When Do Complete Classes Tensorize? Uniformization, Rectangularity, and Minimal E-Process Representations

Complete one-step classes need not remain complete after sequential composition: local dominators may fail to admit measurable predictable choices, while nonrectangular nulls contain global restrictions invisible to local products. We characterize both obstructions. For analytic rectangular conditional nulls, a parameterized one-step class tensorizes into a complete predictable-product class of e-processes if and only if its domination correspondence is strongly Borel pointwise uniformizable; necessity already appears in a two-stage revelation experiment. If the local class is admissible, its products are exactly the globally admissible e-processes and form the unique minimal complete class. We also prove a Borel selection theorem for probability mixtures of continuous kernels over compact parameter spaces. This yields pointwise complete predictable exponential-mixture representations for conditional sub-$\psi$ e-processes and, under natural cumulant and support conditions, identifies their unique minimal complete class. Finally, we characterize the exact rectangularity frontier, exhibit an explicit horizon-two i.i.d.\ counterexample, and construct one horizon-coherent infinite product valid at every bounded stopping time.