Article 2: PEW — Internal Consistency of the PEW Lagrangian: κ² = 8π as the Exact Closure Condition and Prediction of Newton's Constant

The Primordial Energy Wave (PEW) framework models all physical entities as topologicalvortex configurations of a single complex scalar field Ψ = ÷exp(iθ), governed by theLagrangian L = αc(∂μA)² + βA²(∂μθ)² − λA⁻² − μ²A². This paper demonstrates that the twoindependent PEW expressions for Newton's gravitational constant G — one derived fromthe Lagrangian parameters and one from the Planck length definition — are mutuallyconsistent if and only if κ² = 8π, where κ is the vortex stability parameter and β = κ⁻⁶ =(8π)⁻³. The proof is purely algebraic and exact: Gnd ≡ β/(8π·αc²) = (64π²)²/(8π)⁴ = 1 whenκ²=8π. This closure condition reduces PEW to a single free parameter αc = 1/(64π²), withall other constants (G, ħ,…