Even Adjacency in Cubic Bricks: A Proof via Vertex Deletion and Two-Edge Routing

We prove that every cubic brick satisfies Even Adjacency: for any two compatible edges there exists a conformal even cycle containing both. This resolves Open Problem 1 posed by Norine and Thomas (2007) for the class of cubic bricks.The proof combines a direct vertex-deletion argument with three structural pillars: State-Completeness (every gadget state is realized, from bicriticality), Defect-Path-Closure (gadget switches produce conformal connector cycles), and Defect-Routing (a Lock/Non-Lock dichotomy combined with the Good-Port Theorem guarantees edge-spanning defect-path-families). The Lock Case is resolved through a communication-graph contraction and block-cut-tree iteration. The Non-Lock Case uses a No-Side-Branch Lemma (tight-cut obstruction) and a parity–connectivity argument to close the critical block-co-location and single-edge-reachability steps. A fully rigorous alternative proof via Norine–Thomas induction is also provided, with the E3- and E22-lift lemmas proved in complete detail via proxy tables, state-pair classification (10 pairs in 3 types), and a cross-switch resolution theorem. All results are verified computationally on a census of 447 cubic bricks on at most 14 vertices, with zero failures.