OpenAI · 265-01 · 2026-09-24 · Mathematical physics

A two-dimensional area law from a global spectral gap

preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026

Family 265. The two-dimensional gapped area law.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Page curve. Language of radiation or an area law is not a Page curve, not an island, and not a premise of the Hawking sheets. Hawk04-C's finite access analogue is not a Page curve. Hawk04-D's conditional certificate is not a Page curve. Hawk05-A's designed turnover is not a gravitational Page curve. Hawk05-B's finite Haar mean is not a gravitational Page curve and not Family 265's area law. Hawk05-C's conditional extrapolation is not a gravitational Page curve and not Family 265's area law. Hawk06-A's operational island is not a gravitational Page curve and not Family 265's area law. Hawk07-A's operational wormhole is not a gravitational Page curve and not Family 265's area law. Hawk07-B's operational function certificate is not a gravitational Page curve and not Family 265's area law. I-B's prescribed conical spacetime is not Family 265's area law and not a gravitational Page curve. II-B's finite contact certificate is not Family 265's area law and not a gravitational Page curve. III-B's supported laboratory is not Family 265's area law and not a gravitational Page curve. IV-B's represented records are not Family 265's area law and not a gravitational Page curve. V-B's admissible response is not Family 265's area law and not a gravitational Page curve. VI-B's harmonic descent is not Family 265's area law and not a gravitational Page curve. VII-A-ST's plateau intertwiner is not Family 265's area law and not a gravitational Page curve. VII-E-ST's observation package is not Family 265's area law and not a gravitational Page curve. VII-F-ST's slicing certificate is not Family 265's area law and not a gravitational Page curve. VII-J-ST's scheduled certificate is not Family 265's area law and not a gravitational Page curve. VIII-ST's affine-matching certificate is not Family 265's area law and not a gravitational Page curve. IX-A-ST's sharp-admission certificate is not Family 265's area law and not a gravitational Page curve. IX-B-ST's triangle-calibration certificate is not Family 265's area law and not a gravitational Page curve. IX-C-ST's partial-product certificate is not Family 265's area law and not a gravitational Page curve. IX-D-ST's causal-interface certificate is not Family 265's area law and not a gravitational Page curve. Oai276-A records a conditional one-use Holevo audit and does not verify 276-01. A finite Holevo interval is not a Page curve. Oai279-A records a finite prime-list certificate and does not verify 279-01. A classical prime list is not a Page curve. Oai227-A records a finite stationary--quench certificate and does not verify 227-01. A classical autocorrelation is not a Page curve. Oai119-A records a finite Hellinger certificate and does not verify 119-01. A classical Hellinger gap is not a Page curve. Oai265-A does not verify Family 265. Oai265-B is not a laboratory calibration. Oai290-A does not verify Family 290 and does not construct a Page curve. Oai211-A does not verify Family 211 and does not construct a Page curve. Oai280-A does not verify Family 280. Oai280-B does not identify a five-qubit code with that net. Oai280-C's logical-algebra certificate is not a Page curve. Oai282-A's local Ward certificate is not a Page curve. Oai282-B's free Ward action is not a Page curve. Oai069-A's finite-monodromy certificate is not a Page curve. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. Oai119-A · Oai227-A · Oai279-A · Oai276-A · Oai069-A · Oai282-B · Oai282-A · Oai280-C · Oai280-B · Oai280-A · Oai211-A · Oai290-A · Oai265-B · Oai265-A · En07-C · En07-B · En07-A · En06-B · En05-C · En06-A · En05-B · En05-A · En04-A · Hawk04-B · Hawk04-C · Hawk04-D · Hawk05-A · Hawk05-B · Hawk05-C · Hawk06-A · Hawk07-A · Hawk07-B · Hawk04-A · Hawk03-A · Hawk02-E · Hawk02-D · Hawk02-C · En02-C · XVII-A

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 $L^{- 1}$ .

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.

Preprint source and Lean scope

The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.

No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. A gapped area law is not a Page curve. Oai265-A records that separation and does not verify this preprint. Oai265-B is not a laboratory calibration of it. No compilation is asserted.

Page curve
A gapped area law is not a Page curve
Family 265 records an entropy area law for gapped lattice ground states. That law is not a Page curve, not an island, and not a premise of the Hawking sheets. Hawk04-C's finite access analogue is not that law and not a Page curve. Hawk04-D's conditional certificate is not that law and not a Page curve. Hawk05-A's designed turnover is not a gravitational Page curve. Hawk05-B's finite Haar mean is not a gravitational Page curve and not Family 265's area law. Hawk05-C's conditional extrapolation is not a gravitational Page curve and not Family 265's area law. Hawk06-A's operational island is not a gravitational Page curve and not Family 265's area law. Hawk07-A's operational wormhole is not a gravitational Page curve and not Family 265's area law. Hawk07-B's operational function certificate is not a gravitational Page curve and not Family 265's area law. I-B's prescribed conical spacetime is not Family 265's area law and not a gravitational Page curve. II-B's finite contact certificate is not Family 265's area law and not a gravitational Page curve. III-B's supported laboratory is not Family 265's area law and not a gravitational Page curve. IV-B's represented records are not Family 265's area law and not a gravitational Page curve. V-B's admissible response is not Family 265's area law and not a gravitational Page curve. VI-B's harmonic descent is not Family 265's area law and not a gravitational Page curve. VII-A-ST's plateau intertwiner is not Family 265's area law and not a gravitational Page curve. VII-E-ST's observation package is not Family 265's area law and not a gravitational Page curve. VII-F-ST's slicing certificate is not Family 265's area law and not a gravitational Page curve. VII-J-ST's scheduled certificate is not Family 265's area law and not a gravitational Page curve. VIII-ST's affine-matching certificate is not Family 265's area law and not a gravitational Page curve. IX-A-ST's sharp-admission certificate is not Family 265's area law and not a gravitational Page curve. IX-B-ST's triangle-calibration certificate is not Family 265's area law and not a gravitational Page curve. IX-C-ST's partial-product certificate is not Family 265's area law and not a gravitational Page curve. IX-D-ST's causal-interface certificate is not Family 265's area law and not a gravitational Page curve. Oai265-A states a conditional audit of that record and does not verify the proof. Oai265-B's memory certificate is not a calibration of that law. Oai290-A does not turn that law into a channel. Oai211-A's block-tree certificate is not that area law. The authors are OpenAI. The companion does not verify those proofs.
Page curve Oai211-A path-sum certificate Oai290-A typed recovery Oai265-B memory certificate Oai265-A area-law separation En07-C Hawking comparison En07-B endpoint certificate En07-A complete residual En06-B finite transport En05-C tensor-free comparison En06-A layer account En05-B isotropic limit En05-A exact cone En04-A residual response Hawk04-B typed readout Hawk04-C finite access analogue Hawk04-D conditional certificate Hawk05-A classical-record obstruction Hawk05-B finite Haar mean Hawk05-C conditional extrapolation Hawk06-A operational island Hawk07-A operational wormhole Hawk07-B operational functions Hawk04-A complete complement Hawk03-A partner algebra En02-C finite radiation