This note gives a conditional, testable framework for recovering the weak-field (Newtonian/leading-PPN) limit of GR from a single fundamental field (\Phi:\mathbb{R}^{3,1}\to S^3). The emergent metric (g_{\mu\nu}[\Phi]) is defined from principal-symbol (characteristic) factorization, and deviations from GR are controlled by a single parameter (\varepsilon=\max(\varepsilon_{\rm fact},\varepsilon_{\rm cg})) measuring non-factorization and scale-separation/coarse-graining errors. The paper states an explicit Einstein-form closure postulate valid when (\varepsilon\ll1) and lists the concrete computations needed to bound (\varepsilon) in Solar-System configurations.