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The entanglement area law for gapped ground states in two dimensions

Let H=∑XhXH=\sum_X h_X be a Hamiltonian on finitely many spins Cq\mathbb C^q placed on a finite domain Λ⊂Z2\Lambda\subset\mathbb Z^2, with interactions of bounded range RR and norm at most JJ, and a unique ground state Ω\Omega separated from the rest of the spectrum by a gap Δ>0\Delta>0 independent of the system size. Hastings (2007) proved that in one dimension the entanglement entropy SΩ(A)S_\Omega(A) of any interval is bounded by a constant depending only on q,R,J,Δq,R,J,\Delta. The gap-to-area-law conjecture asks for the higher-dimensional analogue; in two dimensions only subvolume or near-linear bounds for frustration-free, locally gapped models were known. Is there a constant C=C(q,R,J,Δ)C=C(q,R,J,\Delta) such that SΩ(A)≤C∣∂A∣S_\Omega(A)\le C|\partial A| for every region AA of every such two-dimensional system?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Quantum many-body systems; entanglement and spectral gaps
Posed by
The gap-to-area-law conjecture, cited by the manuscript to M. B. Hastings (2007) and Arad, Landau and Vazirani (2012)
Year posed
2007
Years open
19y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for fixed q,R,J,Δq,R,J,\Delta there is CC such that for every finite induced domain Λ⊂Z2\Lambda\subset\mathbb Z^2 and every Hamiltonian of range RR and term norm JJ with a unique ground vector Ω\Omega and H−E0≥Δ(1−∣Ω⟩⟨Ω∣)H-E_0\ge\Delta(1-|\Omega\rangle\langle\Omega|), every A⊆ΛA\subseteq\Lambda has SΩ(A)≤C∣∂ΛA∣S_\Omega(A)\le C|\partial_\Lambda A|. Corollary 1.2 gives the vertex-boundary forms. The companion proves that on open L×LL\times L grids with nearest-neighbour interactions the ground state has a PEPS approximation of bond dimension CLcCL^c and global error L−1L^{-1}. It does not treat dimensions three and higher, Renyi entropies, or degenerate ground states, and gives no explicit constant.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a companion manuscript of the same date, 'Polynomial PEPS approximation of gapped square-grid ground states', which uses this paper's collar estimates as its analytic input.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture. It assumes only a uniform gap for the full Hamiltonian (no local gap, no frustration-freeness, no commutativity) on any finite induced subgraph of the square lattice, including domains with holes, and bounds the entropy of every subset by a constant times its edge boundary, uniformly in domain size and shape. The release has no Lean formalization for this family. The result is two-dimensional only: the paper makes no claim for Zd\mathbb Z^d with d≥3d\ge3. The constant is existential.

Sources

Changelog1 change

Discussion