The Harmonics of the Integers: Number Theory as the Orbit Dynamics of the Doubling Fold
Maria Smith
Abstract
One axiom. One operation. Zero free parameters. The multiplicative order of 2, cyclotomic cosets, Artin’s constant, and the Riemann symmetry are usually separate theorems. This paper shows they are one thing: the orbit dynamics of the doubling fold. The fold map x ↦ 2x (cast out whole Ones) on a reduced fraction p/q is number theory natively. Odd q gives a pure eternal cycle whose period is the multiplicative order of 2 mod q, independent of p — the period of the binary expansion of 1/q. The φ(q) fractions partition into φ(q)/ord_q(2) orbits — the 2-cyclotomic cosets. Even q = 2^a·(odd) decays for exactly a steps then locks into its odd cycle. The reflection j/q ↔ (q−j)/q is the antipodal involution summing to the One, whose only self-antipode is the half-One 1/2 — the seed of the Riemann critical line. A prime is a single fully-ergodic mode iff 2 is a primitive root mod q, with density Artin’s constant. The integers have harmonics; the fold sounds them. Machine-checked; reproduces from one command. A standalone result within the Smithian Fold Theory of Everything (SFTOE). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory