Two-term dilogarithms, mixed-base Nahm sums, and Fricke symmetry

Abstract

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