Stick Theory Number Mechanics: Primes, Collapse, and the Structural Rewriting of Arithmetic

This paper introduces a structural reinterpretation of arithmetic grounded in recursive identity stability rather than quantitative fusion. In Stick Theory, the number 1 is the only irreducible whole, and primes are defined as recursive singularities that resist decomposition. The classical addition operator is replaced by a priming operator that preserves identity within shared recursive fields. A new Collapse Identity Equation models the ratio of recursive behavior to identity retention, allowing for a field-based understanding of number coherence. This framework provides a structural explanation for the divergence of the Riemann Zeta Function and reclassifies composite numbers as unstable recursive overlays. The work formalizes a field-theoretic alternative to traditional number theory, grounded in collapse geometry and recursive memory fields.