Fibonacci system of equations: part 1

In this recent unanswered question I had asked about the probability of exactly one negative solution in a $$3\times 3$$ system of equations with Fibonacci-like coefficients, denoted by $$F_i$$.

With $$\theta_i$$ as defined in that question; that is, the concatenation of $$1.F_i$$, how can we prove that the sign of $$F=\theta_i\theta_{i+5}-\theta_{i+1}\theta_{i+4}$$ is positive only when $$5\mid i$$?

If this is disproved, is there an alternative pattern that $$\text{sgn}(F)$$ satisfies?

Note that I have defined $$F_2=2$$ not $$1$$ for the purpose of having different values for each $$i$$.

I ask this because when we find the inverse of that $$3\times 3$$ matrix, we end up with the expression in the denominator for $$X,Y,Z$$ so if we want to explore the case $$X,Y\ge0$$ and $$Z<0$$, we need to know its sign to progress on with removing the denominator.

Here are some numerical details I've found: $$\begin{array}{c|c}i&1&2&3&4&5&6&7&8&9&10\\\hline\operatorname{sgn}(F)&-&-&-&-&+&-&-&-&-&+\end{array}$$

• my guess: it is negative only when $F_{i+1}$ has more digit than $F_{i}$, that happens for $i=5,10,15$, for example. You should test it for $i=19$, since it is the first time that happens for non-multiple of 5 index – Exodd Sep 16 '18 at 10:46

For $$i=19$$ where $$5\not\mid 19$$, $$F$$ is positive.
If this is disproved, is there an alternative pattern that $$\text{sgn}(F)$$ satisfies?
I have not seen any pattern for $$i\le 30$$.
$$\begin{array}{c|c}i&11&12&13&14&15&16&17&18&19&20\\\hline\operatorname{sgn}(F)&-&-&-&-&+&-&-&-&+&-\end{array}$$
$$\begin{array}{c|c}i&21&22&23&24&25&26&27&28&29&30\\\hline\operatorname{sgn}(F)&-&-&+&+&-&-&-&-&+&-\end{array}$$