AI 中文总结
本文研究两项Rogers二对数关系对秩二Nahm系统的算术约束，确定特定族中的允许步长，证明指标13系统的两项恒等式并精确再现Fricke振幅，同时指出相关乘积求值仍为猜想。
AI 中文摘要
我们研究两项Rogers二对数关系对具有混合分母步长的秩二Nahm系统所施加的算术约束。单项互补方程在每个代数次数中给出到二次数据的一致转换。在一个指定的十记录最简三次族内，正耦合、正定性和整数值指数在m>2时恰好留下步长(1,3)、(1,13)和(1,31)。在最小相容模31处，所需支撑不能是乘法子群陪集。二次例子包括已知乘积恒等式和两个系统，对于这两个系统，首次径向修正排除了每个具有有理线性项的单个模和。对于指标13系统，我们证明了一个两项Rogers恒等式，并从先前证明的二维Fricke变换的矩阵中恢复其参数。其尖点增长与所得鞍点作用一致，独立计算精确再现了所有三个前导Fricke振幅。相关的两项乘积求值仍是猜想性的；其小秩和维数与同一模处经典秩五、六分量Andrews-Gordon构造形成对比。一个混合符号三次族说明了算术限制的极限。
英文摘要
We study the arithmetic constraints that two-term Rogers dilogarithm relations impose on rank-two Nahm systems with mixed denominator steps. Monomial complement equations give a uniform conversion to quadratic data in every algebraic degree. Within a specified ten-record simplest-cubic family, positive coupling, positive definiteness, and integer-valued exponents leave, for m > 2, exactly the steps (1,3), (1,13), and (1,31). At the smallest compatible modulus 31, the required support cannot be a multiplicative subgroup coset. Quadratic examples include known product identities and two systems for which the first radial correction excludes every individual modular sum with rational linear terms. For the index-13 system, we prove a two-term Rogers identity and recover its arguments from the matrix of a previously proved two-dimensional Fricke transformation. Its cusp growth agrees with the resulting saddle action, and an independent calculation reproduces all three leading Fricke amplitudes exactly. The associated two-summand product evaluations remain conjectural; the small rank and dimension contrast with the classical rank-five, six-component Andrews-Gordon construction at the same modulus. A mixed-signature cubic family illustrates the limits of the arithmetic restriction
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