This research presents a first-principles derivation of the hadronic charge radius of the K^+ meson using the framework of the Unified Vibrational Force Theory (UVFT). Instead of relying on the phenomenological parameters typically found in Quantum Chromodynamics (QCD), this work derives the spatial extent of the kaon directly from the dynamics of a single degenerate scalar field Sₕ. Abstract and Results Mechanism of Spatial Equilibrium: The paper identifies a Vibrational Phase Transition where the high-frequency oscillation of the kaon (= mK c²) activates a nonlinear elastic response in the vacuum. Vacuum Compression Energy (VCE): We introduce the VCE, an expansive pressure arising from the degenerate higher-derivative term ₀ Sₕ² (² Sₕ) ², which counteracts the contractile pressure of the field's bare mass. Analytical Prediction: The derived equilibrium formula RK = LK (64K/9⁴) ^1/9 yields a charge radius of 0. 560 fm, matching the PDG 2024 experimental value with absolute precision. Hierarchy Resolution: The work solves the energy hierarchy problem between the Planck scale and the hadronic scale through a self-consistency condition that determines an anomalous mass-scaling exponent = -0. 4719. Significance The results demonstrate that the femtometric scale of matter is a direct consequence of the same geometric stiffness ₀ that governs galactic rotation and cosmological structure. This research provides a critical mathematical link in the unification of General Relativity and Quantum Mechanics by showing how a single vibrational field adaptively rescales its response across 19 orders of magnitude in energy.
Alejandra Borgiani (Sun,) studied this question.