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Lexical Reference Frames: Interpreting LM Residual Streams in Native Semantics

Jian Gu, Hongyu Zhang, Chunyang Chen, Aldeida Aleti

arXiv:2610.22169v1cs.CLcs.LG

TL;DR

Intermediate-state methods either use fixed lenses or summarize point clouds, rather than reading states through the model’s context-dependent remaining layers. Lexical reference frames approximate that native reading with a chosen amount of attention between positions and typical attention outputs for the rest. The paper reports that first-order frames recover lexical readouts from mid-depth and support early exit, steering, and attribution.

  • Problem

    Fixed lenses read every state at a layer identically, while point-cloud statistics describe state distributions rather than how the model uses states.

    Source: §1 · p. 1

  • Method

    Lexical reference frames read a state through the model’s remaining layers with a chosen number of attention rounds between positions, replacing remaining attention with typical outputs estimated from unlabelled text.

    Source: §1 · p. 2

  • Results

    From the middle layers, LEXFRAME[1] recovers native lexical readouts beyond every lens and supports early exit, steering, and more faithful attribution.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The paper concludes that reading states through the remaining layers and their context offers an alternative to lenses for interpreting intermediate states.

    Source: §5 · p. 9

Abstract

from arXiv · show

The vocabulary of a language model is its native semantics: the embedding and the unembedding fix the meaning of a state at the two ends of the residual stream. Between them, existing methods either reuse a fixed lens or treat the states as a point cloud, and neither reads a state the way the model does. We argue that every hidden state has a native reference frame, induced by the layers that read it next and adaptive to the context. Lexical reference frames approximate it by reading a state through the model's own remaining layers with a stated number of rounds of attention between positions and a typical attention output for the rest, without fitting to the model's output. No lens, with fixed directions per layer, can exceed the square root of the native frame's coherence, 0.30 and 0.35 at mid-depth of SmolLM3-3B and OLMo3-7B, whereas the first-order frame, LexFrame[1], reaches 0.97 and 0.97 and reads a state within 0.16 nats of the model's output. Its coefficients attribute predictions more faithfully than attribution patching, its lexical basis steers a prediction while sparing other tokens, and its confident readings would let an early exit skip 0.45 and 0.42 of the layers, where lenses skip almost none. In this way, lexical reference frames make every hidden state of the model readable, attributable and steerable.

1 Introduction

Intermediate-state methods either apply a fixed lens or describe states as a point cloud, while the model’s own lexical reading depends on its remaining layers and prefix context. Lexical reference frames approximate that reading through the model’s layers with a chosen amount of attention between positions, and the paper reports that first-order frames recover lexical readouts from mid-depth and support early exit, steering, and attribution.

1 Introduction

  • Fixed lenses ignore prefix context, while point-cloud statistics describe state distributions rather than how the model uses individual states.The remaining layers read each state in its prefix context, so the lexical directions can vary across layers and contexts.Source: §1 · p. 1
  • At layer 16 of OLMo3-7B, the logit-lens and Jacobian-lens directions deviate by 75 and 77 degrees, and four layers before output both still deviate by at least 40 degrees.Figure 1 reports median angles over states and tokens after every fourth layer.Source: §1 · p. 2, Figure 1 · p. 2
  • LEXFRAME[n] reads a state through the model’s remaining layers with n rounds of attention between positions, replacing the rest with typical layer attention outputs estimated from unlabelled text.LEXFRAME[0] reads what a state holds by itself, LEXFRAME[1] what it draws from its prefix, and LEXFRAME[∞] is the native frame; the method is not fitted to model output.Source: §1 · p. 2
  • From the middle layers, LEXFRAME[1] recovers native lexical readouts beyond every lens and supports early exit, steering, and more faithful attribution.One attention round closes most of the gap left by LEXFRAME[0]; the readings settle on the model’s prediction early enough to skip more than two fifths of the layers.Source: §1 · p. 2

2 Related Work

Prior work reads intermediate states with lenses, summarizes their point-cloud geometry, or studies directions and components associated with predictions. This paper relates reference frames to all three questions and evaluates whether they capture how the model uses its states.

Language model lenses

  • Lenses apply the output frame, fit a layer-specific map, use remaining-layer Jacobians, or patch states; these approaches read states without their context.The cited lens methods include the logit lens (nostalgebraist, 2020), tuned lens (Belrose et al., 2023), and Jacobian lens (Gurnee et al., 2026).Source: §2 · p. 3

Point clouds and manifolds

Directions and components

3 Lexical Reference Frames

Lexical reference frames express hidden states in vocabulary coordinates: fixed at the residual stream’s input and output interfaces, but adaptive to layer and context between them. Finite-order frames approximate the model’s native reading by limiting how information passes among positions.

