Arithmetic–Spectral Vacuum Superposition: A Prime-Indexed Framework for Emergent Quantum Structure

This preprint proposes an arithmetic-spectral vacuum framework in which quantum superposition is interpreted as a structural property of an underlying vacuum substrate. The model represents the vacuum as a discrete spectral state space whose elementary modes are indexed by prime numbers. Physical states are described as superpositions over these prime-indexed modes, while observables, effective time, geometry, entanglement, uncertainty, and measurement are interpreted as emerging from filtered and stabilized structures within this state space. The framework introduces a realizability filter, a structural capacity functional, and a coherence functional in order to distinguish formal mathematical configurations from physically persistent ones. The work is presented as a speculative mathematical framework, not as an experimentally established fundamental theory. Its purpose is to provide a structured research program for exploring whether quantum phenomena may be reformulated as emergent features of a deeper arithmetic-spectral organization of the vacuum.