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Weight bound
Among the higher coverings \(\Cov_r(S)\) E0007, the partitions \(\Part_r(S)\) are precisely those of minimal weight, with weight equal to \(|S|\).
(1) Lemma. For any covering \(H \in \Cov_r(S)\) with \(r \geq 1\), \(\mathrm{wt}(H) \geq |S|\), with equality if and only if \(H \in \Part_r(S)\).
Proof. Ad \(r = 1\)) \(\Cov_1(S) = \Part_1(S) = \set{S}\) and \(\mathrm{wt}(S) = |S|\).
Ad \(r = 2\)) \(\mathrm{wt}(H) = \sum_{T \in H} |T| \geq |\bigcup H| = |S|\), with equality iff the blocks are pairwise disjoint.
Ad \(r \geq 3\)) Each \(K \in H\) lies in \(\Cov_{r-1}(\lf(K))\), since \(K \in \KP_+^{r-1}(\lf(K))\) by recursion on levels. By induction, \(\mathrm{wt}(H) = \sum_{K \in H} \mathrm{wt}(K) \geq \sum_{K \in H} |\lf(K)| \geq |S|\), with equality iff each \(K \in \Part_{r-1}(\lf(K))\) and the leaf supports partition \(S\). In the equality case \(K \mapsto \lf(K)\) is injective (the \(\lf(K)\) are nonempty and disjoint), so \(H\) matches the recursion defining \(\Part_r\) E0007; conversely each \(H_B \in \Part_{r-1}(B)\) has \(\lf(H_B) = B\).
(2) Validation (AI review, 2026-07-19, claude-fable-5, pass). All three cases, both directions of the equality criterion, edge cases (\(S = \emptyset\) vacuous, singletons, repeated leaf supports, \(r \geq 1\) necessary). No issues found.
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