Gauge-Invariant Matter on a Dynamical Graph

We formulate gauge and chiral matter on a quantum graph whose potential edges and plaquettes belong to a fixed finite ambient complex. Each edge has an inactive gauge-singlet sector and an active \(L^2(K)\) sector for a compact gauge group \(K\). Haar averaging supplies a normal conditional expectation onto the gauge-invariant local algebra and fixes a neutral rank-one creation channel. We construct nonvanishing gauge-invariant graph moves and a self-adjoint graph-changing Hamiltonian, with finite-dimensional matter and an ambient-metric Lieb--Robinson estimate. A counterexample excludes a uniform bound based solely on the pre-move active-graph distance. Integrating out matter yields a convergent heavy-fermion Wilson-walk expansion, effective-resistance edge gains, and entropy bounds for gapped finite-species ensembles. With specified ambient Wilson--Dirac data, a uniform gap gives locality, stability, and conditional index preservation under controlled graph transitions. The statements concern a finite ambient model and do not construct a continuum gauge theory on arbitrary quantum graph superpositions. Reproducibility checks and explicit assumptions accompany the proofs.