The proof of Theorem 5.1 in Chang (1996) shows that the assumptions that P uv is invertible and not a permutation matrix for every edge { u, v } must be satisfied for a model θ ∈ Θ if P$_{θ}$ = P$_{θ}$$_{0}$ for some θ$_{0}$ ∈ Θ, by virtue of the form of P$_{θ}$$_{0}$ . The set of strand-symmetric matrices is closed, so the assumption that each such P uv is strand-symmetric must be satisfied for all θ ∈ Θ. Finally, if compositional asymmetry is violated at any internal node for a θ ∈ Θ, then for at least one pair of terminal nodes a and b , the eigendecomposition of S ( J ab ) − $^{1}$SJ ab will have at least one repeated eigenvalue. That is, if P$_{θ}$ = P$_{θ}$$_{0}$ for some θ$_{0}$ ∈ Θ, then for θ we must have that all internal nodes satisfy compositional asymmetry. So far we have shown that if P$_{θ}$ = P$_{θ}$$_{0}$ for θ$_{0}$ ∈ Θ and θ ∈ Θ then the assumptions of Corollary 3.1 are satisfied by both models and so they must have the same rooted topology. It remains to show that if P$_{θ}$ = P$_{θ}$$_{0}$ for θ$_{0}$ ∈ Θ and θ ∈ Θ then the transition probability matrices for θ$_{0}$ are the same as for θ . We will show that the inductive steps of Theorem 3.1 still apply in this context. As the assumptions of Corollary 3.1 are satisfied for θ , Lemma 3.1 can be safely applied, so the only remaining question is the ordering of rows of the transition probability matrices. Take a transition probability matrix P from the model θ ∈ Θ and the matrix P$_{0}$ on the corresponding edge from θ$_{0}$ ∈ Θ. Firstly assume P$_{0}$ ∈ M . For some permutation matrix R , P = RP$_{0}$ , but if R = I , RP$_{0}$ / ∈ M as M is strongly reconstructible from rows. We must also check that RP$_{0}$ / ∈ S$_{ϵ}$ ( M ). By construction, S$_{ϵ}$ ( M ) ⊆ S ( M ) and RP$_{0}$ / ∈ S ( M ), so RP$_{0}$ / ∈ S$_{ϵ}$ ( M ) and P = P$_{0}$ . Next assume that P$_{0}$ ∈ S$_{ϵ}$ ( M ). Again P = RP$_{0}$ , but RP$_{0}$ / ∈ M . In this case the order of rows of P is decided by the order of those on neighbouring edges, and must be the same for θ and θ$_{0}$ . It is therefore possible to modify the proof of Theorem 3.1 such that it ensures that for θ$_{0}$ ∈ Θ and θ ∈ Θ, P$_{θ}$ ̸ = P$_{θ}$$_{0}$ if θ = θ$_{0}$ We are therefore able to apply Lemma 4.1 and the model can be estimated 22