Positions of Instances of b in Fibonacci String

Theorem
Let $S_n$ denote the $n$th Fibonacci string.

Let $F_n$ denote the $n$th Fibonacci number.

Let $k \in \Z$ such that $k \le F_n$.

Then the $k$th letter of $S_n$ is $\text b$ :
 * $\left\lfloor{\left({k + 1}\right) \phi^{-1} }\right\rfloor - \left\lfloor{k \phi^{-1} }\right\rfloor = 1$

where:
 * $\left\lfloor{\, \cdot \,}\right\rfloor$ denotes the floor function
 * $\phi$ denotes the golden mean.