Geometric Attention Mechanism (GeoLLM): Production-Grade Architecture for O(N log N) Context Retrieval

The scalability of Large Language Models (LLMs) is fundamentally constrained by the O(N^2) time and memory complexity of standard Transformer attention mechanisms, which force discrete token sequences into continuous Euclidean spaces. This technical documentation introduces the Geometric Attention Mechanism (GeoLLM), a production-grade architecture built on discrete p-adic topology and the SL(3, Z) algebraic group. By replacing the dense O(N^2) attention matrix with a topological geometric inner product governed by the Tribonacci constant (approx. 1.8393), the framework natively compresses token history into a geometrically stable state. This paradigm shift yields exact context retrieval with O(N log N) time complexity and flat O(N) memory scaling, entirely eliminating the VRAM explosion associated with long-context windows. Benchmark results demonstrate an 86x inference speedup over standard attention at 8,192 tokens. This 30-page blueprint provides the complete engineering specification required for large-scale training and inference. It details the underlying mathematical proofs and explicit pseudocode for: • Hardware-level parallelization and vectorization. • Gradient stability and numerical precision protocols. • Memory-efficient Streaming and Sparse Geometric Attention. • Elimination of standard KV-cache bottlenecks. Access Note: This document contains proprietary production-level architectural designs and is published under a restricted CC BY-NC 4.0 license. Access to the full manuscript is granted upon request for purposes of enterprise integration, commercial licensing, and strategic API deployments.