Negative Definition and Positive Specification in Viability Geometry
Year: 2026
Type: article
Abstract: We present a structural note on viability geometry for open systems under perturbation. Let $S$ denote a measurable state space, let $C\subset S$ denote collapse or baseline-inadmissible states, and let $A_0=S\setminus C$ denote the baseline admissible set. Where dynamics are specified, let $K_0=\operatorname{Viab}(A_0)$ denote the corresponding viability kernel. We distinguish negative constraints, which exclude collapse or baseline inadmissibility, from positive specifications, which impose additional persistent predicates such as target configurations, output requirements, organizational patterns, or sustained behaviours. The central observation is constraint-geometric. If a positive specification $M\subset S$ is non-redundant relative to $A_0$, then the instantaneous admissible set contracts from $A_0$ to $A_M=A_0\cap M$. Under dynamics, the corresponding specified viability kernel is $K_M=\operatorname{Viab}(A_0\cap M)$, and in general $K_M\subseteq K_0\cap M$, with equality requiring additional invariance assumptions. This contraction should not be confused with collapse: exit from $A_M$ or $K_M$ may be specification failure while the system remains non-collapsed inside $A_0$ or $K_0$. The note also records a systems-theoretic interpretation. Positive specification creates additional failure conditions and, when ambient dynamics or perturbations tend to drive trajectories away from the specified subset, may require regulation, repair, control, dissipation, monitoring, or structural rigidity. This maintenance burden is not derived from set inclusion alone; it requires a cost functional or model-specific dynamics. The contribution is therefore not a new physical law, thermodynamic principle, or first-passage theorem, but a classificatory synthesis connecting viability theory, first-exit reasoning, admissibility, and persistence through the distinction between collapse avoidance and continued satisfaction of positive specification.
Author: Wadje, Rajendra
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7127978638
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