Methodology: Generalized Structural Continuum Hypothesis (GSCH)

Proposes the Generalized Structural Continuum Hypothesis (GSCH): a structure that appears in discrete form — as integer order, mode index, or finite partition — should first be tested for a lossless, dynamically closed continuous re-integration before it is declared essentially discrete. Gives three conditions for a legitimate continuous re-integration (recoverability, dynamical compatibility, invariant preservation) and a decision criterion for essential discreteness (the essential-discreteness witness), and compresses the decision procedure into a three-layer test (index continuation, hierarchy re-integration, lossless dynamical closure). Closes with two cases from the Navier–Stokes pure-continuous route — integer derivative order can be lifted to a Gevrey carrier, and interaction order can be re-integrated via a generating functional — showing that a discrete index alone does not yet constitute an essential-discreteness witness, and lays the methodological foundation for every subsequent Pure-Continuous round.