Surface Geometry of Prime Gap Spectra: Curvature, Twist, and Shape-Preserving Structure in Multifractal Prime Gaps

AbstractWe investigate the multifractal singularity spectra \( f(\alpha) \) of log-normalized prime gaps across multiple scales by stacking them along the logarithmic scale axis \( u = \log_2 \varepsilon \). This construction yields a two-dimensional surface \( S = \{(\alpha, u, f(\alpha, u))\} \) in three-dimensional space, which we refer to as the Prime Gap Spectrum Surface.Using differential geometry, we compute the Gaussian curvature \( K(\alpha, u) \), mean curvature \( H(\alpha, u) \), and twist intensity \( T(\alpha, u) \). Our analysis reveals that the surface forms a weakly twisted, dome-like geometric structure with predominantly positive (elliptic) curvature in the central region and narrow negative (hyperbolic) curvature bands near the flanks. The surface exhibits strong shape-preserving behavior across more than three orders of magnitude in scale, with remarkably stable curvature statistics and localized twist.Comparative analysis with randomized surrogate models (shuffled, phase-randomized, and AR(1)) demonstrates that the prime-gap surface possesses significantly stronger curvature magnitude and twist intensity — typically 3 to 5 times larger — indicating the presence of intrinsic, non-random arithmetic structure in the distribution of prime gaps.These findings establish a novel geometric perspective on prime number distribution and suggest that multifractal spectra of prime gaps encode a coherent, weakly twisted geometric surface reflecting deep number-theoretic correlations.Keywords: Prime gaps, Multifractal analysis, Singularity spectrum, Surface geometry, Gaussian curvature, Twist intensity, Prime Information Geometry