Teleonomical intelligence across scales: A categorical and partial differential equation framework for biology and artificial systems.

Living systems maintain characteristic structures and functions despite constant disturbance. Examples range from cellular homeostasis and tissue morphogenesis to organismal behaviour and social organisation. This capacity suggests that biological systems actively preserve certain invariants across multiple levels of organisation. In this work we introduce a mathematical framework for describing such behaviour, which we call teleonomic intelligence: the ability of a system to use information in order to maintain or restore structural invariants under changing conditions. The proposed approach represents each organisational level by a dynamical model that combines local physical processes with a measure of deviation from a target invariant. These models are coupled across scales so that information can propagate between levels: lower levels provide evidence about the current state of the system, while higher levels impose constraints that guide recovery when disturbances occur. The resulting dynamics naturally produces stabilising behaviour in which systems attempt to minimise invariant violations while remaining compatible with their underlying physical processes. We demonstrate the framework using pattern-forming reaction-diffusion systems subject to noise and structural perturbations. The model recovers disrupted patterns and preserves their global organisation even after substantial disturbances. Additional examples illustrate how the same principles extend to topological and multi-component consistency constraints. The framework provides a unified mathematical language for analysing how biological systems maintain structure across scales, and it suggests design principles for artificial systems that remain robust when environments change.