he Prime Tree: A Geometric Framework for Riemann Zeta Zeros withNumerical Verification

Abstract

AbstractWe propose the Shui Prime Tree —a geometric framework embedding the distribution of primes into three-dimensional Euclideanspace. Each prime is mapped to a node on a helical curve, , where the radiusencodes the Riemann critical line , the vertical coordinate is base- , and the angular phase encodes the prime's Fourierfrequency. We derive the analytic form of the induced metric, show that the Gaussian curvature is predominantly negative (hyperbolic), and ind that the total curvature integral satisfies . FFT analysis of primes (up to ) shows that the angular powerspectrum peaks match the first 25 nontrivial zeros of , with median deviation ; the zero-spacing variance deviates from the GUErandom-matrix value by only . A quantization scheme yields probability density , consistent with the PrimeNumber Theorem. This work provides reproducible numerical evidence for a geometric approach to the Riemann Hypothesis.