Phase-Model Paper 00B: Derivation Framework from a Phase-Amplitude Field to Canonical Equation Classes

This paper presents a compact derivation framework for the strongest defensible version of the phase-first program without overstating what is proved. Starting from a complex coarse-grained phase-amplitude field, the paper derives exact continuity and phase-dynamics equations and shows how several canonical equation classes emerge as projection regimes rather than unrelated ansatzes. These include the Schrodinger/Gross-Pitaevskii, Madelung-type hydrodynamic, complex Ginzburg-Landau, and Kuramoto-type forms. The paper also formulates a gauge-coupled extension through local phase invariance and presents a relativistic phase-based interpretation in which proper time, spacetime interval, and local frequency are mutually translatable. A central contribution is the explicit separation between exact results, projected limits, and conjectural extensions, so that the phase-first program is stated in a falsifiable and non-overstated form.