HH-SAE: Discovering and Steering Hierarchical Knowledge of Complex Manifolds

Rare semantic innovations in high-dimensional, mission-critical domains are often obscured by dense background contexts, a challenge we define as textitfeature density conflict. We introduce the Hybrid Hierarchical SAE (HH-SAE) to resolve this by factorizing manifolds into a nested hierarchy of Contextual (L0), Atomic (f1), and Compository (f2) tiers. Evaluating across disparate manifolds, HH-SAE demonstrates superior resolution by ``fracturing'' administrative clinical labels into physiological modes and achieving a peak cross-domain zero-shot AUC of 0.9156 in fraud detection. Path ablation confirms the architecture's structural necessity, revealing a 13.46% utility collapse when contextual subtraction is removed. Finally, knowledge-steered synthesis achieves a +9.9% AUPRC lift over state-of-the-art generators, proving that HH-SAE effectively prioritizes high-order mechanistic innovation over environmental proxies to enable high-precision discovery in high-stakes environments.