From a Local Quantum Medium to a Patched Effective Theory: RG Closure and an Explicit Hedgehog Defect in Emergent Gravity

We formulate a clean micro-to-EFT branch for a single-medium approach to emergent gravity. The microscopic theory is built from a local condensate sector, a rotor $O(3)$ sector, a gapped dissipative bath, and a stress channel whose collective variable generates the effective metric. Within this setting we make four claims. First, the inertial completion of the nodal sector and the infrared scalar $\chi$ acquire direct microscopic origin. Second, a patched low-energy action with fields $(g_{\mu\nu},u^\mu,n^A,\chi)$ emerges after integrating out gapped modes. Third, in the window $H_0\ll k\ll M_*$ the Wilsonian flow closes on the patched operator family up to $O(\partial^4/M_*^2)$, with the deep invariant $\Xi=\adeep/(c^2\sqrt{\Lambda_{\mathrm{eff}}})$ preserved at leading order. Fourth, the rotor sector supports an explicit localized defect prototype. We present a hedgehog-sector configuration with integer boundary degree, derive its leading gradient-plus-stabilization scaling, and show that the quartic stabilization enforces a finite defect radius $R_{\mathrm{node}}\sim\sqrt{\kappa_4/\kappa_n}$ and mass scale $M_{\mathrm{node}}\sim C(Q)\sqrt{\kappa_n\kappa_4}$. The paper does not claim a full particle spectrum or a non-perturbative proof of RG universality. Its goal is narrower: to turn the statements ``micro-model'', ``RG closure'', and ``nodes as matter'' into one coherent and externally legible theorem-level branch.