This paper shows that a wide class of lattice models that encode non-invertible symmetries—so-called anyon chains—become much simpler and essentially unique once you allow an infinite-dimensional extra space at every site. Concretely, after "stabilization" (adding an infinite-dimensional ancilla at each lattice site), every anyon chain can be written locally as a plain tensor product of infinite-dimensional Hilbert spaces. From that fact the authors deduce two main consequences: any unitary fusion category (a mathematical object that encodes a broad class of quantum symmetries) can be realized as a symmetry on a tensor-product Hilbert space with infinite-dimensional factors, and any two anyon chains that implement the same fusion-category symmetry become related by a locality-preserving, symmetry-compatible unitary after stabilizing. In short, there is a single stable equivalence class of kinematical realizations of each fusion category on the lattice once infinite-dimensional ancillas are allowed.

What the authors did is a mathematical analysis using the language of operator algebras. They encode lattice kinematics by quasi-local C*-algebras, with local subalgebras assigned to intervals of the chain, and they formalize locality-preserving changes of variables as bounded-spread isomorphisms (a precise version of saying an operator is moved only a uniformly bounded distance). Because infinite-dimensional ancillas introduce additional weak topologies, the paper treats local algebras as W*-algebras (von Neumann algebras) when doing stabilization. Using these tools they prove a factorization theorem: stabilized anyon chains admit a local tensor-product decomposition. They then prove a uniqueness theorem: any two stabilized realizations with the same fusion-category symmetry are connected by a symmetry-compatible, locality-preserving unitary. They also derive a corollary about boundary operator algebras of Levin–Wen (string-net) models: after stabilization, two such boundary algebras are bounded-spread isomorphic exactly when the bulk topological orders match.