What two disjoint regions can jointly inherit: a realization theorem

Fix a quantum channel and two disjoint regions X, Y of its output. Operator-algebra quantum error correction assigns to each region a largest algebra of input observables it can exactly recover. We ask which pairs of such algebras occur. Our contribution is a realization theorem: for every pair of block-size vectors p, q and every nonnegative integer matrix m_ij with no zero row or column, an explicit channel has exactly those algebras as the maximal recoverable algebras of two disjoint outputs. Maximality is what makes this non-trivial: it forbids the channel from leaking anything else to either side. For a correctable-private pair the analogous realization is known (Kribs, Levick, Nelson, Pereira and Rahaman). The necessary direction is assembled from known material and we say so. Joint measurability forces the two algebras to commute. The decomposition of the code space into sectors C^{p_i} (x) C^{q_j} (x) C^{m_ij}, with the two algebras acting as diagonal copies and dim = sum_ij p_i q_j m_ij, is the Bratteli inclusion matrix of a commuting pair, which under complementary recovery is the one used in holographic quantum error correction. The identification of what both regions recover with the connected components of the bipartite graph of the nonzero m_ij is, at multiplicity one, a theorem of Vanrietvelde, Mestoudjian and Arrighi on partitions of C*-algebras; our residue is the extension to higher multiplicity, which their partitions exclude. Together: the quantum part of what two regions share is partitioned along the edges of a graph and never copied, the classical part surviving in both is exactly the connected components, and no admissible pattern is vacuous. Two disjoint heirs of a shared factor M_d inherit factors M_p, M_q only if pq divides d, sharpening pq <= d, so a shared qubit does not split. We also state the approximate counterpart, via the universal-cloning frontier.