The R(Dst) Self-Consistency Equation

The function R(Dst) — the density-dependent recursion resistance that governs propagation speed in the Cohesion UFT — has been characterised by five geometric constraintsand explored through a parametric family, but its explicit functional form has remainedOpen Problem 1 of the series. The field equations paper established that R(Dst) is thedispersion relation of the recursion Lagrangian in a curved background. This paperderives the self-consistency equation that R(Dst) must satisfy: a nonlinear ordinary differential equation arising from the requirement that the recursion field, its stress-energytensor, the background metric, and the local density all be mutually consistent. Theequation is:dRdDst=R(Dst) R(Dst) − R0 Dst · R(Dst)· G(Dst),with boundary conditions R(0) = R0, R(∞) = 1, and R′ > 0 everywhere. The functionG is determined by the stress-energy tensor of the recursion field. Three densitythresholds already established in the series — Dcrit (inflation), Dacc (electroweak), andDGUT (grand unification) — provide calibration constraints that, combined with theODE, uniquely determine R(Dst) everywhere. Open Problem 1 advances from 20% to50% complete: the equation is derived; its explicit solution requires computing G andmatching the three calibration points.