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 legitimately,with correlation function I_cross = I_H·cos²(θ/2) derived from theconnection 1-form on S³. The equivalence principle follows from the localphase speed C̃_loc = C̃₀·Ã/Ã_vac. A new interpretation of C̃₀ is proposed: it is the topological condensationthreshold — the maximum Mother Wave speed beyond which no stable vortex canexist — distinct a priori from c. The agreement C̃₀ ≈ c to 99.98% is aresult of the theory, not a postulate. A new falsifiable prediction, m_H ≈ 2656 MeV/c², is obtained for theradial fluctuation mode of Ã, independent of all free parameters. Selectedmolecular observables (H₂O bond angle, H₂ ground-state energy) arereproduced within the same framework. All results are classified byepistemic status; open problems are explicitly identified.