Is spacetime absolutely or just most probably Lorentzian?

Pre-gauging the cosmological scale factor $a(t)$ does not introduce unphysical degrees of freedom into the exact FLRW classical solution. It seems to lead, however, to a non-dynamical mini superspace. The missing ingredient, a generalised momentum enjoying canonical Dirac (rather than Poisson) brackets with the lapse function $n(t)$, calls for measure scaling which can be realised by means of a scalar field. The latter is essential for establishing a geometrical connection with the 5-dimensional Kaluza-Klein Schwarzschild-deSitter black hole. Contrary to the Hartle-Hawking approach, (i) The $t$-independent wave function $\psi(a)$ is traded for an explicit $t$-dependent $\psi(n, t)$, (ii) The classical FLRW configuration does play a major role in the structure of the 'most classical' cosmological wave packet, and (iii) The non-singular Euclid/Lorentz crossovers get quantum mechanically smeared.