Formalizing autopoiesis: Toward a Categorical-Thermodynamic Calculus of Closure

The concept of autopoiesis, introduced by Varela and Maturana (1974) to characterize the organizational closure shared by all living systems, has remained an open challenge for rigorous mathematical treatment. This paper traces Varela's formalization program from Spencer-Brown's calculus of distinctions through the Arithmetic of Closure and Form Dynamics, to the emergence of the Eigenform as a fixed-point model of biological identity. We analyze the methodological gap between this deterministic algebraic framework and the stochastic dynamics characteristic of living systems, and identify it as the central open problem motivating contemporary post-Varelian research. As a programmatic proposal rather than a completed formal system, we outline a Categorical-Thermodynamic Calculus of Closure that would integrate lambda-calculus fixed-point operators, topological category theory, and variational free energy minimization into a common setting for autopoietic systems under thermodynamic noise. We position this proposal within the contemporary cartography of autopoiesis, restrict its scope to autopoiesis in its strict organizational sense, and maintain a clear distinction between autopoietic organization and its formal representation. Rather than claiming a finished formalization, we identify the primitives, open problems, and evaluation criteria that such a formalization would require, and discuss the direction this sets for theoretical biology, cognitive science, and the foundations of autonomous systems.