A Rigid Sixth-Order Modular Differential Equation: Hypergeometric Descent, Binomial–Kummer Reciprocity, and a Resonant-Prime Conjecture

This preprint studies a rigid sixth-order modular linear differential equation and the arithmetic behavior associated with its resonant primes. The differential equation is shown to admit a uniquely determined accessory parameter, and the corresponding rank-six local system is proved to be irreducible and rigid. Explicit cyclic-vector transformations and a four-step middle-convolution descent reduce the system to a Gauss hypergeometric equation and provide an explicit description of its Frobenius structure. On the arithmetic side, the paper establishes a binomial–Kummer reciprocity law and relates the resonance behavior of the differential equation to a quadratic Kummer class. Exact identities involving Cartier coefficients, Gauss–Jacobi sums, and Frobenius data are also proved. These results lead to a conjectural arithmetic description of the resonant primes. The vanishing of the marked MLDE resonance obstruction is predicted to be governed by the Kummer class. Extensive exact computations support this conjecture, while the remaining marked Frobenius comparison is isolated as the principal open problem. The record includes the manuscript, LaTeX source, supplementary data, machine-readable results, and computational code.