3.1 Preliminaries

  • The input and output lexical bases are fixed across texts, while intermediate states have no predefined basis and must be read using a frame adaptive to layer and context.The embedding assigns input states to labels; the unembedding maps final states to vocabulary logits.Source: §3.1 · p. 3, §3.2 · p. 3

3.2 Native Reference Frame

  • A lexical reference frame pulls the output frame back through the remaining layers, making each state’s readouts and basis depend on both its layer and prefix context.Its coefficients yield a next-token distribution; differences in coefficients can attribute predictions to layer updates.Source: §3.2 · p. 4

3 Lexical Reference Frames

  • Finite-order lexical reference frames transport a state through the model’s remaining layers, limiting attention exchanges between positions and replacing withheld attention with typical outputs estimated from unlabelled text.LEXFRAME[∞] recovers the model’s native frame; finite orders approximate its lexical readouts and basis.Source: §3.3 · pp. 4–5

3.3 Orders of Reference Frames

  • LEXFRAME[0] withholds prefix information, transporting the state through remaining layers with typical attention outputs while retaining a nonlinear, state-dependent reading.It lacks only information the remaining layers would gather from other positions.Source: §3.3 · pp. 4–5
  • LEXFRAME[1] lets the state attend to its prefix, while prefix positions evolve independently; it withholds only attention exchanges among those positions.The difference from LEXFRAME[0] measures the contribution of this round of attention.Source: §3.3 · p. 5
  • Higher orders add exchanges within the prefix, allowing information to travel through more positions until LEXFRAME[∞] recovers the model’s native frame.Frames are exact wherever at most n layers remain, so all orders agree near the output.Source: §3.3 · p. 5

4 Experiments

Experiments compare lexical reference frames with lenses and point-cloud distances, then test whether their coordinates support useful interventions. Across the comparisons, first-order frames track the model’s readings more closely than fixed lenses and provide useful steering and attribution signals.

4.1 Comparison with Lenses

  • At mid-depth, LEXFRAME[1] reaches cosine 0.97 and 0.97 against lens bounds of 0.30 and 0.35; its divergence is 0.16 and 0.15 nats.The divergence figure shows that higher frame orders reduce divergence at nearly every layer, while lenses remain several nats above frames until near the output.Source: §4.1 · p. 6, Figure 4 · p. 6
  • LEXFRAME[1] would allow an early exit to skip 0.45 and 0.42 of layers, compared with at most 0.04 for a lens.Its top-token agreement reaches 0.86 and 0.86 across states, and its calibration error is 0.027 and 0.018 against the model’s 0.018 and 0.023.Source: Table 1 · p. 6, §4.1 · p. 7

4.2 Comparison with Point-Cloud Geometry

  • No point-cloud distance exceeds rank correlation 0.28 for predicting state-replacement effects, while LEXFRAME[1] readout divergence reaches 0.94 and 0.95.The local share of readout norm in leading cloud directions peaks at 9.2% and 2.9%, versus 3.1% and 1.6% for a random direction.Source: Table 2 · p. 7, §4.2 · pp. 7–8

4.3 Coordinates at Intermediate Layers

  • The lexical basis identifies the label whose logit rises most in 0.98 and 1.00 of cases, while the LEXFRAME[1] Jacobian basis reaches 0.84 and 0.97.The LEXFRAME[1] readout reaches 0.54 and 0.73, showing that a direction measuring a label differs from one that moves the output toward it.Source: Table 3 · p. 8, §4.3 · p. 8
  • LEXFRAME[1] transported states yield coefficients within 0.16 and 0.15 nats of model logits, whereas coefficients on the layer state or a fixed-map image are several nats away.Linearising the LEXFRAME[∞] readout against the layer state correlates only 0.10 and 0.02 with the logit.Source: Table 4 · p. 8, §4.3 · p. 8

4.4 Steering and Attribution

  • The lexical basis makes the cleanest steering flips: it changes other tokens by 0.009 and 0.004 nats and yields clean flips for 0.73 and 0.85 of predictions.LEXFRAME[1] readout flips 0.92 and 0.94 of predictions, nearly matching LEXFRAME[∞] at 0.95 and 0.97, but the basis disturbs other tokens less.Source: §4.4 · p. 8, Table 5 · p. 9
  • LEXFRAME[1] coefficient differences correlate with measured attribution effects at 0.88 and 0.90, above LEXFRAME[∞] linear attribution at 0.62 and 0.68.Across single positions, the median correlations are 0.97 and 0.93; the lenses reach at most 0.20.Source: Table 6 · p. 9, §4.4 · p. 9

5 Conclusion

The conclusion frames intermediate states as readable through a context-adaptive native reference frame. Lexical reference frames approximate that reading and support attribution and steering in the two evaluated model families.

