{"schema":"postcutoff/event@1","as_of":"2026-10-18T22:10:00+02:00","url":"https://postcutoff.com/e/2026-10-16-openai-2d-area-law-lean-formalized-axiom/","md":"https://postcutoff.com/e/2026-10-16-openai-2d-area-law-lean-formalized-axiom/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-16-openai-2d-area-law-lean-formalized-axiom","date":"2026-10-16","date_precision":"day","short_title":"OpenAI's 2D area law proof Lean-verified by Axiom","deck":null,"takeaway":"One of the physics claims in OpenAI's October maths release, an entropy area law for gapped 2D lattice ground states (open since Hastings' 1D proof in 2007), now has a complete Lean 4 proof with a frozen statement layer and comparator check, produced by AI agents under a human auditor.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"AxiomMath/AreaLaw2D (Lean 4 formalization, README)","url":"https://github.com/AxiomMath/AreaLaw2D","type":"code","group":"primary","domain":"github.com"},{"n":2,"title":"AreaLaw2D HISTORY.md (commit history of the development)","url":"https://github.com/AxiomMath/AreaLaw2D/blob/main/HISTORY.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"OpenAI: A two-dimensional area law from a global spectral gap (paper PDF, openai/math)","url":"https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":4,"title":"openai/math CONTENTS.md (family 265 description)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"official","group":"primary","domain":"github.com"},{"n":5,"title":"LionSR/TNLean (Sirui Lu's tensor-network library in Lean)","url":"https://github.com/LionSR/TNLean","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"Isaac Kim on X: quantum claims in OpenAI's release","url":"https://x.com/Isaac__kim/status/2107607429437657563","type":"discussion","group":"reactions","domain":"x.com"}],"official":5,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Axiom Math","OpenAI"],"title":"OpenAI's AI-generated proof of a 2D entanglement area law fully formalized in Lean by Axiom (AxiomProver tooling + Codex), with no sorry","summary":"On Oct 16, 2026 Axiom Math published AxiomMath/AreaLaw2D, a Lean 4 formalization of \"A two-dimensional area law from a global spectral gap\" (family 265 of OpenAI's openai/math release, dated Sept 24, 2026). Theorem 1.1 (entanglement entropy of the unique ground state of a gapped finite-range Hamiltonian on any finite square-lattice domain is bounded by a constant times the boundary size), Corollary 1.2 and every intermediate proposition are proved with no `sorry`, depending only on Lean's three standard axioms, and checked against a separate challenge statement with the Lean comparator. The README says the proofs were \"produced by Axiom's agent tooling and OpenAI Codex, directed by Ruiz Chen\". OpenAI's own catalogue had listed no Lean proof for this family. The formal proof covers the statement layer Axiom wrote from the paper; no human expert review of the paper is reported yet.","key_facts":["Repo: github.com/AxiomMath/AreaLaw2D, created 2026-10-16, last pushed 2026-10-17; Lean v4.33.0, Mathlib v4.33.0, Physlib","Formalized: Theorem 1.1 (area law on finite induced subdomains of the square lattice from a global spectral gap), Corollary 1.2, Propositions 4.3, 9.6, 10.2, 11.2 and the lemmas of Sections 2, 3 and 11","Audit: 'The build contains no sorry. Every result, including OpenAI.area_law, depends only on the axioms propext, Classical.choice and Quot.sound'; a Challenge/Solution/Comparator pair restates Theorem 1.1 importing only Mathlib and Physlib","Statement layer (OpenAI/Statements.lean) 'was frozen before any proof was written'; deviations from the paper's conventions are logged in docs/oai265/Memory.md (e.g. proofs over ℂ transferred to any RCLike field)","AI role (README, quoted): 'The proofs were produced by Axiom's agent tooling and OpenAI Codex, directed by Ruiz Chen, who froze and audited the statement layer and checked each result independently.'","Builds on Sirui Lu's TNLean (tensor networks in Lean), 'whose independent formalization of this paper covers Sections 2 to 4 and much of the geometry of Section 11', and on Axiom's internal quantum-arealaw library begun in June 2026 for the Brandão–Horodecki 1D area law","HISTORY.md: the oai265 statement layer was started 2026-10-12; the full development took about four days","Paper: OpenAI, 'A two-dimensional area law from a global spectral gap', Sept 24, 2026 (openai/math family 265, 'Area laws and tensor networks for two-dimensional gapped systems'); the same family claims polynomial-bond-dimension PEPS approximations, which this repo does not formalize","Context: the 1D area law was proved by Hastings (2007); a general 2D area law for gapped systems is a long-standing open problem (only special cases such as 2D frustration-free systems, Anshu–Arad–Gosset, were known). Isaac Kim listed '1. Proof of area law in 2D' first among the 'shocking' quantum claims in OpenAI's release"],"key_numbers":[],"tags":["physics","quantum-information","area-law","many-body","lean","formal-verification","axiomprover","codex","openai-math-release","ai-for-math"],"science":{"field":"physics","subfield":"quantum many-body physics / quantum information (entanglement area laws)","problem":"Area law for entanglement entropy of ground states of gapped local Hamiltonians in two dimensions","result":"Lean 4 proof, with no sorry, of OpenAI's theorem that the unique ground state of a globally gapped finite-range Hamiltonian