The excursion probability P(d) for power-of-two depths d = 2^k (k ≥ 1) is exactly 5/8 − 1/2^(d+1) and converges to 5/8.
The user is sharing a new mathematical paper regarding a binary difference-triangle construction. The paper specifically proves that for power-of-two depths, the excursion probability P(d) converges to 5/8.