The Transverse Slicing Operator: Curvature Dynamics and the Emergence of General Relativity (Part V of the ODD Series)

We derive the dynamics of 4D spacetime curvature as a rigorous mathematical consequence of the Transverse Slicing Operator ˆQ within the 5D Operator-Derived Dimension (ODD) framework. The Einstein Field Equations (EFE) are obtained explicitly without axiomatic postulation.By projecting the 5D Anti-de Sitter (AdS5) bulk curvature onto the domain wall, computing the First Fundamental Form, and applying the Gauss-Codazzi equations, we demonstrate that the bulk curvature and the domain wall tension perfectly cancel. We explicitly calculate the extrinsic curvature jump via the Israel Junction Conditions to prove domain wall stability. This exact geometric cancellation yields a strictly flat 4D Minkowski vacuum with a vanishing effective cosmological constant (Λ4D = 0), natively resolving the Cosmological Constant Problem. Matter confined to the slice by the P¨oschl-Teller zero-mode produces an exactly normalized stress-energy tensor, leading to fully emergent General Relativity.