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 χ acquire direct microscopic origin. Second, a patched low-energy action with fields (gμν,u^μ,nA,χ) emerges after integrating out gapped modes. Third, in the window H₀ k M_* the Wilsonian flow closes on the patched operator family up to O(∂⁴/M_*²), with the deep invariant Ξ=/(c²√Λeff) 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 Rnode~√κ₄/κₙ and mass scale Mnode~ C(Q)√κₙκ₄. 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.