Asymptotic Equivalence of Bit-Based Prime Complexity and Shannon Entropy

\begin{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 $\mathfrak{B}(n)$ of an integer $n$ is defined as the total number of prime factors with multiplicity, coinciding with the classical prime Omega function $\Omega(n)$. 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:\[\lim_{n \to \infty} \mathbb{E}[\Omega(n)] =\ln2 \lim_{n \to \infty} \logtwo \log n = \ln2\lim_{n \to \infty} I(p_n)\]where $I(p_n)$ 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.\end{abstract}