The Arithmetic Jump: A Branch-Free Index Inversion for 3D Arrays

This work presents a compact arithmetic formulation for inverting row-major linear indices into three-dimensional coordinates. The formulation defines a bijective and reversible mapping based solely on integer division and modulo operations and avoids iteration and control-flow constructs. A traversal-based reconstruction strategy and the arithmetic formulation are evaluated on Graphics Processing Unit (GPU) hardware across multiple volumetric configurations. The experimental results show that arithmetic index decomposition yields uniform execution behavior, low run-to-run timing variability, and constant per-thread execution cost under massively parallel execution. The observed differences follow from GPU architectural characteristics, particularly sensitivity to control-flow divergence. The formulation provides a portable reference model for multidimensional index inversion suitable for parallel kernels and hardware-oriented implementations.