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

This paper introduces a deterministic, integer-only coordinate framework for spatial enumeration on discrete lattices (Z² and Z³). 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 (r³) to O (r²), 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.