Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.
| Object | What it means | Instance here |
|---|---|---|
| \(X\) | The ambient vector space or module. It specifies the objects that might be generated. | \({\mathbb Q}^6\) with two chains of length three. |
| \(S\subseteq X\) | The selected seed family. A family can contain vectors with components in several packets. | \(\{e_0+f_0\}\) or \(\{e_0,f_0\}\). |
| \(\AA\) | The fixed unital algebra of admissible operators. It is not chosen anew for each target. | \({\mathbb Q}[N]\) versus \({\mathbb Q}\langle N,P_e,P_f\rangle\). |
| \({\operatorname{Hull}}_{\AA}(S)\) | The smallest invariant linear subspace containing the seeds. | \({\operatorname{span}}\{e_j+f_j:0\le j\le2\}\). |
| \(P_\alpha\) | A finite family of pairwise annihilating idempotents whose sum is \(I\). | Chain projections, oblique projections, or residue masks. |
| \(X_\alpha\) | The packet subspace \(P_\alpha X\). It can have dimension greater than one. | One chain of dimension three. |
| \(T_{\beta\alpha}\) | The block \(P_\beta T P_\alpha\): input \(\alpha\), output \(\beta\). | A numerical entry when each coordinate packet has dimension one. |
| \(F/I_s\) | A module of operator words modulo relations that kill the seed. | A quotient of left modules. It need not be a quotient algebra. |
| \(U\) | An invertible change of realization coordinates. | A shear that makes orthogonal packets oblique. |
| \(\Phi\) | An intertwiner that need not be invertible. | Projection from two chains to the first chain. |
| \(\tau\) | The topology used when taking the closure of an algebraic hull. | The \(\ell^2\) norm for a bilateral shift. |
Read products from right to left
In \(P_\beta T P_\alpha x\), first select the input packet, then apply \(T\), then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.
Where to begin
For seed rank, start with Visual 3. For exact reconstruction, start with Visual 7. For graphs, use Visual 15 and Visual 16. For completion in infinite dimensions, use Visual 18 and Visual 19. The scope of the formal proofs is collected on Visual 22.