The Cohesion UFT Field Equations

Previous papers in this series identified the Klein-Gordon equation, the Einstein fieldequations, and the QCD Yang-Mills equations as limit cases of the Cohesion UFTrecursion dynamics, but derived these limits from the established equations of particlephysics and general relativity rather than from the Cohesion UFT itself. This papercloses that gap. The recursion medium is described by a complex scalar field Φ(x, τ )whose dynamics are governed by a Lagrangian density derived from two ingredientsalready in the series: the energy relation E = pr (which gives the kinetic structure)and the asymptote theorem R0 > 0 (which gives the potential structure). The resultingLagrangian L =12(∂µΦ)∗(∂µΦ) −R02|Φ|2is the scalar field Lagrangian with a mass termset by the positive asymptote R0. The Euler-Lagrange equation gives the Klein-Gordonequation □Φ + R0Φ = 0 — not borrowed from particle physics but derived from therecursion Lagrangian. The stress-energy tensor of Φ in the high-density, low-gradientlimit reproduces the GR stress-energy tensor, and the Einstein field equations emergeas the consistency condition for the background metric. For the three-component slipphase field Φ = (Φr, Φg, Φb), the gauge-covariant Lagrangian with the SU(3) connection(already derived from local phase freedom) gives the Yang-Mills equations for the gluonfield — QCD derived from the recursion Lagrangian without external input. Therunning of the strong coupling αs follows from the density-dependence of the recursionfield’s dispersion relation, giving the QCD β-function coefficient b3 = −7 from therecursion geometry. General relativity and QCD are no longer inputs to the CohesionUFT; they are outputs.