Unary Theta Solutions and Gaussian Ray-Class Density for a Resonant Sixth-Order Modular Differential Equation

This preprint studies a resonant sixth-order modular differential equation and develops an explicit connection between its solution space, unary theta series, and arithmetic over the Gaussian integers. The logarithm-free resonant case leads to a factorization through the Euler derivative, and the resulting rank-five equation admits an explicit eta-weighted unary-theta basis with a modular Wronskian. After a suitable twist, the associated system has finite monodromy, giving an arithmetic interpretation of its behavior in positive characteristic. For the corresponding rank-six lifting problem, the first characteristic-p obstruction is reduced to a coefficient of a distinguished theta solution. Using theta identities and the arithmetic of Gaussian integers, the vanishing of this obstruction is characterized by explicit Gaussian residue classes. Ray-class equidistribution then gives a relative natural density of one quarter in each of the relevant rational congruence classes. The distinguished theta series also has a CM interpretation in terms of Hecke theta series. A finite quartic Kummer comparison is included as an additional arithmetic observation, while its general reciprocity interpretation remains open.