Linear loops with visible and hidden coordinates

Exact statement

An $s$ -state rational linear loop is $z(n+1)=Mz(n)$ with $M\in\mathbb Q^{s\times s}$ . Its first $d$ coordinates form the visible orbit $u(n)$ , and a polynomial invariant $p$ requires $p(u(n))=0$ for every $n\ge0$ . Nontriviality means that the visible orbit contains infinitely many distinct points.

StatusPrecisely stated; homogeneous updates and literal visible coordinates are explicit

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

Context

Hidden coordinates enlarge the finite-dimensional shift-invariant space available to the visible coordinate sequences.

Definitions

The state and update matrix are rational.
The visible variables are the first d state coordinates, not an arbitrary output map.
A nontrivial loop visits infinitely many distinct visible states.

Proof or evidence

The model is reconstructed directly from the source statement and fixes the conventions needed by the dimension results below.

Model and quantifiers →

Verification notes

The homogeneous-update, coordinate-output, ground-field, and nontriviality conventions were checked separately.

Limitations

Allowing affine updates or a separate output map changes the relevant dimension count.

Open work

Keep homogeneous and affine updates, literal coordinates and output maps, and infinite time and infinite orbit distinct.