Theorem: Absorption tail and mean

If $1-P_{j,n}(0)\ge r_{\ast }>0$ for every nonterminal sector and opportunity, then

Terms $i>t$ are zero; for $r_{\ast }=1$, the tail is $1_{\{t<K\}}$. Absorption occurs almost surely, not necessarily by a deterministic finite horizon.

Proof. Couple the population recursion to independent uniform variables. A value at most $r_{\ast }$ forces a sector decrease of at least one. The waiting time for $K$ Bernoulli-$r_{\ast }$ successes dominates absorption. Its tail is the displayed binomial tail and its mean is the sum of $K$ geometric means $1/r_{\ast }$. This auxiliary probability construction does not measure the coherent quantum history.