The same letter can have different roles in different realizations. The table fixes the roles used here. Every small matrix in the guide is explicitly supplied. No diagonalization or classification theorem is inferred from a picture.
| Object | Role | Worked instance |
|---|---|---|
| \(P_{N,j}\) | Retain coefficients with address \(j\) modulo \(N\). | Even/odd masks, then mod-\(4\) and mod-\(8\) masks. |
| \(\mathcal A_N,\mathcal S_N\) | Split into range-restricted packet vectors, then sum them. | A permutation/interleaving of eight coordinates. |
| \(J,C\) | Exact refinement and regrouping. In the defect chapter, \(C\) may instead be any supplied coarsening map. | A permutation in V5 and a shear in V22. |
| \(T_{ba}\) | Input packet \(a\), output packet \(b\): \(P_bTP_a\). | Blocks of a finite cyclic filter. |
| \(D,H,A\) | \(T=D+H=D(I+A)\), with \(A=D^{-1}H\). | Explicit integral \(2\times2\) matrices. |
| \(w(e),W(\gamma)\) | Positive certified edge gain and its additive word weight. | \(w(a)=1\), \(w(b)=2\). |
| \(L\) | Precision depth in a quotient such as \({\mathbb Z}/2^L{\mathbb Z}\). | \(L=5\) gives arithmetic modulo \(32\). |
| \(A(t),Z(t)\) | Polynomial graph adjacency and unsigned walk-count series. | Two parallel edges remain distinct. |
| \(E,\Lambda,s\) | A rational projector, a specified lattice, and a clearing exponent. | \(E_+\) has local weight \(1\) at \(2\). |
| \((s_i)\) | Directional weights of a supplied Smith diagonal. | \((1,2)\) modulo \(8\) or \((0,1,3)\) across depths. |
| \(\Delta\) | The failure of an independently chosen operator to intertwine: \(CT_f-T_cC\). | One nonzero entry \(1/16\). |
Suggested reading routes
For exact coordinates, read Visual 3 through Visual 8, then the finite filter on Visual 20. For weighted inversion and counting, read Visual 9 through Visual 13. For arithmetic separation and congruence fibers, read Visual 14 through Visual 19. The final pages distinguish nonlinear chart geometry, error propagation, and the formal contract.
Read this as
Operator words carry signed matrix values. Graph walks carry unsigned counts. Smith normal form concerns linear congruences. The Smith impedance chart on V21 is a different, nonlinear construction.