The running example begins \(1,\tfrac12,\tfrac12,\tfrac{11}{24},\tfrac{61}{144}\). Its first four outputs give
Read the diagram in words
The two columns of the input delay matrix are independent in the running example. Their paired next columns identify a transition on that input span. Coverage of a larger unknown state space is not established by a finite observed rank.
The determinant \(\det Z_-=1/4\) establishes coverage of the declared two-dimensional delay space. The resulting recurrence is
For general paired states, a linear fit exists exactly when \(\ker Z_-\subseteq\ker Z_+\). It is unique on \(W={\operatorname{im}} Z_-\), and maps \(W\) to itself exactly when \({\operatorname{im}} Z_+\subseteq W\).
The mathematical point
These data identify the transition on their input span. The known generating system separately certifies that this span is the required full predictive state space.
Scope
Finite sampled rank does not exclude a larger unexcited state space. Finite continuation agreement does not establish unrestricted discovery. Perturbations additionally require a conditioning bound.
Formal route
DelayClosure.lean; DataSpan.lean; see the exact module map at the end.