Pate proved ordinary irreducible-immanant permanental dominance through order $13$ and identified $(4,4,3,3)$ as the sole remaining order-$14$ case, with $(5,4,3,3)$ and $(3^5)$ forming the order-$15$ frontier. These three cases are settled here; consequently $d_λ(A)/f^λ\le \operatorname{per}(A)$ for every partition $λ\vdash n$ with $n\le15$ and every complex Hermitian positive-semidefinite matrix $A$. The argument also yields results beyond this finite frontier: an exact four-term bridge for $(4,4,3,3)$, the uniform family $(m,4,3,3)$, a two-parameter family $(a,b,3,3)$ for $a\ge b\ge4$ and $5a\ge8b$, and a long-first-row criterion for arbitrary fixed tails. These results arise from explicit specializations of Pate's $W$-function positivity framework using partial swaps, Young projectors, Pieri–content identities, and branching data. For $(3^5)$, an exact Farkas certificate shows that the central-projector partial-swap cone is insufficient; a branching-refined one-swap construction escapes this obstruction and yields a positive $106+19$-witness certificate. Boundary-compression and node-moving results further describe the reach and limitations of the local-filter method. All finite certificates are checked by exact integer or rational arithmetic and are supplied as ancillary material. The order-$14$ bridge is additionally formalized and kernel-checked in Lean 4 for all complex Hermitian positive-semidefinite matrices, including the exact coefficient normalization and the deduction of $(4,4,3,3)$ permanental dominance from four explicitly stated Pate inequalities.