This manuscript advances the quadrature comparison framework for the Báez-Duarte formulation of the Nyman–Beurling criterion. After Mellin normalization, the discrete Gram matrix becomes translation-invariant in logarithmic coordinates, with spectral density qfull xi equals 2 pi times zeta of one half plus i xi divided by one fourth plus xi squared. A continuous proxy variational problem is introduced, and discrete–continuous energies are compared through oscillatory quadrature estimates. The analysis reduces the original weighted density problem to punctured frequency windows around critical zeros, sharpening the global density obstacle into a local interpolation summability problem near punctures. In addition, a regularized target family is introduced, and its L2-convergence to the original weighted target is proved. This regularization provides evidence that interpolation errors near punctures may be controlled under suitable assumptions, offering a potential route around the singular structure of critical zeros. All other analytic steps—including Mellin–Plancherel reduction, quadrature comparison, punctured-window reduction, exact local reproduction, and far-region interpolation—are established within the present framework. The remaining open problem is the summability of adaptive interpolation errors near punctures, now supplemented by a regularization strategy that strengthens the conditional program. Keywords: Riemann Hypothesis, Báez-Duarte criterion, Nyman–Beurling, Mellin–Plancherel, quadrature comparison, punctured-window reduction, interpolation gap, regularization, L2-convergence