Golden-ratio growth of Conway's subprime closure

Let $s(m)$ be the Conway subprime function define the binary operation on the natural numbers $x\circ y=s(x+y)$, and denote by $C_n$, $n\ge 0$, the sequence of subsets of natural numbers defined by $C_0=\{1\}$, and $C_{n+1}=C_n\cup (C_n\circ C_n)$. We prove the conjecture by Caragiu, Vicol and Zaki that $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\mathrm{\mid }{C}_{n+1}\mathrm{\mid }}{\mathrm{\mid }{C}_n\mathrm{\mid }}=\frac{1+\sqrt{5}}{2}\mathrm{.}$$ The underlying mathematical proof in this paper was constructed with some algorithmic assistance from GPT-6 Astra and its correctness has been formally verified using the Lean 4 proof assistant.