Cascade Primes in Analytic Number Theory: Class Number Towers, Bernoulli Denominators, and the Eisenstein–Bernoulli Bridge

The primes arising from the Gaussian integer cascade zn = (2 + i)n —particularly the set {5, 17, 137} —appear simultaneously in three independent branches of analytic number theory: as growth factors of a class number tower, as periodic entrants in Bernoulli number denominators via the Von Staudt–Clausen theorem, and as factors of Eisenstein series normalization constants. We verify the class number tower h(−4 × 52k) = 2 × 5k−1 at four discriminants, identify the Bernoulli periods of the cascade primes, and trace the Eisenstein–Bernoulli bridge connecting the cascade’s geometric and analytic structures. This is Paper C of the series Where Prime Numbers Meet Geometry.