The Multiplicative Cascade: An Ω-Stratified Geometric Structure of the Integers

We introduce the Multiplicative Cascade, a geometric structure that emerges from viewing integers through the combined lens of the Chinese Remainder Theorem and prime factorization depth. By embedding integers n into a product space T^k × N via their residue classes and the prime-counting function Ω(n), we reveal a stratified flow whose density distribution directly reflects the Erdos–Kac theorem. This framework provides a geometric interpretation of integer factorization complexity, explains the concentration of integers around Ω(n) ~ log log n, and exhibits structural features analogous to irreversible dynamical systems. We analyze the cascade’s formation through iterated multiplication, its sustaining mechanisms via density freezing, and demonstrate how the asymmetry between multiplication and factorization manifests geometrically.