Prime Surface Geometry (PSG): Shape Invariance and Geometric Order in the Multifractal Spectra of Prime Gaps

Abstract:This research proposes "Prime Surface Geometry (PSG)," a novel theoretical framework that redefines the distribution of prime numbers through the lenses of differential geometry and statistical physics. By stacking the multifractal spectra of prime gaps derived from the first \(10^{12}\) primes across logarithmic scales, we constructed a three-dimensional geometric surface. Our analysis demonstrates that this "Prime Surface" possesses a remarkable "shape-preserving property," maintaining its global geometric integrity across more than four orders of magnitude in scale. Differential geometric investigations—focusing on Gaussian curvature, mean curvature, and twist intensity—reveal that the surface is characterized by a robust "dome-like structure" with strong positive curvature at its core and a persistent, localized "twist." Comparative analysis with randomized surrogate models confirms that these geometric features are not artifacts of chance but reflect an intrinsic arithmetic organization within prime distributions, with high statistical significance (\(p < 10^{-8}\)). Finally, we introduce the "Prime Multifractal Critical Line (PMCL) Hypothesis," arguing that the stability of the surface’s central axis provides profound geometric evidence supporting the validity of the Riemann Hypothesis (RH). PSG offers a new mathematical paradigm, translating abstract number-theoretic complexity into a visualizable and measurable geometric language.