A Control-Law Extension of the Curvature Adaptation Hypothesis in Hierarchical Transport Networks

Abstract We introduce a control-law interpretation of the Curvature Adaptation Hypothesis (CAH) using a synthetic hierarchical transport model in which a global control parameter, (γ), regulates access to distal shortcut routes. The model evaluates transport efficiency, maintenance cost, congestion, and graph geometry under two regimes: a fixed-edge regime, in which the number of distal shortcuts is held constant while their effective routing weights are modulated, and a growing-edge regime, in which both shortcut number and shortcut accessibility vary with (γ). In the fixed-edge regime, repeated simulations reveal a robust intermediate optimum in a composite objective (J), indicating that neither minimal control nor unconstrained maximal shortcut permissiveness is optimal. This optimum is tracked more clearly by the lower tail of the curvature distribution (q₁0) than by mean curvature, suggesting that the most relevant geometric signal lies in the selectively negative routing substructure of the network. Comparison with the growing-edge regime shows that the optimum is not solely a densification artifact, although edge growth substantially reshapes the curvature trajectory at high (γ). Broader parameter sweeps further show that the existence of the optimum is robust, while its precise control and curvature coordinates depend on graph architecture and maintenance cost. These results extend CAH from a theory of geometric transition into a candidate control framework in which adaptive transport systems regulate access to curvature-dependent routing regimes under competing energetic and structural constraints. Summary This study extends the Curvature Adaptation Hypothesis (CAH) from a descriptive theory of geometric phase transition into a control-law framework. Using a synthetic hierarchical transport model with distal shortcut routes regulated by a global control parameter, γ, the paper asks whether adaptive transport systems should regulate access to curvature-dependent routing regimes under competing pressures from efficiency, congestion, maintenance cost, and integration. In the fixed-edge regime, repeated simulations reveal a robust intermediate optimum in the composite objective (J), showing that neither minimal nor unconstrained maximal shortcut permissiveness is optimal. The results further show that this optimum is tracked more clearly by the lower tail of the curvature distribution than by mean curvature, and that it cannot be reduced to a trivial effect of graph densification alone. Across broader parameter sweeps, the existence of the optimum remains robust, although its precise location depends on architecture and maintenance cost. Taken together, the findings recast curvature adaptation as a candidate thermodynamic control principle for hierarchical transport systems. Related Works Pender, M. A. (2026). Dynamic Curvature Adaptation: A Unified Geometric Theory of Cortical State and Pathological Collapse. Zenodo. https: //doi. org/10. 5281/zenodo. 18615180 Pender, M. A. (2026). The Manifold Chip: Silicon Architecture for Dynamic Curvature Adaptation via Dual-Gated Analog Shunting. Zenodo. https: //doi. org/10. 5281/zenodo. 18717807 Pender, M. A. (2026). Geometry-Aware Plasticity: Thermodynamic Weight Updates in Non-Euclidean Hardware. Zenodo. https: //doi. org/10. 5281/zenodo. 18761137 Pender, M. A. (2026). Computation as Constrained Transport: A Geometric Perspective on Information Processing. Zenodo. https: //doi. org/10. 5281/zenodo. 19216884