This paper studies non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken, the particle spectrum often has degeneracies dictated by the non-invertible symmetry. The authors deduce a procedure to determine the allowed multiplets of these degeneracies. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. The results are illustrated in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z₂ odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum. The paper discusses fusion categories, module categories, and degeneracies in (1+1) dimensional QFTs. It introduces the concept of open sectors and their degeneracies, and explains how kernels and cokernels can be used to determine the existence of degeneracies. The paper also discusses particle degeneracies in massive QFTs, showing that non-invertible symmetries can enforce degeneracies in the particle spectrum. The authors apply these results to the Tricritical Ising model deformed by the φ_{2,1} operator, showing that the massive particle spectrum has a threefold degeneracy. They also consider the Tricritical Ising model deformed by the φ_{1,3} operator and minimal models deformed by the φ_{1,3} operator, showing that the non-invertible symmetry implies a multiplet of 2(n-2) degenerate single particle states consistent with predictions from integrability. The paper concludes that the degeneracies are robust and do not require integrability or other special features of an RG flow. Instead, they hold for any model with spontaneously broken Fibonacci symmetry.This paper studies non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken, the particle spectrum often has degeneracies dictated by the non-invertible symmetry. The authors deduce a procedure to determine the allowed multiplets of these degeneracies. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. The results are illustrated in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z₂ odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum. The paper discusses fusion categories, module categories, and degeneracies in (1+1) dimensional QFTs. It introduces the concept of open sectors and their degeneracies, and explains how kernels and cokernels can be used to determine the existence of degeneracies. The paper also discusses particle degeneracies in massive QFTs, showing that non-invertible symmetries can enforce degeneracies in the particle spectrum. The authors apply these results to the Tricritical Ising model deformed by the φ_{2,1} operator, showing that the massive particle spectrum has a threefold degeneracy. They also consider the Tricritical Ising model deformed by the φ_{1,3} operator and minimal models deformed by the φ_{1,3} operator, showing that the non-invertible symmetry implies a multiplet of 2(n-2) degenerate single particle states consistent with predictions from integrability. The paper concludes that the degeneracies are robust and do not require integrability or other special features of an RG flow. Instead, they hold for any model with spontaneously broken Fibonacci symmetry.