TFPT scientific source excerpt
Original repository path: documents/tfpt_1_architecture_e8/sections/005-source.tex
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000166 | \subsection{The $\mu_4$ glue and the $5/4+3/4=2$ discriminant theorem (proved)}
000167 | The two factors have equal discriminant groups and $E_8$ is unimodular, so the glue index is exactly
000168 | $|\mu_4|$:
000169 | \begin{equation}
000170 |   \operatorname{disc}(D_5)=\operatorname{disc}(A_3)=\mathbb Z_4,\qquad
000171 |   [E_8:D_5\oplus A_3]=\sqrt{\tfrac{\det D_5\det A_3}{\det E_8}}=\sqrt{\tfrac{4\cdot4}{1}}=4=|\mu_4| .
000172 | \end{equation}
000173 | The discriminant-form norms sum to the $E_8$ root norm, and --- strikingly --- they are pre-existing
000174 | TFPT constants:
000175 | \begin{equation}
000176 |   \boxed{\ \begin{aligned}
000177 |     &q(D_5)=\tfrac54=\frac{\dtwo}{\dtop^2},\qquad q(A_3)=\tfrac34=\xi_{\mathrm{tree}},\\
000178 |     &q(D_5)+q(A_3)=2=|\text{$E_8$ root}|^2
000179 |   \end{aligned}\ }\qquad\mE\ \veri{v1\_e8\_glue.py}
000180 | \end{equation}
000181 | This is literally the $E_8$ spinor-root split $\tfrac12(\pm)^8=\tfrac12(\pm)^5\oplus\tfrac12(\pm)^3$,
000182 | $5\cdot\tfrac14+3\cdot\tfrac14=2$.