A new type of Yang-Baxter equation (YBE) for
R-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the
ZN symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for
N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated
R-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter
R-matrix---realized as an intertwiner of cyclic representations of
Uq(sl2) at a root of unity---and Shastry's two-parameter
R-operator obtained from the decorated YBE.
Integrating infinitesimal generators of local dynamics into finite-time evolution is generally harder than extracting a local Hamiltonian from a quantum gate. For integrable one-dimensional quantum many-body systems, a recent bootstrap constructs one-parameter
R-operators directly from Hamiltonians satisfying the Reshetikhin condition. However, this condition excludes important models such as the Hubbard model, whose two-parameter
R-operator was obtained by Shastry from a decorated Yang--Baxter equation obeyed by its free-fermion sector. Here, I extend the bootstrap to
R-operators constrained by both the ordinary and decorated Yang--Baxter equations. Their joint expansion produces additional Hamiltonian-level necessary conditions and recursively determines candidate higher-order terms, subject to two-site locality at each order. I also derive compatibility and factorization conditions under which a Shastry-type superposition solves the
RLL relation after deformation by an onsite involution. The construction recovers the known two-parameter
R-operators of the Hubbard and XYh models, whose full Yang--Baxter equations follow from established tetrahedral Zamolodchikov algebra relations. For a superconducting Hubbard deformation, the scalar equations are generically incompatible, placing it outside this framework. These results provide diagnostics and a systematic route to fermionic
R-operators beyond the Reshetikhin condition. General sufficiency, convergence of the formal bootstrap, and sufficiency of the deformation conditions remain open.