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Phys-3-A2 · revision r1 · on Phys-3

OpenAI claim summary for Phys-3

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The area theorem treats all finite induced square-lattice domains, arbitrary finite-range uniformly bounded interactions, fixed finite site dimension and a unique ground vector with a global gap Delta. Its entropy bound uses exactly the number of lattice edges crossing A and one size-independent constant depending only on those fixed parameters. Within these assumptions, this is a claimed affirmative answer to the stated question.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Entanglement Area Law for 2D Gapped Quantum Systems
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Entanglement Area Law for 2D Gapped Quantum Systems

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Unique ground states of gapped finite-range square-lattice Hamiltonians are claimed to have entropy bounded by a constant times the boundary size for every site subset.

Mathematical statement of the claimed result

Result 1.
On a finite induced square-lattice domain $\Lambda$, place $\mathbb C^q$ at each site and consider \[ H=\sum_{\varnothing\ne X\subseteq\Lambda,\ \operatorname{diam}_{d_\Lambda}X\le R}h_X, \qquad h_X=h_X^*,\quad\|h_X\|\le J, \] with one term per support, fixed integers $q\ge1,R\ge0$, and $J>0$. If the normalized ground vector $\Omega$ is unique up to phase and \[ H-E_0I\ge\Delta(I-|\Omega\rangle\langle\Omega|),\qquad\Delta>0, \] the claimed constant $C(q,R,J,\Delta)<\infty$ satisfies \[ -\operatorname{Tr}\rho_A\log\rho_A\le C|\partial_\Lambda A| \qquad(A\subseteq\Lambda),\quad \rho_A=\operatorname{Tr}_{\Lambda\setminus A}|\Omega\rangle\langle\Omega|. \] The boundary counts nearest-neighbor edges crossing from $A$ to its complement, and $C$ is independent of domain, subset, and admissible interactions.

Outline of the argument

Marginal tail estimates start with boxes, and positive constraints truncate states near finite sets. Conditional operator bounds and symmetric-replica entropy metrics are transported through finite metric trees, converting the global gap into a uniform area estimate for arbitrary induced domains.

Formal verification

Family 265 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources