Quantised Geometric Structure in Planetary Architectures: A 13σ Test of the UST φ-Harmonic Scaling Law

This study provides a decisive empirical validation of the Unified Substrate Theory (UST) harmonic scaling law by analysing semi-major axes in 115 multi-planet exosystems from the NASA Exoplanet Archive. The UST prediction—that orbital architectures should organise into logarithmic “harmonic ladders” with inter-orbit spacings converging to integer multiples of ln(φ), where φ is the golden ratio—is evaluated using the ladder model ln(an) = A + Δn and the harmonic distance statistic S. Real planetary systems exhibit an exceptionally tight alignment with φ-quantised spacings (Sdata = 0.2475), far outside the 2000-realisation permutation null distribution (Snull = 1.0522 ± 0.0610), yielding a 13.2σ separation and p = 5×10⁻⁴.

A Solar-System–specific BIC analysis reveals a three-ladder architecture—Mercury–Venus, Earth–Mars, Jupiter–Neptune—and predicts three unoccupied harmonic shells (1.35 AU, 2.32 AU, 2.94 AU) matching dynamically meaningful gaps in the asteroid belt and the Earth–Mars transition region. The results demonstrate that planetary orbital radii are not randomly distributed in logarithmic space but follow a quantised geometric order consistent with the UST harmonic law. This constitutes the strongest large-scale evidence to date of φ-governed discrete-scale organisation in planetary systems.