5 Conclusion

  • A hidden state’s native reference frame is fixed at the residual stream’s ends and adaptive between them.Lexical reference frames approximate it to a stated order without fitting to the model’s output.Source: §5 · p. 9
  • In two model families, LEXFRAME[1] recovers lexical readouts and outputs from mid-depth, beyond methods using one map per layer.The conclusion also reports that it attributes and steers predictions.Source: §5 · p. 9
  • The frame explains why fixed lenses fail at mid-depth and increasingly so along training, while point-cloud geometry reveals little about how the model uses a state.The paper’s concluding recommendation is to read states through the remaining layers, in context and in the model’s native semantics.Source: §5 · p. 9

A.1 Implementation Details

The implementation uses published model checkpoints and WikiText-103, with calibration and evaluation on separate text splits. The frames are estimated without training, and evaluation uses fixed sampling and bootstrap intervals.

A.1 Implementation Details

  • SmolLM3 checkpoints roughly triple their training-token counts, while OLMo3 checkpoints match token counts and include a randomly initialized model and a four-billion-token checkpoint.Calibration uses 256 training-split sequences of 128 tokens; evaluation uses 64 test-split sequences of 128 tokens.Source: §A.1 · pp. 12–13
  • Typical attention outputs are estimated in closed form from one calibration-text forward pass, with each layer’s ridge chosen by two-fold cross-validation.The lexical basis is computed with an explicit 32-bit Jacobian in one test and with 30 LSMR steps in steering.Source: §A.1 · p. 13
  • Experiments use fixed-seed samples and the same test sequences, with 95% percentile bootstrap intervals from at least 300 sequence-level resamples.The substitution test was repeated on a second disjoint test-split portion.Source: §A.1 · p. 13

A.2 Artifacts and Computations

The project used released SmolLM3-3B and OLMo3-7B checkpoints and WikiText-103, and the authors disclose AI assistance in writing and code preparation.

A.2 Artifacts and Computations

  • The experiments use Apache 2.0-licensed model checkpoints and WikiText-103, with about 103 million training tokens and 0.25 million test tokens.The project used about 500 GPU hours, including about 100 hours for final reported experiments.Source: §A.2 · p. 13

A.3 AI Use Disclosure

  • AI assistants supported paper writing and code preparation; the authors checked the text, code, and results and take responsibility.Source: §A.3 · p. 13

B Additional Results

Across checkpoints and context lengths, LEXFRAME[1] stays close to the model’s readings where fixed lenses diverge, while fitted attention terms preserve that fidelity. The appendix also reports settled-depth and worked-example results, and introduces reference-frame properties.

B Additional Results

  • LEXFRAME[∞] mid-depth cosine with the final-layer readout is 0.07 and 0.26 in the base models, and declines over training as the frame moves away from the output frame.At every trained checkpoint, LEXFRAME[1] remains within 0.11 nats of the randomly initialized OLMo3 output, versus 0.52 nats for the logit lens.Source: B Additional Results · pp. 13–14
  • LEXFRAME[1] agrees with the model’s top token on at least 0.85 and 0.86 of states at every trained checkpoint, while the logit lens declines to 0.04 and 0.15.In SmolLM3, divergence grows from 4.82 to 29.06 nats for the logit lens but moves only from 0.15 to 0.16 for LEXFRAME[1].Source: B Additional Results · pp. 13–14
  • LEXFRAME[1] exceeds 0.95 cosine with LEXFRAME[∞] from layers 16 and 15, while the mid-depth Jacobian lens reaches only 0.20 and 0.23.LEXFRAME[2] exceeds 0.95 from layers 10 and 9; fixed directions approach the frame only near the output.Source: B Additional Results · p. 14, Figure 7 · p. 14
  • LEXFRAME[1] remains close over longer contexts: its divergence rises to 0.25 and 0.18 nats, and it agrees with the model’s top token on 0.84 and 0.86 of states.The order of methods is unchanged, and divergence grows slowly with context length.Source: B Additional Results · p. 14, Figure 8 · p. 15
  • LEXFRAME[1] readings stay settled across 0.82 and 0.81 of layers, and its confidence-based exit leaves 0.45 and 0.42 of layers unused, versus at most 0.04 for lenses.LEXFRAME[1] itself saves no computation because it transports the state through later layers.Source: B Additional Results · p. 15, Table 7 · p. 15
  • Replacing the fitted attention terms with a mean or zero output worsens every LEXFRAME[1] measure, while four calibration sequences yield divergence of 0.41 and 0.34 nats.With the usual 256 calibration sequences, divergence is 0.16 and 0.15 nats; across three disjoint calibration texts it stays between 0.160 and 0.163 for SmolLM3 and 0.146 and 0.151 for OLMo3.Source: B Additional Results · p. 15, Table 8 · p. 15

