True Spherenetics - A Unified Scale-Invariant

For nearly a century, 3D morphological characterization relied on decoupled geometrical metrics first introduced by Hakon Wadell: volumetric macro sphericity (\Psi, 1932) and projected micro roundness (1935). While useful for sedimentological classification, the lack of a unified scalar state metric and a formal Greek symbol twin for local curvature smoothness left a fundamental void in multi-scale spatial analysis.

This paper formally introduces Sphereocity (\Phi) as the direct local counterpart to sphericity (\Psi), establishing the dimensionless scalar identity True Spherenetic (\Sigma_{\text{True}} = \Psi \cdot \Phi). By unifying global volumetric compactness with local sub-voxel boundary smoothness into a single scale-invariant operator (0 < \Sigma \le 1.0), we resolve scale-mismatch anomalies inherent in traditional single-tier metrics.

Additionally, we define a complete morphological arithmetic operator suite—comprising the Sharpness Index (\Xi = \Psi / \Phi), Elongation Factor (\Theta = \Phi / \Psi), and Morphological Bias (\Delta = \Psi - \Phi). We evaluate analytical bounds across Euclidean continuous manifolds, regular Platonic polyhedra (Tetralis, Hexalis, Octalis, Pentalis, Icosalis), discretized voxel lattice meshes, and relativistic spacetime boundaries (Schwarzschild and extremal Kerr black hole horizons). True Spherenetic provides a scale-invariant framework for spatial fidelity across computational geometry, discrete lattice physics, and theoretical cosmology.