Algebraic Structures as the Foundation of Quantum Theories: Generalizing the Number System Operator (NSO) Paradigm

We propose a unified framework where algebraic structures, rather than geometry, serve as the foundation of quantum theory. We derive the Hilbert space, canonical operators, and the uncertainty principle from an underlying algebra. The Number System Operator (NSO) model emerges as a concrete realization, linking precision (k), numerical base (B), and measurable quantities. We prove the non-commutativity of NSO projectors, introducing a “digital noise” as a new, testable form of uncertainty. We analyze its scale relative to Planck units and demonstrate Lorentz covariance of the NSO procedure, ensuring compatibility with special relativity. A roadmap for experimental verification is provided, suggesting high-precision spectroscopy, atomic clocks, and gravitational-wave interferometers as potential tests. This framework transforms conceptual ideas into a computable and physically relevant theory, bridging foundational physics, computation, and measurable predictions.