= Solution
Every function $h$ with <Lipschitz constant> at most $1$ yields a feasible pair $(h,-h)$, and replacing $h$ by $-h$ reverses the objective. Hence
$$
\gamma(P,Q)=\sup_{\|h\|_L\leq1}\left(\int h\,dP-\int h\,dQ\right)\leq m_L(P,Q).
$$
Conversely, the <Lipschitz regularization of transport potentials> in part iii dominates every feasible pair by $(h,-h)$ with $\|h\|_L\leq1$, so $m_L\leq\gamma$. Together with the preceding parts this proves the <Kantorovich–Rubinstein theorem> here in all four requested forms:
$$
\boxed{m_d(P,Q)=W(P,Q)=m_L(P,Q)=\gamma(P,Q)}.
$$
The seminorm convention is essential. If instead $\|h\|_L$ meant $\|h\|_\infty+\operatorname{Lip}(h)$, the equality would fail: on a two-point space at distance $2$, with point masses at the two different points, $W=2$ whereas the latter unit-ball supremum is $1$. Indeed, an oscillation $a$ needs supremum <norm> at least $a/2$ and <Lipschitz constant> $a/2$, so the sum <norm> bounds $a$ by $1$, attained by values $-1/2,1/2$. This distinguishes the <Lipschitz> seminorm used above from a full bounded-<Lipschitz> <norm>.
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