TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--03_em_flavor_source/sections/001-section-kernel-numerics-and-electromagnetic-closure.tex
Original file SHA-256: 5fdbdc1b7d8af7f85e5fc2435f567402b3dbcb96938951e58e633cf42549c7e2
Inclusive original lines: 9-45
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000009 | \begin{theorem}[Primitive seam generator]
000010 | \label{thm:primitive-seam-generator}
000011 | The primitive relative seam loop defines the positive generator
000012 | \[
000013 | [u_\Sigma]=1\in K^1(S^1)\cong \mathbb Z.
000014 | \]
000015 | Hence the associated seam spectral flow is
000016 | \[
000017 | SF(U_\Seam)=1,
000018 | \qquad
000019 | \Delta_\Seam=2\pi.
000020 | \]
000021 | In the relative Chern normalization, the seam constant is
000022 | \[
000023 | \cthree:=\frac{\Delta_\Seam}{16\pi^2}=\frac{1}{8\pi}.
000024 | \]
000025 | Moreover the seam-even base opening is the normalized complement of the carrier quadratic norm,
000026 | \[
000027 | \phibase
000028 | :=
000029 | \frac{1-\gamma}{\pi}
000030 | =
000031 | \frac{1-\tr_EY^2}{\pi}
000032 | =
000033 | \frac{1}{6\pi}.
000034 | \]
000035 | \end{theorem}
000036 | 
000037 | \begin{proof}[Proof sketch]
000038 | The primitive reconstruction yields one elementary relative seam class and therefore one primitive
000039 | generator of the seam $K^1$ lattice. Along a single positive seam circuit the spectral flow changes
000040 | by one, so the associated loop has class $[u_\Sigma]=1$ and spectral flow $SF(U_\Seam)=1$. The
000041 | primitive circuit closes after one full turn, giving $\Delta_\Seam=2\pi$ and hence
000042 | $\cthree=\Delta_\Seam/(16\pi^2)=1/(8\pi)$. The carrier theorem gives
000043 | $\gamma=\tr_EY^2=5/6$, so the seam-even opening is the normalized complement of the carrier
000044 | quadratic norm rather than a new independent numerical input.
000045 | \end{proof}