Walking on the Pythagorean Tree

This is a preprint of the manuscript: "Walking on the Pythagorean Tree" Abstract:Starting from the prime 3, we construct a deterministic walk through the primes by following primitive Pythagorean triples. At each step we either advance along the hypotenuse when it is prime, or detour through the longer leg of the most recent successful triple when the forward hypotenuse is composite. The procedure generates a sequence of primes (3, 5, 13, 37, 613, 93637, …). The walk organises the forward arrows p → (p² + 1)/2 into a directed forest on the primes. Chains starting from different primes coalesce after sharing a common forward step. Additionally, the cumulative turning angle along the chain from 3 empirically stabilises near a constant Δ ≈ 1.51181…, whose mathematical nature (existence as a limit, closed form) remains open. The manuscript has been submitted to a peer-reviewed journal and is currently under review.