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 $\mathit{\text{feature}}\mathit{\text{density}}\mathit{\text{conflict}}$. We introduce the $\mathbf{\text{Hybrid}}\mathbf{\text{Hierarchical}}\mathbf{\text{SAE}}\mathbf{\text{(HH-SAE)}}$ to resolve this by factorizing manifolds into a nested hierarchy of $\mathbf{\text{Contextual}}$ ($L_0$), $\mathbf{\text{Atomic}}$ ($f_1$), and $\mathbf{\text{Compository}}$ ($f_2$) tiers. Evaluating across disparate manifolds, HH-SAE demonstrates superior resolution by $\mathbf{\text{“fracturing”}}\mathbf{\text{administrative}}\mathbf{\text{clinical}}\mathbf{\text{labels}}\mathbf{\text{into}}\mathbf{\text{physiological}}\mathbf{\text{modes}}$ and achieving a peak $\mathbf{\text{cross-domain}}\mathbf{\text{zero-shot}}\mathbf{\text{AUC}}\mathbf{\text{of}}\mathbf{\text{0.9156}}\mathbf{\text{in}}\mathbf{\text{fraud}}\mathbf{\text{detection}}$. 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.