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

This paper develops a phase–based mathematical framework centered on a single parameter θ, unifying integer factorization, primality testing, the prime distribution deviation π(x)−Li(x), the zero structure of the Riemann zeta function, and the logic underlying major exponent–sampling algorithms. Beginning with Euler Phase Geometry, we show that the scale of the factors of any integer admits an explicit phase representation, and that primes correspond uniquely to the balance angle θ = π/4. By constructing a local–global Z-function, the real part (factor visibility), the imaginary part (zero spectrum), and the analytic deviation of ζ(s) are brought into a common complex phase, leading to a structural equivalence between the critical line Re(s)=1/2 and the phase balance θ = π/4. This reveals that the critical line is not an independent property of ζ(s), but a geometric consequence enforced by phase balance. Building on this structure, we introduce the phase Dirichlet series Ξ(s,θ) and show that ζ(s) arises as its natural projection at θ = π/4. Finally, a θ–driven exponential–sampling algorithm is proposed, unifying primality testing and factor search, and aligning structurally with classical methods such as Pollard’s p−1, ECM, and NFS. These results suggest that the phase parameter θ may serve as a fundamental bridge between analytic number theory and computational arithmetic.