Abstract This paper resolves the P vs NP problem through a geometric reinterpretation of computational complexity. The central instrument is a custom three-colour aperiodic tiling — the Borg Tiling — constructed on a manifold of hexagonal gear-and-rotor cells. The three colours correspond to three rotational phases in $\mathbb{Z}/3\mathbb{Z}$. The green phase $\phi = 0$ is the *null state*: the identity element, carrying no problem content, imposing no constraint, propagating freely across all available space. The teal and olive phases are the *problem states*: non-null rotational configurations encoding computational constraints. The null state does not identify problems. It identifies the absence of problems. When the null state has propagated across all available space on the plane, what remains — the teal and olive configurations it could not reach — are the problems. Each such configuration is an *isolation*: a valid, internally consistent problem state enclosed entirely by the null state. The enclosure is the certificate. When the boundary of an isolation is entirely green, the problem is sealed and identified. The framework operates through two complementary coupling laws. The *Summation Law* governs null-state propagation. The *Mismatch Law* governs problem encoding. The two laws are two computational pathways, not two inconsistent definitions. The NP-complete problem that maps directly and naturally onto this framework is graph 3-colouring. A graph is the tiling. Vertices are cells. Edges are adjacencies. A valid 3-colouring is a globally consistent phase assignment under the Mismatch Law. A non-3-colourable region is an isolation, sealed by the null state in polynomial time. Multiple isolated problems on the same plane are independent and potentially relatable via functions defined on the null-state medium between them. P = NP because every NP instance, encoded as a graph 3-colouring problem in the Borg Tiling, is either directly resolvable by null-state propagation or identifiable as an isolation by the closure of its green boundary — both in polynomial time.