Establish an exhaustive global-to-local correspondence for product symmetries

Establish an exhaustive global-to-local correspondence between nearest-neighbour Hamiltonians commuting with local product-bond operators of the form w_jw_{j+1} and Hamiltonians conserving the corresponding global product charge G=∏_j w_j.

Background

The paper extends its scalene Yang–Baxter construction beyond additive charges to product-bond operators w_jw_{j+1}. When w is invertible and each local Hamiltonian density commutes with the corresponding product-bond operator, the construction yields conservation of the global product charge G=∏_j w_j. The appendix provides a scalene realization of this local commutant condition and derives [H,G]=0.

What remains unresolved is the converse direction: determining whether every nearest-neighbour Hamiltonian with the prescribed global product symmetry can, possibly after an appropriate modification of its local densities, be represented by densities commuting with the corresponding product-bond operators. Such a result would parallel the additive-charge global-to-local theorem established earlier in the paper.