Cosmographic reconstruction of the quintessence potential in scalar-tensor gravity

We generalise the cosmographic reconstruction programme of Chakraborty, Dunsby & Scherrer to scalar-tensor gravity. Working in the Jordan frame with a general non-minimal coupling $F(Φ)$, we derive exact closed-form expressions for the potential slope $λ_U$, the coupling slope $λ_F$, and their curvature parameters $Γ_U$ and $Γ_F$ in terms of the cosmographic parameters $(q,j,s)$, the Planck-mass running $α_M = d\ln F/d\ln a$ and its derivatives. We express $α_M$ and its derivatives in terms of a set of coefficients $g_n$ describing the time variation of the gravitational constant $G\propto F^{-1}$, casting all four quantities in fully observable form. We then evaluate the reconstruction against current data, including DESI~DR2 baryon acoustic oscillation measurements and Lunar Laser Ranging constraints on $\dot G/G$, propagating the cosmographic uncertainties through to the reconstructed potential. All expressions reduce analytically to those of Chakraborty, Dunsby & Scherrer in the minimally-coupled limit, which we verify symbolically. We find that the potential slope $λ_U$ is recovered at the few-tenths level while the curvature $Γ_U$ is essentially unconstrained. We further show that the positivity of the scalar kinetic term is equivalent to $w_{\rm eff,0}\ge-1$, so that the region of cosmographic parameter space closed to minimally-coupled quintessence is exactly the phantom region, and that the non-minimal coupling reopens it through a single coefficient $g_2$ of the gravitational tower. Because Lunar Laser Ranging forces $|g_1|\lesssim2\times10^{-4}$ while the reconstruction is sensitive to $g_2\sim O(0.1)$, the framework presupposes that $α_M$ is passing through zero at the present epoch, as expected on the Damour--Nordtvedt least-coupling attractor.