Emergence of Flavor Mixing, CKM Hierarchy, and CP Violation from a Resonant Brane Model (ERB) – Preprint & Stability Proof

This paper presents version 10 of the Emergent Resonant Brane (ERB) model, which defines elementary particles as resonant oscillation nodes within an elastic vacuum brane (tension ε₀ ≈ 0.1099). The central element of this framework is the identification of a fundamental 7.02% geometric asymmetry ("gap") between the up- and down-quark sectors, acting as the mechanical driver for global phase synchronization. We mathematically derive that the system enforces a topological phase-lock at γ = 2π, enabling the definition of a stable 4-pi fermion state. We demonstrate the existence of an algebraic coupling between the dispersion exponent β of the whip-core and the mass-scaling exponent α in the form of α · β = γ = 2π. This relation allows for the direct calculation of the fine-structure constant αₑₘ as a mechanical amplitude ratio with 99.9% accuracy relative to experimental values. Furthermore, the observed hierarchy of the six quark masses—including the extreme scaling factor of 8 · 10⁴ between the up and top quarks—is attributed to an exponential r2π dependence. While the topological structure is strictly dictated by 2π, we identify a remaining numerical divergence as the effect of a transducer operator T. This operator mediates between the "naked" Bessel geometry and the effective inertia of the brane. The consistency of the model is validated through a simultaneous fit of both isospin sectors, where the "honest limit" of the theory is discussed as a necessary consequence of vacuum warping effects.