Propagation-State Transformation Matrix in Extended Classical Mechanics

The propagation-state formalism recently introduced within the framework of Extended Classical Mechanics (ECM) provides an alternative physical interpretation of the relativistic Doppler effect by distinguishing the normalized propagation state from the observed propagation state while preserving the mathematical validity of the relativistic Doppler relation. The present work extends that conceptual interpretation by developing an explicit algebraic transformation connecting normalized and observed propagation states through a propagation-state transformation matrix. Starting from the standard relativistic Doppler equation together with the ECM propagation-state definitions, the derivation establishes an algebraic relationship between the normalized propagation velocity, the observed propagation velocity, the propagation-state deviation, the frequency ratio, and the relativistic Doppler factor. This formulation leads naturally to a propagation-state transformation operator that maps the normalized state vector into the observed state vector while preserving the mathematical consistency of the Doppler relationship. The resulting transformation matrix provides a compact mathematical representation of propagation-state evolution and establishes the propagation-state deviation as an independent ECM quantity that may be evaluated directly from measurable frequency and wavelength relationships. The formalism further distinguishes the standard relativistic limit, where the propagation-state deviation vanishes, from the generalized ECM case in which measurable propagation-state evolution is represented by a non-zero deviation. Rather than replacing the relativistic Doppler formulation, the proposed matrix formalism supplements it with an alternative physical interpretation based upon propagation-state dynamics. The transformation operator introduced here provides a mathematical foundation for subsequent ECM investigations involving phase evolution, frequency dynamics, wavelength transformation, effective mass formation, and quantum-to-cosmological propagation-state transitions.