# Verification report

Complete self-audited proof of the sharp two-sided selection theorem for every finite monic real-rooted family with a GLOBAL common interlacer and a common trace. The optimal uniform dilation is2/sqrt(3) in every degree>=3 and one in degrees1,2. A constant-diagonal Hermitian principal-submatrix corollary is proved. This is one complete scoped theorem, not closure of all questions in OWR-13674-019. No general-tree, new paving/signing, or quantitative-distance theorem is asserted.

Written proof: remove common multiplicities and endpoint factors; nonnegative residues have a common affine part; endpoint constraints yield a probability measure with prescribed mean; exact square-completion identities give an affine majorant of both barrier costs; a single score selects both endpoint signs. Repeated roots, zero residues, zero-weight deletion, degrees1/2, and radius0 are handled explicitly. The cubics x^3-3x+/-2 give the sharp obstruction, including a zero-diagonal matrix realization. The principal-submatrix corollary uses Cauchy interlacing and the characteristic-polynomial derivative identity.

Exact checker: two universal multivariate polynomial identities after clearing denominators;780 rational residue families in degrees2..14, family sizes2..7, and1560 endpoint-sign checks in ordered Q(sqrt(3)); direct cubic factorizations and permutation determinants for +/- adjacency(K3). These finite checks support the proof, not replace it.
Checker SHA-256: 24a0e987cded5f78a04cb2fb55cc6e84a568d90db6caade46bdf31f903e012a3.
Certificate SHA-256: 84ed8cf1ac69df2dcdbac145d675d9138506cbf6ddf5cb50a97d0c17c96e863d.

The original question was visually checked in the source PDF. Primary-source comparisons include MSS common interlacing, Ravichandran-Srivastava multi-paving, and Lin-Zhou October2026 spectral recovery. No matching sharp fixed-trace theorem was found in that bounded search; absolute priority is not certified. Independent review and proof-assistant verification were not performed. No mathematical gap was found in the detailed originating-system audit.

The five-page English PDF compiled successfully in the built-in editor and with Tectonic. All pages were visually inspected. Only the author's name is prominent; affiliation and contacts are a footnote; AI assistance is disclosed in prose.