  A corollary to the Reeh-Schlieder theorem is proved: that the time-ordered
Vacuum Expectation Values and the S-matrix of a regularized Lagrangian quantum
theory can be approximated by a local operator that uses nonlinear functionals
of a locally supported source function. For the Wightman axioms, this suggests
a modification that takes the algebra of measurement operators not to be
generated by an operator-valued distribution. The use of operator-valued
nonlinear functionals of a source function introduces many abstract fragments
of the source to give a well-defined top-down construction of interacting
quantum fields, in contrast to a bottom-up blocking and scaling construction or
to analyzing response to changing renormalization scales. The construction can
also be thought of as solving a localized inverse problem for the interacting
dynamics or as a generating function for multi-point bound state fields.
