Euler Phase Geometry: A Unified Structure for Integer Factorization, Prime Bias, and Riemann Zeros

This paper proposes a unified structure centered on a phase variable, designed to connect integer factor structures, prime distribution biases, and the phase-balance property of Riemann zeros. Based on Euler phase geometry, we establish an isomorphic correspondence among factor visibility, the structure of the critical line, and local--global deviations, and we construct a phase Dirichlet series and a phase Z-function, thereby placing zero behavior, factor scale, and phase balance points within a single geometric framework. On the basis of this structure, we introduce a phase-compression system consisting of four core assumptions: a logarithmic-scale constraint on phase sets, the uniqueness of the zero-residual phase corresponding to each factor, a lower-bound condition for phase density, and the controllability of exponential phase growth. Under these assumptions, we provide a strictly deterministic unified algorithm that performs composite factorization of large integers in polynomial time, and, under the same conditions, yields deterministic primality certification. The algorithm exhaustively scans all phase witnesses and therefore, within the hypothesis system, its outputs (factors or "Prime (Certified)") possess full determinacy. The structure developed in this work has a dual significance: on one hand, it offers a mathematical viewpoint that unifies factor structures, zero structures, and phase geometry; on the other hand, it establishes a clear and testable structural framework under which polynomial-time factorization and primality certification can be rigorously discussed. The geometric system constituted by phase number theory provides a clear and extensible path for studying the relationships among factors, zeros, and phase structures. The algorithm exhaustively scans all phase witnesses. Within this self-consistent hypothesis system, this yields a strictly deterministic procedure solving the composite factorization problem in polynomial time O((log N)^3.585), thereby producing outputs---either non-trivial factors or "Prime (Certified)"---that possess full determinacy.