18. A complete output-only identification example
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Return to the rational system from Section 13, now with an additional independent hidden coordinate:
The third coordinate never influences the observed output. The first coordinate alone is not closed, but
has rank two and \(O_3\) has the same rank. Proposition 14 gives an exact two-dimensional delay state. Choose \(x_0=(1,0,q)^\top\) for any \(q\). The first four observations are
A learner receives those four outputs and the declared delay length \(h=2\). It does not receive \(L\), \(C\), \(O_2\) or \(q\). From consecutive observed windows it forms
Since \(\det Z_-=1/4\), these windows span the declared two-dimensional state. Exact arithmetic gives
The learned recurrence is
It predicts \(y_4=61/144\), which was not used to identify the coefficients. Proposition 14 and the verified rank condition on the true observation maps establish why this recurrence continues for every initial state of this particular system. The four samples alone establish the fitted recurrence only within the declared closed two-state model class. Proposition 17 forbids confusing those two conclusions.
The reference script separates the roles. One routine fits only the observed samples. A different check, with access to the generating system, verifies the theoretical assumptions and compares subsequent outputs. This prevents hidden access to the true operator from being presented as learning.
The same example connects back to the exact memory kernel. Section 13 expressed the output through a decaying convolution of its history. Here the history is summarized by a two-component recursive state. Neither representation stores the independent third coordinate. Memory and state realization describe the same observable behavior in two different ways.
