{ "abstract": "A Lean+mathlib proof of Nivat's conjecture: Given a finite alphabet A, we consider configurations c: Z^2 -> A, which are arbitrary maps that can be thought of as colorings of the 2D lattice. Writing P_c(m, n) for the number of distinct m-by-n block patterns occurring in a configuration c, Nivat conjectured that if P_c(m, n) <= mn for some positive m and n, then c is periodic; that is, there exists nonzero h such that c(v+h) = c(v) for all v. The proof is due to GPT-6 Pro.", "authors": [ { "name": "Boon Suan Ho" } ], "challenge_render": { "artifact_path": "renders/PALOMAR-2026-09-14-000003-v1/8901ee040bc907a000b2a255104c0f74daa68514c669f4301e88ee959c5f52cd/", "artifact_tree_sha256": "8901ee040bc907a000b2a255104c0f74daa68514c669f4301e88ee959c5f52cd", "entrypoint": "Challenge/index.html", "format": "verso-html", "landrun_commit": "811cfff51ceaf3d9843708aa6d22e9b84ccac8b4", "rendered_at": "2026-09-14T04:54:20Z", "renderer_commit": "a013555a88a0fc9ec910a09ea833dc9cc338db35", "verso_commit": "3bdedf29bada13d8103e6c979001c51dcee210c8" }, "classification": { "arxiv": [ "math.DS", "math.CO" ], "msc2020": [ "37B10", "68R15" ] }, "first_registered_on": "2026-09-14", "formalization": { "challenge_path": "Challenge.lean", "comparator_config_path": "comparator.json", "definition_names": [], "formalization_metadata_path": "formalization.yaml", "lakefile_path": "lakefile.toml", "lean_toolchain": "leanprover/lean4:v4.33.0", "permitted_axioms": [ "propext", "Quot.sound", "Classical.choice" ], "project_dependencies": [ { "name": "mathlib", "repository": "leanprover-community/mathlib4", "revision": "db584cd6d46c92f209a44c0f1c829460d327499d" }, { "name": "plausible", "repository": "leanprover-community/plausible", "revision": "b7eb3304aeae834b12dda98993a37f6a41f6f0bb" }, { "name": "LeanSearchClient", "repository": "leanprover-community/LeanSearchClient", "revision": "5f4d51b81cbd3f6b32b156bfad9056621a040404" }, { "name": "importGraph", "repository": "leanprover-community/import-graph", "revision": "16f02aa7642864af59f1ff0e384a015994db9118" }, { "name": "proofwidgets", "repository": "leanprover-community/ProofWidgets4", "revision": "4be2e3d5087eeb272cf5a8853b8f9dd025ef5957" }, { "name": "aesop", "repository": "leanprover-community/aesop", "revision": "3448c0bcc5ce01b2d1546e483ec3620e32df3d0e" }, { "name": "Qq", "repository": "leanprover-community/quote4", "revision": "92c15be17b7caf78c2ad767ec40f89052d908d81" }, { "name": "batteries", "repository": "leanprover-community/batteries", "revision": "4488d40d070b9700d4d5a6aa342f0d40c31b2a2d" }, { "name": "Cli", "repository": "leanprover/lean4-cli", "revision": "6130a47896ce867c6a4a55373441e59e565bad0f" } ], "solution_path": "Solution.lean", "theorem_names": [ "NivatSubmission.nivat" ] }, "id": "PALOMAR-2026-09-14-000003", "preservation": { "archive_owner": "PalomarArchive", "archived_at": "2026-09-14T04:54:56Z", "receipt_sha256": "f78db283f9b4c4ce11894db2e0583e53a6cbfaef7e22c48309b45ae95f299a5f", "repositories": [ { "commit": "0d94547e934418c9d1b75fcbb7d6af493e86759d", "fork_repository": "PalomarArchive/boonsuan--nivat--8f5b1fc73411", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/0d94547e934418c9d1b75fcbb7d6af493e86759d", "source_repository": "boonsuan/nivat" }, { "commit": "3448c0bcc5ce01b2d1546e483ec3620e32df3d0e", "fork_repository": "PalomarArchive/leanprover-community--aesop--01b46b609077", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/3448c0bcc5ce01b2d1546e483ec3620e32df3d0e", "source_repository": "leanprover-community/aesop" }, { "commit": "4488d40d070b9700d4d5a6aa342f0d40c31b2a2d", "fork_repository": "PalomarArchive/leanprover-community--batteries--1b4c776305c9", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/