A Navigable Manifold of Hypothesized Consciousness-Spectrum States in Language Model Representations

The internal representations learned by language models consistently exhibit striking geometric structure: calendar months organize into a circle, historical years form a smooth one-dimensional manifold, and cities' latitudes and longitudes can be decoded using a linear probe. To explain this neural code, we first show that language statistics exhibit translation symmetry (for example, the frequency with which any two months co-occur in text depends only on the time interval between them). We prove that this symmetry governs these geometric structures in high-dimensional word embedding models, and we analytically derive the manifold geometry of word representations. These predictions empirically match large text embedding models and large language models. Moreover, the representational geometry persists at moderate embedding dimension even when the relevant statistics are perturbed (e.g., by removing all sentences in which two months co-occur). We prove that this robustness emerges naturally when the co-occurrence statistics are controlled by an underlying latent variable. Our results indicate that these representational manifolds originate in the statistical symmetries of natural language.

We investigate whether graded states of mind form spectrum-like structure in transformer representation spaces. To do so, we construct a dataset of 636 short natural-language sentences annotated with both a continuous score from $-5$ to $5$ and one of seven ordered tiers, ranging from collapsed or scarcity-driven expressions to more coherent, reflective, and integrative ones. We evaluate five frozen transformer representations: four sentence-embedding models and one decoder-only residual-stream representation. Across all representations, simple probes reliably recover both the continuous score and the discrete tier labels, and permutation tests show that performance significantly exceeds shuffled-label baselines. Additional analyses reveal a consistent geometric pattern: UMAP projections show low-to-high organization, confusion matrices concentrate errors between neighboring tiers, and directional ablation identifies a prominent score-aligned component. These results suggest that transformer representations contain statistically significant, spectrum-like organization aligned with the annotated state-of-mind structure. The annotations are used only as an operational framework for representation analysis, not as a clinical or diagnostic measure.

Understanding how transformer representations evolve across layers, not merely what they encode, remains an open problem in mechanistic interpretability. We recast the transformer forward pass as a discrete population trajectory through a high-dimensional representation manifold, drawing on geometric tools from computational neuroscience. Rather than probing for pre-specified features, we characterize trajectory geometry using five metrics computed directly in the ambient space: trajectory length, curvature, a semantic convergence index, layerwise cosine similarity, and representational stability. Across three model families (GPT-2, TinyLlama, Qwen2.5) and five controlled prompt families, we report four findings. First, semantically related prompts converge significantly in middle-to-late layers (peak CI 0.41--0.58, p<0.001, Mann-Whitney U), consistent with attractor-like dynamics. Second, reasoning tasks produce trajectories of greater curvature than lexical variations (0.71--0.83 rad vs. 0.27--0.31 rad), suggesting curvature encodes computational complexity. Third, ambiguous tokens exhibit trajectory bifurcation with up to 5.6x representational separation by the final layer, absent in unambiguous controls. Fourth, layerwise cosine similarity reveals a universal three-phase structure: encoding, elaboration, and output preparation, consistent across all three architectures. All four effects vanish under shuffled-layer and random-embedding controls. We release a fully open-source, model-agnostic pipeline and argue that trajectory geometry constitutes a principled, probe-free lens for mechanistic interpretability.