Theorem: Exact engineered membrane–FPP reduction

The functional $\mathcal{M}$ of Definition def:membrane is undirected square-lattice FPP with i.i.d.\ $\mathrm{Exp}(\lambda )$ weights. Its macroscopic norm is $\mu$. Conditional on hyp:exp, its limiting cost ball is strictly convex with $C^1$ boundary.

For a finite simply connected planar domain and two dual boundary points separating specified boundary arcs, the minimum of the compatible vertex-cut entropies equals the corresponding restricted dual shortest-path cost.

Proof. Every dual edge is in bijection with a primal edge. The local entropy identity in Theorem thm:quench preserves the sampled independent exponential costs exactly. The admissible dual paths are undirected nearest-neighbour paths. This proves equality of the optimization problems and hence of their macroscopic norms.

In the finite domain, a compatible cut has an interface joining the specified boundary points, possibly with extra closed interface components. Positivity permits deletion of the latter. Every simple compatible dual joining path separates the prescribed arcs and determines a vertex cut. Theorem thm:quench identifies its entropy with the path cost. Minimizing on both sides gives the finite-domain statement. Boundary edges must use the specified domain convention; no full-plane conclusion is inferred from an unspecified boundary graph.

The last regularity statement is Corollary cor:regularity.