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

This version extends the previously published record with two new results; the original thesis and structure are unchanged. The paper establishes that every continuous function appearing in a QGD derivation — integrals, derivatives, transcendental functions — 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. It derives the Preonic Truncation Theorem (the decimal expansion of any metric quantity terminates at the preonic distance scale, an ontological consequence of Axiom 1, not a measurement limitation) and the Constructive Finiteness Principle (every integral is a finite Riemann sum, every derivative a finite difference, every transcendental function a finite series truncation), and shows both principles jointly dissolve three classical continuum pathologies — ultraviolet divergences, the self-energy infinity, and the cosmological constant problem — without new mechanisms, cutoffs, or renormalisation. New in this version: two results extending the finite-prescription programme down to the base level beneath continuous notation — the integers themselves. A Physical Cost Invariance theorem establishes that the number of preonic binding events required to construct any positive integer v is exactly v−1, invariant under every construction path, derived from the indivisibility and non-duplicability of preons⁺ (a tree connecting v nodes always has v−1 edges, regardless of shape). A Minimum Momentum Cost theorem sharpens this: applying the Law of Discrete Momentum Change to each binding event shows the total physical momentum cost of a construction is bounded below by (v−1)·c̃, an inequality tight only for sequential (chain) construction — so while every construction path takes the same number of steps, they do not take the same physical momentum, and sequential construction is uniquely cheapest. Both results are proved in full and checked against worked examples at multiple scales. Companion to: P12 (Mathematics as Subset of Physics), P16 (The Physicality of Logic), P21 (The Breakdown of Continuum Physics), P27 (Physically Derivable Set Theory), P28 (The Limits of Mathematical Freedom).