λ-oper reformulation of conditions (4.11)–(4.12) for general q and d

Derive a discrete λ-oper formulation equivalent to the conditions (4.11) and (4.12) for the q-oper (4.10) with apparent singularities for general deformation parameters q and d, thereby extending the λ-form description beyond the special cases treated in Appendix C.

Background

Appendix C relates two forms of Bethe equations—folded and unfolded types—and develops a λ-oper reformulation in a special case (e.g., d = 1). While a λ-form equivalence is shown for specific settings, the general case involving the full deformation parameters remains unresolved.

Completing this λ-oper reformulation would unify oper descriptions of Bethe equations across parameter regimes and solidify the connection between discrete λ-opers and q-opers with apparent singularities.