The learner text and PA1–PA16 prove the averaging properties and Lemma 2.1 under the stated scalar entire Taylor, finite-dimensional compactness, Cauchy–Schwarz, smooth-cutoff, polynomial-root, uniform-continuity and Lebesgue/Fubini hypotheses. Nonzero constants and degree drops are included. Compact support in the punctured ball is asserted for n>=1; n=0 has separate unit mass and K={0}.