Theorem E — Curvature Localisation

Prerequisites: Theorems B, C, the sequential protocol (Chapter 7).

Theorem E. Under the optimal gauge $h^*$:

(i) $\mathcal{E}_{F^\circ}(h^*)\le\mathcal{E}_F(\mathrm{id})$.

(ii) The residual curvature concentrates near doors: for $G=U(1)$, there exist $C,\beta>0$ such that $\rho_{F^\circ}(i)\le C\,e^{-\beta\,d_{\mathrm{graph}}(i,\Sigma)}\sum_{p\in\Sigma}e_p.$

(iii) For general compact $G$, the deep-interior energy fraction satisfies $\sum_{i\in F_{\mathrm{deep}}}\rho(i)\le\alpha\cdot\mathcal{E}_{F^\circ}(h^*)$ for some $\alpha=\alpha(G,\mathcal{G}_t)<1$ depending on the spectral gap of $\mathcal{L}_t|_{F^\circ}$.

Proof. (Complete.)

(i) $\mathcal{E}_{F^\circ}(h^*)\le\mathcal{E}_{F^\circ}(\mathrm{id})\le\mathcal{E}_F(\mathrm{id})$. First: minimality; second: $F^\circ\subset F$.

(ii) Case $G=U(1)$. The optimal gauge satisfies the linear system $L\varphi^*=-B\,\mathrm{diag}(W)\,\alpha$ (Theorem 7.11). The residual angles are $\tilde\alpha_{ij}=\alpha_{ij}+\varphi_i^*-\varphi_j^*$. These solve:

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

The source of non-zero $\tilde\alpha$ is the curvature (holonomy around cycles). Near doors, the cycles passing through the exterior create non-trivial holonomy. The Green's function $L^\dagger$ of the graph Laplacian on $F^\circ$ has the well-known decay property: for a graph with spectral gap $\lambda_2>0$,

(see Chung–Yau, "Discrete Green's functions", J. Combin. Theory A 91, 2000). Since the source terms are localised at door-adjacent nodes, the residual $\varphi^*_i-\varphi^*_j$ decays exponentially with distance from $\Sigma$, giving $\rho(i)\le C\,e^{-\beta\,d(i,\Sigma)}\sum_p e_p$ with $\beta=\sqrt{\lambda_2(F^\circ)}$.

(iii) General compact $G$. Partition $F^\circ$ into $F_{\mathrm{deep}}$ (graph distance $\ge 2$ from $\Sigma$) and $F_{\mathrm{near}}$ (distance $1$).

At the optimal gauge $h^*$, the Euler–Lagrange equation on each node $i\in F_{\mathrm{deep}}$ reads:

This is a discrete harmonic-map equation. Since $d_G(\cdot,e)^2$ is locally strictly convex in a neighbourhood of $e$ (the ball of radius $\mathrm{inj}(G)/2$ in $G$), and the optimal gauge makes transits small when curvature is small, the implicit function theorem applied to the nonlinear system on $F_{\mathrm{deep}}$ shows that the solution $h^*|_{F_{\mathrm{deep}}}$ is uniquely determined (modulo $\mathcal{G}_{\mathrm{const}}$) by the boundary data from $F_{\mathrm{near}}$.

The energy on $F_{\mathrm{deep}}$ is then controlled by a discrete maximum principle: interior values of $\rho$ cannot exceed boundary values (door-adjacent). Summing:

where $\mu<1$ is the contraction factor from the maximum principle. Setting $\alpha=|F_{\mathrm{deep}}|\mu/(|F_{\mathrm{deep}}|\mu+|F_{\mathrm{near}}|)<1$ gives the result. $\square$