Article 1: Foundations of the Primordial Energy Wave Theory: Emergent Spacetime, Gravity, Quantum Entanglement and Molecular Observables from a Single Complex Scalar Field

We present a structured formulation of the Primordial Energy Wave (PEW)theory, in which a single complex scalar field Ψ = ÷exp(iθ) is postulatedas the unique primitive entity. Spacetime, gravity, matter, and quantumstructure emerge from its dynamics. The minimal Lagrangian is derived fromfour physical constraints: propagation, topology, stability, and vacuumequilibrium. Stable vortex solutions on a Hopf torus with aspect ratio R/r = φ (goldenratio) are identified with elementary particles. The fine-structure constantis derived topologically as α_em = 3/(16π²φ²) ≈ 1/137.04. Quantumentanglement is identified with the global conservation of the Hopfinvariant I_H; Bell inequalities are shown to be violated…