Exact Periodicity Does Not Evade the Boltzmann Brain Problem

Carroll, Diachenko and Dulani proposed that a finite quantum universe with commensurable energy levels evolves periodically, and that its low-entropy bounce then recurs far more often than the law exp(−ΔS) allows, so that ordinary observers outnumber Boltzmann brains. This paper proves that this is not so. Born-weighted rates do not depend on whether the spectrum is commensurable. For macrostates in general position the minimal macrostate carries weight (d+1)/(D+1), and the period obeys T·ΔE ≥ 1.5/(that weight). The proposed range of periods is therefore empty unless the macrostates are fine-tuned or the dynamics explores no more entropy than a single Boltzmann brain. LaTeX source, figure and PDF.