  It is shown that it is impossible to construct a free theory of fermions on
infinite hypercubic Euclidean lattice in even number of dimensions that: (a) is
ultralocal, (b) respects the symmetries of hypercubic lattice, (c) chirally
nonsymmetric part of its propagator is local, and (d) describes single species
of massless Dirac fermions in the continuum limit. This establishes
non-ultralocality for arbitrary doubler-free Ginsparg-Wilson fermionic action
with hypercubic symmetries ("strong non-ultralocality"), and complements the
earlier general result on non-ultralocality of infinitesimal
Ginsparg-Wilson-Luscher symmetry transformations ("weak non-ultralocality").
