Primes as Triangles: Frobenius Endomorphisms and Pythagorean Triple Generation on the Lemniscatic Elliptic Curve

Every rational prime p ≡ 1 (mod 4) splits in the Gaussian integers as p = πp·π̄p. The primary Gaussian factor πp = a + bi generates a primitive Pythagorean triple (|a² − b²|, 2|a||b|, p). We verify this for all 24 split primes up to 200 through direct point-counting on the lemniscatic elliptic curve Elem: y² = x³ − x, confirming the Hecke identity ap = 2·Re(πp) at every prime. The cascade zn = (2 + i)n is the distinguished subsequence that stays inside the single prime p = 5 and its powers. This is Paper B of the series Where Prime Numbers Meet Geometry.