TFPT scientific source excerpt
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000285 | 
000286 | \textbf{Step 2: Defect sector reweighting in the Euclidean partition function.}
000287 | We treat the discrete correction as a defect sector weight. An additive deformation of the
000288 | defect action by $\alpha$ implies exact exponential reweighting:
000289 | \begin{lemma}{Defect Sector Reweighting}{defectreweight}
000290 | Let the Euclidean partition function split into sectors
000291 | \begin{equation}
000292 |     Z(\alpha)=Z_{\mathrm{triv}}(\alpha)+Z_{\mathrm{def}}(\alpha),
000293 | \end{equation}
000294 | with defect weight $\delta_{\mathrm{top}}(\alpha)=Z_{\mathrm{def}}(\alpha)/Z_{\mathrm{triv}}(\alpha)$.
000295 | If the defect action is deformed additively,
000296 | \begin{equation}
000297 |     S_{\mathrm{def}}(\alpha)=S_{\mathrm{def}}(0)+\alpha\,\mathcal{J},
000298 | \end{equation}
000299 | then
000300 | \begin{equation}
000301 |     \delta_{\mathrm{top}}(\alpha)=\delta_{\mathrm{top}}\,e^{-\alpha\mathcal{J}}.
000302 | \end{equation}
000303 | For a degree-$d$ cover with identical sheet contributions, $\mathcal{J}=d\,\mathcal{J}_1$.
000304 | We identify the per-sheet normalization with the spectral-flow unit, $\mathcal{J}_1=\mathrm{SF}$,
000305 | so $\mathcal{J}=d\,\mathrm{SF}$. For the orientable double cover, $d=2$ and $\mathrm{SF}=1$,
000306 | so the exponent is fixed to $k=2$ and
000307 | $\delta_{\mathrm{top}}(\alpha)=\delta_{\mathrm{top}}\,e^{-2\alpha}$. If $\mathrm{SF}\neq 1$,
000308 | this corresponds to a different topological sector with a different $k$. \hfill QED
000309 | \end{lemma}
000310 | 
000311 | \begin{tcolorbox}[colback=blue!3, colframe=blue!40, 
000312 |     title={\textbf{Lemma (Cover-Degree Scaling of Deck-Invariant Response)}}, fonttitle=\bfseries\small]
000313 | Let $p:\widetilde{M}\to M$ be a degree-$d$ covering and let $\mathcal{J}$ be a deck-invariant
000314 | source functional. Then
000315 | \begin{equation}
000316 |     \int_{\widetilde{M}}\mathcal{J} = d \int_{M}\mathcal{J}.
000317 | \end{equation}
000318 | If the response of $\phiz$ is built from such an integrated source, the effective
000319 | weight acquires a factor $d$. For the orientable double cover, $d=2$.
000320 | \end{tcolorbox}
000321 | 
000322 | \textbf{Step 3: Double cover fixes the exponent.}
000323 | The defect-sector lemma and cover-degree scaling fix the exponent to $2$ for the
000324 | orientable double cover. Adding the baseline yields the exact response:
000325 | \begin{equation}