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

We present the "Relativistic Walker" model, a nonlinear hydrodynamic framework demonstrating that quantum mechanics, relativity, and gravitation emerge from the stability constraints of a Lorentz-covariant scalar vacuum.We model the fundamental particle as a pulsing topological soliton ("breather") and prove that energetic coherence is achieved only when the coupling strength aligns with the Euler stability basin (g ≈ e • σ 2.5 √ Ω), identifying a first-principles origin for the Planck relation.Applying Mach's Principle, we derive an inverse-square vacuum decay law (D ∝ t -2 ) from the principle of global flux conservation.This decay naturally resolves the Vacuum Catastrophe (10 -122 ) as a direct consequence of the universe's expansion age (T R ≈ 8 × 10 60 ).Furthermore, we establish a Machian coherence limit for matter-the Effective Vortex Envelope (r p ≈ 3.2 fm)-which scales with the volumetric expansion of the vacuum to match the range of the strong interaction.This scaling reveals a fundamental Entropic Identity (10 121 ), showing that the number of supported modes in the universe is strictly determined by the square of its expansion age.Finally, high-precision 3D SI-calibrated spectral simulations confirm that the overlapping pressure depletion fields of these stable walkers recover the Newtonian inverse-square law (1/r 2 ) at macroscopic scales, while precluding gravitational singularities through fluiddynamic saturation at the core.This work offers a rigorous, deterministic bridge between nonlinear wave mechanics, atomic quantization, and unitary cosmology.