The original tensor entries lie in B_(P0,ell), represented as the constant U coordinates of A_(P0). Their contraction is a finite sum of products of those entries, so it remains in that subring. After the recursive computation returns the correct residue, extract its constant U coordinate for every retained power of V; all nonconstant U coordinates are zero modulo 2^P0. Do not sum the V coefficients here: that is a separate operation required by particular counting encodings, not by tensor contraction itself. The extension enabled interpolation without changing the requested output ring.