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Phys-49-A1 · revision r1 · on Phys-49

OpenAI claim summary for Phys-49

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.

Odd open spin-one chains with endpoint field $3/5$ are claimed to have gap above $\log(10)/392$ from the stated length onward, with selected infinite-volume index $-1$. 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. A gap in the open spin-one chain with fixed same-sign boundary fields
  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 gap in the open spin-one chain with fixed same-sign boundary fields

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Odd open spin-one chains with endpoint field $3/5$ are claimed to have gap above $\log(10)/392$ from the stated length onward, with selected infinite-volume index $-1$.

Mathematical statement of the claimed result

Result 1.
For a spin-one open chain on $2L+1$ sites, with coupling one and equal endpoint fields, set \[ H_L(h)=\sum_{j=-L}^{L-1}\boldsymbol S_j\cdot\boldsymbol S_{j+1} -h(S_{-L}^z+S_L^z),\qquad\sum_a(S^a)^2=2I. \] Let $E_0\le E_1$ be its two lowest eigenvalues counting multiplicity. At the fixed field $h=3/5$, the claimed bound is \[ E_1(L,3/5)-E_0(L,3/5)>\frac{\log10}{392}\qquad(L\ge960). \] No endpoint-to-endpoint bond is included.

Outline of the argument

Endpoint fields enter boundary vectors of a spatial transfer operator. Periodic concentration estimates are extended to boundary-weighted moments, and finite initialization with quadratic improvement gives the uniform gap; the index conclusion then uses the cited index theorem.

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