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Phys-59-A1 · revision r1 · on Phys-59

OpenAI claim summary for Phys-59

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.

Uniformly gapped square-grid ground states are claimed to admit PEPS approximations with polynomial bond dimension and normalized vector error at most $L^{-1}$. The question retains these same parameters and conventions. 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. Polynomial-bond PEPS approximation of every gapped square-grid ground vector
  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

Polynomial-bond PEPS approximation of every gapped square-grid ground vector

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Uniformly gapped square-grid ground states are claimed to admit PEPS approximations with polynomial bond dimension and normalized vector error at most $L^{-1}$.

Mathematical statement of the claimed result

Result 1.
Fix local dimension $q\ge2$ and $J,\Delta>0$. On an $L$-by-$L$ square grid, let \[ H=\sum_vh_v+\sum_eh_e,\quad\|h_v\|,\|h_e\|\le J,\qquad H-E_0I\ge\Delta(I-|\Omega\rangle\langle\Omega|), \] with site/edge-supported Hermitian terms and a unique normalized ground vector $\Omega$. The claimed constants $C,c>0$, depending only on $q,J,\Delta$, give a nonzero projected entangled-pair state $\Phi$ for every $L\ge2$, with maximum bond dimension $CL^c$, satisfying \[ \min_{\theta\in\mathbb R}\|\Phi/\|\Phi\|-e^{i\theta}\Omega\|\le L^{-1}. \] The assertion is existence of the PEPS, without an efficient construction guarantee.

Outline of the argument

Separated-region information bounds lead to adaptive finite-rank constraints on small patches. Distributed density networks are compressed using encoded frames and controlled ownership changes; a planar distribution protocol assembles the PEPS without assuming gaps for subregions.

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