Euclidean Transport of Multiplication

This note extends Euclidean decomposition to products of arbitrary multiplicative rank. It defines the quotient–remainder array A_d(n,q,r) for d-fold products and shows that Euclidean re-expression under the base change n -> n+1 is governed by a rank-independent transport map, with R ≡ r−q (mod n+1).The reverse Euclidean re-expression is recorded for a fixed nonnegative integer, and the array at the new level is decomposed exactly into inherited interior mass and newly admitted boundary mass. Thus transport is rank-independent, while occupancy and multiplicity remain rank-dependent.