Boltzman Brain-Death

The Boltzmann brain hypothesis dissolves on two independent grounds. First, the standard count $N_{\text{BB}} = r_{\text{BB}}\,T$ is a hybrid. It takes a constant fluctuation rate from one description of de Sitter cosmology (the 8D covariant phase space, in which time is a coordinate) and multiplies it by the unbounded integration time of another, the $(6{+}1)$D non-comoving picture, in which time is external. Neither description supports the product internally. The divergence is a coordinate artifact. Performed honestly within one frame, the phase-space integration over the de Sitter horizon yields $P \sim 10^{-8.7 \times 10^{12}}$. Second, a Boltzmann brain is not a single particle. It is a joint configuration of order $10^{14}$ informationally distinct cells. In a closed box of side $N$, the per-step probability that $k$ cells form a brain-shape connected cluster is bounded by the Klarner--Rivest lattice-animal count combined with Gibbs indistinguishability, $P_0 \leq k!\,(2eD)^k / N^{(k-1)D}$, exponentially small in $k$. The closed-box ideal is recurrent: Kac's lemma gives a mean recurrence time $\gtrsim 10^{2.22 \times 10^{13}}$ interaction steps: extremely long but finite. Any single physical departure from the closed-box-static idealization (slow thermal leakage at one missed step per universe lifetime, or cosmological expansion at one site added per universe lifetime) collapses the cumulative probability to $\lesssim 10^{-2.22 \times 10^{13}}$ even at $T \to \infty$. Either route suffices to block the standard divergence. Together they show that Boltzmann-brain dominance is not a consequence of recurrence alone, but of combining recurrence with an unlicensed temporal or observer-moment multiplier.