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 acquire direct microscopic origin. Second, a patched low-energy action with fields (g_, u^, nA, ) 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 =/ (c²₄₅₅) 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₍₎₃₄₄/ₙ and mass scale M₍₎₃₄ C (Q) ₙ₄. 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.