Flash-Jacobian: Regime Maps of Depth-Dependent Geometry, Semantic Entanglement, and Bounded Factuality Access in LLMs

A strong global form of the Linear Representation Hypothesis assumes that high-level concepts are encoded as linear directions in large language model (LLM) representations. We introduce Flash-Jacobian, a scalable GPU system for extracting cluster-representative Jacobians (CRJs), and use it to map where linear access works in Qwen-3.5-4B, Llama-3.2-3B, and Phi-3-mini. First, CRJ fidelity follows a robust inverted-U depth profile: a branch-selective, family-dependent middle-layer corridor is followed by a late basis-sensitive collapse band. Second, descriptively clean clusters do not yield clean global semantic control: principal-direction interventions are dominated by boundary suppression, while local conditional policies provide only weak trust-region-local gains. Third, factuality is bounded by access family: tested raw-state linear probes are near chance, selected-layer probes recover benchmark-sensitive signal, and the strongest detectors are probe-rich composite stacks rather than score-free geometry. In the full pooled raw-coordinate detector family, leave-model-out transfer collapses; under stricter metadata-free held-out-model audits, selected probe-rich subfamilies still transfer on held-out Qwen and Phi-3 and remain above chance on held-out Llama, while geometry-only features stay near chance. Flash-Jacobian is therefore best read as a scalable regime diagnostic and trust-region guide, not as a universal concept or truth readout.