15. From hidden state to an accessible predictive state
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Let a finite-dimensional autonomous linear process have hidden state \(x_t\in\mathbb R^d\) and observed output \(y_t\in\mathbb R^p\):
The observation map need not be injective. For a positive integer \(h\), define the observation map
The coordinates of \(O_hx_s\) are future observations relative to time \(s\). They are not available at time \(s\) merely because this matrix is written down. At time \(t\geq h-1\), however, the same vector is the observed history
Thus a forward observation map becomes a causal state after a declared warm-up interval. The time index is part of the contract.
Proposition 14: Exact causal delay closure
The following statements are equivalent:
There is a linear map \(D_h:V_h\to V_h\) satisfying \(O_hL=D_hO_h\).
\(\ker O_h\subseteq\ker(O_hL)\).
\(\operatorname{rank}O_{h+1}=\operatorname{rank}O_h\).
When these statements hold, \(D_h\) is unique on \(V_h\). The history state obeys \(z_{t+1}=D_hz_t\), and its last output block is \(y_t\). Its iterates therefore determine every subsequent observed output. If a positive integer \(d\) bounds the hidden dimension, \(h=d\) is sufficient for closure, though a smaller positive \(h\) may suffice.
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Proof
The equivalence of the first two statements is the linear factorization criterion: define \(D_h(O_hx)=O_hLx\) and require independence from the representative \(x\). The rows of \(O_hL\) are \(CL,\ldots,CL^h\). All except the final block already occur in \(O_h\). The kernel condition is therefore equivalent to \(CL^h\) being in the row space of \(O_h\), which is exactly the rank equality. It also implies \(O_hLx\in V_h\) because this vector is \(O_h\) applied to \(Lx\). Uniqueness follows because every element of \(V_h\) has a representative. The history recurrence follows directly. Cayley-Hamilton expresses \(L^d\) as a linear combination of lower powers, so \(CL^d\) is in the row space of \(O_d\). \(\square\)
This does not recover hidden variables in \(\ker O_h\). It identifies the quotient needed for the declared outputs. An independent hidden component can remain unobservable forever without preventing exact output prediction.
The rank equality here is a property of the true observation maps. A plateau in a finite empirical data matrix is not the same statement. It may reflect unexcited directions rather than a closed observation space. Section 17 makes the difference exact.
For several actions \(L_u\), a fixed delay vector does not automatically have one autonomous transition matrix. Products of the actual action matrices determine its observation map, which can change with the action history. One needs an action-aware state, a common closed observable family, or a separately justified belief-state construction. Proposition 14 is not silently generalized to arbitrary controlled or stochastic dynamics.