Permanent-on-top bounds in order four

It is proved that the permanent is the largest eigenvalue of the Schur power matrix for every real symmetric positive semidefinite matrix of order four and every complex Hermitian positive semidefinite matrix of order four and rank at most two. For general complex Hermitian positive semidefinite matrices of order four, a factor 19/18 bound and further sector bounds are obtained. The complex rank-two assertion holds in all cases exactly up to order four. The remaining complex order-four problem is reduced to one determinant inequality and remains open. The proofs combine representation-theoretic reductions with exact rational polynomial certificates. The deposit contains the manuscript, its LaTeX source, and the unchanged exact certificates and checking programs from companion release v1: https://github.com/michaeliu4/permanent-on-top-order-four/releases/tag/v1.