We propose a holonomy-based framework for discretizing curvature on graphs equipped with local symmetric positive-definite metrics. Each vertex carries a fibre metric \(g_i\), and each directed edge carries a reversible metric-compatible transport \(F_{ij}\). The ordered product around an oriented triangular loop \(\mathcal C\) gives a holonomy \(H_{\mathcal C}\), whose normalized logarithm \(Ω_{\mathcal C}=-s_{\mathcal C}^{-1}\operatorname{Log}(H_{\mathcal C})\) is used as a finite-loop curvature observation. Thus the construction discretizes the geometric principle that infinitesimal holonomy is controlled by curvature, rather than treating holonomy as a heuristic feature. Since \(Ω_{\mathcal C}\) lies in the \(g_i\)-orthogonal Lie algebra, it is not itself a velocity of an SPD metric. We therefore introduce two aggregation mechanisms: a commutator with a symmetric response matrix, producing symmetric Ricci-type metric responses, and an incidence-aware covariant divergence of curvature-induced edge fluxes, reflecting the relation between trace and covariant divergence. The resulting responses are locally orthogonal-gauge equivariant and can drive exponential updates that preserve positive definiteness. We also give a reversible metric-compatible parametrization of edge transports, allowing orthogonal edge factors, loop scales, weights, and response matrices to be learned while respecting the graph geometry. Known-geometry calibrations on the unit sphere test the holonomy--curvature relation, curvature preservation under nontrivial local metric representations, and the empirical recovery of edge transports from local observations.
Geometric architectures are often motivated by internal mechanisms, but accuracy alone does not show whether predictions use them. In Sheaf Neural Networks (SNNs), edge transports form a connection whose cycle products define holonomy. We ask whether training changes triangle holonomy, whether predictions rely on the learned connection, and whether holonomy drives triangle counting. We use basis-independent loop readouts with identity interventions and shortcut controls. On high-homophily GraphUniverse graphs, triangle counting increases the mean SO(2) triangle rotation in Neural Sheaf Propagation (NSP) from 0.010 to 0.388 radians, while community detection ends at 0.029 radians. With more data, learned SO(2)--NSP outperforms Identity NSP, and replacing its transports after training increases error further. However, ridge regression is more accurate, diagonal maps improve without continuous rotation, and fixed-degree models develop rotation without improved counting. Thus, NSP can learn and rely on a nontrivial connection, but our experiments do not show that triangle holonomy drives its predictions.
This paper tests whether holonomy concentrates on active sparse-autoencoder (SAE) feature planes in Gemma 2 2B, a concrete operationalization of the broader semantic-concentration prediction. Holonomy is measured at the final-token layer-12 to layer-13 residual-stream readout by carrying a local frame around small loops using the instrument's restricted-Jacobian transport rule, then normalizing the resulting rotation by enclosed area. The design, materiality threshold, analysis, and verdict rules were preregistered and frozen before the analysed measurements were inspected. The prediction was falsified in reverse: active-feature planes carried less holonomy than matched mixed-feature controls, with an adjusted log contrast of -0.29439 and 95% interval [-0.43989, -0.14889]. A magnitude-only explanation was not supported in this design, while the three-way ordering across random, mixed-feature, and active-feature planes was undefined at matched magnitude because common support failed. Post-freeze diagnostics at the same readout supported the area law on a small validation subset, bounded matched-center displacement under a simple paired regression, and identified transport distortion as a live mechanism or confound. The result is therefore a narrow, auditable operational reversal, not a causal claim that meaning suppresses holonomy. The cause remains open, with activation-strength geometry, degree of feature engagement, dictionary geometry, matched-center displacement, activation-manifold proximity, and transport shear as live alternatives.