Framed Quantum Graphity: Exact Ground States and Locality Obstructions

We construct a finite permutation-invariant quantum graph model with an exactly classifiable geometric ground sector. Colored incoming and outgoing pointers, commuting-permutation constraints, and adjacency matching select framed Cayley quotients \(Cay( Z^d/Λ,\{± e_a\})\) of full-rank sublattices \(Λ≤ Z^d\), when connectedness is imposed by a global projector or a spanning-tree certificate. The quotient systole controls the radius of exact Euclidean colored balls. We derive sector counts, primitive-translation order laws, character spectra, and growing-systole and continuum-torus limits. We also prove limitations on local diagonal energies: bounded-range neighborhood penalties cannot detect all global quotient sectors, and bounded connectedness penalties compete unfavorably with disconnected-sector entropy. Exact permutation-orbit degeneracies obstruct a unique labeled ground frame without relational selection or symmetry breaking. A phase-register model and a coherent move term provide explicit finite benchmarks. The construction establishes finite emergent-locality mechanisms and obstructions, not Lorentzian Einstein dynamics or a universal continuum limit.