Theorem E — Curvature Localisation

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

Theorem E. Under the optimal gauge $h^{\ast }$:

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

(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 ${\sum }_{i\in F_{\mathrm{deep}}}\rho (i)\le \alpha \cdot {\mathcal{E}}_{F^{\circ }}(h^{\ast })$ for some $\alpha =\alpha (G,{\mathcal{G}}_t)<1$ depending on the spectral gap of ${\mathcal{L}}_t{|}_{F^{\circ }}$.

Proof. (Complete.)

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

(ii) Case $G=U(1)$. The optimal gauge satisfies the linear system $L{\phi }^{\ast }=-B\mathrm{diag}(W)\alpha$ (Theorem 7.11). The residual angles are ${\tilde{\alpha }}_{ij}={\alpha }_{ij}+{\phi }_i^{\ast }-{\phi }_j^{\ast }$. 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 $L^{†}$ 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 ${\phi }_i^{\ast }-{\phi }_j^{\ast }$ 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 $h^{\ast }$, 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 $h^{\ast }{|}_{F_{\mathrm{deep}}}$ 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. $\square$