Toward a Theory of Viability Domains: Formalization of Viability Domains, Constraints, and the Harmony Norm

Abstract

This article proposes the next stage in the formalization of Vitology following the development of the comparative methodology introduced in G15: a transition from the general concept of viability toward a research candidate for a Theory of Viability Domains. The central idea is that the viability of a complex system may be more productively investigated not as correspondence to a single fixed optimal state, but as the capacity of a system to preserve, recover, or form a set of states and relations compatible with specified conditions of continued existence, functioning, regeneration, adaptation, and development. Mathematical Viability Theory is considered as an external formal foundation. Within this framework, the viability of dynamical systems is investigated relative to explicitly defined constraints, while the viability kernel identifies the set of initial states from which at least one admissible evolution exists that keeps the system within the constraint set (Aubin, 1990; Aubin, Bayen, & Saint-Pierre, 2011). In control theory, invariant safe sets and control barrier functions provide a related formal approach by representing safety as the preservation of system trajectories within specified sets of states (Ames, Xu, Grizzle, & Tabuada, 2017). The article preserves the fundamental distinction established in G15: Viability Kernel ≠ Vitological Viability ≠ Harmony Norm. G15 introduced the working concept of the Viability Coordination Domain as a multidimensional domain of states and relations compatible with specified viability conditions, and subsequently the Dynamic Viability Coordination Domain as its time-dependent refinement. G16 develops these constructions and proposes a formal architecture that distinguishes between state–relation space, viability constraints, mutual compatibility conditions, a harmony-compatible domain, a viability domain, a viability boundary, a viability margin, dynamic changes in the domain, and transitions between domains. The following research relation is proposed: Viability Constraints + Compatibility Conditions → Viability Domain together with the more complete sequence: State and Relations → Constraints → Compatibility → Viability Domain → Boundary → Margin → Dynamics → Testing. Particular attention is devoted to the formalization of the Harmony Norm. The Harmony Norm is not identified with a single equilibrium point or with a viability kernel. Instead, it is proposed as a principle for defining conditions of mutual compatibility among critical variables, relations, flows, and levels of a system. This makes it possible to represent a viability domain as the intersection between direct state admissibility conditions and conditions describing compatibility among system components and between the parts and the whole. The article does not claim that a universal Theory of Viability Domains has already been established. The proposed construction is assigned the status Proposed Theory / Theory Candidate and requires further formalization, comparison with simpler models, testing on independent systems, counterexample analysis, and evaluation of predictive and diagnostic added value. Keywords: Vitology; viability; viability domain; viability theory; Harmony Norm; constraints; state space; viability kernel; Mathematical Viability Theory; viability boundary; viability margin; compatibility; dynamic viability domain; adaptation; regeneration; robustness; systems theory; formalization; interdisciplinary research.