Reconstruction Between Generalized Hybrid Metric--Palatini Gravity and $\mathrm{\Phi }(R,\phi ,X)$ Theories

We develop a local reconstruction framework between $\mathrm{\Phi }(R,\phi ,X)$ theories with linear dependence on $X$ 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 $\mathrm{\Phi }(R,\phi ,X)$ and $f(R,\mathcal{R})$ functions that describe the same regular Einstein-frame two-scalar sector. Starting from a given $\mathrm{\Phi }(R,\phi ,X)$ model, we derive the equation that determines the compatible hybrid functions $f(R,\mathcal{R})$ 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 $\mathrm{\Phi }(R,\phi ,X)$ theories, parametrized by the kinetic coupling. Explicit examples illustrate the reconstruction procedure, its domain of validity, and the translation of model parameters between the $\mathrm{\Phi }(R,\phi ,X)$ and $f(R,\mathcal{R})$ formulations.