Off-Axis, On Purpose: Where a Transformer Computes Concepts and Why it Does So

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 between 75 and 96° off it at every depth. Moving attention's values onto the read-out is 64–84× more damaging than a matched random rotation, and the damage runs entirely through cross-token mixing: the off-axis subspace insulates composition from the vocabulary. Beneath that phase the frame is itself turning: one rotation, fitted to the token stream and never to the concepts, carries the whole concept constellation from layer to layer at R2 = 0.93, and the turn is the antisymmetric part of the layer's linear response. In the second phase the answer arrives on-axis, late—first becoming the top decoded token at layer 9 of 12 on average—and by addition rather than by turning accumulated content onto the read-out.

Pressing every layer onto the read-out instead, the pressure the early-exit line applies, 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 training seeds collapse, because a model told to null its read-out projection obeys most cheaply by discarding dimensions. Inserting one fixed rotation at the boundary between the two phases lands it instead, seven of eight seeds at baseline quality with twice the baseline's effective dimensionality. A sparse rotation, one the surrounding weights can absorb without changing the function, converges on all nine seeds against five of nine for ordinary unpressured training, and does so with half the dense device's effective dimensionality, close to the baseline's own. 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 downstream quality unchanged.

The paper addresses two questions the lens literature does not answer: Why is the working content off the vocabulary axis? And how does the answer come to be on it? The usual response treats the off-axis condition as an obstacle or defect to be corrected. Work on the depth alignment sweep frames it as a structural consequence of residual connections and proposes to mitigate it; the early-exit line applies the opposite pressure directly, training intermediate layers to be decodable through a shared read-out. Both take for granted that content nearer the read-out is content better placed.

This paper argues the opposite. The off-axis position is doing a job, and the job is insulation. Attention composes values across tokens by averaging them. If those values carried vocabulary predictions, averaging them would blur the predictions together, and the blur would land on the axis the model is trying to write its answer to. Holding the values orthogonal to the read-out puts the mixing somewhere the unembedding discards, so composition costs the forming prediction nothing. We measure the cost of removing that insulation: rotating attention's values onto the read-out is 64–84× more damaging than a matched random rotation of the same magnitude, and the damage is entirely in the cross-token path.

That insulated phase ends, and how it ends is the second thing we measure. Asked whether the answer is turned out of the accumulated off-axis content or written fresh, the geometry answers plainly. Across the commit span a best-fit rotation captures only R2 rot = 0.38 of the change, 64% of the committed state is content orthogonal to any rotation of what came before, and the residual norm grows by 1.74×, which a rotation cannot do. The answer is added rather than turned. It also arrives late: the final prediction first becomes and remains the top decoded token at layer 9.2 on average, and at 39% of positions only at the last layer. A 12-layer model therefore computes in two phases, an off-axis one that composes and an on-axis one that writes, with a boundary between them that is a real location in the network.

If the geometry is functional, the next question is whether it can be imposed rather than observed. It can, but only in a particular way. Asking for the geometry through the loss, by driving the residual to exactly 90° off the read-out through the first half of the network and onto it through the second, is a coin flip: six of eight seeds collapse, because a model told to null its read-out projection can obey most cheaply by throwing away dimensions. Handing the model a frame instead of demanding one removes the strain. A single fixed rotation inserted into the residual stream at the phase boundary takes convergence from two of eight to seven of eight, at baseline quality, with a concept-phase workspace about twice the baseline's effective dimensionality. The model was never unable to compute off-axis; it was unable to build the frame and hold the pressure at once.

What the frame has to be turns out to be almost nothing. A sparse rotation that the network's normalization can absorb reaches the same quality with a workspace half the size of the dense one, so the dimensionality of the workspace is not what carries the result. Nor does the particular choice matter: twenty-five runs across thirteen distinct rotations train to the same model, and two baselines trained from different seeds, aligned by their read-outs, hold their concepts in frames about 90° apart, indistinguishable from unrelated. The 90° frame is a free gauge, fixed neither by the task nor by the device.

A free gauge is also a gauge that can be prescribed. Prescribing a randomly drawn basis in advance and asking each concept-phase layer to hold its second moment diagonal in it produces models that adopt the frame we chose, uniformly across the concept phase and robustly to deflating the largest directions, with downstream quality unchanged. The off-axis phase is therefore not only a place where things happen to end up. It is a place we can put them.

The paper characterizes the two-phase geometry of a 12-layer transformer and gives a functional account of it: attention writes 75–96° off the read-out at every depth, that position insulates cross-token composition at a measured 64–84×, the commit is additive rather than rotational, and the frame the concept phase computes in turns rigidly with depth for reasons traced to the layer's linear response. It quantifies the cost of removing the off-axis position: forcing every layer on-axis reaches parity by collapsing the workspace, from about twenty-five effective dimensions to fourteen, and does so invisibly, since perplexity, LAMBADA and BLiMP all stay inside the baseline range. It shows that prescribing the geometry through the loss alone is unreliable, that supplying a frame makes it routine, and that a cheap absorbable frame does the same job while converging on every seed we ran, against five of nine for ordinary unpressured training. Finally it shows the frame is both immaterial to the model and ours to prescribe.

