Sharp Kernel Penalties for Canonical Parseval Approximation

For a closed-range synthesis operator F, let P_F be the projection onto the orthogonal complement of its kernel. We study the metrics d_(p,kappa)(F,G) = (||F-G||^p + kappa^p ||P_F-P_G||^p)^(1/p), for finite p >= 1. The polar factor is universally a nearest Parseval synthesis operator among partial isometries with arbitrary ranges exactly at the penalty threshold kappa >= 1. Every minimizer then has the original kernel. We parametrize all minimizers and their ranges by a contraction, an orthogonal defect isometry and an operator inequality. Over complex Hilbert spaces we obtain an exact spectral uniqueness criterion and describe when distinct minimizing subspaces occur. A rank-collapse example excludes the same universal property for metrics inducing the operator-norm topology. The case p=2, kappa=1 is the existing kernel metric of Corach, Maestripieri and Mbekhta; the norm lower bound is a special case of Chiumiento's polar-factor theorem. This unrefereed preprint makes no absolute priority claim. Source dataset record: OWR-720-010 (ulamai/UnsolvedMath v1.6.0). The manuscript discloses OpenAI Codex assistance.