A Phase–Hilbert Framework for Arithmetic: Euler Totient Geometry, Prime Eigen–Angles, and $p$-Adic Shifts

This work proposes a phase-theoretic and Hilbert-space framework for arithmetic, centered on the Euler totient function and the geometry of roots of unity. The core idea is to interpret primitive roots of unity as independent phase directions and to view $\varphi(n)$ as a phase dimension governing the internal structure of arithmetic at scale $n$. Within this framework, additive and multiplicative arithmetic operations are represented as unitary rotations on the unit circle, with particular emphasis on the arithmetic shift $+1$ as a minimal prime eigen–angle rotation. Extending this viewpoint, the paper develops a $p$-adic phase model in which integers are represented as infinite coordinate vectors, and arithmetic carries are interpreted as inter-level phase transfers across a hierarchy of refined eigen–angles. This leads to a natural correspondence between cyclotomic towers, $p$-adic digit structure, and multi-scale phase decompositions. The resulting model provides a unified geometric interpretation of additive, multiplicative, $p$-adic, and quantum Fourier–theoretic structures, without claiming new classical number-theoretic results. Instead, the contribution lies in a coherent conceptual framework that reframes familiar arithmetic operations as phase symmetries in Hilbert space. An appendix records the early conceptual formulation of the theory, preserving the original geometric intuition that motivated the formal development presented in the main text.