SL₂-Tilings Do Not Exist in Higher Dimensions (mostly)

We define a family of generalizations of $\operatorname{SL}_2$-tilings to higher dimensions called $\boldsymbol{\epsilon}$-$\operatorname{SL}_2$-tilings. We show that, in each dimension 3 or greater, $\boldsymbol{\epsilon}$-$\operatorname{SL}_2$-tilings exist only for certain choices of $\boldsymbol{\epsilon}$. In the case that they exist, we show that they are essentially unique and have a concrete description in terms of odd Fibonacci numbers.