Alena-Rainich-Codazzi Branch Geometry and Integral Twistor-Link Zero Modes: Structural Reconstruction of Standard Model Multiplets

This article develops an Alena-Rainich-Codazzi (ARC) branch geometry for traceless gauge-side stress tensors of non-null Rainich type. In the admissible sector, the stress-balance condition is represented by a Codazzi-closed anisotropic Lorentzian branch metric whose residual scalar acts as a Codazzi multiplier. The resulting branch has shear-free geodesic Rainich principal null directions and admits explicit warped examples. Compact-leaf obstructions are treated through regular integral link da