First MAP positions at Fibonacci-number differences

Prove that the first position of a maximum-length MAP in the Fibonacci word satisfies i(F_{2n+1})=F_{2n+3}-2 and i(F_{2n})=F_{4n}-1.

Background

The paper obtains complete formulas for several families of differences near Fibonacci numbers, but notes that the case d=F_n remains unresolved because the maximal length A(F_n) is not eventually constant and therefore cannot be handled by the authors’ automatic encoding approach.

Empirical evidence nevertheless suggests explicit formulas for the earliest starting positions of maximum-length MAPs at odd- and even-indexed Fibonacci differences.

References

Empirical evidence generated by suggests making the following Conjecture \ref{CONJ:idfn}. Unfortunately, because A(F_n) is not eventually constant, as will be shown in the next section, we once again run into the limitation of 's inability to handle the multiplication of two variables. \begin{conj}\label{CONJ:idfn} We have i(F_{2n+1}) = F_{2n+3} - 2, and i(F_{2n})=F_{4n}-1.

— Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words  (2501.05830 - Joshi et al., 10 Jan 2025) in Section 4.2, subsection “Investigating MAPs in the Fibonacci word in Walnut,” Conjecture 4.8