Beyond Legendre Dynamics: Cascaded Noether Invariants, Energy Propagation, and Vorticity Analysis on 3D Response Manifolds

These papers present a novel three-step geometric and field-theoretic framework for analyzing coupled differential systems, centering on the Operator Splitting Cascade. By applying Emmy Noether's theorem layer-by-layer, we derive localized conserved charges (I_1) and establish rigorous energy bounds via Cauchy-Schwarz inequalities to prevent finite-time divergence. Furthermore, the global response manifold is modeled as a 3D vector field, where curl vorticity directly quantifies non-conservative inter-layer power flux and ensures strict topological phase stability within a bounded invariant torus (T^3).