Finite-rank kernel realization and completion

We consider when local finite-rank positive definite kernels comefrom a single global finite-rank kernel. The first part treats thecase where the global kernel is fixed and shows that the missing mixedblocks control when a global rank-$r$ realization exists and howrank grows when it does not. The second part treats the completionproblem, where only local kernels are given. In that setting, rank-preservingcompletion becomes a unitary patching problem on overlap generatedsubspaces. Forests always patch, full overlaps are governed by cycleconditions for the induced unitaries, and partial overlaps lead tofinite-dimensional subspace compatibility conditions. We also givea rank bound for finite completions obtained by joining two completedpieces along their common feature subspace.