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A two-dimensional area law from a global spectral gap
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Abstract
We prove an entropy area law for the unique ground state of a finite-range Hamiltonian on any finite induced subgraph of the square lattice. A lower bound on the spectral gap of the full Hamiltonian and fixed bounds on the local dimension, interaction range, and interaction strength suffice. For every set of sites, its entanglement entropy is bounded by a constant times the number of edges crossing its boundary, independently of the size and shape of the domain.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Tomotaka Kuwahara, The Rényi area law conjecture is false below α=1 in high dimensions (arXiv:2610.15199, 10 Oct 2026). Constructs uniformly gapped finite-range Hamiltonians in dimension ≥2 with volume-law Rényi-α entanglement for α<1. A "Note added" cites this paper as "a proposed proof" of the von Neumann area law and says that, assuming it, Rényi area laws follow for all α≥1; the author states the results are independent of it.
Formalization
- LionSR/TNLean, Tracking: two-dimensional area law from a global spectral gap (issue #8735) (GitHub, opened 7 Oct 2026, closed 18 Oct 2026). The maintainer's closing comment reports Theorem 1.1 as
TNLean.PEPS.AreaLaw.uniformAreaLawand Corollary 1.2 asuniformAreaLaw_vertex_boundary_bounds/rectangularEntropyAreaLaw, using only the standard axioms; the planning is disclosed as prepared with OpenAI Codex. [AI-generated per source] - AxiomMath/AreaLaw2D, Release v1.0: complete formalization of the 2D area law (GitHub, 17 Oct 2026). An independent Lean 4 development that states it proves the main theorem and every proposition of this paper with no
sorry. Its README says the proofs were produced by Axiom agent tooling and OpenAI Codex under Ruiz Chen, who audited the statement layer, and lists deviations from the paper's conventions indocs/oai265/Memory.md. [AI-generated per source] - TNLean, Safe-Box Entropy and the Sublinear Collar Bound (formalization scope note, 2026-10-15). The note records the formal treatment of Propositions 9.5–9.6, including removal of intermediate entropy hypotheses and extension to real clearance parameters, and labels its scope restriction resolved.
- LionSR, Area law: two ordered families of separated lattice regions (TNLean issue #8758, opened 2026-10-07). This closed formalization task tracks Proposition 11.2’s two-family geometric decomposition for arbitrary cuts in finite induced domains.
- hz-xiaxz in LionSR/TNLean, Formalize the two-dimensional area law and polynomial PEPS approximation from a global spectral gap (issue #8732) (GitHub, 7 Oct 2026). Proposes the formalization and states that the Lean code in the OpenAI release proves neither headline theorem; it reports the only substantive endpoints as a special case of Proposition 3.3 and Lemma 5.1. [family-level]
- TNLean contributors, Tensor-network theory, formalized in Lean 4 (Lean 4 repository, accessed 2026-10-21). The repository hosting the family-specific formal developments describes a predominantly agent-produced formalization workflow and notes that some library results remain unfinished or assumed as axioms. [AI-generated per source] [family-level]
- LionSR/QICLean, doc(paper-gaps): update the area-law and PEPS gap notes after the TNLean proofs (PR #856) (GitHub, merged 18 Oct 2026). Updates three "paper-gap" notes on the area-law and PEPS manuscripts to resolved status, citing the TNLean declarations. [family-level]
- LionSR/TNLean, Tracking: explicit constants in the area law and the polynomial PEPS approximation (issue #9638) (GitHub, opened 18 Oct 2026, closed). Aims to make the existential constant C(q,R,J,Δ) of Theorem 1.1 explicit; the thread describes it as a tower of three exponentials and marks some estimates as heuristic. [family-level]
- LionSR/TNLean, Snake matrix product state for square-grid ground states from the two-dimensional area law (issue #9553) (GitHub, 16 Oct 2026). Proposes deriving from Theorem 1.1 an exp(O(L))-bond-dimension MPS approximation along a snake ordering of the L×L grid; the maintainer has put it on hold.
Related prior literature mentioned by others
- Anurag Anshu, Itai Arad and David Gosset, An area law for 2D frustration-free spin systems (arXiv:2103.02492, 2021; STOC 2022). The Binary Verse AI exposition uses this locally gapped, frustration-free vertical-cut result to contextualize the manuscript’s claim under a full-system gap for arbitrary regions.
Possibly relevant literature (found by the agent, not checked)
- J. Eisert, M. Cramer, M. B. Plenio, Area laws for the entanglement entropy – a review (Rev. Mod. Phys. 82, 277 (2010)). A standard review of rigorous lattice area-law results in one and higher dimensions, relevant to the paper's historical account of the gap-to-area conjecture (Section 1.2); it is not in the bibliography.
News and blogs
- Scott Aaronson, The Mathocalypse (Shtetl-Optimized, 7 Oct 2026). A post on the release that lists "Area law for 2D gapped Hamiltonians" among its highlights, calling it "One of the main open problems in Hamiltonian complexity."
- Azmat, 2D Area Law: How OpenAI’s Proof Connects the Global Gap to Entanglement (Binary Verse AI, 2026-10-13). This exposition outlines the proposed area-law argument and distinguishes the companion’s polynomial-bond PEPS existence claim from efficient tensor construction or contraction. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 265
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026,
author = {{OpenAI}},
title = {{A two-dimensional area law from a global spectral gap}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf}{OAI:A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026}},
year = {2026}
}Family 265: Area laws and tensor networks for two-dimensional gapped systems
Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open $L\times L$ squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L−1.
Other papers in this family:
- Polynomial PEPS approximation of gapped square-grid ground states · September 24, 2026