Researchers have identified fundamental obstructions in inverting curvature-dependent conformal transformations, a type of spacetime metric rescaling crucial in modified gravity theories. These transformations, such as $\widetilde g_{\mu\nu}=F(R[g])g_{\mu\nu}$ where $F>0$ and $F_R\neq0$, are used to relate different "frames" or representations of gravity, like the Jordan and Einstein frames. The study reveals that while the forward transformation is local, its inverse is not generically local, which has significant implications for quantum frame equivalence and the formulation of effective actions.

The core issue lies in the fact that recovering the original metric from the transformed one requires a differential constraint. The complete inverse metric tangent map contains a nonpolynomial projector, implying that no differentiable finite-jet inverse (a formula involving only finitely many derivatives at the same point) exists on an open set of unrestricted metrics. This means that, unlike standard conformal transformations, the relationship cannot be simply inverted point-by-point. However, it is suggested that branchwise functional inverses may exist after boundary or Cauchy data are specified.

The work uses $f(R)$ gravity as an explicit realization, where its local scalar-tensor representation in the Einstein frame acts as a "parent" theory. The metric-only Einstein-side description, however, requires a differential section governed by a normal operator. The authors derive the pulled-back classical Hessian and its off-shell embedding term, and in the quadratic model, show how constrained Gaussian elimination produces the scalaron nonlocal kernel, the corresponding normal determinant for the displayed measure, and the zero-mode compatibility condition. These results provide a precise framework for assessing curvature-dependent frame transformations in modified gravity and clarify their implications for effective actions, semiclassical analyses, and quantum frame equivalence.