Vector–Operator Reconstruction Theory

This work introduces the Vector–Operator Reconstruction Theory (VORT), a general mathematical framework for lifting nonlinear dynamical systems into Hilbert spaces, evolving them through linear or nonlinear operator families, and performing a structured pullback that preserves admissibility and physical meaning. The theory combines classical tools—Koopman operators, semigroup evolution, monotone operator methods, convex projections, and set-valued analysis—with new admissibility structures and class-invariant filters. VORT provides a unified reconstruction procedure valid for finite- and infinite-dimensional systems, offering a domain-agnostic pathway for model reduction, stability analysis, and operator-theoretic interpretation across physics, engineering, and applied mathematics.