The ARC engine serves as Auxy's core diagnostic framework, systematically probing AI models across Association, Relevance, and Citation dimensions to evaluate how brands exist in model weights and live search results. It evaluates bidirectional entity-brand relationships through prompt probing (Entity-to-Brand, Brand-to-Entity, and Query-to-Brand) to determine memory strength, while running binary yes/no relevance tests across multiple sample passes to quantify query-level domain authority. Furthermore, the engine tracks live web-grounded citations from Gemini, OpenAI, and Anthropic, extracting sources, query fan-outs, and sentiment while applying a matryoshka hierarchy algorithm to resolve sub-brands and attribute competitive visibility.