Continuous Mathematics as Finite Prescriptions: The Role and Limits of Calculus in a Discrete Physical Framework

This paper establishes a systematic account of how continuous mathematics functions within the Quantum-Geometry Dynamics (QGD) / Minimally Physically Derivable Theories (MPDT) framework. The central thesis is that every continuous function appearing in a QGD derivation is a finite prescription: a compact notational device generating physically meaningful outputs only at the resolution the preonic substrate permits, not an ontological claim that the real number line is physically instantiated. The paper identifies three levels at which this principle operates — ontological, computational, and notational — and derives two structural consequences. The first is the Preonic Truncation Theorem (Theorem 1): the decimal expansion of any metric quantity derived in QGD terminates after at most K = ⌊−log₁₀ (x) ⌋ significant figures, where x is the metric length of one preon (−) (the fundamental preonic distance, Pathway 2 of P14). This is an ontological consequence of Axiom 1 — the indivisibility of preons (−) — not a measurement limitation. The proof proceeds through three lemmas: integer representation (every spatial metric quantity is an integer count of preons (−) times x) ; sub-preonic indistinguishability (any two configurations differing by less than one preon (−) are physically identical) ; and finite meaningful decimal expansion (at most K decimal places carry physical content). Two corollaries follow: a precision chain bounding the precision of all derived QGD quantities by min (prec (k), prec (x), prec (c⃗), prec (m̃) ) ; and a finite upper limit for momentum integrals (pₘax = N₊ · m̃ · c⃗, finite by Axiom 2). The second consequence is the Constructive Finiteness Principle: every integral in a QGD derivation is a finite Riemann sum over a discrete set of resolvable values; every differential equation is a large-N limit of a preonic difference equation; every transcendental function is a finite series truncation at the preonic precision bound. The large-N approximation is valid and accurate at all scales where the physical domain contains many preonic units, and continuous notation remains the correct and efficient tool in that regime. The paper shows that these two principles dissolve three classical pathologies of continuum physics at their common root, requiring no new mechanism, no imposed cutoff, and no renormalisation. Ultraviolet divergences do not arise because the momentum integral terminates at pₘax (finite by Axiom 2), not at infinity. The electrostatic self-energy infinity does not arise because the radial integral has lower limit x, not zero. The cosmological constant problem does not arise because the mode sum is a finite sum over N₊ preonic degrees of freedom, not a sum over infinitely many continuous field modes. In each case the dissolution is the finite prescription interpretation applied consistently: the integral is a finite prescription, and reading it as one removes the domain extension that generates the pathology. A practitioner's protocol is provided specifying how QGD equations should be written and read: continuous notation is correct and efficient; numerical outputs are physically meaningful to at most K significant figures; and the large-N approximation is explicitly valid throughout the macroscopic and nuclear regimes. The paper positions itself precisely against Bishop constructive analysis (stronger: termination in fact at a physically determined bound, not merely constructibility in principle), Weihrauch computable analysis (orthogonal: physical content of output, not computational complexity), and effective field theory cutoffs (categorically different: x is the scale of spatial constitution, not the scale of theoretical incompleteness). The paper is the positive, practitioner-level complement to P28 (Limits of Mathematical Freedom) and P21 (Breakdown of Continuum Physics), operationalises the P12 thesis (Mathematics as Subset of Physics) at the equation level, and adds the QGD-specific precision bound absent from P27 (Physically Derivable Set Theory). It establishes the portfolio-wide reading protocol for the continuous mathematics used throughout the QGD/MPDT programme. Part of the Quantum-Geometry Dynamics / Minimally Physically Derivable Theories programme. Full programme: https: //doi. org/10. 5281/zenodo. 19584666