8. Memory is an exact alternative to adding state
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Refinement and memory are two ways to represent the influence of unresolved components. This can be derived rather than merely proposed as an architectural option.
For fixed linear dynamics and an orthogonal split, write
and define the restricted blocks
Thus \(z_{t+1}=Az_t+Bq_t\) and \(q_{t+1}=Cz_t+Dq_t\).
Proposition 10: Exact projected memory and a conditional tail bound
For every \(t\geq0\),
If \(\|D\|\leq\rho<1\) and \(\|z_s\|\leq Z\) over the relevant history, omitting terms with \(k\geq H\) changes the one-step right-hand side by at most
Dropping the initial unresolved-state term adds at most \(\|B\|\rho^t\|q_0\|\).
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Proof
Iteration of the unresolved recurrence gives
Substitute this into the retained recurrence and change indices. Bound the omitted terms by the geometric series. \(\square\)
The memory kernel is \(BD^kC\). It records a path from retained variables into unresolved variables, evolution there, and return. The first kernel \(BC\) is precisely the return path appearing in the two-step composition defect. The term \(BD^tq_0\) is deterministic when \(q_0\) is specified. It becomes uncertain only relative to incomplete information or a probability model for \(q_0\).
This is the elementary linear instance of projection-memory reduction, associated with Mori-Zwanzig theory [R3]. A finite hand-chosen power-law or exponential memory feature is an approximation class for such effects, not a general derivation of the exact kernel.
The tail bound compares right-hand sides using the same resolved history. Replacing that history by recursively generated predictions introduces additional rollout error. Stability of that recursion needs its own bound. Also, a spectral-radius condition alone does not imply \(\|D\|\leq\rho\) in the norm already being used, especially for nonnormal operators.