We present AXIOM (Adaptive eXecution with Intelligent Operations Memory), a novel sparse routing architecture for cost-efficient large language model inference. AXIOM routes incoming queries across three model tiers — Surface, Reasoning, and Deep — using a 128-dimensional structural encoder and a hierarchical resolver with dynamic coalition formation and non-local graph communication, requiring no preference data, no GPU infrastructure, and no ML frameworks. Implemented in pure Rust with 1,205,376 parameters, AXIOM achieves 100% routing accuracy on simple queries, 58.0% overall accuracy across 200 benchmark queries, and 56.6% cost reduction compared to routing all queries to a frontier model, with a mean routing latency of 1,311 microseconds. The primary architectural contribution is a sparse computation graph supporting four distinct traversal directions — forward, lateral, feedback, and temporal — enabling non-local communication between routing nodes that no existing LLM router provides. We identify and characterise the structural encoder ceiling and propose attention-based extensions as future work.