For complete input and output projector families,
Refining either family subdivides these transfer blocks without changing their total. Orthogonal packet families also give the corresponding squared-norm decomposition.
Read the diagram in words
The transfer array records how every input packet contributes to every output packet. The accompanying curves distinguish the mixed contribution in a square from the square of the omitted amplitude. Retaining every block is an exact decomposition; dropping blocks needs a separate error estimate.
For a polynomial operation, multilinearity produces every ordered packet interaction:
In the scalar square, the mixed contribution is \(2uv\). With \(u=1\), it is linear in omitted amplitude \(v\), whereas the omitted packet square is quadratic.
The mathematical point
A nonlinear operation acts on interactions between packets. Reconstruction energy alone does not bound an omitted packet's influence on a future quantity.
Scope
Complete refinement and truncation are different operations. Truncation requires an explicit omitted-interaction estimate on its declared domain. Projectors from different atlases need not commute.
Formal route
PacketTree.lean; TreeEnergy.lean; Transfer.lean; Polynomial.lean.