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For a bosonic Abelian anyon theory, a Lagrangian subgroup of Abelian anyons is a subgroup of types of anyons closed under anyon fusion, with trivial topological spin for every member and trivial mutual braiding among members. Every anyon outside the subgroup must braid nontrivially with at least one member; pairwise compatibility alone is insufficient.

These conditions describe a maximal condensable collection for an appropriate fully gapped boundary. In the toric code, the vacuum together with the electric anyon gives one choice, and the vacuum together with the magnetic anyon gives another. Electric and magnetic anyons cannot both belong to the same collection because their mutual braiding is nontrivial. Additional physical context, including a nonchiral bulk and any imposed symmetry, matters when using this data to construct a boundary.^[ref]

The block matrix isIf both $q\in \mathcal{L}$ and $q^{\prime }$ condense, choose their boundary excursions so that the two string operators cross once. Their commutator is their mutual full-braiding phase:Both operators act as the identity on every ground state, so consistency requiresWriting $q^{\prime }=(q_{\mathrm{top}}^{\prime },q_{\mathrm{bot}}^{\prime })$, this saysSet $l_{\mathrm{top}}=M^{-1}{q}_{\mathrm{bot}}^{\prime }$ and choose $l_{\mathrm{bot}}=0$. Then $Kl=(0,q_{\mathrm{bot}}^{\prime }{)}^T$, andTherefore every additionally condensable anyon has the formThe $Kl$ term is a local particle in the anyon lattice of an Abelian Chern--Simons theory, so the condensate is maximal modulo local excitations, as required for a Lagrangian subgroup of Abelian anyons.