Surprises in Lorentzian path-integral of Gauss-Bonnet gravity

In this paper we study the Lorentzian path-integral of Gauss-Bonnet gravity in mini-superspace approximation four spacetime dimensions and investigate transition amplitude from one configuration to another. Past studies motivate us on imposing Neumann boundary conditions initial as they lead stable behaviour fluctuations. The is computed exactly while incorporating non-trivial contribution coming sector gravity. A saddle-point analysis involving usage Picard-Lefschetz methods allow gain further insight nature amplitude. Small-size Universe Euclidean which shown by exponentially rising wave-function. It reaches a peak after wave-function becomes oscillatory indicating an emergence time phase Universe. We also notice interesting hypothetical situation when independent completely, happens cosmological constant coupling have particular relation. This however doesn't imply that momentum left arbitrary it needs be fixed value chosen demanding regularity at stability