From Geometric Constraint to Field-Level Coupling Emergence: A Structural Derivation of the Fine-Structure Constant

This record contains the paper and computational scripts for a structural derivation of the electromagnetic fine-structure constant α within the ENSO (Emergen None-local Scaling Order) Framework. The central result is α = gap / (RG · (π² − 1/φ)), where gap = 24 − φ³/RG² and RG = ln(φ)/ln(π). Both geometric quantities were derived in prior timestamped ENSO publications independently of α; no parameter was tuned to match the experimental value. The formula reproduces α to 0.19% accuracy against CODATA 2018. A second independent encoding, α ≈ (14 − πφe)/25, achieves 0.0077% accuracy using all three ENSO trinity constants. A pre-registered null test (1,000,000 draws from a 40-constant pool excluding π, φ, e) yields p = 9.5 × 10⁻⁵, confirming the identity is statistically rare. A constrained grammar search over 241,950 candidates with a complexity penalty and Pareto frontier analysis confirms the trinity result is Pareto-optimal among axiomatically admissible expressions. The geometric coefficient π² − 1/φ is shown to equal π² − φ + 1 exactly, via the golden ratio identity φ² = φ + 1, revealing non-arbitrary algebraic structure in the denominator. Files included: Fine_Structure_Constant_Refined.pdf — full paper (23 pages) Fine_Structure_Constant_Refined.tex — LaTeX source validate_alpha_calculation.py — primary derivation verification (mpmath, 50 d.p.) trinity_null_test.py — pre-registered null test, 1,000,000 draws, seed 20260226 enso_grammar_search.py — constrained grammar search with Pareto analysis, seed 42 README.txt — script documentation, requirements, disclaimer All scripts require Python 3.8+ and mpmath. A broader survey testing whether α is a universal "unit of incompleteness" across transcendental products did not survive null testing at any threshold; that conjecture is explicitly withdrawn. The paper states what is established, what is not, and how every result can be independently verified. Licenses Creative Commons Attribution Non Commercial No Derivatives 4.0 International For further information about the ENSO Framework, please contact Eric Needham:ensotheory1@gmail.com