A Gate-and-Kernel Framework for Baryogenesis on Descendant Backgrounds

Three companion case studies on the Hamada mechanism, causal fermion systems, and loop quantum cosmology suggest a common structural question: when has an asymmetry formula earned the descendant language it uses? This note formulates a gate-and-kernel answer. In a first descendant regime where linear evolution is meaningful, the observable asymmetry is written as a gated transport law, \({{\frac{dY}{d\tau} = {{L{(\tau)}Y} + {G{(\tau)}P{(\tau)}S_{prim}{(\tau)}}}},{{\eta_{B}{(\tau_{f})}} = {e_{B}^{\top}Y{(\tau_{f})}}}}\). The transport identity itself is standard variation of parameters; the useful claim is the decomposition around it. Gates are built from calibrated witness maps for ordering, geometry, current basis, thermal support, EFT control, and, when needed, transport-state admissibility. Their thresholds are inherited from the control inequalities of the chosen reduction rather than fitted to the desired baryon yield, which makes the framework prospective as well as retrospective. The binary gate is an admissibility audit; when a chosen reduction resolves a crossover, the same witness data can be refined into an activation weight without inventing a gate profile by hand. A baryon-channel support diagnostic then yields a three-category classification of published formulas. The kernel representation is placed inside a five-layer architecture linking admissibility criteria, background realization, primitive source and transfer, descendant transport, and final readout. Tests include the Hamada/CFS/LQC triad, standard thermal leptogenesis, resonant leptogenesis, and an inflationary helicity-to-leptogenesis example. The result is a reusable audit and model-building scaffold for asymmetry generation on descendant backgrounds, including emergent, inflationary, and post-inflationary settings.