The scaling construction is minimal for its orbit

Exact statement

For the scaling orbit $x(n)=2^n$ , $y(n)=f(2^n)$ used in FPRD-LL-T02, the minimum coordinate-state dimension is exactly $r_f=|\{1\}\cup\operatorname{supp}(f)|$ .

StatusProved from block-Hankel rank and Vandermonde independence

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

Context

The result certifies the displayed orbit rather than the polynomial graph as a whole.

Proof or evidence

The occupied exponential sequences are independent, so r(u)=r_f; x and y are independent, so q(u)=2; FPRD-LL-T01 gives s_min=r_f.

Minimality of the constructed orbit →

Verification notes

The orbit-specific quantifier was retained throughout.

Limitations

A different infinite orbit on the same graph may have smaller realization dimension.

Open work

Use finite Hankel minors as checkable certificates for proposed orbit dimensions.