Every model is a decoder-only transformer of the GPT-2 small class: 12 layers, d model = 768, 12 attention heads of width 64, a feed-forward inner width of 3072, GELU activations, pre-layer-normalization blocks, learned positional embeddings, no biases, and input and output embeddings tied. The vocabulary is the GPT-2 byte-pair vocabulary at 50,257 tokens and the context length is 2048. This gives 125,143,296 parameters, of which 40,170,240 are embedding (token and positional) and 84,934,656 are the attention and feed-forward blocks.

Training is on OpenWebText, approximately 8.9 billion tokens, for one epoch, with a held-out validation split of 120 million tokens. Optimization is AdamW at β = (0.9, 0.95), weight decay 0.1, peak learning rate 3 × 10−4 with 2000 warmup steps and a cosine decay to a tenth of peak, batch size 16 at the full 2048-token context, gradient clipping, and bf16 mixed precision. One epoch is 272,687 steps. Runs execute on a single H200 and differ only in the training seed unless stated otherwise.

A substantial fraction of from-scratch runs fail, and that attrition is characteristic of the setting rather than of our recipe. Large-model training logs report that from-scratch optimization diverges as a matter of course: the PaLM loss spiked roughly twenty times despite gradient clipping and recovered only by restarting from an earlier checkpoint and skipping batches, and the OPT-175B log records at least 35 manual restarts from hardware failures. At our exact scale the rate has been measured directly: a GPT-2 124M model trained on OpenWebText in bf16 diverges on about one run in ten within five percent of the training budget. We do not repair a failed run by hand, and we judge each by a downstream zero-shot metric, so this instability appears in our numbers rather than being engineered away.

Three measurements recur throughout the paper. The participation ratio counts how many directions a representation uses, computed as PR = (Σλi)2 / Σλi2, where λi are the eigenvalues of the centered covariance of the residual activations. The orthogonal Procrustes fit asks whether the network reaches a later state by turning an earlier one or by adding to it, using the rotation minimizing ∥XR − Y∥ F over orthogonal R, with the closed form R = UVT from the singular value decomposition XTY = UΣVT. The write angle to the read-out sends each sublayer's additive write through the final layer normalization and the tied unembedding, the logit lens, and measures the angle between the resulting vocabulary-space vector and the model's final logits, averaged over token positions.

Nine models of the GPT-2 small class are trained on OpenWebText, identical but for the random seed. Five of nine seeds produced a usable model. The unusable runs share no single failure mode: one collapses to near rank-one in its early layers (participation ratio one to five), another inflates through the middle of the network (above one hundred thirty), and write angles turn obtuse, past 100°, pointing away from the read-out, in each failure and in none of the converged runs.

The attention write sits 75–96° from the final logits at every one of the twelve layers: attention never turns toward the vocabulary. The feed-forward write holds near 85° through the body of the network and swings toward the read-out over the last four layers, reaching about 55° at the output. The residual participation ratio is moderate through the middle of the network, dropping to roughly ten, and expands sharply toward the output, to seventy or more at the final layer.

The reason for this off-axis organization is in what attention does. Attention mixes across token positions: every position's output is a weighted average of the sequence's values. Were that averaging to run on the read-out axis it would blur the prediction, smearing one position's next-token distribution into its neighbours'. Held orthogonal to the read-out, the same mixing moves information between positions in a subspace the unembedding discards, and the prediction is left untouched. The off-axis workspace is therefore protective: it insulates the read-out from the blur that cross-token averaging would otherwise inflict.

The cost of violating it is measurable. On a trained baseline we rotate each attention head's value vectors toward the read-out by a small angle and let attention mix them as usual. Averaged over the five converged models, each against four draws of a random rotation of the same magnitude, the damage this does to the next-token loss is 64× that of the random control at a 10° rotation (range 38–82 across models) and 84× at 20° (range 71–107). A matched control isolates the cause: apply the identical rotation but strip the cross-token averaging, each position keeping its own rotated value, and the asymmetry collapses to 1.05× and 1.14× at those same two angles. Read-out-aligned values are no worse than randomly rotated ones until they are mixed; the damage is entirely in the mix.

Through the body of the network the feed-forward write is just as far off the read-out as attention's, 84.5° against 84.9° averaged over layers 1–7, the per-layer means of both spanning 82–88°; the two separate only at the commit, where attention holds near 82° while the feed-forward path swings to 64°. A feed-forward layer mixes nothing across token positions. It has no blur to protect against, so its off-axis position cannot be insulation in the sense measured above. The two facts fit together without a second mechanism. The off-axis subspace is a shared workspace, and insulation is the reason that workspace has to be off-axis at all: were it on the read-out, attention's averaging would blur the prediction at the cost the rotation experiment measures. Once it sits off-axis, everything that writes into it writes off-axis, the feed-forward path included, because that is where the content it reads and adds to already lives.

