An explicit pair realizing every marginal polynomial growth exponent

For every real $\gamma\geq0$, a finite family of real matrices is constructed whose joint spectral radius is one and whose maximal norm of a product of length $n$ is bounded above and below by positive constant multiples of $n^\gamma$, at every positive integer $n$. For $0<\gamma<1$, two matrices of order six suffice. A rank-two projection reduces arbitrary switching words to products of two-dimensional triangular matrices. A telescoping identity and Hölder's inequality give the upper bound for all words; a prescribed word with logarithmic gap length gives the lower bound at every length. Two matrices with dyadic-rational entries realize every nonnegative rational exponent. Rational pairs also realize some irrational exponents.