Needhamian Geometry: A Universal Mathematical Framework Across Quantum, Terrestrial, and Astronomical Scales

Needhamian Geometry presents a unified mathematical framework that reveals an elegant structure underlying physical reality across scales—from subatomic particles to planetary geometry to stellar distributions. This framework is built on two core principles: The Log(13) transformation, which allows precise mapping between different physical scales while preserving mathematical structure. The π/ϕπ transition ratio (α ≈ 1.42037), which governs recursive shell structures and dimensionless ratios across systems. Spanning over 40 orders of magnitude, the paper presents compelling evidence that particle mass ratios, Earth's geometric alignments (via the Origin Line), and the CAROL stellar shell distribution all conform to the same mathematical laws. Predictions made through this framework outperform conventional models and suggest a fundamental organizing principle in nature. By shifting from absolute units to dimensionless ratios, this work reduces prediction errors by over 75%, offering unprecedented precision across disciplines. This may represent a major step toward a unified theory of physical structure. This version includes full mathematical derivations, methodology, and statistically testable predictions. Technical formatting corrections only; content unchanged from original May 16, 2025 version. If you would like more information about the ENSO Framework please contact Eric Needham: ensotheory1@gmail.com