We formulate a boundary-residual-closure mechanism to systematically construct finite-parameter open XXZ Hamiltonians featuring non-diagonal boundary data. By retaining the standard trigonometric XXZ R -matrix, we promote the auxiliary boundary entries to operator-valued generators and establish the sixteen reflection-residual components as the defining relations of a boundary extension algebra. Reducing this algebraic system to its regular Hermitian scalar sector endogenously generates a non-diagonal six-vertex boundary representative, ensuring that the required amplitude and phase emerge prior to the formulation of the double-row transfer object. Subsequently, we embed this residual-generated representative into a mixed detuned double-row seed. By extracting the regularity-point logarithmic derivative of this composite structure, we derive an explicit Hermitian open XXZ Hamiltonian with a transverse left boundary field. Ultimately, this exact mathematical framework ensures the resulting system strictly recovers the standard diagonal open XXZ chain as its zero-amplitude anchor.
Tao et al. (Fri,) studied this question.