A counterexample to Scottish Book Problem 155

We construct real Banach spaces X and Y and a bijection U: X → Y which preserves all pairwise distances on every closed ball of radius 1/4 but is not a global isometry. The proof starts from an explicit injective map that preserves every distance at most 1/2 and shortens one longer distance. We then enlarge the source and target through a transfinite recursion that inserts every missing target point while preserving both injectivity and this distance threshold. This preprint is a revised paper based on “AI-Assisted Research Report: A counterexample to Scottish Book Problem 155” (https://doi.org/10.5281/zenodo.22168214). The PDF is also available on the author’s website: https://yoshito-ishiki-math.github.io/preprints/scottish.pdf.