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.

I've uploaded a new short paper developing a binary difference-triangle construction that grew out of my experiments with Gilbreath-style difference triangles. The construction begins with 00, 11 → 0 10 → 1 01 → 2 followed by iterated absolute differences. For power-of-two depths d = 2^k, k ≥ 1, the paper proves the exact excursion probability P(d) = 5/8 − 1/2^(d+1), and therefore P(d) → 5/8. The argument uses an extremal-range characterization, a two-exception classification, and parity properties of binomial coefficients via Lucas's theorem. I'd especially welcome attempts to find a flaw in the argument, or pointers to earlier work involving an equivalent construction. @dpiponi @johncarlosbaez Paper, proof, and supporting material: https://doi.org/10.5281/zenodo.22777679 #Combinatorics #Mathematics #DifferenceTriangles #ExperimentalMathematics

I've uploaded a new short paper developing a binary difference-triangle construction that grew out of my experiments with Gilbreath-style difference triangles. The construction begins with 00, 11 → 0 10 → 1 01 → 2 followed by iterated absolute differences. For power-of-two depths d = 2^k, k ≥ 1, the paper proves the exact excursion probability P(d) = 5/8 − 1/2^(d+1), and therefore P(d) → 5/8. The argument uses an extremal-range characterization, a two-exception classification, and parity properties of binomial coefficients via Lucas's theorem. I'd especially welcome attempts to find a flaw in the argument, or pointers to earlier work involving an equivalent construction. @dpiponi @johncarlosbaez Paper, proof, and supporting material: https://doi.org/10.5281/zenodo.22777679 #Combinatorics #Mathematics #DifferenceTriangles #ExperimentalMathematics