Asymptotic Equivalence of Bit-Based Prime Complexity and Shannon Entropy

{abstract}We propose a novel information-theoretic interpretation of arithmetic structure by introducing a finite-bit formalism grounded in constructibility and information invariance. In this framework, the bit content B(n) of an integer n is defined as the total number of prime factors with multiplicity, coinciding with the classical prime Omega function Ω(n). We show that the average bit content over integers up to n is asymptotically equivalent to the Shannon information of detecting a prime number near n. Specifically, we prove that:\[limn → ∞ E[Ω(n)] =ln2 limn → ∞ log n = ln2limn → ∞ I(p_n)\]where I(pₙ) denotes the Shannon information associated with the probability of primality near n. This result reveals a structural correspondence between the arithmetic complexity of integers and the informational cost of primality, offering a new unifying perspective on classical number theory, logic, and information. The interpretation also suggests that bit-invariant arithmetic encodes asymptotic entropy, bridging constructible mathematics with probabilistic information theory.{abstract}