First Principles of the Relativistic Walker: A Hydrodynamic Derivation of Quantization, Inertia, and Mass-Energy Equivalence

We present the "Relativistic Walker" model, a hydrodynamic framework demonstrating the emergence of quantum mechanics and relativity from scalar field dynamics.We simulate the radial stability of a pulsing vacuum excitation ("breather") and demonstrate that energetic stability requires the coupling strength to follow a geometric scaling law proportional to the square root of the frequency.This constraint forces the emergence of the Planck relation (E = ℏ ef f Ω) as a hydrodynamic stability condition.Furthermore, we analytically derive inertia as the hydrodynamic drag of the vacuum fluid, recovering the mass-energy equivalence (E = mc 2 ) as the effective dispersion relation for a wave packet, consistent with classical wave mechanics.This framework offers a rigorous mathematical bridge between classical fluid dynamics and the fundamental constants of modern physics.