One Constant Rate Asked to Inflate and to End: Guth (1981)

Guth's 1981 model holds the universe in a metastable false vacuum. The vacuum energy drives exponential expansion. The phase transition to the true vacuum is supposed to finish the job by bubble nucleation, percolation, collision, and thermalization. The model has one dial for both jobs. That dial is beta, written β = Γ/H⁴. Γ is the bubble nucleation rate per unit volume. H is the expansion rate. In the 1981 model both are constants. So β is a constant. This paper prices the two demands on that one constant. The price is computed, not argued. Inflation needs β ≤ 1.6667×10⁻³ for a Hubble volume to survive 60 e-folds at the 1/e level. It needs β ≤ 1.756×10⁻³ for a clean survival at S(60) ≥ 0.9. Completion needs β ≥ 0.24 at the literature percolation threshold as commonly quoted — the upper end of the range §3.3 carries as stated — or β ≥ 0.716 at the toy volume-decline threshold. The two windows do not meet. They miss by a factor of 407.86 at the strict ends. At the inflation edge, β = 1/60, the false vacuum still grows. The toy growth exponent is g = 2.9302 per e-fold. That is 97.673% of the pure de Sitter rate. Waiting does not fix the sign. Guth and Weinberg said the same thing in 1983, in print, about their own model. At slow nucleation the bubbles never percolate. They form finite clusters. The large-scale thermalization the 1981 scenario needs does not take place. Verdict, in full: a reason was found, and the question remains open (not solved). No stronger claim than that verdict is made. No exit mechanism is claimed. The deficit stands as counted: 2 opposed requirements, 1 constant dial, 0 exit equations in the 1981 paper. A boundary, stated here so nothing in this paper can be read past it: the vacuum catastrophe remains unsolved, and no derivation in this program has produced the observed vacuum energy. This paper does not touch that problem. It claims no progress there.