Erdős Problem 1122: Additive Functions and Logarithmic Growth

### What is the conjecture

Let $f:\mathbb{N}\to\mathbb{R}$ be a completely additive function, meaning $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$ (i.e., $\gcd(a,b)=1$). Define the set $A=\{n\geq 1: f(n+1)

Start by reviewing the repository's existing formalized conjectures and the source at erdosproblems.com/1122. Define the completely additive function and the asymptotic condition, then determine how the decreasing-index density should be represented. Done means the conjecture is added as a Lean formalization with its required supporting infrastructure.

Written by the indexing model from the issue text.