Arithmetic Properties of Iterated Gaussian Integer Powers: The (2+i)n Cascade

We study the arithmetic of the power sequence zn for the Gaussian integer seed z = 2+i. The sequence generates Pythagorean triples at every level via the identity N((2+i)n) = 5n, where N denotes the Gaussian norm. The fourth power J = (2+i)4 = -7+24i produces the primitive Pythagorean triple (7, 24, 25). We establish a parity alternation theorem: Re(zn) and Im(zn) have strictly opposite parity for all n ≥ 0. We derive a closed-form quartic polynomial f(a) = 9a4 - 2a2 + 1 characterizing the flanking norm construction on thin Gaussian primes (a+i), and show that f(2) = 137. The argument structure arg(zn) = n * arctan(1/2) achieves approximate rotational closure at n = 4, when the cumulative rotation first exceeds pi/2. All results are elementary and verified computationally against LMFDB and independent algebra systems. No physical interpretation is offered or implied; the paper is purely number-theoretic.