Title: Beyond Turing: Topological Closure as a Foundation for Cognitive Computation
Primary area: learning theory
Keywords: Cognitive computation; topological closure; cycle formation; memory-amortized inference; order invariance

Abstract: Classical models of computation, epitomized by the Turing machine, are grounded
in \emph{enumeration}: syntactic manipulation of discrete symbols according to
formal rules. While powerful, such systems are intrinsically vulnerable to
Gödelian incompleteness and Turing undecidability, since truth and meaning are
sought through potentially endless symbolic rewriting. We propose an
alternative foundation for non-enumerative computation based on
\emph{topological closure} of semantic structures. In this view, cognition operates by promoting
transient fragments into closed cycles, where $\partial^2=0$ ensures that only
invariants persist. This shift reframes computation from \emph{syntax} to
\emph{structure}: memory and reasoning arise not by enumerating all possibilities,
but by stabilizing relational invariants that survive perturbations and
generalize across contexts. We formalize this principle through the
dot–cycle dichotomy: dots or trivial cycles ($H_0$) serve as high-entropy scaffolds for
exploration, while nontrivial cycles ($H_1$ and higher) encode low-entropy invariants
that persist as memory. Extending this perspective, we show how
Memory-Amortized Inference (MAI) implements an anti-enumerative principle by storing
homological equivalence classes rather than symbolic traces, yielding robust
generalization, energy efficiency, and structural completeness beyond
Turing-style models. We conclude that \emph{topological closure} provides a
unifying framework for perception, memory, and action, and a candidate
foundation for cognitive computation that transcends the limits of enumeration.