Deterministic Drift-Free Calculus on Rational Grids Exact Newton–Leibniz Theorem on the Grid: Finite Differences and Quadratures without Arithmetic Error (Elimination of IEEE 754 Artifacts)

Deterministic Drift-Free Calculus on Rational Grids: Exact Newton–Leibniz Theorem on the Grid, Finite Differences and Quadratures without Arithmetic Error (Elimination of IEEE 754 Artifacts) We develop a deterministic formalism of differential and integral calculus on uniform rational grids of the form h = 1/B. The central idea is the explicit separation of three error components: the sampling error E_samp (discretization of a continuous object), the quantization error E_quant (projection of values onto the rational value grid M_s of scale s), and the arithmetic error E_arith (IEEE 754 rounding or ambiguous integer division semantics for negative numbers). We show that a broad class of grid operators — finite differences and quadratures — can be realized without division during computation by transferring all denominators to the scale of the result, achieving E_arith = 0 exactly. We introduce a theorem on the minimal implementation scale for general difference stencils via reduction of the denominator by gcd(m, B^k), and formalize a catalogue of boundary stencils on [0,1] for derivatives of order k ≥ 1 and k ≥ 2, with a unified common-scale rule via lcm. Bounds on E_samp are derived for central and higher-order finite difference operators as functions of solution regularity. Bounds on E_quant are derived for general stencils, revealing an amplification law of order B^k/s for derivatives of order k. The theory is completed by a rescaling-policy analysis for iterative algorithms, combining an error budget (stability of perturbation accumulation) with overflow control (growth of numerators and scale). We establish conditions for geometric stabilization of the re-quantization error in contractive time-stepping methods (ODE/PDE), including a criterion for selecting the rescaling target scale from a prescribed error budget. An exact discrete Newton–Leibniz identity is proved, showing that the Riemann sum of the forward difference recovers the boundary increment without any scale growth. The experimental part verifies bitwise determinism of the integer-arithmetic core, empirically confirms all analytical bounds across 540 derivative tests and 18 quadrature cases with zero violations, and demonstrates geometric stabilization of the rescaling error in contractive iterations versus unbounded growth in the non-contractive regime.