On global reconstruction in generalized Rastall gravity and the invertibility of the Ricci scalar.

For the static and spherically symmetric case, we study the global consistency of the inverse reconstruction of generalized Rastall theory, G αβ +λH(R)g αβ = κ T αβ , when the geometry is fixed before the generalized Rastall function H(R). The analysis addresses the existence of a global inverse r=r(R) and the compatibility among the different radial branches when several radii produce the same value of the Ricci scalar curvature R. To illustrate the relevance of the analysis, we consider the regular Fan--Wang family with vanishing independent Schwarzschild mass. It is shown that R(r) is globally invertible if and only if μ=3 and 0<ν ≦ 1. The case ν=1 admits the elementary inverse r(R)=(24M F q 2 /R) 1/5 -q, whereas for 0<ν<1 the inverse is unique even though, in general, it cannot be expressed explicitly. It is also shown that the Hayward geometry employed by Lin and Qian and identified by (μ,ν)=(3,3) possesses two physical inverses over R min < R<0; the functions reconstructed on the two branches differ for ω ≦ 1/3, so the reconstruction is multivalued. For ω ≠ -1,1/3, a global single-valued function H(R) exists throughout the regular Fan--Wang family if and only if μ=3 and 0<ν ≦ 1. In this sector, H ∈ C 1 \ C 2 for 0<ν<1 and H ∈ C 0 \ C 1 for ν=1.