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

Abstract

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 mathematical structure of the interface itself. Three results are central. 1. The channel counts 5, 21, 57, 137 are consecutive members of one classical object: the lattice-point counts N_n (5) generated by powers of the Jacobi theta-function. The ball carries a second grading by support (number of non-zero coordinates), yielding a closed form that fixes the finite list of permitted channel counts. 2. The interface radius postulate m=5 reduces to a self-consistency condition m=N_1 (m) with exactly two solutions. A second axiom—the interface supports the minimal asymmetric anchor pair (1, 2)—selects m=5 uniquely (Selection Theorem). 3. The support grading becomes an arity grading—a coupling terminating on v anchors carries the support-v class—under two structural axioms, equivariance of the channel index under the interface symmetry and blindness to unoccupied directions, together with a single fibre-completeness postulate shown to have content at exactly one arity. A mutual two-ended connection in three dimensions therefore carries 36 channels; a coupling specifying no partner carries the whole ball (137). The interface is interpreted as an oscillating boundary restated in Lorentz invariants: electric-dominated configurations are QD-leaning, magnetic-dominated are SD-leaning, and the null class (light) lies on the boundary. The speed of light is expressed as maximum transduction frequency times minimal interface length. 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