Connection and Metric Reconstruction from Gauge-Covariant Relational Kernels Local and Global Reconstruction, Standard Recovery, No-Go Results, and the Reconstruction-Residue Program

We develop a reconstruction framework in which local connection, curvature, holonomy, and positive-semidefinite metric data are extracted from a sufficiently regular complex relational kernel transforming under the endpoint gauge law . For kernels nonvanishing near the diagonal, the first derivative of the phase in the second endpoint defines a local Abelian connection, which obeys. Compatible local kernels patch to a global line-bundle connection, curvature, and holonomy. When is Hermitian and positive semidefinite, diagonal normalization gives , and the mixed logarithmic Hessian of the normalized magnitude defines a gauge-invariant positive-semidefinite relational metric, The full complex relational Hessian decomposes. For state-overlap kernels the construction exactly recovers the Berry connection, Berry curvature, Fubini–Study metric, and quantum-geometric tensor. We also prove local reconstruction stability away from kernel zeros, show that reconstruction is generally noninjective because only finite diagonal jets determine the local geometry, and establish that a smooth second-order scalar-kernel Hessian yields Riemannian rather than genuinely non-Riemannian Finsler structure. The paper separates proven reconstruction, standard recovery, interpretive extension, and open proper-extension targets.