Boundary-Generated Overlap Measures under Principal Jacobi Extensions

Let Hₙ be a finite real symmetric Jacobi matrix, let Hₙu=λ u, and embed u by zeros into a larger principal Jacobi extension H. The squared overlaps of the embedded vector with the eigenvectors of H define a probability measure ν on shifted energies. For a nonzero bridge residual, we show that this overlap measure is generated exactly by the local spectral measure ρ at the first added coordinate: if β is the squared residual norm, then x²,dν(x)=β,dρ(x). The possible atom of ν at $x=0$ is recovered as the residual probability mass required by normalization; in the zero-residual case ν=δ₀. Consequently, every centered overlap moment of order at least two is a bridge residual factor times a rooted return moment of the extended Jacobi matrix. This yields a finite-radius moment ladder: the moment of order +2 depends only on the radius-/2 neighborhood of the bridge, and on a path the first appearances of a distant edge and diagonal coefficient occur at exactly determined orders. We then combine these locally generated moments with a finite truncated-moment linear program to obtain optimal, independently checkable lower bounds for the overlap mass carried by selected nearby eigenvalues. The optimization formalism itself is classical; the contribution is the deterministic mechanism that supplies its moments from a principal Jacobi extension. We include a closed-form four-site example, degeneracy and zero-residual cases, an alternative Stieltjes-transform proof, and a numerical pilot on 255 embedded eigenvectors. The results are finite-dimensional and deterministic; no localization or large-volume limit is asserted.