Prime-Guided Walks on the Pythagorean Tree: Turning Mechanism, Forest Structure, and a Family of Deficit-Angle Constants

We define a deterministic walk on the tree of primitive Pythagorean triples, guided by primality. Given an odd prime A, its unique primitive triple is (A, (A²−1)/2, (A²+1)/2). If the hypotenuse is prime, the walk continues forward; if composite — typically due to a mod-5 obstruction — a turning mechanism redirects through the companion leg to a new prime hypotenuse. Starting from 3, this generates the prime sequence {3, 5, 13, 37, 613, 93637, 1179973, 9608116477, 12404716588557373, ...}, verified to 9 terms. We prove that in every primitive triple, the hypotenuse–leg gap equals (m−n)² or 2n² for Euclid parameters (Gap Structure Theorem), enabling an optimized O(√B) search. Chains from different starting primes form a directed forest with deterministic merging (Forest Structure Theorem). Each chain defines a deficit-angle constant Δ = arg(∏(bₙ + aₙ·i)) via Gaussian integer products, converging doubly-exponentially; for the chain from 3, Δ ≈ 1.51181304949354 rad. Numerical evidence suggests Δ is a new transcendental constant. This repository contains the preprint, complete Python implementation, and OEIS submission template.