Proton Mass as a Silver-Ratio Beat — The √2 Fixed Point of Vacuum Interference

We show that the proton mass emerges as a silver-ratio beat--the interference minimum between topologically distinct soliton modes in the Vacuum Energy Quanta Field (VEQF). When Wang's empirical relation 2 m_u^2 = m_d^2 is interpreted as the frequency ratio f_d / f_u = sqrt(2) within VEQF's mass-frequency correspondence (m = h f / c^2), the proton mass formula m_p = 4 K |f_d - f_u| yields the silver ratio conjugate (sqrt(2) - 1) as its core scaling factor. This irrational constant is not a fitted parameter but the mathematical signature of the Principle of Least Tension (P4): sqrt(2) represents the unique fixed point where TCI = 2 and TCI = 3 soliton modes coexist with minimal ripple entropy in the vacuum field. The topological invariants (integer winding numbers) remain strictly preserved while the dynamical frequencies adopt the irrational ratio required for optimal interference. This fixed-point structure parallels the Koide relation in the lepton sector, suggesting a unified vacuum spectroscopy where square-root mass hierarchies reflect fundamental frequency geometry. We further demonstrate that Wang's proposed proton generations arise naturally as uniform scaling of soliton frequencies across vacuum energy densities, with sqrt(2) invariant under the transformation f -> a * f. The work constitutes a parameter-free theoretical synthesis: the silver ratio is derived as the necessary consequence of least-tension interference between the proton's u-d-u soliton stack, transforming Wang's empirical observation into a geometric prediction of vacuum topology.