Purple Numbers: Foundations — A Signal Calculus on the Integers, the Composition of Zero and Infinity, and the Determination Structure

This paper gives the Purple Mathematics framework a mathematical foundation. It defines the Purple Number System: a signal calculus in which every positive integer is a spectrum over the primes — each prime a pure tone, each composite a chord, the unit a silence — and in which the framework's two founding assertions become exact statements: zero is not empty (it is the full spectrum, carrying every prime to infinite order), and infinity is composed (it is the archimedean place, the second and last kind of pole a number system can have). The system is presented axiomatically and then shown consistent by exhibiting a model built entirely from classical objects — the free commutative monoid of the fundamental theorem of arithmetic, Steinitz's supernatural numbers, the p-adic valuations with v_p(0) = ∞, the product formula, and Ostrowski's theorem — so that the new system can never contradict established mathematics, while remaining free to extend it. On this foundation the paper constructs the framework's original layer: the neighbourhood calculus (reach, dig, determination weight, and wall as operators on signals), the reception pairing with the zeros of the Riemann zeta function, and a dictionary translating every primitive of Purple Mathematics — poles, determination, inheritance, identity, deviation, Balance Boundary — into an exact object of the system. The paper closes with a research program of five open problems native to the system, including the determination constant, the excursion law, and the joint law of archimedean and finite determination. All classical content is attributed by name; the contribution of this paper is the axiomatization, the assembly, the calculus, and the program. Foundational paper in the Purple Mathematics series. No new theorems about the integers are claimed in the foundational layer; the classical mathematics invoked (fundamental theorem of arithmetic; Hensel's valuations; Steinitz's supernatural numbers, 1910; Ostrowski's theorem, 1916; the product formula; the Riemann–von Mangoldt explicit formula) is attributed in Section 10. The original contributions are the axiomatization of the Purple Number System with its consistency theorem, the determination-independence postulate (Axiom 6), the neighbourhood calculus, the dictionary of framework primitives, and the five-problem research program.