Euclidean Decomposition in Plane Arithmetic

This note extends Euclidean Decomposition into plane arithmetic by allowing products, bases, or both to pass from Z to Z[i].The resulting plane-arithmetic readings are compared relative to the standard Gaussian frame and the specified nested square windows. Among the non-Core extensions, the Gaussian-product, integer-base reading uniquely retains the Core arrow n -> n+1 without additional level-by-level choices. In this retained reading, the remainder chamber changes from an interval to a square, while multiplicity, support, modular shadows, transport, boundary admission, first appearance, and weighted visibility extend to Gaussian products. The products generated by the nested first-quadrant square windows exhaust exactly the Gaussian integers with nonnegative imaginary part, and native product data such as norm, direction, and multiplicative phase become intrinsic.