Rows are targets and columns are sources throughout. If \(E_{ji}\) is a matrix unit, then \(E_{ji}e_i=e_j\). The notation \(e_i\) means a basis vector. A named graph edge is written \(a,b,c,d\) or \(e,f\) when its role is explicit.
| Object | What it records | First worked page |
|---|---|---|
| \({\operatorname{src}},{\operatorname{tgt}}:E\to V\) | The ordered endpoints of each distinct edge, including parallel edges. | Visual 3 to Visual 4 |
| \(L_e=c(e)E_{{\operatorname{tgt}}(e),{\operatorname{src}}(e)}\) | One operator supported on a packet. | Visual 3 |
| \(L_w=L_{e_n}\cdots L_{e_1}\) | Execute the written word from left to right. | Visual 4 to Visual 5 |
| \(T=\sum_eL_e\) | Add all edge operators before taking powers. | Visual 4 to Visual 7 |
| \(F_a\) | A synthesis frame from chart \(a\) into the ambient space. | Visual 9 |
| \(\mathcal A_a,\mathcal S_a\) | Inverse analysis and synthesis maps, which may be maps of sets. | Visual 10 |
| \(J_{b\leftarrow a}\) | Canonical coordinate transport, forced by the exact charts. | Visual 9 to Visual 10 |
| \(\Gamma_{b\leftarrow a}\) | Additional connection data, with reversibility stated explicitly. | Visual 11 to Visual 12 |
| \(\Delta_{c,b,a}\) | The difference between transport through two steps and direct transport, a map from \(C_a\) to \(C_c\). | Visual 12 |
| \(H_{c,b,a}\) | Loop transport based at \(a\), an automorphism of \(C_a\). | Visual 13 |
| \(G_a\) | An invertible gauge change at each chart. | Visual 14 to Visual 15 |
| \(\Pi_\mu{\operatorname{Ob}}\Pi_\nu\) | Block from input to output of \({\operatorname{Ob}}=H-I\). | Visual 16 to Visual 18 |
| \(r(x,p)\) | A residual at packet \(p\) for the named candidate \(x\). | Visual 19 |
| \(t,\operatorname{inputs}(t)\) | A finite operation tree and its syntactically used input labels. | Visual 20 |
Reading routes
For transfer, read V3 to V8. For the finite geometry models, read V9 to V18. For candidate certificates and nonlinear scope, read V19 to V22. V23 puts the matrix mechanisms together. V24 gives the formal boundary and replay guide.
Read this as
A \(C_a\to C_c\) defect and a \(C_a\to C_a\) holonomy are not the same typed map. Returning the defect to the base fiber requires a specified inverse transport.