Spectral Arithmetic over Orthomodular Lattices

Spectral families provide a representation of observables in quantum logic. This paper develops an addition operation for bounded spectral quantities over orthomodular lattices to examine the relation between familiar arithmetic behavior and lattice structure. The operation extends the established addition of bounded observables. Under this operation, bounded spectral quantities form an abelian group precisely when the lattice is Boolean. Conjugating the operation by spectral negation yields a dual sum, and equality of the primal and dual sums characterizes modularity. A modular non-Boolean example separates these two regimes concretely.