Theorem E — Curvature Localisation

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1. #1

   (ii) Case . The optimal gauge satisfies the linear system (Theorem 7.11). The residual angles are . These solve:

   $$\tilde{\alpha }=(I-B^T{L}^{†}B\mathrm{diag}(W))\alpha =:\Pi \alpha .$$
2. #2

   The operator is the projection onto the cycle space of the graph. The residual curvature at node is:

   $$\rho (i)=\sum _{j\sim i}{W}_t(i,j){\tilde{\alpha }}_{ij}^2.$$
3. #3

   The source of non-zero is the curvature (holonomy around cycles). Near doors, the cycles passing through the exterior create non-trivial holonomy. The Green's function of the graph Laplacian on has the well-known decay property: for a graph with spectral gap ,

   $$|L^{†}(i,j)|\le \frac{C_0}{\mathrm{vol}(F^{\circ })}\cdot e^{-\sqrt{{\lambda }_2}{d}_{\mathrm{graph}}(i,j)}$$
4. #4

   At the optimal gauge , the Euler–Lagrange equation on each node reads:

   $$\sum _{j\sim i}{W}_t(i,j){\nabla }_{h(i)}{d}_G(g_t^h(i,j),e{)}^2=0.$$
5. #5

   The energy on is then controlled by a discrete maximum principle: interior values of cannot exceed boundary values (door-adjacent). Summing:

   $$\sum _{i\in F_{\mathrm{deep}}}\rho (i)\le \frac{|F_{\mathrm{deep}}|}{|F_{\mathrm{near}}|}\cdot \mu \cdot \sum _{i\in F_{\mathrm{near}}}\rho (i)$$