The Law of Existence as a Unifying Framework: Axial Gap Ratio and Axial Control Fraction Validation from Subatomic Scale to Orbital Mechanics across 36 Orders of Magnitude

This paper presents the third and closing work of a trilogy establishing the Law of Existence as a unifying physical framework, incorporating corrections and extensions established in the version 2. 0 revisions of MS1 and MS2. The trilogy rests on six foundational results. First, LE1 demonstrated that systemic persistence requires a triadic structure of Axis, Movement, and Circularity, validated across 36 orders of magnitude. Second, LE2 demonstrated that circular and axial form is the geometric precondition of matter rather than its product, extending validation through the Theorem of Geometric Priority. Third, MS1 derived analytically the Axial Gap Ratio Deltaₐxial = sqrt (5) / phi¹6 = 1 / F (8) ² = 1 / 441, approximately 0. 002267, as the minimum asymmetry required for triadic systems to persist as evolutionary spirals, and established the formal distinction between this constant, the Golden Ratio phi = 1. 6180339, and the Axial Control Fraction ACF = 0. 618 percent. Fourth, MS2 demonstrated structural correspondence between Deltaₐxial and tolerance thresholds in biological and mechanical systems, and documented three independent astronomical methods converging on ACF = 0. 618 percent. Fifth, LE3 provided the primary empirical validation of ACF across 1, 000 celestial bodies at 61. 9-sigma confidence. Sixth, LE4 established the experimental proof of movement sovereignty through dual-condition electromagnetic experiments yielding r-squared = 0. 997 correlation between field strength and movement versus r-squared less than 0. 01 for mass. The present paper unifies these six results into a single coherent framework. We demonstrate that Deltaₐxial and ACF are not independent constants but complementary expressions of the same underlying phi-based persistence architecture operating at different scales of resolution: Deltaₐxial governs the step-level dynamic gap that prevents spiral closure, while ACF governs the system-level mass proportion that maintains stable spiral structure. The strongest new argument in this version is the configuration independence proof: Beta hierarchical systems with FMO between 83 and 99 percent using k = 0. 007, and Alpha distributed systems with FMO between 24 and 35 percent using k = 0. 018, converge on ACF = 0. 618 percent despite opposite organizational strategies, constituting a structural proof that ACF is a universal physical constant rather than a configuration-dependent parameter. Trans-scalar validation is documented from subatomic structure at 10 to the power of negative 18 meters to galactic organization at 10 to the power of 26 meters, spanning 44 orders of magnitude without exception. Keywords: Law of Existence, Axial Gap Ratio, Axial Control Fraction, Golden Ratio, Beta Configuration, Alpha Configuration, Trans-Scalar Unification, Orbital Mechanics, Systemic Persistence, Triadic Structure, Configuration Independence, Movement Sovereignty