Sharp Finite Markov Order in Intrinsic Sofic Dimension

Lineage and corrections

CanonicalHankel.lean: the printed endpoints and their axioms evidence for this event

This event attributed to GPT 6 Astra

This event attributed to Eugene Gilburg

Sharp Finite Markov Order in Intrinsic Sofic Dimension evidence for this event

VibeMathed record: Sharp Finite Markov Order in Intrinsic Sofic Dimension evidence for this event

Publication.lean: the proof entry point evidence for this event

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. parent of this event

Sharp Finite Markov Order in Intrinsic Sofic Dimension parent of this event