GABRIEL'S HORN, MICHAEL'S HORN, AND THE ARITHMETIC GROUND OF THE HARMONIC SERIES

This paper presents the resolution of Torricelli's duality through thePrime Field Architecture of the Arithmetic Modulation Field of M = 30= 2 × 3 × 5. The resolution rests on a single foundational discovery:the composite spectrum of the integers IS complete. Every compositeinteger belongs to exactly one of two structurally determined compositeclasses — the E₂ Layer (even composites, generated by the boundaryprime 2) or the O₃ Fiber (odd composites, generated by the boundaryprimes 3 and 5). This complete account of the composite spectrum makesthe prime-capable class P₁ visible, bounded, and exact for the firsttime in the history of mathematics.