Spontaneous Crystallisation of Q3 Octahedra from Unstructured Qubit Networks under Simulated Quantum Annealing

We demonstrate via simulated annealing that an unstructured network of N qubits, subject only to a degree-3 regularity constraint and spectral energy minimisation, sponta- neously partitions into ⌊N/8⌋copies of the Q3 hypercube graph — the unique 3-regular, vertex-transitive graph on 8 vertices supporting a distance-4 error-correcting code realis- able as the face-adjacency graph of a regular octahedron in three dimensions. Over 100 independent trials from random initial conditions with N = 24, perfect Q3 crystallisation occurs in 94% of runs. We prove that Q3 is the unique optimal target by ruling out all competing graphs on independent geometric and coding-theoretic grounds: the Petersen graph fails both the convex polyhedral embedding test and the 4-cycle (distance-4 parity check) requirement. Frustrated configurations (N ̸≡0 (mod 8)) produce high-energy partial clusters that cannot close their parity-check circuits, providing a discrete model of quantum vacuum fluctuations. When inter-cluster bonding is permitted, the isolated octahedra spon- taneously form bridge connections, assembling into a connected lattice network. The code and all simulation data are publicly available for independent reproduction.