Dynamic Vacuum Theory: A Unified Mechanical Framework From Quantum Potentials to Galactic Rotation Curves (With First-Principles Derivation of χᵥ)

This paper presents a comprehensive, multi-scale mathematical formulation of the Dynamic Vacuum Theory (DVT) and the Galactic Pressure Law (GPL)[cite: 1, 2, 3, 4]. The physical vacuum is modeled not as an empty spacetime background but as a dynamic, non-zero density superfluid medium characterized by localized pressure (Pᵥ) and density (ρᵥ) distributions[cite: 1, 2, 3, 4]. Gravitational interactions, macroscopic cosmic expansion, and quantum-scale phenomena emerge naturally as direct consequences of hydrodynamic pressure gradients within this medium. The framework is systematically validated across three distinct physical regimes: microscopic quantum wave-particle duality, macro-scale planetary orbital mechanics (Mercury perihelion precession at 43 arcseconds/century), and galactic-scale rotation curves (Andromeda, M31)[cite: 1, 2, 3, 4]. Crucially, a singular Universal Vacuum Permeability constant (χᵥ) calibrated on M31 observations independently predicts the rotation profile anomalies of the Triangulum Galaxy (M33) with greater than 95% mathematical accuracy[cite: 1, 2, 3, 4]. In this definitive version, χᵥ is successfully derived from first principles as χᵥ = (2/π) × G, where G represents Newton's gravitational constant and 2/π arises as a fundamental geometric projection factor from 3D spherical fields onto 2D galactic disc distributions[cite: 1, 2, 3, 4]. This parameter-free derivation effectively transitions DVT from a phenomenological model to an exact, falsifiable mechanical law, rendering dark matter and geometric spacetime curvature hypotheses redundant[cite: 1, 2, 3, 4].