Complex-Time Contour Principle for Lorentzian Minisuperspace Path Integrals

Lorentzian minisuperspace path integrals reduce the gravitational path integral to an oscillatory integral over the lapse N with integrand exp ( iS( N) ). In such settings, contour prescriptions (including iϵ, Wick-type deformations, and Picard–Lefschetz thimbles) determine which saddle points contribute and, near Stokes lines, can lead to apparent ambiguities. We propose a Complex-Time Contour Principle (CTCP), motivated by a complex- time geometry T=t+iϕ and a past-fixed / future-pending (pf/fp) decomposition, to formulate simple physical requirements for admissible complex lapse contours: (i) future-regularity (no exponential blow-up) at asymptotic ends and singular structures, (ii) future-positivity fixing the sign of the fp component, and (iii) physical homology to preserve Lorentzian boundary conditions. For actions with FRW-type structure S( N)=aN3 +bN−A/ N+O( 1) (a>0, A>0), CTCP yields: (a) an asymptotic convergence wedge 0 0 near N=0; and (c) a restriction of the contributing saddle sector to those thimbles compatible with both endpoint constraints. CTCP thus provides a time-structural interpretation of contour prescriptions and a route to resolving Stokes-type ambiguities by physical regularity conditions. CTCP is not an alternative to Picard–Lefschetz theory: for a specified Lorentzian contour it leaves the semiclassical weighting unchanged, while supplying a time- structural criterion for admissible contour classes and for resolving Stokes-type ambiguities.