Spectral chart selection and average trace-class distortion

We construct genuine finite negative-type metrics with n points whose first uniform-average distortion into the trace class S_1(H) is at least c sqrt(log n)/(log log n)^(9/2), uniformly over all complex Hilbert spaces H. The construction uses the multichart Hilbert kernel of the stipulated pinned uniform-sparsest-cut source. A dimension-independent matrix selection replaces scalar coordinate selection: scalar moment matching supplies spectral cutoffs, and a positive Gaussian dual pairing averages them without requiring commutation. The argument tracks chart count, rounding mesh and triangle repair; finite-rank compression, Hermitian dilation and uniform copies preserve the full statement. The final release includes attribution corrections and bounded primary-reading limits. Worldwide priority is not claimed.

AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review or worldwide-priority certification. Identified OpenAI source inputs at revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb are stipulated true. Lean verifies scoped supporting results only, not the full theorem. Original prose: CC BY 4.0; original code and Lean: MIT; third-party rights preserved. See the supporting archive for precise assumptions, coverage, provenance and corrections.