  Lenia is a continuous generalization of Conway's Game of Life. Bert Wang-Chak
Chan has discovered and published many seemingly organic dynamics in his Lenia
simulations since 2019. These simulations follow the Euler curve algorithm
starting from function space initial conditions. The Picard-Lindel\"of Theorem
for the existence of integral curves to Lipschitz vector fields on Banach
spaces fails to guarantee solutions, because the vector field associated with
the integro-differential equation defining Lenia is discontinuous. However, we
demonstrate the dynamic Chan is using to generate simulations is actually an
arc field and not the traditional Euler method for the vector field derived
from the integro-differential equation. Using arc field theory we prove the
Euler curves converge to a unique flow which solves the original
integro-differential equation. Extensions are explored and the modeling of
entropy is discussed.
  Keywords: arc fields; discontinuous vector fields; integro-differential
equations; entropy models
