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000198 | \begin{axiom}{Gauss--Bonnet Normalization with APS Gluing Term}{GB}
000199 | The geometric normalization on the orientable double cover $\widetilde{M}$ is fixed by the
000200 | spectral gluing term associated with the $\mathbb{Z}_2$ identification along the seam $\Gamma$.
000201 | Cutting along $\Gamma$ yields a manifold-with-boundary $\widetilde{M}_\Gamma$ with components
000202 | $(C_1,C_2,\Gamma_+,\Gamma_-)$ and a Dirac operator equipped with APS boundary conditions on
000203 | $C_1,C_2$ and a unitary matching operator $U_\Gamma$ on $\Gamma_\pm$. The $\eta$-invariant gluing
000204 | formula produces a discrete seam contribution $\Delta_\Gamma$, and for the minimal nontrivial
000205 | $\mathbb{Z}_2$ gluing class one has $\Delta_\Gamma = 2\pi$.
000206 | 
000207 | \tcblower
000208 | \textbf{Status:} Standard APS/$\eta$-gluing framework; the discrete seam term follows from the
000209 | minimal $\mathbb{Z}_2$ matching class (Appendix~\ref{app:seam}).
000210 | 
000211 | \textbf{Consequence:} Two physical boundary cycles ($2\pi+2\pi$) plus $\Delta_\Gamma=2\pi$ give
000212 | total curvature $6\pi$, fixing
000213 | \begin{equation}
000214 |     \phitree = \frac{1}{6\pi}.
000215 | \end{equation}
000216 | \end{axiom}
000217 | 
000218 | \noindent\textbf{Why this is well-defined.} The seam is encoded as a gluing condition, not as an
000219 | ad hoc boundary. In APS index theory, the gluing formula for $\eta$-invariants adds a discrete
000220 | spectral-flow term determined by the matching operator $U_\Gamma$. This makes the seam
000221 | contribution quantized and computable, rather than a geometric guess.
000222 | 
000223 | \begin{tcolorbox}[colback=green!3, colframe=green!50!black, 
000224 |     title={\textbf{Proof Sketch for $\phitree = 1/(6\pi)$ (via $\eta$-gluing)}}, fonttitle=\bfseries\small]
000225 | \textbf{Setup:} Cut the orientable double cover along $\Gamma$ to obtain
000226 | $\widetilde{M}_\Gamma$ with boundary components $(C_1,C_2,\Gamma_+,\Gamma_-)$.
000227 | Impose APS boundary conditions on $C_{1,2}$ and gluing by a unitary operator
000228 | $U_\Gamma$ between $\Gamma_+$ and $\Gamma_-$.
000229 | 
000230 | \textbf{Step 1 (Boundary operator):} On the seam, take the tangential Dirac operator
000231 | \begin{equation}
000232 |     D_\Gamma = i\,\frac{d}{d\theta}, \qquad \theta\in[0,2\pi).
000233 | \end{equation}
000234 | 
000235 | \textbf{Step 2 (Minimal gluing class):} The minimal M\"obius class is modeled by the
000236 | unitary phase
000237 | \begin{equation}
000238 |     U_\Gamma(\theta)=e^{i\theta}.
000239 | \end{equation}
000240 | 
000241 | \textbf{Step 3 (Spectral flow):} Interpolate the boundary condition
000242 | $\psi(\theta+2\pi)=e^{i\varphi}\psi(\theta)$ for $\varphi\in[0,2\pi]$. The eigenvalues are
000243 | \begin{equation}
000244 |     \lambda_n(\varphi)=n+\frac{\varphi}{2\pi}, \qquad n\in\mathbb{Z},
000245 | \end{equation}
000246 | so exactly one eigenvalue crosses zero as $\varphi$ runs from $0$ to $2\pi$:
000247 | $\mathrm{SF}(U_\Gamma)=1$.
000248 | 
000249 | \textbf{Step 4 (Gluing term):} The APS gluing theorem yields
000250 | $\eta(\widetilde{M}) = \eta(\widetilde{M}_\Gamma) + 2\,\mathrm{SF}(U_\Gamma)$,
000251 | so the seam contribution is $\Delta_\Gamma=2\pi$.
000252 | 
000253 | \textbf{Step 5 (Normalization):} The two physical boundary cycles contribute $2\pi+2\pi$,
000254 | and the seam adds $2\pi$, so $R_{\text{total}}=6\pi$. The stationarity condition
000255 | $\varphi\cdot R_{\text{total}}=1$ then gives $\phitree=1/(6\pi)$.