Continuous Euclidean Transport

This note records Euclidean transport under continuous variation of a positive real base. For a fixed nonnegative real number, it gives the exact base-change formula for its quotient and remainder coordinates and recovers the discrete transport invariant under the unit change n -> n+1.On every open interval of constant quotient, the Euclidean remainder moves locally linearly with (d/dy)R_y(x) = -Q_y(x), while quotient changes occur at discrete thresholds. Thus continuous refers to variation of the base parameter, not to global continuity of the remainder function.