General 1:1 transformations

When the transformation $T({\bf x})$ is dimension-preserving and 1:1, there is no energy statistic or reference hypothesis needed since the J-function simplifies to the determinant of the Jacobian matrix of transformation $T({\bf x})$, $J({\bf x}; T) = \vert{\bf J}\vert,$ where

$${\bf J}_{i,j}= \frac{ \partial z_i}{\partial x_j}.$$

Re-synthesis of ${\bf x}$ is through the inversion of the 1:1 transform.