bibkey: lee2026longrangeswap authors: Eunghyun Lee year: 2026 title: Integrability of multispecies long-range swap models with species-dependent interpolation doi: 10.1088/1742-5468/ae80b5 url: https://arxiv.org/abs/2604.12136v1 claim: “Remark 3.1 conjectures that every reduction operator 𝔄_k is invertible for every species-dependent parameter μ_i in [0,1].” strata_touched:
- D5/S3/StatisticalMechanics/LongRangeSwap/LeeCollisionExit
- D5/S3/StatisticalMechanics/LongRangeSwap/LeeReductionInvertibility license: citation-only triage: anchor
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DOI: 10.1088/1742-5468/ae80b5
Source: https://arxiv.org/abs/2604.12136v1
Crossref identifies the author, title, DOI, and publication in Journal of
Statistical Mechanics: Theory and Experiment, 2026, 073102.
Equation (7), source label 138am42, page 10, defines the local matrices B and B′.
Lemma 3.4, equation (26), page 17, gives the adjacent coefficients
ℬ_i and ℬ′i. Equation (28), source label 1122am331, defines
𝔄_0 = I and 𝔄_k = I − ℬ{k+1} 𝔄_{k−1}^{−1} ℬ′_k.
Remark 3.1 immediately after it, page 18, states:
These results lead us to conjecture that 𝔄_k is invertible for all parameters μ_i ∈ [0,1].
Section 2.3, page 7, states:
The invertibility of the general case (μ_i ∈ (0,1) with arbitrary species composition) remains open due to the complexity of the operators arising in the reduction procedure.
The public encoding uses species Fin N, words Fin n → Fin N, and real
matrices indexed by words. For N ≥ 1, n ≥ 2, j ≥ 1, and j + k + 1 ≤ n,
claim requires IsUnit (A μ j k) for every parameter vector with
0 ≤ μ a ≤ 1. The adjacent-slot entry rule is the paper’s tensor product
in word coordinates, with zero-based slot j + i − 2 corresponding to the
paper’s one-based slot j + i − 1.
Proposition 2.2, “Integrability under invertibility”, and the Yang–Baxter result explain the consequence for the source’s reduction to two-particle interactions. The spectral-radius observation in Remark 3.1 is a separate numerical statement and is not an assumption of the invertibility proof.