Weighted Root Deletions and Coefficientwise Toeplitz Positivity

For independent indeterminates \(a,x_1,\ldots,x_N,y_1,\ldots,y_N\), we prove that the sequence \(b_k=a e_k(X)+\sum_i y_i e_k(X\setminus x_i)\) is coefficientwise totally nonnegative: every minor of its upper Toeplitz matrix has nonnegative integer coefficients. In particular, this holds for the coefficients of \((uD_z+v)\prod_i(1+x_i z)\), 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.