Companion C Integration Map
The Companion C visual atlas supplies finite decompositions and independent numerical controls for the Python library. Each adopted construction needs a package-native convention, typed interface and explicit certificate scope.
Shared mathematical structure
The shared operator framing is
\[
P_\alpha=U^{-1}E_\alpha U,
\qquad
x=\sum_\alpha P_\alpha x,
\qquad
T=\sum_{\alpha,\beta}P_\beta T P_\alpha,
\]
and the general analysis-synthesis obligation
\[
SA=I
\]
on the declared object or model space.
This framing is reflected in the stable semantic contract for complete decomposition, polyphase refinement, reconstruction, and certificates.
Cross-checks relevant to the current package
The Companion C numerical ledger includes finite checks for:
- packet synthesis
- operator-block synthesis
- polyphase synthesis
- two-to-four refinement
- mixed-radix packet synthesis
- dyadic and ternary residue refinement
- perfect-reconstruction filter banks
- Fourier and wavelet coordinate synthesis
- coordinate transport
- CRT packet synthesis
- overlap-add synthesis
- quantum and modal synthesis
The overlapping polyphase, mixed-radix, lattice, persistence and reconstruction identities are represented in package-native behavioral fixtures.
Material retained as research or documentation
The following Companion C areas should remain outside the stable API until separately reviewed:
- standalone plotting and page-layout helpers
- duplicate Fourier, wavelet, DCT, DST, Walsh, Gabor, and sampling implementations
- visual-only diagnostics without package certificates
- PDE, multigrid, Krylov, and low-rank examples that do not yet map to one stable interface
- heuristic comparison plots and model-specific RMSE claims
Promotion rule
A Companion C construction may enter the stable package only after it has:
- a package-native mathematical convention
- a typed public interface
- exactness or numerical certificate semantics
- deterministic regression tests
- a persistence decision
- API tier classification
- documentation that states its finite scope
The stable API admits a construction only after its mathematical and software contracts have been reviewed.