An Intrinsically Discrete Continuum Model: The Formal Number System Z[μ,μ−1] and Discrete Analysis Thereon

This preprint presents an intrinsically discrete continuum model based on the formal number system Z[μ,μ−1], developing a self-consistent discrete calculus and proving the discrete Fundamental Theorem of Calculus. It establishes a standard part map to connect the discrete framework with classical continuous concepts (e.g., π) and provides a geometric realization for the discrete intuition of interval measure. The model offers a rigorous mathematical tool for discrete spacetime research in quantum gravity and addresses foundational questions of discreteness and continuity in mathematics.