Can a polynomial be positive while hiding negative coefficients? Yes. Squaring a difference gives a simple example. This research candidate asks for the stronger property: every individual coefficient must be nonnegative. It studies elementary symmetric polynomials, built by adding products of distinct variables. Multiply the term of degree k by one affine factor, then arrange the resulting sequence along the diagonals of a Toeplitz matrix. The manuscript supplies a uniform proof that every finite minor has nonnegative monomial coefficients. There is no limit on the number of variables or the order of the minor, and gaps between selected rows and columns are allowed. The central construction connects the sequence to a diagonal matrix times a rank-one perturbation of the identity. Classical representation theory turns its coefficients into squared lengths. Crucially, separate perpendicular layers isolate the degrees, so the argument controls each coefficient rather than just positive numerical inputs. A two-parameter version includes its zero endpoint by a polynomial identity. The archive contains the full proof, exact finite checks and an explicit counterexample to a stronger claim about a different polynomial basis. Those finite checks are diagnostics, not a substitute for the universal argument. The theorem concerns one affine factor, not the broader multiple-factor conjecture. The paper credits earlier mathematical methods, and historical priority remains uncertain. This is an Evidence Press unrefereed candidate released on 7 September 2026. The full paper, evidence and limitations are linked. The voice is AI-generated. This audio is a communication aid, not additional mathematical evidence.