Across the commit span (layers 8–11) a single best-fit rotation captures R2 rot = 0.38 of the change, while 64% of the committed state is content orthogonal to any rotation of the input. The residual norm grows by 1.74× over the same span; a rotation preserves norm, whereas additive writing grows it. The off-axis representation does move as a near-rigid body from layer to layer, but that turn stays within the off-axis subspace and never delivers the answer, which arrives as a separate on-axis write. We measure the rigidity with a rotation that is never fitted to the concepts: the polar factor of the residual cross-covariance, determined from the token stream alone (N ≈ 250,000 states in 768 dimensions, so the estimate is fully determined). That single rotation carries the concept constellation from one layer to the next at R2 = 0.93, against 0.76 for leaving the constellation unchanged, and the figure holds between 0.933 and 0.938 across every model we measure.

Consistently, the prediction surfaces late: decoded through the logit lens, the layer at which the final token first becomes and remains the top choice averages 9.2, and 39% of positions resolve only at the final layer. The same measurement on the other boundary says the model enters the workspace the way it later leaves it. Across layers 1–6, where the residual moves off the read-out rather than onto it, a best-fit rotation captures R2 rot = 0.33 [0.26–0.39], 81% of the resulting state is fresh content, and the norm grows 1.59×. Going off-axis is written rather than turned, and marginally more so than the commit.

If any content were carried by clean rotation, it should show up somewhere as a set of positions that turn by roughly 90° while adding little. There is no such set. Fitting a separate rotation for each of the 44 token types with enough positions to constrain one, over 704,512 positions in each of the five baselines, the strongest candidates are function words and punctuation, tokens whose own embedding is swamped by context and which therefore sit furthest off-axis. Even these reach only R2 rot = 0.20 on average, and 0.50 for the single best basket in any run, at a mean angle of 82° with 83% of the content freshly written. Rotation is therefore not an operation this network performs, in either direction and at no depth we can find it.

Attributing the final logits to each sublayer write, with the final normalization shared so that the writes sum to the logits exactly, the answer is written almost entirely by the last few feed-forwa…

Based on this paper, here are the specific improvements you can make to AI systems:

Insert a single fixed orthogonal rotation into the residual stream at the layer where the model transitions from concept computation to answer writing (layer 6 of 12 in this architecture). This improves training reliability from 55% convergence (5/9 seeds) to 88% (7/8 seeds) with dense rotations, and 100% (9/9) with sparse signed-permutation rotations. The improved system trains to the same quality as unmodified baselines but does so reliably rather than as a coin flip.

Instead of dense Haar-random rotations, use block-diagonal 2×2 quarter-turn rotations (signed permutations of coordinate pairs). These are absorbable by layer normalization and surrounding weights, eliminating training spikes and achieving 100% convergence across all seeds tested. The system reaches baseline quality with roughly half the effective dimensionality of dense-rotation models, making it computationally cheaper to train around.

Draw a fixed orthogonal basis B at random before training, and add a per-layer penalty that keeps each concept-phase layer's residual second moment diagonal in B. This produces models that adopt the prescribed frame uniformly across all concept-phase layers, with downstream quality unchanged. The improved system allows direct layer-by-layer comparison between models without alignment steps, since they share a common internal coordinate system.

When training models with early-exit pressure or other read-out-alignment objectives, compute and report the participation ratio (effective dimensionality) of intermediate representations. The paper shows this metric drops from 25 to 14 dimensions under on-axis pressure while perplexity, LAMBADA, and BLiMP remain unchanged—so it is the only instrument that detects the geometric distortion. An improved system would surface this metric automatically to catch silent representational degradation.

If you need intermediate layers to be decodable, do not penalize KL divergence between intermediate and final distributions—this collapses quality (LAMBADA drops from 0.271 to 0.142). Instead, penalize the cosine angle between centered logit vectors. The angle-constrained system reaches near-baseline quality (LAMBADA 0.258 vs 0.271) while still achieving the target geometry, because it avoids forcing attention to write anti-aligned corrections that cancel the feed-forward answer.

Recognize that the off-axis frame is a free gauge—the task fixes the answer axis but not the 90° frame. Two baselines from different seeds hold concepts in frames 90° apart, indistinguishable from random. By prescribing the frame in advance (via a fixed rotation or basis penalty), you eliminate the search that causes training collapse. The improved system separates discover a stable frame from hold the frame under pressure, leaving only the latter for gradient descent to solve.

The paper shows the answer is written additively (residual norm grows 1.74×) rather than turned out of accumulated content (best-fit rotation captures only R2=0.38). An improved system can exploit this by structuring late layers to add fresh on-axis content rather than attempting rotational transformations, which the network does not perform at any depth in either direction.

Since the frame is immaterial to the model (25 runs across 13 distinct rotations give identical quality), you can choose it for interpretability. An improved system can be trained with a basis chosen for human legibility (e.g., aligned with semantic features), making internal representations directly inspectable without post-hoc probing. The system will compute in that frame with no quality cost, turning interpretability from a discovery task into a design choice.

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.