New Quantum Sectors from Relational Spectral Geometry Revised Manuscript Incorporating Referee Comments

We develop a rigorous spectral framework for twisted connection (magnetic) Laplacians on arbitrary finite graphs, showing that Hilbert spaces derived from their spectra naturally exhibit holonomy–twisted sector structures. This provides a unified mathematical description encompassing Aharonov–Bohm effects and Bloch–Floquet theory on general graph topologies, including irregular and disordered graphs without translational symmetry. We prove gauge covariance, spectral continuity with respect to twist parameters, and a necessary and sufficient holonomy criterion for the existence of zero modes. A central quan- titative result is a spectral band theorem showing that eigenvalues shift by at most 2W(θ) under holonomy insertions, yielding explicit stability bounds and quantitative zero–mode ob- struction criteria. We further show that on the 2–core of supercritical Erdős–Rényi random graphs the spectral gap persists with high probability under sufficiently small twists, and we de- rive high–probability spectral stability bounds for random holonomy perturbations using matrix concentration inequalities. We establish a general non–Abelian zero–mode theorem for arbitrary unitary representations, identifying the kernel of the connection Laplacian with the global fixed subspace of the holonomy representation; the SU(2) fundamental case appears as a natural corollary with dim ker = 2. Motivated by these results, we formulate an open problem on the holonomy–induced stability threshold governing when random twists preserve or destabilize the spectral gap on random graph cores. On the interpretive level, we derive coarse–graining conditions under which smooth spacetime geometry and U(1) gauge structure can emerge from discrete transport dynamics. In this picture, spatial variation of twist acts as an effective transport–frustration density that can be interpreted as a macroscopic source term for emergent curvature. The results extend Aharonov–Bohm and Bloch–Floquet phenomena from periodic lattices to arbitrary, including random, graph topologies and provide testable predictions for analog quantum simulators, including cold atoms, photonic lattices, and superconducting circuits. In particular, the spectral band theorem yields quantitative thresholds for experimentally resolving holonomy–induced sector splitting. This clarifies the precise sense in which quantum kinematics can be formulated independently of background spacetime and suggests a spectral–geometric route toward emergent gravitational dynamics. Connections to graph machine learning and geometric AI are outlined in Appendix B.