Fibonacci recurrence conjecture for the auxiliary function $u$

Let $a_0=1$, $a_1=2$, and define $a_n=3a_{n-1}-a_{n-2}$. For ${\boldsymbol{i}}=(i_1,i_2)\in\langle a_n\rangle\times\langle a_{n+1}\rangle$, let $u(\alpha({\boldsymbol{i}}))$ be the auxiliary integer-valued function associated with the corresponding ${\boldsymbol{q}}$-vector, where ${\boldsymbol{r}}={\boldsymbol{x}}=(a_{n+1},a_n)$. Fibonacci auxiliary-function conjecture. (1) The value of $u(\alpha({\boldsymbol{i}}))$ is independent of $n$. (2) For all $i_1\geq 0$,

$$u(\alpha(i_1,0))=\left\lceil i_1\left(\frac{1+\sqrt{5}}{2}\right)^2\right\rceil.$$

(3) For all $i_2\geq 0$,

$$u(\alpha(0,i_2))=2i_2-\left\lfloor i_2\left(\frac{1+\sqrt{5}}{2}\right)\right\rfloor.$$

The conjecture predicts explicit boundary values and stability in $n$ for the function governing the Fibonacci-family examples.