A Power-of-Two Excursion Theorem for Schimmel Triangles

This paper introduces the Schimmel triangle, a directed binary absolute-difference construction motivated by computational experiments with opposing difference triangles. Given a binary word w, its directed transition row T(w) is defined by 00,11 → 0, 10 → 1, and 01 → 2, after which absolute adjacent differences are iterated to a single terminal value. For every power-of-two depth d = 2^k with k ≥ 1, the paper proves that the probability, over uniformly distributed binary words of length d + 2, of obtaining terminal value 0 is P_d = 5/8 − 1/2^(d+1). Consequently, along power-of-two depths, P_(2^k) converges to 5/8. The proof consists of an extremal-range lemma, an exact classification of the two inputs producing terminal value 2, and a parity argument establishing an exact terminal-1 probability of 3/8. Exhaustive computations at finite depths are included as independent verification and are not premises of the theorem. The Schimmel triangle is presented as a directed binary variant in the classical Proth–Gilbreath absolute-difference tradition. The power-of-two result is a theorem applying to the infinite family of depths d = 2^k; it is not merely a finite computational observation. The result does not assert a new theorem concerning the distribution of prime numbers.