  This paper proposes a general framework for nonperturbatively defining
continuum quantum field theories. Unlike most such frameworks, the one offered
here is finitary: continuum theories are defined by reducing large but finite
quantum systems to subsystems with conserved entanglement patterns at short
distances. This makes it possible to start from a lattice theory and use rather
elementary mathematics to isolate the entire algebraic structure of the
corresponding low-energy continuum theory.
  The first half of this paper illustrates this approach through a persnickety
study of (1 + 1)D continuum theories that emerge from the $\mathbb{Z}_K$ clock
model at large $K$. This leads to a direct lattice derivation of many known
continuum results, such as the operator product expansion of vertex operators
in the free scalar CFT. Many new results are obtained too. For example,
self-consistency of the lattice-continuum correspondence leads to a rich, novel
proposal for the symmetry breaking structure of the clock model at weak
coupling, deep in the BKT regime. This also makes precise what one means by
"continuous" when saying that continuous symmetries cannot be broken in (1+1)D.
  The second half of this paper is devoted to path integrals for continuum
theories of bosons and fermions defined in this finitary formalism. The path
integrals constructed here have both nonperturbative lattice definitions and
manifest continuum properties, such as symmetries under infinitesimal rotations
or dilatations. Remarkably, this setup also makes it possible to generalize
Noether's theorem to discrete symmetries.
