Sharp Two-Sided Root Selection for Common-Interlacing Polynomials with Fixed Trace

Let a finite family of monic real-rooted polynomials have a common interlacer and the same sum of roots. If a positive weighted average has all its roots in an interval with center c and radius R, then one member has all its roots in the concentric interval of radius 2R/sqrt(3). The factor 2/sqrt(3) is optimal in every degree at least three, even for characteristic polynomials of zero-diagonal real symmetric matrices. The optimal factor in degrees one and two is one. The proof uses nonnegative residues, a probability measure with a prescribed mean, and an exact affine majorant. Weak interlacing and repeated roots are included. A corollary selects a principal submatrix of a constant-diagonal Hermitian matrix using the root interval of the derivative of its characteristic polynomial. This is a complete scoped theorem in the direction of the two-extreme-eigenvalue question in OWR-13674-019. It requires a global common interlacer; it does not assert the same factor for arbitrary interlacing trees, a new paving/signing bound, or closure of the separate quantitative real-rootedness question. AI-assisted, self-audited preprint; no independent review, formal verification, or absolute-priority certification is claimed.