4. Packets, refinement and transfer blocks
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4.1 Packets belong to a declared resolution
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Within one orthogonal atlas, let \(E_i\) be pairwise orthogonal coordinate selectors with \(\sum_iE_i=I\). Pull them back by a unitary map:
A packet is the component \(P_ix\) together with its projector, domain, atlas and resolution. A bare coefficient block without this information is not a fully specified packet.
Projectors from different atlases need not commute and must not be treated as one larger mutually orthogonal family. Weighted frame packets can sum to the identity while failing to be idempotent. These are different structures.
Proposition 3: Complete refinement preserves reconstruction
Suppose a finite rooted tree has root projector \(I\), and every internal projector is the sum of pairwise orthogonal child projectors. Every complete antichain \(\Lambda\), meeting each root-to-leaf path exactly once, satisfies
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Proof
Replace an internal node by the sum of its children. Repeated replacement reaches any complete antichain and preserves the total projector. Different selected nodes have disjoint descendant branches, hence orthogonal ranges. The norm identity is Pythagoras. \(\square\)
This is a conservation law for information under complete refinement. Deleting a child, storing only its energy, or imposing a coefficient threshold changes the contract. For a retained subset \(J\) of an orthogonal decomposition, the lost signal is exactly \(\sum_{i\notin J}P_ix\), and its squared norm is the sum of the omitted packet energies.
The finite best-basis recursion
minimizes an additive cost over the declared finite tree [K1, R1]. The recurrence itself does not make a nonadditive predictive objective additive. If a prediction cost couples two leaves, a scalar cost at each node is generally insufficient for global optimization. Packet construction and cost evaluation must also be included in an implementation's complexity account.
4.2 Operations, not just signals, decompose
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Let \(L:X\to Y\) be linear, with orthogonal input projectors \(P_i\) and output projectors \(Q_j\). Its transfer blocks are \(L_{ji}=Q_jLP_i\).
Proposition 4: Transfer reconstruction and refinement
One has
If \(P_i=\sum_rP_{i,r}\) and \(Q_j=\sum_sQ_{j,s}\) are complete refinements, then
For another linear map \(M:Y\to Z\) and output projectors \(R_k\),
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Proof
Insert the appropriate identity decomposition on the input, output or intermediate space. Use \(Q_j^2=Q_j\) in the composition formula. \(\square\)
This is the algebraic reason to make transfer blocks first-class objects. The same operation can be expressed at several resolutions, while the coarse and fine expressions remain linked by exact identities. Approximations acquire explicit omitted-block terms rather than an unnamed representation error.
4.3 Nonlinear operations need interactions
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For a nonlinear map, \(F(\sum_iP_ix)\) generally differs from \(\sum_iF(P_ix)\). Applying a linear transfer formula to such a map silently loses interactions.
Proposition 5: Polynomial packet interactions
Let
where each \(B_m:X^m\to Y\) is multilinear. Then
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Proof
Substitute \(x=\sum_iP_ix\) in every argument and distribute by multilinearity. \(\square\)
For example, \((u+v)^2=u^2+2uv+v^2\). A grammar containing only independent packet squares misses \(2uv\). For nonpolynomial maps, one needs direct nonlinear semantics or a local expansion with a stated remainder bound. The word algebraic does not remove that obligation.