证明
Lemma 19. The following statements hold for all sufficiently large n . At…Lemma 25 ([Walsh estimates]). Let all weights below have absolute value at most…Lemma 44 ([Gram normalization]). Let a plus slots and b minus slots in one…观点
By Lemma 19, the image laws are uniform laws on the indicated independent vectors with their same-endpoint Gram entries prescribed. Relative to the product of the two separate batch laws, the union law adds only the mutual Gram entries and joint injectivity. The latter may be omitted at an exponentially small error because the number of columns is bounded. Expand the indicators of the mutual Gram entries into binary characters. Their values, which are determined by the nominal inputs, enter this expansion only as signs. In a nontrivial character, after identical descriptions of an unordered slot pair have been collapsed, the coefficient matrix between the two batches is nonzero. Its bit rank is at least N . The individual batch Gram densities can be included in the two separate bounded weights.Lemma 25 ([Walsh estimates]) bounds this character by a constant times F__∞2^ -N/2 . There are only boundedly many characters. Applying the same expansion with F=G=1 controls the normalization, while Lemma 44 ([Gram normalization]) bounds all internal normalization factors. The zero character gives the product of separate expectations.