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 $rho-3P=4Lambda$. At the mini superspace level, a two-particle variant of th
e 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 ${a^2,aphi}simsqrt{4n_1+1}pmsqrt{4n_2+1}$.
