Exact state dimension for a fixed visible orbit

Exact statement

Let $V_u=\operatorname{span}_{\mathbb Q}\{S^ju_k:j\ge0,1\le k\le d\}$ , let $r(u)=\dim V_u$ , and let $q(u)=\dim\operatorname{span}_{\mathbb Q}\{u_1,\ldots,u_d\}$ . If $r(u)<\infty$ , the least state dimension of a rational linear loop whose first $d$ coordinates are exactly $u_1,\ldots,u_d$ is $d+r(u)-q(u)$ .

StatusProved with matching lower bound and construction

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

Context

The usual realization dimension is r(u) when outputs are supplied by a separate linear map. Requiring d literal output coordinates adds the redundancy cost d-q(u).

Hypotheses and scope

All coordinate sequences are rational-valued.
The visible shift span has finite dimension.

Proof or evidence

Any s-coordinate realization has shift span of dimension at most q(u)+s-d, giving the lower bound. A basis of V_u completed from the visible sequences gives a matching rational state update.

Coordinate-realization formula →

Verification notes

Both inequalities and the effect of linearly dependent visible coordinates were rechecked. The construction now keeps all d visible coordinates literally and adds only a complement of their span; independent Vandermonde rank tests are recorded at /research-artifacts/result-review-2026-09-01-batch-18/independent-output.json

Limitations

The result minimizes a specified orbit, not all orbits contained in a polynomial variety.

Open work

Relate the coordinate-cost formula to standard minimal-realization statements with an external output map.