Prove
Lemma 27 ([Opposite field moves]). Arbitrarily long homogeneous shortcut chains…Lemma 26 ([Finite prime complexity]). Let x_1,…,x_s be positive real numbers…Proposition
Color by the two cover categories, the endpoint family, and whether a split and merge occur. There are finitely many colors. The endpoint characteristic is common because every step is in defining characteristic. If both an increasing and a decreasing exceptional field move occur, Lemma 27 ([Opposite field moves]) applies. Otherwise all exceptional primes in a pair ratio have the same sign. The product of the two bounded rational factors belongs to a fixed finite set, and at most two exceptional degrees occur. A bounded exceptional degree has uniformly bounded prime complexity as well. Apply Lemma 26 ([Finite prime complexity]) to x_i=f_i . We may therefore assume that all endpoint fields and families agree. The exceptional degrees at the individual covers are now bounded integers. Indeed, with no opposite field moves their product, or its reciprocal, is a member of the fixed finite set of bounded twisting ratios, because f_j/f_i=1 . Each positive degree divides the numerator or denominator of one of these finitely many rationals. Fix a lower endpoint i . We show that its upper natural dimension has only boundedly many possible values. For block and parabolic shapes, the ambient dimension is the active dimension, or twice that dimension when its dual is present, plus the dimensions of the omitted blocks. The dimensions of nonabelian omitted blocks are determined, with bounded ambiguity, by the bounded-cardinality pool of inactive types. Soluble omitted blocks have bounded dimensions. A tensor shape multiplies the active dimension by a product of the omitted dimensions. A replicated base multiplies it by a replication degree from its bounded-cardinality pool. A form change has the same natural dimension, with the bounded natural-module exceptions included in the family color. Field restriction multiplies dimension by one of the bounded degrees just obtained, and a twisted-tensor step raises it to one of those degrees. Thus one geometric cover gives a bounded list of possible outputs from a fixed input. Two covers give a list whose size is at most the product of the two bounds. For a split followed by a merge the second step has the same simple label, so gives no new natural dimension at all. All labels now have the same family and field. Their strict growth in order makes their natural dimensions distinct. There cannot be more upper endpoints than the bounded number of possible outputs from the fixed lower endpoint. This contradicts arbitrary length.