An Integer-Only Orthotropic Lattice Enumeration Framework and Asymptotic Convergence of Discrete Rational π

Abstract

This paper introduces a deterministic, integer-only coordinate framework for spatial enumeration on discrete lattices (Z^2 and Z^3). By resolving origin-coordinate parity ambiguities, we derive an orthotropic boundary constraint governed by major and minor axes 4r ± 1. This construction yields a scale-dependent rational fraction for discrete π that eliminates floating-point drift and converges asymptotically to transcendental π via a damped harmonic oscillation. Using an O(r^{d-1}) row-collapse algorithm via integer square roots, the framework computes exact lattice point occupancy without evaluating volumetric interiors. Empirical hardware benchmarks executed via pure integer-ALU configurations confirm that by collapsing volumetric evaluation from O(r^3) to O(r^2), the relative computational speedup scales linearly as O(r). This dynamic efficiency divergence yields a 75x reduction in execution time at r=400, accelerating to a 360x reduction when evaluating a 268-billion point discrete domain in 3 dimensions (r = 2000). For high-resolution industrial and commercial applications, this O(r) scaling trend translates to massive, compounding computational savings as domain bounds increase.