Fibonacci-word MAP maximum

Joshi and Rust, Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words, arXiv:2501.05830v2, Theoretical Computer Science 1050 (2025), 115391, Conjecture 4.24:

For all d >= 1, (A(d)-1)/d < sqrt(5)/tau.

Here tau=(1+sqrt(5))/2 and A(d) is the global maximum positive length of a monochromatic arithmetic progression of difference d in the zero-indexed infinite Fibonacci fixed point 0 -> 01, 1 -> 0. Both symbols and every nonnegative starting position are included.

The source gives exact expressions for A(d) in the small- and large-step regimes (Theorems 4.6 and 4.15), but leaves this strict uniform comparison with the golden ratio as Conjecture 4.24.

The strict inequality must hold at every positive difference, including the boundary d=1 and arbitrary starts. A real estimate alone does not establish strictness at the nearest integer threshold.

The repository's infinite goldenWord is read through the golden mechanical letter at the i+1 fractional coordinate. In the small-step regime the opposite letter's interval prevents wraparound; in the large-step regime two successive jumps decrease the position within alternating clusters. The quadratic norm of an integer approximation to d*tau gives a positive integer lower bound. Integer spacing sharpens the small-step real estimate; the large-step estimate applies for d>=7, with the remaining six denominators resolved by exact floor identities and fractional windows. Lengths are bounded and a length-one progression exists, so their supremum is an attained global maximum rather than a formula-defined surrogate.

Any positive difference, start, letter and finite monochromatic progression whose length violates the strict bound would contradict the result. A mismatch between the repository's Boolean letters and the source's zero/one indexing would instead refute the source correspondence.

• Formal source: D5/S1/Words/FibonacciMapBound.lean.
• Definition: goldenMAPMaximum; conclusion: result.
• The result's axiom closure is propext, Classical.choice, Quot.sound.

theorem. The result addresses the literal global Conjecture 4.24 and no claim about other words or asymptotic sharpness.

The bounded source audit through 17 September 2026 found no later qualifying proof or refutation; exhaustive publication priority is unverified.