Is the following proposition true?
Proposition: Suppose $\mathbb{C}\{x_1,\ldots,x_{d_1}\}/(f_1,\ldots,f_{k_1}) \cong \mathbb{C}\{y_1,\ldots,y_{d_2}\}/(g_1,\ldots,g_{k_2})$ are isomorphic complex space germs of dimension $n$. Let $r_1, r_2$ be the respective ranks of the Jacobian matrices $\partial f_i/\partial x_j$ and $\partial g_i/\partial y_j$. Then $d_1 - r_1 = d_2 - r_2$.
More briefly, is the difference between the codimension and the rank of the Jacobian an analytic invariant? If so, could somebody please point me to a reference/proof?
Yes it is an algebraic invariant. Since both the rings are isomorphic, they have same embedding dimension which is defined as the minimal number of generators in the maximal ideal of a local ring.