5. Predictive closure is the central contract
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Let \(\gamma:X\to Z=\gamma(X)\) be a summary and let \(F_u:X\to X\) describe a deterministic transition for action \(u\). The reduced description closes when there is \(\bar F_u:Z\to Z\) such that
This is an equality of functions on the declared domain, not a fit statistic.
Proposition 6: Exact quotient criterion
Such a reduced transition exists if and only if
for every admissible \(x,x'\) and every action. When it exists it is unique on \(\gamma(X)\).
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Proof
Necessity follows by applying \(\bar F_u\) to equal summaries. For sufficiency, define \(\bar F_u(z)=\gamma(F_u(x))\) for any representative with \(\gamma(x)=z\). The stated condition makes this well-defined and also proves uniqueness. \(\square\)
This criterion also applies when \(X\) is a latent physical state and \(\gamma\) is an observation map. Two physically different states can produce identical images and different futures. Exact image reconstruction does not resolve that hidden-state ambiguity. A history, a belief state or additional observations may be needed.
For a stochastic kernel, the corresponding condition is equality of the distributions of the next summary for states with the same current summary. Section 7 makes this finite criterion explicit.
5.1 The obstruction in the linear case
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Suppose \(X\) is a Hilbert space, \(P\) is an orthogonal projection, \(Q=I-P\) and \(x_{t+1}=Lx_t\). The retained state is \(z_t=Px_t\). Then
Proposition 7: Closure defect and compositional defect
The retained state closes for every initial signal if and only if \(PLQ=0\). If it does not close, the error in using only \(PLP\) is
For two linear operators \(L\) and \(M\),
Consequently, compressing two operations separately can discard a path that leaves the retained subspace and later returns.
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Proof
If \(PLQ=0\), the displayed recurrence depends only on \(z_t\). Conversely, apply Proposition 6 to \(x\) and \(Px\), which have the same retained state, to obtain \(PLQx=0\) for every \(x\). The norm bound is submultiplicativity. The compositional identity follows by inserting \(I=P+Q\) between \(L\) and \(M\). \(\square\)
The relevant quantity is not omitted energy alone. A very small omitted packet can have a large effect if its transfer into retained packets is strongly amplified. Conversely, an energetic packet can be irrelevant to a specified prediction if every corresponding return block is zero.
A residual is therefore typed. Reconstruction residual, coefficient-consistency residual, predictive residual, closure residual and statistical uncertainty should not be collapsed into one scalar without a specified objective.