From DeWitt initial condition to cosmological quantum entanglement

A gravity-anti-gravity (GaG) odd linear dilaton action offers an eternal inflation evolution governed by the unified (cosmological constant plus radiation) equation of state $ρ-3P=4Λ$. At the mini superspace level, a 'two-particle' variant of the no-boundary proposal, notably 'one-particle' energy dependent, is encountered. While a GaG-odd wave function can only host a weak Big Bang boundary condition, albeit for any k, a strong Big Bang boundary condition requires a GaG-even entangled wave function, and singles out $k=0$ flat space. The locally most probable values for the cosmological scale factor and the dilaton field form a grid ²,aϕ\~√4n₁+1±√4n₂+1.