C Properties of Reference Frames

  • The appendix characterizes reference frames as expressing hidden states in the model’s native semantics and considers frame sides, lens and order closeness, and order complexity.For a prefix state xℓ(t), the stated setup conditions its reading on earlier states xℓ(s), where s < t.Source: C Properties of Reference Frames · p. 16

C.1 Lexical Readout and Lexical Basis

A lexical reference frame has a covector that reads a label’s coefficient and a vector that represents a minimal state change toward that label’s output direction. At intermediate layers, the coefficient is a transported value, not an inner product with the state.

C.1 Lexical Readout and Lexical Basis

  • The two frame sides are a covector measuring how a small state change alters a label’s coefficient and a vector giving the smallest change whose first-order output effect best matches the label vector.Using Tℓ(1) instead of Tℓ gives the basis of LEXFRAME[1].Source: §C.1 · p. 16
  • The vocabulary exceeds the state dimension, so unembedding rows span states without forming a basis; the canonical dual vectors satisfy P_v s_v w_v⊤ = I.A normalized last state is the sum of its logits times these vectors, and the first-order relation holds at intermediate layers up to the ridge in J+.Source: §C.1 · p. 16
  • At intermediate layers, a label coefficient is not an inner product with the state: normalization largely removes the state-rescaling response, with measured correlations of 0.10 and 0.02.The coefficient is a value of the transported state, not a linear function of the layer state.Source: §C.1 · p. 16

C.2 Coherence Bound and Exactness

A state-independent direction is bounded by the coherence of the model’s native lexical readouts. LEXFRAME[n] makes the withheld attention explicit and becomes exact when no more than n layers remain.

C.2 Coherence Bound and Exactness

  • For many states, a state-independent direction’s mean cosine with the lexical readouts is bounded by √κ, attained by their mean direction.The bound belongs to the native frame, not to the estimator.Source: §C.2 · pp. 16–17
  • LEXFRAME[n] computes each order with one more round of attention between positions, using order n −1 prefix states.Order 0 replaces attention with a typical attention output.Source: §C.2 · p. 17
  • LEXFRAME[n] is exact wherever at most n layers remain, and LEXFRAME[∞] returns the model’s output at every layer.Orders differ from the model only through attention between positions more than n layers above the state.Source: §C.2 · p. 17

C.3 Complexity Analysis

Reading lexical frames and fitting lenses have different forward-pass costs. While per-state lens reads are cheaper, fitting the Jacobian lens can be expensive relative to frame calibration and reading.

C.3 Complexity Analysis

  • Reading every layer of a sequence at order n costs about (n+1)(L+1)/2 forward passes.Calibration takes one forward pass over calibration text plus one ridge regression per layer.Source: §C.3 · p. 17
  • A Jacobian-lens fit costs about Pd backward passes, while a fitted lens reads each state with one matrix product.With LEXFRAME[1], the lens becomes cheaper only after about Pd/(L + 1) sequences are read at every layer; its fit must be repeated at each checkpoint.Source: §C.3 · p. 17
  • Fitting the Jacobian lens took 18 and 44 times as long as calibrating frames and reading 64 test sequences at orders 0 to 2.Costs are counted in forward passes, independent of hardware.Source: §C.3 · p. 17

D Discussion

Lexical reference frames read hidden states through the remaining layers and the context those layers see. They separate a state’s label direction from how much of the label it holds, while making interpretations comparable in the model’s vocabulary.

D Discussion

  • A frame reads a hidden state through the remaining layers, which determine its meaning in the model’s vocabulary.Lenses use one fixed map per layer, while point clouds do not provide a reading.Source: D Discussion · p. 17
  • Separating label direction from label amount shows how context affects readings; LEXFRAME[0] transports states without prefix attention, while a layer skip adds prefix attention without transport.The orders make context use a controlled variable for measuring what each attention hop contributes.Source: D Discussion · p. 17
  • The shared vocabulary lets frames express states across layers, checkpoints, and models without a trained component requiring a separate faithfulness argument.A frame calibrated in one pass can follow a model through training, fine-tuning, or editing; applications are left to future work.Source: D Discussion · p. 17

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