The Self-Foundational Planar Integration Framework: Emancipation from the Gamma Singularities and Geometric Derivation of Prime Frequencies under 90° Isometric Rotations

This paper establishes a novel, self-foundational mathematical framework—Planar Complex Analysis—that emancipates prime number theory from the traditional algebraic constraints of the Riemann zeta function and the singularities of the gamma function. Historically, Bernhard Riemann (1859) utilized the gamma function Γ(s) as an obligatory algebraic bridge to transform discrete series into a continuous integral, paying a topological penalty by introducing infinite singularity poles in the negative domain and executing a 180° mirror-inversion (the negative sign −z) that crushed the semicircle geometry. This work bypasses the classical framework entirely by substituting the gamma function with a smooth, entire planar function e^{−z²}. Embedded within a three-dimensional spatial manifold, this entire function maintains a strictly parallel, non-zero radius (0 < r < 2π) that bypasses nodal collisions. We rigorously prove that under a stable 90° orthogonal isometric rotation matrix (Rz), the hidden imaginary depth is transformed bijectively into a pure cosine phase without phase leakage. The geometric constant of the modified Hankel semicircle (π) deterministically locks the damping exponent at the critical half-line (γ = 0.5). Furthermore, we demonstrate that this system is non-derivative and self-generative; it does not consume pre-calculated Riemann zeros, but instead derives the critical frequencies (Tn) autonomously via spectral resonance peaks and inverse Fourier transforms of discrete prime energy spikes.