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Phys-77-A1 · revision r1 · on Phys-77

OpenAI claim summary for Phys-77

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 pure spin-one antiferromagnetic Heisenberg chain is claimed to have a size-independent positive gap on even periodic rings. 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. A uniform gap in the periodic spin-one antiferromagnetic chain
  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

A uniform gap in the periodic spin-one antiferromagnetic chain

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The pure spin-one antiferromagnetic Heisenberg chain is claimed to have a size-independent positive gap on even periodic rings.

Mathematical statement of the claimed result

Result 1.
For spin-one matrices satisfying $[S^a,S^b]=i\varepsilon_{abc}S^c$, $\sum_a(S^a)^2=2I$, use the even periodic ring \[ H_L=\sum_{j=0}^{L-1}\boldsymbol S_j\cdot\boldsymbol S_{j+1},\qquad\boldsymbol S_L=\boldsymbol S_0. \] Let $\gamma_L$ be the difference between its first two distinct energies. The source claims a unique ground state for even $L\ge60$, and \[ \gamma_L>\frac4{105}\log\frac{80}{79}\quad(L\ge60),\qquad \gamma_L>\frac{\log20}{784}\quad(L\ge2304), \] for even lengths. Therefore $\liminf_{L\to\infty,\ L\text{ even}}\gamma_L\ge\log20/784>0$. Simultaneously similar realizations with the same spin normalization have the same spectral conclusion.

Outline of the argument

Auxiliary twists express partition functions as spatial moments. Two purity comparisons yield a gap criterion, and two finite initializations start an iterative improvement that makes the criterion uniform in ring size.

Formal verification

Family 268 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