Weighted Root Deletions and Coefficientwise Toeplitz Positivity

For independent indeterminates a, x_1,...,x_N, y_1,...,y_N, we prove that the sequence b_k = a e_k(X) + sum_i y_i e_k(X without x_i) is coefficientwise totally nonnegative: every minor of its upper Toeplitz matrix has nonnegative integer coefficients. In particular, this holds for the coefficients of (u D_z + v) product_i(1+x_i z), coefficientwise in u,v,X. The proof realizes the sequence as sums of bordered principal minors of a star-shaped Gram pencil. A maximal-weight compression in a Schur module expresses the necessary Schur-complement characters as traces against orthogonal projections. An additional letter-content grading separates the independent root weights and yields explicit squared-norm coefficient certificates. This resolves the derivative-plus-constant branch of an AIM total-positivity question, not its other operator conjectures. The note is unrefereed and makes no absolute priority claim.

Unrefereed preprint. AI-assisted tools supported research, computation, proof development and manuscript preparation; the author remains responsible for the final text. No independent peer review or formal verification is claimed. The result is a complete theorem for independently weighted single-root-deletion polynomials, not a complete solution of all of AIM-LINEAR_ALGEBRA-0012. The prior anonymous affine result is credited in the manuscript.