Sharp Finite Markov Order in Intrinsic Sofic Dimension

For a stationary finite-alphabet law $\mu$, let $$n=\dim_{\mathbb R}\mathcal H_p$$ be the intrinsic real Hankel dimension of its cylinder-probability function. If $\mu$ has any finite Markov order, then $$\operatorname{ord}(\mu)\le \binom n2.$$ The proof passes to a reduced $n$-dimensional linear representation and uses Holland's criterion that $k$-step Markovity is equivalent to every length-$k$ transition product having rank at most one. Applying $\Lambda^2$ turns this into vanishing of products on a space of dimension $\binom n2$; a uniform nilpotence argument then forces vanishing after $\binom n2$ steps. Sharpness is attained for every $n\ge2$. The construction gives a stationary sofic process with a nonnegative rational presentation of minimal dimension $n$ and exact order $\binom n2$. One realization uses $$\binom n2-1+n^2$$ output symbols.

For a stationary finite-alphabet law $\mu$, let $$n=\dim_{\mathbb R}\mathcal H_p$$ be the intrinsic real Hankel dimension of its cylinder-probability function. If $\mu$ has any finite Markov order, then $$\operatorname{ord}(\mu)\le \binom n2.$$ The proof passes to a reduced $n$-dimensional linear representation and uses Holland's criterion that $k$-step Markovity is equivalent to every length-$k$ transition product having rank at most one. Applying $\Lambda^2$ turns this into vanishing of products on a space of dimension $\binom n2$; a uniform nilpotence argument then forces vanishing after $\binom n2$ steps. Sharpness is attained for every $n\ge2$. The construction gives a stationary sofic process with a nonnegative rational presentation of minimal dimension $n$ and exact order $\binom n2$. One realization uses $$\binom n2-1+n^2$$ output symbols.