The Bigollφ Method a Pythagorean Unification of Atomic Radius Ratios via the Discriminant Triangle

Abstract

Thomas A. Husmann

We present the Bigollφ Method: a single Pythagorean formula that predicts the van der Waals to covalent radius ratio for 97 elements (Z = 3–99) with zero adjustable parameters. The method is derived from the critical Fibonacci Hamiltonian—a 233- site Aubry–André–Harper lattice at self-dual coupling V = 2J with frequency α = 1/φ—whose Cantor spectrum encodes the Py- thagorean triple (√5) ² + (√8) ² = (√13) ². This identity maps directly onto the Dirac energy-momentum relation E² = p²c² + m²c⁴.

The resulting universal expression, ratio = √ (1 + (θ(Z) × BOS) ²), collapses seven previously distinct empirical modes into one equation. θ(Z) is constructed from two constants extracted from the spectrum: √φ (the oblate geometry factor) and W/6 (one- sixth the universal gap fraction). Five topological gates handle the silver vertex and relativistic regimes.

On the 90 elements with reliable experimental data, the method achieves 100% of predictions within 10% error (mean absolute error 3.9%). Across the full set of 97 elements the mean error is 4.5%. All constants derive rigorously from the single axiom φ² = φ + 1. The discovery process itself followed Fibonacci's Elchataym (method of two false positions) via GPU optimization.

The Bigollφ Method offers the first closed-form, zero-parameter unification of atomic radius ratios across the entire periodic table.

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