New properties of the $\phi$-representation of integers

We prove a few new properties of the $\phi$-representation of integers, where $\phi =(1+\sqrt{5})\mathrm{/}2$. In particular, we prove a 2012 conjecture of Kimberling. As software assistants, we used the Walnut theorem-prover, and in one proof, ChatGPT 5.