Class Number Growth in Imaginary Quadratic Orders at Gaussian Prime Norm Discriminants

This is the authors' abstract. We don't add key points for this paper.

We prove that the class numbers of imaginary quadratic orders at cascade-determined discriminants form a geometric tower: h(−4 × 52k) = 2 × 5k−1 for k ≥ 1, with growth factor N(2+i) = 5—the seed norm. Class numbers at cascade-component conductors are themselves cascade quantities: h(7) = 4 = |Z4|, h(17) = 8 = max(Bit+), h(137) = 68 = 4 × 17. The class number map sends cascade numbers to cascade numbers in a closed, self-reinforcing network.