Certified Recovery of Magnetic Linking Numbers from Finite Noisy Measurements: Deterministic Error Bounds and Elliptic-Loop Examples

This work develops a deterministic framework for recovering the integer linking number of two closed curves from a finite number of imperfect magnetic-field measurements. Building on the relationship between the Biot–Savart law, Ampère’s law, and the topological linking number, the paper establishes an explicit error bound accounting for magnetic measurement noise, positional uncertainty, tangent-vector errors, numerical quadrature, and arithmetic errors. The central result provides a sufficient condition under which rounding the computed circulation recovers the exact integer linking number. The framework is demonstrated using two explicitly parametrized elliptic-loop configurations. Rigorous analytical estimates and interval-arithmetic verification establish certified recovery of linking numbers −1 and −2 using 512 measurement nodes under specified error tolerances. The paper also discusses computational certification, sensitivity to geometric perturbations, and the assumptions required for applying the method to physical measurements. A supplementary document and reproducibility materials accompany the manuscript, providing detailed calculations, verification procedures, and computational checks. The contribution is a deterministic finite-measurement certification method, rather than a new topological invariant or a new derivation of Ampère’s law.