Prime Spirals as Spectral Interference Patterns of Riemann Zeta Zeros

This paper establishes an observable connection between the distribution of prime numbers visualized in the Sacks spiral and the spectral geometry of Riemann zeta zeros. We prove that the multiple spiral arms observed in prime number visualizations are interference patterns generated by the imaginary parts of zeta zeros. Through analytical derivation and numerical verification, we demonstrate that each zero $\gamma_n$ generates a distinct logarithmic spiral, and their superposition creates the discrete set of arms visible in the Sacks spiral. This work synthesizes spectral theory, analytic number theory, and geometric visualization to reveal that prime spirals are holographic projections of the Riemann zeta function's non-trivial zeros.