Flexible Static Cosmology

This is the fourth paper in the $K\left(t\right)$ series and introduces the broader framework as Flexible Static Cosmology (FLSC). The framework was first introduced as the operational redshift kernel $K\left(t\right)$ , which reproduced the classical background redshift relations within a flexible static ontology and preserved a Planck spectrum exactly under collisionless transport. This paper supplies the local gravitational closure in ordinary general relativity. Under the stated bulk hypotheses, compatibility between the $K\left(t\right)$ transport law, null-geodesic photon propagation, and covariant number conservation forces geodesic, shear-free deformation. The spatial metric factorizes as $h_{ij}=K^2{\gamma }_{ij}$ , with $\theta =3\dot{K}/K$ , and the spatial Einstein equations require constant sectional curvature. Robertson-Walker geometry is therefore derived step by step rather than assumed. The same deformation rate gives $pK=\text{constant}$ , $T\propto K^{？1}$ , $n_{\gamma }\propto K^{？3}$ , and ${\rho }_{\gamma }\propto K^{？4}$ .