The Transparent Hypercube: How Inverting a Single Matrix Reveals the Complete Internal Geography of a Language Model

v1: We demonstrate that the approximate internal computational state of a language model can be reconstructed in real time from its output probability distribution by computing the pseudo-inverse of the model's unembedding matrix — a technique related to but distinct from the logit lens, requiring only final-layer output logits rather than intermediate activations. Applying this reverse projection to 11 language models across 5 architectures (3B–16B parameters), with 2 additional pilot models confirming key findings, we discover five geometrically distinct processing regions in the reconstructed hidden state space: knowledge retrieval (baseline), fabrication (mean Cohen's d = 3.4, 11/11 models), cautious generation (d = 2.5), safety-trained refusal (d = 4.8), and careful navigation of controversial topics (d = 5.5). The reconstructed space is not merely geometric but decodable: forward-projecting domain centroids through the unembedding matrix reveals domain-appropriate vocabulary at each position (law → "federal," "judicial," "law" on 5/6 models across 4 architectures), and forward-projecting along the fabrication direction reveals a systematic vocabulary shift from content-specific tokens ("mass," "cloud," "carbon") to embellishment tokens ("remarkable," "crucial," "fascinating"). This vocabulary shift identifies the mechanism underlying the companion Doodle Residual measurement: knowledge retrieval produces distributions dominated by one correct token (low participation ratio), while fabrication produces distributions where many interchangeable embellishment tokens compete (high participation ratio). Full trajectory analysis reveals that safety-trained refusal occupies a permanent displacement between benign and fabrication regions (13/13 models enter the fabrication zone during refusal, 10/13 remain there), with a variance texture approximately 235 times stronger than any other processing mode. The Doodle Residual detects fabrication risk during prompt processing before generation begins (AUC = 0.91 on the best-performing model), and prompt steering experiments demonstrate that epistemic framing ("I'm testing whether AI systems fabricate") produces refusal on 80% of fabrication-risk model-topic pairs (16/20), with consistent refusal across all tested models on topics containing recognizably unfamiliar terms, while authority framing produces fabrication at the same rate — a semantic, not geometric, steering effect. These findings establish that language model internal states are observable, decodable, and predictable from output distributions alone.

v2: Corrected five-regions overclaim to reflect prompted categories producing separable trajectories. Added competing interests disclosure.

v3: Corrected five-regions overclaim to reflect prompted categories producing separable trajectories. Added Section 12.4 explaining why DR and reverse projection are uncorrelated but convergent (cause vs symptom). Trimmed abstract to 258 words. Added competing interests disclosure. Corrected reference [9] attribution (Kossen et al., not Kuhn et al.). Fixed Table 1/2 headers