Frequency-Dependent Quantum Geometric Scaling: First-Principles Derivation of Hadronic and Deuteronic Radii, Quantum Circuitry, and the Yang-Mills Confinement Horizon

We present a dimensionally invariant quantum geometric framework resolving subatomic interaction boundaries and nuclear spatial extents from first principles [1, 3]. By replacing temporal parametrization with a characteristic quantum excitation frequency (ν) in the denominator, we formulate a mass-frequency-dependent spatial boundary equation r(n, m, ν) that resolves classical dimensional mismatches while preserving exact spatial scaling. Utilizing the Darkon fundamental gravitational constant (G_D = 6.51408576 × 10⁻¹¹ m³kg⁻¹s⁻²) [1], the framework correctly identifies the proton charge radius (r_p ≈ 0.8413 fm) at mode n = 10, predicts the neutron radius (r_n ≈ 0.8425 fm) via rest-frame Compton frequency ratios, and naturally accounts for the deuteron nuclear charge radius (r_d ≈ 2.1416 fm) at mode n = 7 under total nuclear rest-frame excitation [5, 6]. Extending the binary scaling hierarchy to mode n = 37 yields an ultraviolet confinement horizon at r_37 ≈ 1.1627 × 10⁻¹⁹ m, providing a physical, finite lower bound for the non-Abelian Yang-Mills mass gap (Δ > 0) [2, 8]. Furthermore, we derive scale-invariant quantum capacitance (C_n) and quantum inductance (L_n), confirming the precise electrodynamic continuum linking standard free-space wave impedance (Z_EM ≈ 376.73 Ω) at mode n = 1 with boundary-bounded gravitational wave impedance (Z_grav ≈ 188.37 Ω) at mode n = 0 [4].