on any finite induced square-lattice domain has entanglement entropy at most C × (number of boundary edges), with C independent of domain size and shape.","open_since":"2007","ai_system":["OpenAI internal model (original proof)","Axiom agent tooling","OpenAI Codex"],"human_role":"AI-generated proof (OpenAI release); formalization by AI agents directed by Ruiz Chen (Axiom), who froze and audited the Lean statement layer and checked each result","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean 4 (no sorry, standard axioms, comparator-checked against a challenge file); the correspondence between the Lean statements and the physics claim rests on Axiom's audited statement layer; no peer review"},"status":"pending","shock":"A claim that physicists called shocking went from AI-written preprint to full machine-checked proof in about three weeks, mostly by AI agents."},"body_md":"## What happened\n\nOpenAI's Oct 6 maths release (see the 722-manuscripts entry) included family 265, \"Area laws and tensor networks for\ntwo-dimensional gapped systems\". Its paper \"A two-dimensional area law from a global spectral gap\" (dated Sept 24, 2026) claims an\nentropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced subdomains of the square\nlattice, assuming only a uniform spectral gap of the full Hamiltonian and bounds on the local interactions. Unlike several other\nfamous claims in the release, OpenAI's catalogue gives no Lean link for this family.\n\nOn Oct 16 Axiom Math published AxiomMath/AreaLaw2D, which proves Theorem 1.1, Corollary 1.2 and all the exported intermediate\npropositions in Lean 4 with no `sorry`. The statement layer was written first and frozen. A separate challenge file restates the\nmain theorem with every needed definition copied in, and the Lean comparator accepted the solution. According to the README,\nthe proofs were \"produced by Axiom's agent tooling and OpenAI Codex, directed by Ruiz Chen, who froze and audited the statement\nlayer and checked each result independently\". Parts of the supporting theory were ported from Sirui Lu's TNLean and QICLean\nlibraries. TNLean has its own partial, independent formalization of the same paper (Sections 2–4 and much of Section 11).\n\nThe commit history (HISTORY.md) shows Axiom's internal quantum-arealaw library started in June 2026 with the Brandão–Horodecki 1D\nresults. The OpenAI-specific work began on Oct 12 and the main theorem was finished by Oct 16. The working notes log every deviation\nfrom the paper's conventions, for example a citation defect in a hypothesis docstring and the choice to prove over ℂ and transfer to any RCLike field.\n\nNo Axiom blog post, X announcement or Hacker News thread about the repo was found by Oct 18 evening. We found it through the sweep's GitHub\nwatch of AxiomMath.\n\n## Why it matters\n\nExperts listed the 2D area law first among the \"shocking\" quantum claims in OpenAI's release: Hastings proved the 1D case in 2007\nand the general 2D gapped case has been open since. A full machine-checked proof removes doubt about the logic, though\nnot about whether the formal statement matches the physics claim. That match rests on Axiom's audited statement layer, which\nphysicists can now read. It is also a case of AI agents verifying another lab's AI-generated mathematics within weeks of\npublication.","disputed":[{"label":"Verification","text":"Formal proof in Lean 4 (no sorry, standard axioms, comparator-checked against a challenge file); the correspondence between the Lean statements and the physics claim rests on Axiom's audited statement layer; no peer review","sources":[]}],"related":[{"id":"2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover","url":"https://postcutoff.com/e/2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover/","date":"2026-10-12","date_precision":"day","short_title":"Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized)","deck":null,"takeaway":"It is one of the first results to build directly on OpenAI's claimed quasi-Riemann breakthrough.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Axiom Math"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":84,"official":13,"filed":"2026-10-07","updated":"2026-10-18","orgs":["OpenAI"]},{"id":"2026-09-06-collatz-positive-density-log-time-mazur-codex","url":"https://postcutoff.com/e/2026-09-06-collatz-positive-density-log-time-mazur-codex/","date":"2026-09-06","date_precision":"day","short_title":"Collatz positive-density convergence, AI-developed, Lean-checked","deck":null,"takeaway":"The first linear lower bound for Collatz-convergent integers (earlier bounds were X^0.84, then X^0.90): an explicit positive proportion of n < X reach 1 within 10.46 ln n steps. Developed mainly by AI agents, Lean-checked twice, not yet peer reviewed. The Collatz conjecture itself stays open.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-18","updated":"2026-10-18","orgs":["ProofAtlas","OpenAI","Axiom Math"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-18","type":"filed","text":"Created (README)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-18","run":null,"sources_read":null,"updated":"2026-10-18","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":169,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":108,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":199,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":138,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-2d-area-law"}