TFPT scientific source excerpt
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000017 | \textbf{Step 2: Defect sector reweighting in the Euclidean partition function.}
000018 | We treat the discrete correction as a defect sector weight. An additive deformation of the
000019 | defect action by $\alpha$ implies exact exponential reweighting:
000020 | \begin{lemma}{Defect Sector Reweighting}{defectreweight}
000021 | Let the Euclidean partition function split into sectors
000022 | \begin{equation}
000023 |     Z(\alpha)=Z_{\mathrm{triv}}(\alpha)+Z_{\mathrm{def}}(\alpha),
000024 | \end{equation}
000025 | with defect weight $\delta_{\mathrm{top}}(\alpha)=Z_{\mathrm{def}}(\alpha)/Z_{\mathrm{triv}}(\alpha)$.
000026 | If the defect action is deformed additively,
000027 | \begin{equation}
000028 |     S_{\mathrm{def}}(\alpha)=S_{\mathrm{def}}(0)+\alpha\,\mathcal{J},
000029 | \end{equation}
000030 | then
000031 | \begin{equation}
000032 |     \delta_{\mathrm{top}}(\alpha)=\delta_{\mathrm{top}}\,e^{-\alpha\mathcal{J}}.
000033 | \end{equation}
000034 | For a degree-$d$ cover with identical sheet contributions, $\mathcal{J}=d\,\mathcal{J}_1$.
000035 | We identify the per-sheet normalization with the spectral-flow unit, $\mathcal{J}_1=\mathrm{SF}$,
000036 | so $\mathcal{J}=d\,\mathrm{SF}$. For the orientable double cover, $d=2$ and $\mathrm{SF}=1$,
000037 | so the exponent is fixed to $k=2$ and
000038 | $\delta_{\mathrm{top}}(\alpha)=\delta_{\mathrm{top}}\,e^{-2\alpha}$. If $\mathrm{SF}\neq 1$,
000039 | this corresponds to a different topological sector with a different $k$. \hfill QED
000040 | \end{lemma}
000041 | \textbf{Evidence:} \texttt{backreaction\_index\_anchor\_audit}.
000042 | 
000043 | \begin{tcolorbox}[colback=blue!3, colframe=blue!40, 
000044 |     title={\textbf{Lemma (Cover-Degree Scaling of Deck-Invariant Response)}}, fonttitle=\bfseries\small]
000045 | Let $p:\widetilde{M}\to M$ be a degree-$d$ covering and let $\mathcal{J}$ be a deck-invariant
000046 | source functional. Then
000047 | \begin{equation}
000048 |     \int_{\widetilde{M}}\mathcal{J} = d \int_{M}\mathcal{J}.
000049 | \end{equation}
000050 | If the response of $\phiz$ is built from such an integrated source, the effective
000051 | weight acquires a factor $d$. For the orientable double cover, $d=2$.
000052 | \end{tcolorbox}
000053 | 
000054 | \textbf{Step 3: Double cover fixes the exponent.}
000055 | The defect-sector lemma and cover-degree scaling fix the exponent to $2$ for the
000056 | orientable double cover. Adding the baseline yields the exact response:
000057 | \begin{equation}
000058 |     \phiz(\alpha) = \phitree + \deltatop\,e^{-2\alpha}.
000059 | \end{equation}
000060 | 
000061 | \begin{remark}
000062 | \textbf{Series control (no new parameters):} The exact form implies
000063 | \begin{equation}
000064 |     \phiz(\alpha)=\phitree+\deltatop\left(1-2\alpha+2\alpha^2-\frac{4}{3}\alpha^3+\cdots\right),
000065 | \end{equation}
000066 | so all higher coefficients are fixed (e.g.\ $c_2=2$). Numerically, even the
000067 | $\alpha^2$ term shifts $\phiz$ by $\lesssim 10^{-8}$ since
000068 | $\alpha\approx 0.0073$ and $\deltatop\approx 1.2\times10^{-4}$.
000069 | \end{remark}
000070 | 
000071 | \begin{theorem}{Self-Consistent $\alpha$ with Derived Two-Defect Partition}{selfconsistent}
000072 | Including the minimal backreaction response on the orientable double cover \emph{and} the derived two-defect occupancy weight anchored by holonomy degeneracy $g=5$:
000073 | \begin{align}
000074 |     \phiz(\alpha) &= \phitree + \deltatop e^{-2\alpha} + \delta_2 e^{-4\alpha} + \cdots,
000075 |     \label{eq:backreaction}\\
000076 |     \delta_2 &= \frac{g}{4}\,(\delta_{\mathrm{top}})^2 = \frac54\,(\delta_{\mathrm{top}})^2,
000077 |     \qquad (g=5),
000078 | \end{align}
000079 | where $g$ is fixed by SU(5) holonomy degeneracy prior to any $\alpha$ comparison (see theoryv3 audit chain).
000080 | The prefactor $g/4$ follows from the two-defect cluster expansion with the pair symmetry
000081 | factor $1/2$ and the two-sheet pairing factor $2$ (Appendix~\ref{app:repro}).
000082 | \textbf{Evidence:} \texttt{two\_defect\_cluster\_expansion\_audit}, \texttt{g5\_independence\_audit}, \texttt{g5\_trace\_completeness\_audit}.
000083 | the self-consistent solution of the CFE yields the headline value:
000084 | \begin{equation}
000085 |     \boxed{\alpha^{-1}_{\text{TFPT}}(0) = \num[round-mode=places,round-precision=12]{\TFPTalphaInvGOverFourNum}}
000086 | \end{equation}
000087 | with CODATA 2022 diagnostic $z=\num[round-mode=places,round-precision=3]{\TFPTalphaZGOverFourNum}$ (see theoryv3 module \texttt{defect\_partition\_g5\_audit}; artifacts in Appendix~\ref{app:repro}).
000088 | \end{theorem}
000089 | 
000090 | \begin{tcolorbox}[colback=blue!3, colframe=blue!40,
000091 |     title={\textbf{Two-Defect Bookkeeping}}, fonttitle=\bfseries\small]
000092 | The prefactor $g/4$ in $\delta_2 = (g/4)(\delta_{\mathrm{top}})^2$ follows from
000093 | enumeration of all 30 defect-pair configurations across $g=5$ holonomy sectors
000094 | and 2 sheets:
000095 | \begin{itemize}[leftmargin=1.5em, topsep=2pt, itemsep=1pt]
000096 |     \item Pair symmetry factor: $1/2$
000097 |     \item Sheet pairing factor: $2$
000098 |     \item Holonomy degeneracy: $g=5$
000099 |     \item Combined: $\frac{1}{2}\times 2\times g \times \frac{1}{2} = g/4$
000100 | \end{itemize}
000101 | Total: $\sum_{\text{configs}} w_i = g/4 = 5/4$.\quad
000102 | \textbf{Certificate:} \texttt{two\_defect\_enumeration\_certificate} (30 configs, sum check passed).