LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The periodic fundamental monodromy satisfies RTT

Statement

For the chain in Finite periodic fundamental evaluation chain and transfer matrix, let $0, 0^{'}$ be two auxiliary copies and define $M_{0^{'}} (v)$ by the same ordered product as $M_{0} (u)$ . For all $u, v \in C$ ,

$R_{00^{'}} (u - v) M_{0} (u) M_{0^{'}} (v) = M_{0^{'}} (v) M_{0} (u) R_{00^{'}} (u - v) .$

Facts & Assumptions

Given: A finite chain with the local Lax operators and order specified in Finite periodic fundamental evaluation chain and transfer matrix, and the polynomial spectral Yang–Baxter identity.

[F1]

Finite periodic fundamental evaluation chain and transfer matrix: $M_{0} (u) = \prod_{j = 1}^{m} L_{0 j} (u)$ in increasing site order, $L_{0 j} (u) = R_{0 j} (u - a_{j})$ , and different quantum sites are distinct tensor factors.

[F2]

The type-A rational R-matrix satisfies Yang–Baxter and has exact singular parameters: the polynomial $R$ -matrix satisfies the spectral Yang–Baxter identity for all complex parameters.

Proof

1.1givenF1F2algebra

On a fixed site $j$ , apply [F2] to auxiliary spectral points $u, v$ and the site parameter $a_{j}$ . By the indexed-basis convention of [F1], the local relation is $R_{00^{'}} (u - v) L_{0 j} (u) L_{0^{'} j} (v) = L_{0^{'} j} (v) L_{0 j} (u) R_{00^{'}} (u - v)$ . If $j \neq k$ , then $L_{0 j} (u)$ and $L_{0^{'} k} (v)$ act on disjoint auxiliary and quantum tensor factors, so their coefficients commute and their operators commute.

1.2IHF1given

Assume it holds for $m - 1$ , and write $M_{0}^{(m)} = M_{0}^{(m - 1)} L_{0 m}$ and $M_{0^{'}}^{(m)} = M_{0^{'}}^{(m - 1)} L_{0^{'} m}$ .

2.1basestep 1.1

For $m = 1$ , the required equation is exactly step 1.1.

3.1F1step 1.1step 1.2inductiondischarge-induction: step 2.1algebra∎

Moving $L_{0 m}$ past $M_{0^{'}}^{(m - 1)}$ is valid by step 1.1. The induction hypothesis gives $R M_{0}^{(m)} M_{0^{'}}^{(m)} = M_{0^{'}}^{(m - 1)} M_{0}^{(m - 1)} R L_{0 m} L_{0^{'} m}$ . Apply the local relation at site $m$ , then commute $L_{0^{'} m}$ past $M_{0}^{(m - 1)}$ because they act on distinct sites. The result is $M_{0^{'}}^{(m)} M_{0}^{(m)} R$ , proving the induction step and the claim for every finite $m$ .

Depends on

Used by

Finite periodic transfer matrices commute Theorem

Cited to discharge well-definedness by Finite periodic fundamental evaluation chain and transfer matrix.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

L. D. Faddeev, How Algebraic Bethe Ansatz Works for Integrable Model (standard reference, not scraped)
F. Loebbert, Lectures on Yangian Symmetry (standard reference, not scraped)