TFPT scientific source excerpt
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000312 | \]
000313 | \end{lemma}
000314 | 
000315 | \begin{proof}
000316 | Apply \TFPTxref{lem:retained-gamma5-hermiticity}. The nonzero spectrum of each $\mathcal D_f$ occurs in
000317 | paired sets $\pm i\lambda_n$, while the branch mass eigenvalues are strictly positive by
000318 | \TFPTxref{thm:sheet-cp-protection}. Therefore each sector determinant is nonnegative, and so is the
000319 | full sector weight.
000320 | \end{proof}
000321 | 
000322 | \begin{lemma}[Nontrivial topological support on the closed branch]
000323 | \label{lem:nontrivial-topological-support}
000324 | On the closed branch the primitive topological sector has strictly positive weight:
000325 | \[
000326 | Z_1=Z_{-1}>0.
000327 | \]
000328 | \end{lemma}
000329 | 
000330 | \begin{proof}
000331 | By \TFPTxref{thm:boundary-winding-control}, the canonical branch has unit seam winding, so the primitive
000332 | topological class $Q=1$ is nonempty on the admissible branch. Choose an admissible representative
000333 | $\phi_1$ in that class. By \TFPTxref{thm:sheet-cp-protection}, the same branch carries a genuine compact
000334 | determinant angle with primitive winding number one, so $\phi_1$ has finite admissible action and
000335 | belongs to the closed branch sector $Q(\phi_1)=1$. Let $\mathcal U_1$ be a sufficiently small open
000336 | neighborhood of $\phi_1$ inside that sector. The local Boltzmann density is strictly positive on
000337 | $\mathcal U_1$, and \Cref{lem:topological-sector-positivity} gives nonnegative sector weights.
000338 | Therefore
000339 | \[
000340 | Z_1
000341 | \ge
000342 | \int_{\mathcal U_1} e^{-S_{\mathrm{adm}}[\phi]}\,d\mu(\phi)
000343 | >
000344 | 0.
000345 | \]
000346 | The antiunitary sheet symmetry exchanges $Q=1$ and $Q=-1$, hence $Z_{-1}=Z_1>0$.
000347 | \end{proof}
000348 | 
000349 | \begin{theorem}[Unconditional strong-CP closure on the admissible branch]
000350 | \label{thm:strong-cp-closure}