Captured weight and boundary leakage bound the error of sample-based spectral functions
Rayleigh-Ritz on a subspace from sample-based quantum diagonalization yields the spectral function of a probe state; we ask what bounds its error. The captured Born weight w does not do so alone: an analytic counterexample forces any bound indexed on it to its trivial value. On Hubbard rings at broadening $η=0.18t$, the probe's whole support (w=1) still has relative error 0.35--0.43: Rayleigh-Ritz displaces the poles a subspace keeps. The error is bounded instead by the missed weight and the Hamiltonian coupling across the subspace boundary. Given the exact ground state, we prove for any orthogonal projector that the relative $L_1$ error lies between 1-w and $\min\{1+w,(1+\sqrt{w})(\sqrt{1-w}+Λ(η)/η)\}$, with $Λ$, that coupling at resolution $η$, computed from the Ritz pairs and the retained probe. We test it on L-site rings with classical Born draws of an exactly time-evolved probe, or their infinite-shot ranking; our device runs are execution records only. On the ranked subspaces the bound is vacuous at every operating fraction of the resource scan, at all five sizes up to a 10306296-determinant sector: its leakage branch exceeds the trivial bound 1+w by factors 1.66--8.61. It is informative only well above those fractions, where it is calibrated: at L=6--12 it first beats the trivial bound at a true error of (1.2--7.3)$\times10^{-3}$, though an unproven fit in 1-w is tighter on 11 of 14 splits. For the Born-ranked $A(k,ω)$ at 85% of its sector, the bound is 350--900 times the true error. In the determinant basis, zero leakage at full weight needs every symmetry-allowed determinant: the probe's measured Krylov support at L=6--14. Plane-wave orbitals reduce a momentum probe's support to one block, 1/L of an exponentially growing sector. At full weight a second-order certificate follows; below it, where every operating fraction lies, one is open.