The Arithmetic Black Hole Model: A Structural Analogy on the Multiplicative Semigroup of Integers

This work introduces a structural framework for analyzing the multiplicative semigroup of the integers through a notion of arithmetic depth. Each integer n >= 2 is assigned a depth Omega (n), defined as the total number of prime factors counted with multiplicity. This induces a natural stratification of the integers into shells Sₖ = n: Omega (n) = k, forming a layered structure rooted in prime factorization. Within this framework, primes occupy the outermost boundary (Omega (n) = 1), while composite numbers lie at increasing depths according to their multiplicative complexity. A corresponding radial function r (n) = 1/ (Omega (n) + eps) provides a geometric interpretation of inward compression. Additional structural measures, including the divisor function tau (n) and the normalized ratio tau (n) /Omega (n), are used to characterize local richness and distributional behavior within each shell. The model yields several exact structural results, including bounds on divisor-density ratios and extremal configurations within fixed-depth layers. It also connects to the Sieve Firewall framework, where the S1* anticorrelation bound -- verified computationally across 124, 590 integers with zero violations -- provides a structural link to the Riemann Hypothesis. While the framework is entirely arithmetic in construction, it admits geometric interpretations that make the structure visible and explorable. These interpretations are presented as analogical aids and do not assert physical correspondence. The primary contribution is a coherent structural view of multiplicative depth that unifies factorization, layering, and distribution into a single interpretable system.