The Bigolloφ Theorem Three Doors in the Cantor Gate How a Fibonacci Fractal Predicts Structure from Atoms to the Cosmos

Abstract

Thomas A. Husmann

The companion paper on the Bigolloφ Method showed that a single formula rooted in the golden ratio can predict the relative sizes of atoms for 97 elements with stunning accuracy and zero free parameters. Along the way, three distinct quantized values emerged like hidden shelves in the data: 1.146, 1.311, and 1.409. Atoms appeared to “choose” one of these three modes, much like travelers selecting from three doors in a mysterious gate.

This paper asks the bolder question: do these same three modes appear in completely unrelated systems, from molecules and crystals to distant exoplanets and the expansion of the universe itself?

We examined consecutive ratios in homonuclear diatomic bond lengths, diamond-structure lattice constants, the orbital spacings of the TRAPPIST-1 exoplanets, and our own solar system. Nine out of twenty-two ratios landed within 2% of one of the three modes (mean error 0.62%). A Monte Carlo test against 100,000 random triples gave p = 5 × 10^−5. Even more strikingly, the relativistic correction ρ⁶ = φ^1/6 ≈ 1.0835 matches the local-to-CMB Hubble-constant ratio (73.0/67.4) to 0.039%. Kepler statistics independently confirm that planets around Sun-like stars cluster at the middle “crossover” mode while cooler M-dwarfs prefer the baseline mode. Fisher’s combined p-value across all tests is 1.81 × 10⁻¹³.

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