Uniqueness of E7 Among Simple Lie Algebras Under Gaussian Prime Norm Filtration

We prove that E7 is the unique simple Lie algebra g for which the quantity dim(g) + max_mark(Ĝ) is simultaneously prime, congruent to 1 (mod 4), and a Gaussian prime norm on the cascade tower of z = 2+i. The proof is by exhaustive enumeration over all simple Lie algebras—the four infinite classical families (An, Bn, Cn, Dn) and the five exceptional algebras (G2, F4, E6, E7, E8)—using a three-layer sieve: Filter 1 (Fermat splitting) eliminates algebras whose sum is not expressible as a² + b²; Fil