLhttps://lean-lang. org/doc/reference/latest/ValidatingProofs/. Accessed 2026-03-15. W. Li et al. Autoformalizing mathematical statements by symbolic equivalence and semantic consistency. InAdvances in Neural Information Processing Syste...

work page 2026
[10]

URLhttps://papers.nips.cc/paper_files/paper/2024/file/ 6034a661584af6c28fd97a6f23e56c0a-Paper-Conference.pdf. Y. Lin et al. Goedel-Prover: A frontier model for open-source automated theorem proving,

work page 2024
[11]

URLhttps://arxiv.org/abs/2502.07640. Q. Liu et al. Rethinking and improving autoformalization. InProceedings of the International Conference on Learning Representations (ICLR),

work page arXiv
[12]

Patrick Massot

URLhttps://proceedings.iclr.cc/paper_files/paper/2025/file/ fceedf8c9c0ff51f41b9fe0294ef0070-Paper-Conference.pdf. Patrick Massot. leanblueprint: plasTeX plugin to build formalization blueprints. GitHub,

work page 2025
[13]

URLhttps://aclanthology.org/2025.emnlp-main.907.pdf. Z. Z. Ren et al. DeepSeek-Prover-V2: Advancing formal mathematical reasoning via reinforcement learning for subgoal decomposition,

work page 2025
[14]

URLhttps://arxiv.org/abs/2504.21801. P. Song et al. Lean Copilot: Large language models as copilots for theorem proving in Lean,

work page internal anchor Pith review Pith/arXiv arXiv
[15]

Joseph Tooby-Smith

URLhttps://arxiv.org/abs/2404.12534. Joseph Tooby-Smith. HepLean: Digitalising high energy physics.Computer Physics Communica- tions, 308:109457,

work page arXiv
[16]

Joseph Tooby-Smith

doi: 10.1016/j.cpc.2024.109457. Joseph Tooby-Smith. Formalizing the stability of the two Higgs doublet model potential into Lean: identifying an error in the literature,

work page doi:10.1016/j.cpc.2024.109457 2024
[17]

Formalizing the stability of the two Higgs doublet model potential into Lean: identifying an error in the literature

URLhttps://arxiv.org/abs/2603.08139. Floris van Doorn et al. A formalization of Carleson’s theorem in Lean. GitHub,

work page internal anchor Pith review Pith/arXiv arXiv
[18]

URLhttps://arxiv.org/ abs/2509.22819. H. Wang et al. Kimina-Prover preview: Towards large formal reasoning models with reinforcement learning,

work page arXiv
[19]

URLhttps://arxiv.org/abs/2504.11354. Y. Wu et al. Autoformalization with large language models. InAdvances in Neural Information Processing Systems (NeurIPS),

work page arXiv
[20]

URLhttps://arxiv.org/abs/2205.12615. 11 K. Yang et al. LeanDojo: Theorem proving with retrieval-augmented language models. InAd- vances in Neural Information Processing Systems (NeurIPS),

work page arXiv
[21]

URLhttps://arxiv.org/ abs/2306.15626. K. Yang et al. Formal mathematical reasoning: A new frontier in AI,

work page arXiv
[22]

org/abs/2412.16075

URLhttps://arxiv. org/abs/2412.16075. T. Zhu et al. LeanArchitect: Automating blueprint generation for humans and AI,

work page arXiv
[23]

URL https://arxiv.org/abs/2601.22554. 12

work page arXiv