A Geometric Phase Space Construction for Prime Number Distribution: The Log-Spectral-Prime (LSP) Space with Branch Structure and Dual Number Extensions

We present a novel geometric phase space approach for analyzing prime number distribution through the Log-Spectral-Prime (LSP) space. This framework emerges from Euler's identity geometry via logarithmic scaling and complex phase behaviors. Using spiral-hyperbolic coordinates and second-order logarithmic normalization, we construct a prime counting function expressed as πLSP(X) = Σ(p≤X) p·Li₂(1/p) with rigorous justification via L² energy analysis and Parseval's theorem. The asymptotic prime distribution naturally emerges on the critical line σ = 1/2 through purely geometric principles, without invoking the Riemann zeta function. Numerical validation achieves 0.004% relative error at X = 10⁶, and we prove the asymptotic relation πLSP(X) = π(X) + O(log log X). Recent updates to Section 2.5 now include rigorous branch structure analysis using dual number extensions and Archimedean phase geometry. The complex logarithm's multivalued nature is formalized through nilpotent operators with the constraint ε² = 0, providing explicit treatment of holonomy along the imaginary axis. This geometric framework serves as a bookkeeping device for clarifying phase information without requiring additional spectral assumptions. The work offers a complementary geometric perspective on prime distribution through screw theory and controlled branch extension methods within the Euler product framework.