RG Theory Expansion: Einstein Tensor Decomposition with Experimental Validation. Mathematical and physical: discussion of the strengths and weaknesses of the proposed model formalism.

Presents itself a theoretical extension of General Relativity based on a decomposition of the Einstein tensor into three components, coupled by an extended correlation factor f_{corr}. in version 2 of the article ,this formulation is tested against real gravitational wave data from four confirmed events (GW190521, GW150914, GW190412, GW170817), using data from LIGO Hanford, LIGO Livingston, and Virgo detectors. The results systematically show a lower RMS residual for the proposed model, suggesting it may capture additional physical structures within ringdown signals.  in version 3 of the article present a formal extension of General Relativity (GR) based on a geometric decomposition of the metric tensor and the Einstein tensor, where spatial, temporal, and correlation contributions are treated separately. Introducing a dynamic scalar field ϕ(xα), called the correlation factor, which modulates the interaction between space-timecomponents, and we construct a fully covariant formulation. Demonstrating that the modified field equation, is internally consistent, reducing to standard GR in the limit ϕ → 0 or G(c)µν → 0, while preserving energy-momentum conservation even in the perturbative regime. show that the gravitational wave equation for perturbations hµν is modified by terms proportional to □ϕ and ∇µϕ, generating observable physical effects such as gravitational birefringence, modal decoupling, and systematic shifts in the quasinormal parameters(fn, τn, ϕn).Apply the formalism to the analysis of gravitational ringdown signals, showing that a two-quasinormal-mode modeling is naturally justified, with better data agreement compared to GR, and that the corrections induced by ϕ are directly reflected in statistical parameters (reduction of χ2, increase of R2). In symmetric geometries (e.g., Einstein–Rosen metric), verify that the correlation field generates consistent corrections to the cylindrical wave equation. Conclude that the proposed theory represents a natural generalization of GR, consistent with the principles of general covariance, capable of extending classical theory predictions in the dynamical and post-merger regime, with direct implications for gravitational wave phenomenology and potentially for large-scale cosmology. In the Version 4 the strengths and weaknesses of the proposed model are discussed, in Italian and English versions.