Remark: Family 012. Independent largest prime factors of consecutive integers

Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of $n$ and $n+1$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $P^{+}(n)<P^{+}(n+1)$ have density $1/2$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.