Unified Eisenstein Multiplexing (UEM): A Finite-Field Arithmetic Framework for Virtualized Lane Computation

This technical note presents the Unified Eisenstein Multiplexing (UEM) framework, a mathematical architecture that unifies Eisenstein integer CRT packing, Goldberg-Coxeter spherical tilings, and signal modulation primitives (NTT, CDMA, MIMO, Turbo codes, AMC, and ECC) into a single algebraic system. The framework demonstrates that 12 physical arithmetic lanes can be virtually expanded into thousands of logical lanes through three orthogonal multiplexing dimensions: time (multiplicative orbits), code (Walsh-Hadamard bases), and space (linear transformations). A single scalar multiplication over the finite ring 𝓡 = ℤ/(𝓜) executes all operations. Number Theoretic Transforms collapse time-multiplexing penalties from linear to logarithmic complexity. Orthogonal codes break the Carmichael group-order ceiling. Linear transformations eliminate dependency on physical arrangement. Iterative feedback raises effective precision beyond bit-width limits through geometric error decay. Adaptive selection chooses mantissa width and lane count per batch based on statistical measures. Redundant residues provide single-lane error correction through syndrome decoding. Complete algebraic derivations, mode transition costs, retrieval formulas, entropy density calculations, and exchange principles with signal modulation theory are specified. The framework establishes a symmetric isomorphism between finite-field arithmetic and physical-layer signal processing.