Connection, Holonomy, and Reconstruction Fibres of Operator-Valued Comparison Kernels

Connection, Holonomy, and Reconstruction Fibres of Operator-Valued Comparison Kernels E→M be a finite-rank Hermitian vector bundle and let K(x,y):E_y→E_x be a sufficiently regular endpoint-covariant operator-valued comparison kernel. Positive reproducing kernels of this type are already known to induce canonical bundle connections through Hilbert-space classifying maps, while overlap matrices and their polar factors provide established constructions of non-Abelian geometric transport [1], [2], [3]. The present paper develops a complementary endpoint-jet and finite-comparison formulation designed to determine not only which geometric structures are reconstructed from K, but also which kernel information remains undetermined by them. After diagonal whitening, K ̂(x,y)=K(x,x)^(-1/2) K(x,y)K(y,y)^(-1/2), the first source-endpoint jet of a normalized Hermitian kernel is anti-Hermitian and defines a local connection representative. On the near-diagonal full-rank domain, the polar decomposition K ̂(x,y)=U_K (x,y) R_K (x,y) provides a canonical finite unitary comparator. Ordered products of U_K recover path-ordered transport, closed products recover holonomy, and infinitesimal loops recover curvature. In the positive overlap sector, the singular values of R_K recover principal-angle and Grassmannian metric information. The paper’s principal contribution is the reconstruction-boundary and inverse-problem analysis. We distinguish diagonal-whitening failure, finite-comparison rank loss, Grassmannian immersion failure, and gauge-coordinate singularity; establish that exact endpoint transitivity forces flatness; and show that the map from kernels to connection, curvature, metric, and holonomy is generally noninjective. Distinct positive operator kernels can possess identical reconstructed local geometry while differing at higher order and finite separation. This motivates the operator reconstruction fibre and the residual kernel information not fixed by standard bundle geometry. The resulting framework is presented as a reconstruction synthesis and research program, not as a new origin of non-Abelian gauge geometry.