The Concept Allocation Zone: Tracking How Concepts Form Across Transformer Depth

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.

Concept probes extracted from transformer residual streams are only as reliable as the layer from which they are extracted. The common practice of probing at a fixed late layer or at the peak of a separation score function ignores a fundamental structural feature: concept representations undergo substantial directional rotation during their assembly phase, and do not settle into a stable direction until a characteristic handoff layer after the primary Concept Allocation Zone (CAZ). We introduce Geometric Evolution Maps (GEMs), which track the full directional trajectory of a concept through residual stream activations, identify the handoff layer where rotation ceases, and extract the settled probe direction from that layer. Across 23 architectures spanning 70M to 14B parameters and 17 concept types, the entry-to-exit cosine similarity within CAZs has a mean of 0.233, showing that probe direction at CAZ entry does not reliably predict probe direction at exit. Ablation experiments across 391 concept x model pairs (23 models x 17 concepts) show that GEM-extracted probes are at least as precise as peak-layer probes in 268/391 trials (68.5%), and strictly outperform in 259/391 (66.2%). The architecture split is pronounced: MHA models favour the handoff in 173/221 trials (78.3%); GQA models favour the handoff in only 56/119 trials (47.1%). Model-level Wilcoxon: W=214, N=23, p=0.010 (one-sided). An adaptive ablation width rule targets the 79/391 near-final-layer cases: it improves probe quality in 60/79 triggered cases (75.9%), mean gain +7.44pp. A direction-specificity control confirms the ablation effect is concept-direction specific: median 377x suppression rate versus random-direction ablation (99.1% of concept directions beat all 10 random seeds). Reference implementation: rosetta_tools v1.3.1 (doi:10.5281/zenodo.20361433).

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.