Nonexistence of Whirling-Knight Tours at Half Coil Count for $n\equiv 4,6(\mathrm{m}\mathrm{o}\mathrm{d}8)$

A whirling knight’s tour is a Hamiltonian cycle in the digraph of counter-clockwise knight steps about the centre of an $n\times n$ board; its coil count $c$ is the winding number around the centre. We prove that no such tour with $c=n\mathrm{/}2$ exists when $n\equiv 4(\mathrm{m}\mathrm{o}\mathrm{d}8)$ ($n\ge 4$) or $n\equiv 6(\mathrm{m}\mathrm{o}\mathrm{d}8)$ ($n\ge 6$), settling a conjecture of Beluhov. For each residue class we exhibit a closed-form Farkas certificate for infeasibility of a cycle-cover LP relaxation; the two certificates are structurally distinct.