4488d40d070b9700d4d5a6aa342f0d40c31b2a2d", "source_repository": "leanprover-community/batteries" }, { "commit": "16f02aa7642864af59f1ff0e384a015994db9118", "fork_repository": "PalomarArchive/leanprover-community--import-graph--e0a79261b05a", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/16f02aa7642864af59f1ff0e384a015994db9118", "source_repository": "leanprover-community/import-graph" }, { "commit": "5f4d51b81cbd3f6b32b156bfad9056621a040404", "fork_repository": "PalomarArchive/leanprover-community--LeanSearchClient--9de355ab5f53", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/5f4d51b81cbd3f6b32b156bfad9056621a040404", "source_repository": "leanprover-community/LeanSearchClient" }, { "commit": "db584cd6d46c92f209a44c0f1c829460d327499d", "fork_repository": "PalomarArchive/leanprover-community--mathlib4--67b3d817d8a2", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/db584cd6d46c92f209a44c0f1c829460d327499d", "source_repository": "leanprover-community/mathlib4" }, { "commit": "b7eb3304aeae834b12dda98993a37f6a41f6f0bb", "fork_repository": "PalomarArchive/leanprover-community--plausible--fb632f6f7bf4", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/b7eb3304aeae834b12dda98993a37f6a41f6f0bb", "source_repository": "leanprover-community/plausible" }, { "commit": "4be2e3d5087eeb272cf5a8853b8f9dd025ef5957", "fork_repository": "PalomarArchive/leanprover-community--ProofWidgets4--26014acbbaff", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/4be2e3d5087eeb272cf5a8853b8f9dd025ef5957", "source_repository": "leanprover-community/ProofWidgets4" }, { "commit": "92c15be17b7caf78c2ad767ec40f89052d908d81", "fork_repository": "PalomarArchive/leanprover-community--quote4--90d31d153685", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/92c15be17b7caf78c2ad767ec40f89052d908d81", "source_repository": "leanprover-community/quote4" }, { "commit": "6130a47896ce867c6a4a55373441e59e565bad0f", "fork_repository": "PalomarArchive/leanprover--lean4-cli--df175a7fc790", "ref": "refs/tags/palomar/PALOMAR-2026-09-14-000003-v1/6130a47896ce867c6a4a55373441e59e565bad0f", "source_repository": "leanprover/lean4-cli" } ] }, "provenance": { "mathematical_sources": [ { "author_endorsement": "n/a", "authors": [], "contributors": [ { "name": "GPT-6 Pro", "role": "Proof generation and manuscript writing" }, { "name": "Boon Suan Ho", "role": "Prompting and project direction; no human mathematical authorship claimed" } ], "identifier": "paper/nivat.tex", "license": "MIT", "location": "Theorem 1.1 and its proof in Section 6; Sections 1\u20135 and Appendix A", "relationship": "formalizes", "title": "Pattern complexity and Nivat's conjecture", "type": "paper" }, { "authors": [ { "name": "Jarkko Kari" }, { "name": "Michal Szabados" } ], "identifier": "https://arxiv.org/abs/1605.05929v1", "location": "Lemmas 5\u201310 and 32; Theorem 11 and Corollary 12; manuscript Section 3 and Appendix A", "relationship": "adapts", "title": "An Algebraic Geometric Approach to Nivat's Conjecture", "type": "paper" }, { "authors": [ { "name": "Van Cyr" }, { "name": "Bryna Kra" } ], "identifier": "https://arxiv.org/abs/1208.4090v2", "location": "Generating and balanced sets; manuscript Introduction and Section 5", "relationship": "background", "title": "Nonexpansive Z^2 subdynamics and Nivat's conjecture", "type": "paper" }, { "authors": [ { "name": "Michal Szabados" } ], "identifier": "https://arxiv.org/abs/1710.05360v1", "location": "Two-periodic-summand argument; manuscript Section 5", "relationship": "adapts", "title": "Nivat's conjecture holds for sums of two periodic configurations", "type": "paper" }, { "author_endorsement": "n/a", "authors": [ { "name": "Marston Morse" }, { "name": "Gustav