Simple critical zeros in short intervals: stability, parity, localization, Hilbert compression, and spectral defect

Abstract

For a conjugation-invariant finite multiset, let N count copies, S simple real support, O distinct odd-multiplicity real support, Q the squared-kernel pair sum, and D(G)=tr Psi_2(G) the retained simple-real Gram spectral defect. The paper proves the finite inequality (Q-S-D(G))(N-O) >= 2(N-S)^2 by coupling an attributed rank-trace theorem to odd support. The paper also proves that Pearce-Crump's vector-profile Selberg detector localizes to every fixed finite rank. Applying this transfer to the source-certified six-square constant C6 and then to the Hilbert-parity product places the unique positivity onset in (0.5458837,0.5458838). At theta=0.545884, where the rank-three term remains negative, the new lower bound exceeds 2.5541123454645702e-7. At theta=0.5459 it exceeds 0.0000177645181613023236390595079, improving the rank-three term by more than 9.2635430617773560e-7. A radius-optimized spectral transfer strictly improves every positive point of the rank-six curve; its certified gain at theta=0.5459 exceeds 1.7766622541125682e-68. The six-square constant is an attributed source input whose coefficient matrix is not printed, so it is not independently reconstructed here. The results are asymptotic and depend on recent preprints. No effective starting height, global zero-density record, external peer review, or proof of the Riemann hypothesis is claimed. Source code, data, and computational records are available at https://github.com/fsantibanezleal/CAOS_RESEARCH.