Proposition: Permutation-symmetric certificates

For the same rational law, capacity $sM$ suffices to anchor its diagonal reduction, using composability, nullity, and the structural tensor-permutation equivariance, followed by purification locality.

Proof. Take $M$ blocks of $s$ qubits. Let $\mathcal{W}$ contain all sequences $(k_1,\dots ,k_M)$ with exactly $m_k$ occurrences of label $k$, and set

Block permutations are tensor-factor permutations. They preserve the vector and act transitively on its computational support. Equivariance and nullity make its full outcome law uniform on $\mathcal{W}$. Composability gives the first-block marginal

The first-block reduction is diagonal: completions corresponding to different first labels have different remaining multisets and hence disjoint support. Purification locality gives the stated transfer.