A Unified Theory of Dense Number Representation via CRT: From Gaussian to Eisenstein Integers

Abstract

The theoretical superiority of base-3 arithmetic over binary has been recognised since Shannon's foundational work on radix economy, yet ternary logic remains absent from commercial hardware due to fundamental semiconductor constraints. This document presents a comprehensive framework that circumvents this physical barrier by manipulating the discrete state space of binary memory. Two complete architectural frameworks are developed: Gaussian CRT (ℤ[i]), a square-lattice system packing four independent complex numbers into a 64-bit register using the K-trick K² ≡ -1 (mod M), achieving 16–32 real operations per cycle with <0.01% waste; and Eisenstein CRT (ℤ[ω]), a hexagonal-lattice system packing six independent Eisenstein integers using K² + K + 1 ≡ 0 (mod M), achieving 24–48 real operations per cycle with <0.001% waste. The Eisenstein CRT is identified as the practical sweet spot due to hexagonal lattice packing (~15% denser than square), six SIMD lanes (50% more parallelism than Gaussian), two-thirds of all primes splitting, and native alignment with base-3 symmetry. Advanced dynamic enhancements leverage the Euclidean toolkit: GCD-based exact overflow detection, Bezout-based zero-cost ECC, and homomorphic blinding via Lane 6 for DPA resistance. Extreme optimisations—erasure coding, wavefront routing (+30% clock frequency), micro-threaded interleaving (0.000% extraction overhead), stochastic Hamming-weight prediction, and an algebra-aware LLVM compiler pass—reduce total system overhead below 0.005%.