The Arithmetic Interface: Theta-Series Structure and a Selection Theorem for the QD–SD Boundary

The preceding DDC papers established a dual-domain ontology in which spacetime emerges from a timeless Quantum Domain (QD) through transduction across an interface. This paper develops the structure of the interface itself. Its core results are mathematical: The channel counts of the framework—5, 21, 57, 137—are consecutive members of a single classical object: the lattice-point counts N_n=#{x∈Z^n:|x|^2≤5}, generated by powers of the Jacobi theta-function. The entire hierarchy derives from one function. The interface radius postulate R^2=5 is reducible to a self-consistency condition N_1 (m)=m, which has exactly two solutions, and a second axiom—the interface supports the minimal asymmetric anchor pair (1, 2)—selects m=5 exactly (the selection theorem); the remaining open item is the derivation of that axiom from the stability hierarchy. Around these results, the paper assembles the interpretive picture of the interface as an oscillating boundary, reformulated in Lorentz-invariant terms: the QD- and SD-leaning phases correspond to the electric-dominated and magnetic-dominated field classes, and light—the null class—lies exactly on the boundary. The speed of light is expressed as a maximum transduction frequency times a minimal interface length: c=ν_max l_min. A dimensional-transition ladder relating the hierarchy's four counts to the known interactions is stated as an explicit conjecture with two anchored rungs (137↔ electromagnetism, [1]; 57↔ strong sector), two open rungs, a missing weak sector, and stated fail conditions. The paper claims no unification. It claims a precisely specified mathematical object at the framework's core and a set of well-posed open problems on it. Keywords: dual-domain cosmology; interface; theta-series; fine-structure constant; Lorentz invariants; self-consistency