To be analogous to cellular automata one can then require this mapping to be local, in which case if it is continuous it must be just a linear differential operator involving Derivative[n] —and at some level its behavior must be fairly simple.

Closely related to my discussion of the absence of t 2 terms in volume growth for 4D spacetime cones is the statement that if one sets up a small 3D ball of comoving test particles then the volume it defines must have zero first and second derivatives with time.

But inevitably there will be some places in the space where s is stationary (has zero variational derivative) with respect to changes in fields.