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Phys-49 · physics · mathematics · Physics

A gap in the open spin-one chain with fixed same-sign boundary fields

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For spin-one $H_L=sum_{j=-L}^{L-1}$ $S_j\cdot S_{j+1}-(3/5)(S_{-L}^z+S_L^z)$, is the difference between its two lowest eigenvalues, counted with multiplicity, greater than log(10)/392 for every integer $L\geq 960$?

Short English statement

For spin-one $H_L=sum_{j=-L}^{L-1}$ $S_j\cdot S_{j+1}-(3/5)(S_{-L}^z+S_L^z)$, is the difference between its two lowest eigenvalues, counted with multiplicity, greater than log(10)/392 for every integer $L\geq 960$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-49-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-49

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.

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

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