Characterize singular product-symmetry generators and the necessity of local product-bond commutation

Characterize the case of singular one-site operators w and determine whether a nearest-neighbour Hamiltonian with a prescribed global product symmetry must satisfy a local product-bond commutation relation when w is not invertible.

Background

The product-bond construction in the appendix assumes that the one-site operator w is invertible, which permits the definitions A_{12}=w_1^{-1}r_2 and C_{23}=r_2w_3 and establishes equivalence between the scalene relation and commutation with w_1w_3. The same construction shows that locally product-commuting densities generate Hamiltonians conserving G=∏_j w_j.

The unresolved issue has two parts: extending the scalene realization to singular w, and proving a necessity theorem that would recover a local product-bond commutation condition from a prescribed global product symmetry. The paper explicitly leaves both questions unresolved.