Artificial neural networks as discrete complex systems: native vector-ternary training via the leech lattice Λ ₂₄

Modern artificial neural networks operate in vast, continuous parameter spaces that exhibit high informational redundancy. While conventional quantization methods rely on scalar reduction or apply advanced lattice geometries strictly post hoc (Post-Training Quantization), they largely neglect the topological limits of optimal information routing during the active learning phase. In this work, we conceptualize neural network training as a discrete complex system and introduce a theoretical framework for native state-space discretization. By dynamically projecting continuous parameter states onto the Leech lattice Λ ₂₄ (the densest sphere packing in 24 dimensions) directly during optimization, we constrain the network within a vector-ternary logic from initialization. We demonstrate mathematically that this native coupling effectively compresses continuous weights to a theoretical density of 0.76 bits per parameter. We ground this approach in rate-distortion theory, illustrating how the highly symmetric Voronoi cells of the Λ ₂₄ lattice act as optimal geometric filters: rejecting isotropic stochastic gradient noise while preserving coherent structural momentum. Finally, we present empirical network dynamics as a proof of concept, demonstrating stable training convergence within this rigidly constrained discrete space. Ultimately, this manuscript explores the fundamental structural principles and theoretical limits of sub-1-bit neural discretization as a proof of concept. By providing initial empirical evidence that stable syntactic convergence can emerge naturally from these rigid topological constraints, we propose a theoretical foundation for future memory-bound hardware and discrete cognitive architectures.