37. A filtering defect with a packet interpretation
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The relation to the realization calculus is particularly direct before normalization. Let \(R\) be a linear summary of a hidden probability space, let \(\lambda R=\mathbf1^\top\), and let \(B_s\) be a proposed reduced branch operator. For a true belief \(p\), put
The defect \(\delta_s\) is a vector in the reduced representation, not a probability of model failure. It records the failure of the unnormalized instrument to commute with the summary.
Proposition 36: Exact normalized closure defect
Suppose the true and proposed branch likelihoods
are positive, and let \(\widehat z^+=B_sz/b\). Then
If \(M_s\neq0\) and both likelihoods and normalized updates agree on every strictly positive belief, then \(RM_s=B_sR\). Agreement of normalized updates alone need not force this equality.
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Proof
The true unnormalized summary is \(B_sz+\delta_s\), with mass \(a=b+\lambda\delta_s\). Subtract the two normalized vectors and collect terms. Since a nonzero nonnegative branch has positive likelihood on every strictly positive belief, agreement permits multiplication by the common mass. This gives equality of unnormalized summaries on the interior of the probability simplex, whose linear span is the whole hidden vector space. For the last claim, a one-state model can assign different positive branch probabilities while its normalized state always remains the scalar one. \(\square\)
For an orthogonal retained projector \(P\) preserving the mass row, put \(Q=I-P\), \(R=P\) on its image and \(B_s=PM_sP\). Then
This is the unresolved-to-retained transfer block from Part I. Its posterior effect is not simply its norm: normalization applies the oblique map \(I-\widehat z^+\lambda\) and divides by the true branch probability. A component proportional to \(\widehat z^+\) changes only the branch mass and disappears from the normalized defect, while still changing likelihood-based inference over mechanisms.
A certificate should therefore retain both \(\lambda\delta_s\) and the posterior defect. A posterior that looks correct can hide a wrong evidence model. This is the stochastic counterpart of the distinction between reconstruction and predictive closure.