  Motivated by Gutzwiller's semiclassical quantization, in which unstable
periodic orbits of low-dimensional deterministic dynamics serve as a WKB
`skeleton' for chaotic quantum mechanics, we construct the corresponding
deterministic skeleton for infinite-dimensional lattice-discretized scalar
field theories. In the field-theoretical formulation, there is no evolution in
time, and there is no `Lyapunov horizon'; there is only an enumeration of
lattice states that contribute to the theory's partition sum, each a global
spatiotemporal solution of system's deterministic Euler-Lagrange equations.
  The reformulation aligns `chaos theory' with the standard solid state, field
theory, and statistical mechanics. In a spatiotemporal, crystallographer
formulation, the time-periodic orbits of dynamical systems theory are replaced
by periodic $d$-dimensional Bravais cell tilings of spacetime, each weighted by
the inverse of its instability, its Hill determinant. Hyperbolic shadowing of
large cells by smaller ones ensures that the predictions of the theory are
dominated by the smallest Bravais cells.
  The form of the partition function of a given field theory is determined by
the group of its spatiotemporal symmetries, that is, by the space group of its
lattice discretization, best studied on its reciprocal lattice. Already
1-dimensional lattice discretization is of sufficient interest to be the focus
of this paper. In particular, from a spatiotemporal field theory perspective,
`time'-reversal is a purely crystallographic notion, a reflection point group,
leading to a novel, symmetry quotienting perspective of time-reversible
theories and associated topological zeta functions.
