Beyond Mean Curvature: Lower-Tail Routing Structure in Controlled Hierarchical Networks

Abstract In prior control-law simulations, lower-tail Ollivier-Ricci curvature tracked transport optima more closely than mean curvature, suggesting that the lower curvature tail may mark a special routing structure in hierarchical networks. The Curvature Adaptation Hypothesis (CAH) control-law extension showed that the relevant operating point is generally an intermediate, condition-dependent optimum and that lower-tail curvature q₁0 is a more informative geometric correlate of that optimum than mean curvature alone. Here, we test the stronger possibility that the lower curvature tail is not merely a correlate of efficient transport, but a genuine load-bearing routing substructure. Using repeated-seed transport simulations on balanced hierarchical graphs with sparse distal shortcuts, we compare targeted and random shortcut placement across multiple edge budgets. We show that lower-tail edges consistently bear disproportionate transport load, confirming that the lower curvature tail is a real routing structure rather than a descriptive artifact. However, lower-tail depth alone does not explain system performance. Across repeated seeds, targeted placement increases total tail participation most clearly in the sparse regime, whereas random placement more often produces greater per-edge tail burden concentration and higher tail exploitation intensity. These results indicate that the relevant control variable is not lower-tail depth in isolation, but the regime-dependent organization of the lower curvature tail across recruitment and burden concentration. Taken together, the findings refine the geometric-control picture suggested in previous work. The transport optimum is better understood not as a function of average curvature or even lower-tail depth alone, but as a tradeoff between distinct modes of lower-tail organization. This reframes the lower tail from a scalar clue into an organized routing object and supports a broader view of hierarchical transport as governed by structured lower-tail recruitment rather than uniform curvature reduction. Overview This preprint presents a mechanistic refinement of the Curvature Adaptation Hypothesis (CAH) control-law framework. Previous simulations demonstrated that lower-tail Ollivier-Ricci curvature (q₁0) tracks transport optima in hierarchical networks more reliably than mean curvature. This paper investigates the physical meaning of that statistical signal, testing whether the lower curvature tail marks a genuine, load-bearing routing substructure rather than a mere descriptive artifact. Using repeated-seed transport simulations on balanced hierarchical graphs with sparse distal shortcuts, this study compares targeted and random shortcut placement across multiple edge budgets to isolate how traffic distributes across the network’s geometry. Key Findings The Lower Tail is a Routing Substructure: Edge-level analyses show that q₁0-tail edges consistently bear a disproportionate share of routing load, supporting the interpretation that the lower curvature tail marks a genuine load-bearing routing substructure. Organization Supersedes Depth: Lower-tail depth alone does not explain system performance. Two networks can occupy similar q₁0 regimes but still differ meaningfully in routing efficiency and traffic organization. Two Modes of Lower-Tail Organization: Targeted sparse placement tends to broaden lower-tail recruitment (acting as a distributed scaffold), while random placement more often yields concentrated exploitation (acting as a hot backbone). Regime-Dependent Tradeoffs: Global performance emerges from the balance between broad recruitment and concentrated burden, and is shaped by the network’s sparsity and saturation regime. Conclusion Ultimately, this work reframes the lower curvature tail from a scalar geometric clue into an organized routing object. It suggests that geometric control in transport systems is governed by structured lower-tail recruitment rather than uniform, network-wide curvature reduction. Data & Code AvailabilityCode and simulation outputs supporting this study are available in the project repository: https: //github. com/MPender08/beyond-mean-curvature https: //doi. org/10. 5281/zenodo. 19324654 Related Works Pender, M. A. (2026). Computation as Constrained Transport: A Geometric Perspective on Information Processing. Zenodo. https: //doi. org/10. 5281/zenodo. 19216884 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). A Control-Law Extension of the Curvature Adaptation Hypothesis in Hierarchical Transport Networks. Zenodo. https: //doi. org/10. 5281/zenodo. 19270110 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