LLMs are thought to track "belief states," i.e., running probability distributions over the latent variables that govern language (Shai et al., 2024; Sarfati et al., 2026), but so far this has only been comprehensively demonstrated on toy synthetic data and in a few isolated case studies. It has also never been empirically connected to the geometry of LLM features (the concepts interpretability finds in model activations). In this work, we plant a controllable latent variable inside natural-looking text. An LLM teacher writes ordinary text while we "subliminally" steer it along one of K = 8 unrelated sparse autoencoder directions at each token, with the active directions following a ring-shaped Markov chain. A small transformer model trained on this corpus does indeed track the Bayesian posterior belief about our planted latent variable. Moreover, it also arranges the 8 states themselves on a ring, in the exact order of the Markov chain, which is supporting evidence that a concept's geometry can be formed by the statistical dynamics of the latent variable behind it.
A transformer's answer lives on one axis: the direction its unembedding reads. Its intermediate states largely do not, and that off-axis position is usually treated as an obstacle to interpretation. We show it is functional. A 12-layer model computes in two phases. Through the first, every sublayer writes into a subspace held near-orthogonal to the read-out, attention 75 to 96 degrees off it at every depth. Moving attention's values onto the read-out is 64 to 84 times more damaging than a matched random rotation, and the damage is entirely in cross-token mixing: the subspace insulates composition from the vocabulary. Beneath it the frame itself turns rigidly with depth. In the second phase the answer arrives on-axis, late, and by addition rather than by turning accumulated content onto the read-out. Pressing every layer onto the read-out instead, as training for early exit does, matches the baseline on perplexity, LAMBADA and BLiMP while cutting the concept-phase workspace from about twenty-five effective dimensions to fourteen, a change none of those benchmarks register. The geometry can also be imposed, though not by asking for it. Prescribing it through the loss is a lottery: six of eight seeds collapse, because a model told to null its read-out projection obeys most cheaply by discarding dimensions. Inserting one fixed rotation at the phase boundary lands it instead, at baseline quality. A sparse rotation the surrounding weights can absorb converges on all nine seeds, against five of nine for ordinary training. Which rotation is immaterial: twenty-five runs across thirteen distinct ones reach the same quality, and two baselines from different seeds hold their concepts in near-orthogonal frames while agreeing on their read-outs. That freedom is usable: a basis drawn at random and prescribed before training is adopted across the concept phase, with quality unchanged.
Large language models place structured concepts on geometrically faithful manifolds: weekdays lie on a circle, months on another, usually taken to be a fixed world-model the network stores and looks up. We show that context is king: the structure a model actually uses is set by the in-context specification. A declarative rule fixes not only which relations the geometry encodes but its topology type: the same tokens form a cycle or a branching tree on command, built even on arbitrary, meaning-free tokens with no prior to inherit, which a relabeled stored shape cannot do. When the specification conflicts with a strong pretrained prior, the context-set geometry dominates it in capable models, read from the same activations (representational similarity 0.6--0.9 to the imposed structure versus near-zero to the prior), across the priors we test and both families we study (Gemma, Qwen). Activation patching shows the map is causally used, not a probe correlate: swapping one entity's activation for another's makes the model answer with the other entity's successor under the imposed order. A rough map forms readily, present even in small and base models; what scale gates is using it cleanly: clean dominance and the causal crossover emerge only in the larger models (up to Gemma-31B and Qwen-27B) and weaken or reverse below, so a mechanism present in a large model can be absent in a smaller one of the same family. Whether the model builds this geometry anew or reconfigures a stored one we leave open; operationally, the geometry it uses is the one the context specifies.
Existing hypotheses represent a concept in an LLM as a single point, a linear direction, or a Gaussian cluster, yet it remains unclear how and why such structures emerge. Here, we show that concept geometry can be precisely characterized via Laguerre Geometry, in which a concept is defined as a region--a Laguerre-Voronoi cell or a union of cells--allowing us to strictly define, measure, and separate concepts. Building on this formulation, we show that finer-grained concept structures, such as inclusion and hierarchy, are naturally revealed by the Laguerre weights. We then push this geometry inside the transformer. Decomposing each layer into piecewise-linear operators, we show that a token's hidden trajectory is governed by two coupled mechanisms: a static tree of self-contained piecewise-linear flow, and a dynamic transport that hops the trajectory across trees when cross-token attention fires. This decomposition yields Geometric Lens, a training-free, hyperparameter-free method for reading out the exact concept a hidden vector encodes at any layer. We also develop Laguerre Autoencoder, a 2D visualizer that renders both the decision geometry and a model's full reasoning trajectory in one view. Finally, we move beyond explanatory geometry toward actionable interpretability, showing that Geometric Lens recovers the correct factual token when a model is prompted with in-context interference. The code is available on GitHub.