A Discrete Symmetry Model with Observable Signatures

We propose a quantum framework for the early universe in which spacetime is composed of discrete qubits at the Planck scale. The state space is taken to be the 64-dimensional Hilbert space spanned by the elements of the group (Z2)6(Z2)6, interpreted as the fundamental degrees of freedom of spacetime. The Hamiltonian is constructed from the discrete translation generators of this group, and its eigenvalue spectrum is solved exactly, yielding a band structure E(k)=−J∑acos⁡(ka)E(k)=−J∑acos(ka). The coupling constants JJ and λλ are fixed by requiring consistency with a companion geometric projection model, resulting in a parameter-free theory. The continuum limit reproduces the standard Schrödinger equation, while the discrete spectrum leaves observable imprints. We present three testable predictions: (i) a comb-like gravitational wave background in the 0.01-0.1 Hz band accessible to LISA, (ii) a parity-odd signature in CMB B-mode polarization, and (iii) a dark matter candidate of mass ∼108∼108 GeV with only gravitational interactions. The model is falsifiable and provides a concrete microscopic foundation for quantum cosmology.