Can the Universe Change Signature When Gravity is Dynamical?

Abstract: Can a universe undergo a regular Euclidean-Lorentzian signature transition when the gravitational coupling itself is dynamical? We address this question in scalar-tensor gravity with nonminimal coupling $F(φ)R$. Although the spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) sector can be mapped to the Einstein frame for $F&gt;0$ and a nondegenerate scalar redefinition, not all transition properties are frame independent. For a finite, positive, and sufficiently regular conformal factor, the existence and transverse character of the type change are preserved, whereas the extrinsic geometry is not. We therefore use the Einstein frame as a solution-generating representation and impose total geodesy in the physical Jordan frame. We construct two exactly integrable classes of solutions. In the oscillator-ghost-oscillator branch, the scalar field is stationary at the transition and Jordan-frame total geodesy follows under suitable regularity assumptions on the conformal factor. In the critical exponential-potential branch, the scalar is generically nonstationary and total geodesy instead requires \begin{equation*} H_E\big|Σ= \frac12 \left( \frac{d\ln F}{dψ} \right)Σ\dotψΣ. \end{equation*} We also distinguish this condition from the stronger smoothness requirements of a Kossowski-Kriele-type transverse metric. Finally, while the canonical Einstein-frame scalar does not allow effective phantom evolution, the dynamical nonminimal coupling can generate a locally superaccelerating Jordan-frame regime near the transition when $\left(\frac{d\ln F}{dψ}\right)</em>Σ&gt;0$ for the chosen oscillator branch. Thus signature change persists beyond Einstein gravity, with regularity and effective cosmological behavior remaining intrinsically frame sensitive.