Real-Rootedness of Antichain Polynomials for [2] x [m] x [n]

For all positive integers m and n, this manuscript proves that the antichain generating polynomial of the product poset [2] x [m] x [n] has only negative real zeros, proving Conjecture 4.3 of Ding and Dong. A canonical two-gap tail-switching bijection first proves the previously unproved 2 x 2 determinantal formula found experimentally by He, Langner, and Witek for the associated Zhang--Zhang polynomial. After a Mobius transformation, the determinant factors into two controlled combinations of adjacent Jacobi polynomials; interlacing and endpoint signs locate all finite zeros. Status: Public Beta v0.1; internally verified candidate proof; external mathematical review pending.