Variance Governance: Maintaining Structural Openness in Adaptive Systems

Adaptive systems increasingly operate under conditions in which the set of realizable states contracts irreversibly. Existing governance approaches optimize outcomes within a given state space; they do not address the preservation of the space itself. This paper introduces variance governance as a formal framework for maintaining structural openness in adaptive systems. Structural openness is defined measure-theoretically as V (t): = μ (Mₜ), where Mₜ ⊆ Ω is the realizable support at time t. The framework distinguishes V (t), the support measure, from the Shannon entropy Hₜ = H (pₜ) of any distribution defined over that support; the two are formally distinct and related only by the bound Hₜ ≤ log₂ V (t), with support collapse forcing entropy collapse but not conversely. The variance recurrence V (t+1) = (1−aₜ−bₜ) V (t) + aₜ Vₘax is shown to be non-injective at the scalar level (Proposition 1), grounding irreversibility within the dynamics rather than externally. System stability is characterized by Φ (V) = A·ln (1+V) − C·V², with optimum V* derived from the first-order condition. Governance is formalized as an intertemporal optimization over Φ (V (t) ) subject to V (t) ≥ Vₘin. Three falsification conditions are stated. Illustrative applications to AI ecosystems and climate systems are provided without empirical validation. The framework occupies a structural position complementary to the resilience tradition (Holling 1973; Walker et al. 2004) and to classical path-dependence (David 1985; Arthur 1994) ; a fuller information-theoretic treatment of the V/H relationship is developed in a companion manuscript. Additional Notes Version 3. 0. Theoretical anchor paper for the V-channel of the VASTIS framework for irreversible selection in dynamic possibility spaces. Companion to VASTIS 3. 0 (Brexner 2026a, doi: 10. 5281/zenodo. 19372372). A fuller information-theoretic treatment of the V/H relationship introduced in §3 is developed in a companion manuscript (Path-DependentEntropy, in preparation).