bibkey: hou2024rationalqsystems authors: J. Hou, Y. Jiang, Y. Miao year: 2024 title: “Rational Q-systems at Root of Unity I. Closed Chains” doi: 10.21468/SciPostPhys.16.5.129 url: https://arxiv.org/abs/2310.14966 claim: “Appendix C, conjecture (C.2), gives the primitive-state count with infinite Bethe roots.” strata_touched:

• D5/S0/Certificates/Combinatorics/RationalQSystemInfinityCountRefutation license: citation-only triage: anchor

Hou–Jiang–Miao rational Q-systems at a root of unity

J. Hou, Y. Jiang and Y. Miao, Rational Q-systems at Root of Unity I. Closed Chains, arXiv:2310.14966v3 (2024), published as SciPost Phys. 16, 129 (2024), doi: 10.21468/SciPostPhys.16.5.129.

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Appendix C, page 35, states:

Before introducing the algorithm, we make the following conjecture for the number of primitive states with infinite Bethe root(s) by observing the numerical results: [ N^{pri}{±∞}(L, M, n±) = \binom{L}{M − n_±} − \sum_{x=0}^{M−n_±−1} \binom{L}{x}, \tag{C.2} ] when n_± = n_+ = n_− is a solution to (3.15). When there is no solution to (3.15), N^{pri}{±∞}(L, M, n±) = 0.

The surrounding scope says: “We focus on the case with no twist, i.e. κ = 1 and η = iπ/3 for simplicity.” It considers even (L), primitive states with (M \le L/2), and equations (3.15)–(3.17) impose (0 \le n_± \le 2) and (L \equiv 2(M-n) \pmod 6) in this specialization.

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This note records the citation and the verbatim source statement. The kernel-checked refutation is in D5/S0/Certificates/Combinatorics/RationalQSystemInfinityCountRefutation.result.

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• Source: https://arxiv.org/abs/2310.14966
• DOI: https://doi.org/10.21468/SciPostPhys.16.5.129