Topological Kleene Field Theories: A new model of computation

Abstract: In this article, we establish the foundations of a computational field theory, which we term Topological Kleene Field Theory (TKFT), inspired by Stephen Kleene's seminal work on partial recursive functions. Our central result shows that any computable function can be simulated by the flow on a smooth bordism of a vector field with good local properties. More precisely, we prove that reaching functions on clean dynamical bordisms are exactly equivalent to computable functions, setting an alternative model of computation to Turing machines. The use of non-trivial topologies for the bordisms involved is essential for this equivalence, suggesting interesting connections between the topological structure of these flows and the computational complexity inherent in the functions. We emphasize that TKFT has the potential to surpass the computational complexity of both Turing machines and quantum computation.