Reconstruction between generalized hybrid metric-Palatini gravity and Φ (R,φ ,X) theories

We develop a local reconstruction framework between Φ (R,φ ,X) Φ ( R , ϕ , X ) theories with linear dependence on X X and generalized hybrid metric-Palatini gravity. The construction is formulated in vacuum in the Einstein frame, where both formulations can be written as two-scalar theories with the same field-space geometry. The framework provides a practical method for finding Φ (R,φ ,X) Φ ( R , ϕ , X ) and f(R,R) f ( R , R ) functions that describe the same regular Einstein-frame two-scalar sector. Starting from a given Φ (R,φ ,X) Φ ( R , ϕ , X ) model, we derive the equation that determines the compatible hybrid functions f(R,R) f ( R , R ) and show that it has a Clairaut-type structure. We also show that the inverse reconstruction is not unique: a regular hybrid Einstein-frame potential determines a family of compatible Φ (R,φ ,X) Φ ( R , ϕ , X ) theories, parametrized by the kinetic coupling. Explicit examples illustrate the reconstruction procedure, its domain of validity, and the translation of model parameters between the Φ (R,φ ,X) Φ ( R , ϕ , X ) and f(R,R) f ( R , R ) formulations.