Existence of Life in Lenia

TL;DR

This paper proves the existence of solutions for Lenia's complex integro-differential equations using arc field theory, confirming the organic dynamics observed in simulations despite discontinuous vector fields.

Contribution

It introduces arc field theory to establish the existence and uniqueness of solutions for Lenia's integro-differential equations, overcoming limitations of traditional methods.

Findings

Simulations follow arc fields, not Euler curves.

Unique solutions exist despite discontinuous vector fields.

Extensions and entropy modeling are discussed.

Abstract

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