Reading the figure
Native ranks two, four and eight each have trace one. In ambient rank eight their traces are zero, zero and one. Native trace and amplified ambient trace have different roles.
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
Native rank 2: trace 1 In rank 8: trace 0 Native rank 4: trace 1 In rank 8: trace 0 Native rank 8: trace 1 In rank 8: trace 1 Native trace and amplified ambient trace have different roles.
Original figure caption
A trace-one native matrix amplified by an even multiplicity has ambient trace zero.
For odd \(q\), let \(a=(q+1)/2\) and \(z=t_L^a\alpha\). Then
The trace identity is
It sends inner trace 1 to outer trace 1. At native rank \(2^j\), every constructor has trace 1; its next native correction has trace \(1-2\cdot1=1\). In rank 8, the earlier amplified approximations instead have traces \(0,0\), and the final matrix has trace 1.
Reading the diagram
Only the appropriate native trace establishes the scalar-character compatibility at each recursive stage.