One orbit's Hankel rank is not a variety lower bound

Exact statement

The rank certificate for a chosen infinite C-finite orbit in $V(p)$ cannot be promoted to a lower bound for $p$ itself without proving that every admissible infinite orbit in $V(p)$ has at least that rank.

StatusQuantifier failure isolated

External reviewNo documented external or specialist review of this FPRD exposition is recorded.

Context

The hidden-dimension problem minimizes over all admissible infinite orbits on a variety, whereas realization theory minimizes a state space for one fixed orbit.

Proof or evidence

A lower bound for one member of a set does not lower-bound the minimum over the set. The two minimizations have different quantifier order.

Why orbit minimality is not variety minimality →

Verification notes

The graph construction and its rank certificate were re-read with the orbit and variety quantifiers written explicitly.

Limitations

This rules out an inference; it does not exhibit a lower-rank competing orbit for every graph.

Open work

Find a family for which every infinite rational C-finite orbit has growing rank, or prove a uniform compression theorem.