Integer frieze patterns and Tabachnikov's four sign changes

Tabachnikov asked for which rows the differences between the entries of a row of two frieze patterns of the same width must change sign at least four times along the period, and left open whether this holds for all rows when the entries are positive integers. We show that it does not. For every even $n \geq 14$ there are two integer (Conway--Coxeter) frieze patterns of width $n - 3$ whose middle rows are strictly ordered, and for every odd $n \geq 15$ there are two whose rows $\lfloor (n-3)/2 \rfloor$ differ with no zero and exactly two sign changes. The second family lies inside the range $1 \leq k \leq [w/2]$ in which the question is stated. The differences in both families never vanish, so they are counterexamples however zeros are counted. When zeros may count as sign changes, an exhaustive search shows that $14$ is the smallest $n$ with strictly ordered rows and $15$ the smallest with fewer than four sign changes in a row of the stated range. Both families are verified in Isabelle/HOL for all $n$, and the base pairs also in Lean.

Companion site (blueprint): https://karlesmarin.github.io/blueprints/integer-frieze-sign-changes/