The ϕ–5 Cage: Geometric Recursion as Stroboscopic Sampling of a Horn Torus with 5‑Fold Defect

DOI: 10.5281/zenodo.18504145Title: The ϕ–5 Cage: Geometric Recursion as Stroboscopic Sampling of a Horn Torus with 5‑Fold Defect Author: Miles Enoch Tracy contact: milestracy@yahoo.com ORCID: https://orcid.org/0009-0006-7781-8259 DOI DescriptionThis work introduces a new geometric–arithmetic framework that reinterprets discrete recursion as stroboscopic sampling of a continuous, constrained manifold. The central construct—the “ϕ–5 cage”—is formed from the constant \(c = \phi^{-1} + 1/5\) and exhibits a decimal digit sequence under integer multiplication that reveals a parity flip at every fifth term. This flip is interpreted as a Berry‑phase holonomy arising from a topological defect on a horn torus, whose kiss point corresponds to the cusp of a cardioid obtained via \(\pi\)-sampling. The cage demonstrates that recursion is not algebraic iteration but the discrete shadow of bounded, curved motion. When mapped to a Clifford‑algebraic Smith chart, the same geometry forces the zeros of the Riemann zeta function onto the critical line, transforming the Riemann Hypothesis from an analytic problem into a constraint‑satisfaction one. This approach bridges number theory, topology, and signal processing, offering a concrete, elementary representation of geometric phases and topological confinement through simple arithmetic sequences. **Keywords:** geometric recursion, Berry phase, horn torus, ϕ–5 cage, stroboscopic sampling, Riemann Hypothesis, Clifford algebra, Smith chart, decimal digit parity, topological defect, Collatz geometry, quasicrystal arithmetic