An Arithmetic Spectral Model of Dark Matter from Discrete Spacetime Dynamics

This work proposes a conceptual framework in which dark matter is not modeled as a new particlespecies, but rather as an emergent gravitational phenomenon associated with the discrete, arithmetic structure ofspacetime at the Planck scale. The central hypothesis is that dark matter couples preferentially to the spectral andarithmetic degrees of freedom of a discretized spacetime geometry, remaining invisible to local field observablesbut manifesting through global gravitational effects. It formalizes this idea by describing dark matter as amathematical motive realized only through spectral and arithmetic cohomology, whose dynamics are encoded ina Hamiltonian composed of three key components: (i) a driven term generating a discrete time crystal, (ii) anarithmetic interaction defined over finite fields and isocrystal structures, and (iii) a unitary symmetry termimplementing Galois automorphisms on the quantum state space. It is shown that the temporal correlationfunctions of this system naturally generate structures mimicking motivic L-functions, suggesting observableconsequences in long-range cosmological correlations.