RG-S Framework v27: Complete Lagrangian, Minimal MERA Berry Phase, Falsifiability, and Heat-Kernel Derivation of γ = 0.0931

Public version contains strictly astrophysical fundamental theory only. Full Lagrangian with scalar-vector interactions ∝ φ21φ2µφ µ2and engineering mappings is reserved under U.S.provisional patent filings April-July 2026. We present the sanitized v27. The action couples a Z2 symmetron scalar φ1 and amassive Proca vector φ2µ to gravity. In the sanitized sector we take m2eff = m20 + ξR withξ = 1/6 conformal. The Kretschmann scalar K = RµνρσRµνρσ enters only the heat-kernelcoefficient a2, not the tree-level mass, ensuring dimensional consistency. From the 1-loopheat-kernel on Euclidean Schwarzschild we obtain γ = 0.0931 ± 0.0004 for c1 = 3 physicalProca polarizations. The conical-deficit matching gives remnant mass M∗ =pγ/4π MPl =0.0861 MPl with S → 0, T → 0 in the M → M∗ limit. The model predicts a falsifiable GW echo Aecho/Ainit = 0.03 ± 0.01 at delay ∆t = 45 ± 5 ms for ∼ 62M remnant, independent of m0 and temperature at leading order. Topological hysteresis is defined as a Berry phasein a MERA cortex; in the sanitized limit with no explicit φ21φ22coupling the adiabatic displacement vanishes, providing a null test. The coherence bound Nent ≥ 5 × 1053 rules out kpc displacements, consistent with decoherence falsifier.Note on versions: v25 full version with explicit gsp coupling is restricted. v26 sanitized openversion contained a typographical error M∗ =√γ/4π. v27 corrects to M∗ =pγ/4π. Keywords Modified Gravity, General Relativity Extension, Symmetron Field, Proca Field, Fluid Geometry, Galactic Dynamics, Astrophysics, Theoretical Physics, Open Science