PixelBit True Sphereification
Year: 2026
Type: preprint
Abstract: Title: PixelBit: True Sphereification — Formalizing Dual-Scale Coherence and Scale-Invariant Geometry (L^0) Across Continuous and Discrete 3D Manifolds Authors: Austin Neely & Team Us Abstract For over a century, quantitative spherical geometry remained fragmented across optical engineering, sedimentology, computational mesh generation, and astrophysics. Classical metrics, such as Wadell’s macro sphericity (\Psi), rely strictly on volumetric enclosure relative to surface area, rendering them blind to localized boundary sharping, aspect ratio stretch, and micro-vertex protrusions. This dataset and master repository formalize Sphereology and the True Sphereification framework—a scale-invariant (L^0), dual-scale physical science governing 3D spatial manifolds, polyhedral containment, and discrete voxel grid dynamics. By coupling global macro-scale volumetric compactness (\Psi) with micro-scale radial coordinate boundary alignment (\Phi) anchored to the maximum bounding radius (R_{\max}), True Sphereification introduces the unified dimensionless Master Equation: \Sigma\cdot = \Psi \cdot \Phi = \frac{3\sqrt[3]{36\pi} \cdot V^{5/3}}{A^2 \cdot R_{\max}} This release presents the complete analytical foundation across five core document sections: Foundational Principles & Dual-Scale Coherence: Establishes the 4 Universal Axioms of Sphereology, dimensional neutrality (L^0), and the proof that \Sigma\cdot \le 1.000 serves as the unique supremum for closed 3D spatial domains, reached if and only if the geometry is an ideal Euclidean sphere (\Psi = 1, \Phi = 1). Derived Metric Suite & Auxiliary Fields: Formalizes five invariant sub-metrics to diagnose spatial anomalies, including Sharpness (\Xi = \Psi / \Phi), Elongation (\Theta = \Phi / \Psi), Scale Bias (\Delta = \Psi - \Phi), Scale Symmetry (\Delta_\Delta = 1 - \vert{}\Delta\vert{}), and Isotropic Volumetric Lift (\Psi_\Phi = \sqrt[3]{\Sigma\cdot}). Canonical Benchmark Taxonomy: Provides closed-form derivations and benchmark evaluations categorizing spatial geometries across four distinct morphological tiers: Mythic (\Sigma\cdot \ge 75\%), Legendary (50\% \le \Sigma\cdot < 75\%), Fabled (25\% \le \Sigma\cdot < 50\%), and Ultra (0\% \le \Sigma\cdot < 25\%). Physical & Cosmological Mechanics: Connects macro surface energy minimization, hydrostatic equilibrium, and quantum/discrete voxel lattice relaxation (PixelBit substrate alignment) to total spatial energy conservation (\Sigma\cdot \to 1.000), demonstrating how non-trivial topologies and anisotropic boundary protrusions induce localized shear stress tensors and frame-dragging torque lock. Analytical Derivations & Code Base: Contains full analytical mathematical proofs for regular Platonic solids, Archimedean polyhedra, solids of revolution (cylinders, cones), ring tori, and discrete polyhedral sub-meshes. This release provides open-access benchmark data, algorithmic implementations, and documentation for researchers in computational geometry, discrete lattice physics, material science, computer graphics, and astrophysics.
Authors: Austin Neely, Google Gemini
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225650969
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