31. Packets can reconstruct a belief without being beliefs
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Let \(P_1,\ldots,P_m\) be a complete family of linear packet maps on a predictive space, with \(\sum_iP_i=I\). Algebraic reconstruction gives \(z=\sum_iP_iz\). It does not imply that \(P_iz\) is an unnormalized probability state.
Proposition 30: Positive packet decomposition and its boundary
Suppose a cone \(C\) has a strictly positive mass functional \(\lambda\). If every packet map preserves \(C\), then each normalized \(z\in C\) decomposes as a genuine mixture
A zero weight has \(P_iz=0\). In the standard cone \(\mathbb R^n_+\), however, a complete family of cone-preserving idempotents must consist of diagonal coordinate-selection matrices. In particular, a nontrivial Fourier or Haar packet family cannot generally be reinterpreted as a mixture of probability states merely because it is orthogonal and complete.
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Proof
Cone preservation and strict positivity give the weights and the zero-weight claim. Apply \(\lambda\) to the complete decomposition to obtain their sum. Normalization of each nonzero term gives the mixture.
On \(\mathbb R^n_+\), cone preservation is entrywise nonnegativity. Since the matrices sum to \(I\), every off-diagonal entry in every matrix is zero. Idempotence forces each diagonal entry to be zero or one, and completeness makes the selections disjoint and exhaustive. \(\square\)
For a two-state belief \(p=(3/4,1/4)^\top\), the Haar detail projector
produces \((1/4,-1/4)^\top\). It is a valid signed detail packet, but not a subprobability distribution. Its total mass is zero despite being nonzero, which cannot happen inside a cone with a strictly positive mass functional.
This does not prohibit useful spectral packets. It fixes their type. Signed algebraic packets can be used for transfer, approximation and diagnostics. They must be synthesized before probability interpretation unless a separate positive decomposition has been established. That distinction prevents a reconstruction identity from being misused as a Bayesian mixture identity.