A. Hedlund" } ], "identifier": "https://doi.org/10.2307/2371264", "location": "American Journal of Mathematics 60 (1938), 815\u2013866", "relationship": "background", "title": "Symbolic dynamics", "type": "paper" }, { "authors": [ { "name": "Jarkko Kari" }, { "name": "Etienne Moutot" } ], "identifier": "https://doi.org/10.1007/s00224-021-10063-8", "location": "Theory of Computing Systems 67 (2023), 125\u2013148", "relationship": "background", "title": "Decidability and periodicity of low complexity tilings", "type": "paper" }, { "authors": [ { "name": "Mike Boyle" }, { "name": "Douglas Lind" } ], "identifier": "https://doi.org/10.1090/S0002-9947-97-01634-6", "location": "Transactions of the AMS 349 (1997), 55\u2013102", "relationship": "background", "title": "Expansive subdynamics", "type": "paper" }, { "authors": [ { "name": "Manfred Einsiedler" }, { "name": "Douglas Lind" }, { "name": "Richard Miles" }, { "name": "Thomas Ward" } ], "identifier": "https://doi.org/10.1017/S014338570100181X", "location": "Ergodic Theory and Dynamical Systems 21 (2001), 1695\u20131729", "relationship": "background", "title": "Expansive subdynamics for algebraic Z^d-actions", "type": "paper" } ], "related_formalizations": [], "repository_role": "substantive-development", "responsible_maintainers": [ { "name": "Boon Suan Ho" } ], "result_origin": "source-based" }, "registered_at": "2026-09-14T04:54:33Z", "review": { "outcome": "neutral", "policy_commit": "e9c8c238f5695b10f75db7175648a1d0195352c1", "report": { "sha256": "0dc4bd401fe87fc488cb72b38e2d81f9103cdf7de32596c49d89e4aa0ff1ba50" }, "reviewed_at": "2026-09-14T04:31:37Z", "reviewer_models": [ "codex:gpt-5.6-sol" ], "warnings": [] }, "schema_version": 3, "source": { "commit": "0d94547e934418c9d1b75fcbb7d6af493e86759d", "license": { "declared_identifier": "MIT", "detected_identifier": "MIT", "path": "LICENSE", "sha256": "97182fa2b484a3bda028b0eef349d2b2a8e321a4b4234a4f63922546477bcea5" }, "repository": "boonsuan/nivat", "repository_url": "https://github.com/boonsuan/nivat", "tree_url": "https://github.com/boonsuan/nivat/tree/0d94547e934418c9d1b75fcbb7d6af493e86759d" }, "status": "registered", "submission": { "authorization": { "relationship": "maintainer" }, "submission_id": "pl82qo90wiyy" }, "title": "A proof of Nivat's conjecture", "trust": { "challenge_bytes": 1623, "challenge_dependencies": [ { "provenance": "allowlisted", "repository": "leanprover-community/mathlib4" } ], "challenge_imports": [ "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Data.Int.Interval", "Mathlib.Data.Set.Card" ], "challenge_lines": 40, "level": "high", "reasons": [] }, "verification": { "challenge_sha256": "71cdab884cf4a781808b05f381dae0023e1246239a2a74581fa11e4c7feb48c5", "comparator_commit": "575674928e239f5bc452aab72d1dd7b0f1326494", "evidence_path": "evidence/PALOMAR-2026-09-14-000003-v1/147c991264d62357e967ab1faf8dd4b154f612f917a517f68cff424129fd2582/", "evidence_tree_sha256": "147c991264d62357e967ab1faf8dd4b154f612f917a517f68cff424129fd2582", "landrun_commit": "811cfff51ceaf3d9843708aa6d22e9b84ccac8b4", "lean4export_commit": "15f6055e299ad5b89345e533cc2192f4cc00f659", "mechanical_report_sha256": "d788f15192014d266f09a22a9a376574a13686f9946318f2c55c6c171f5c528e", "nanoda_commit": "68d5ca9db226849b41a6fff59d796ff19d0a8840", "repository": "PalomarRegistry/PalomarSubmission", "run_id": 34800364154, "solution_sha256": "0e664b3ea2a2f748eb62825ac3b64924d2eb920d83da5a822b89b6127eecf609", "verified_at": "2026-09-14T02:53:44Z", "workflow_commit": "ef2fa1eadcb246c2346ddba39b52eaa53d4bb763", "workflow_path": ".github/workflows/submission.yml", "workflow_run_attempt": 1, "workflow_url": "https://github.com/PalomarRegistry/PalomarSubmission/actions/runs/34800364154" }, "version": 1 }