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Token-to-token activation continuity in LLMs, and how it changes with depth

updated: 2026-06-30

Literature dive — 2026-06-29.

Question. For a fixed layer, take the sequence of residual-stream vectors h_t (one per token position). How related is h_t to h_{t+1}? Is the trajectory over token positions smooth (slowly varying) or does it have discontinuities/jumps? And how does that smoothness change across layers (early vs. late)?

Scope note. This is the sequence axis (adjacency in token position), not the depth axis (adjacency in layer index). The depth axis is far better studied; see §5 for the one-paragraph version. The honest headline: the sequence axis is comparatively under-studied — the cleanest direct measurement of token-to-token continuity is a single 2026 paper (Barenholtz). That gap is itself a finding.

What the question maps onto in the literature

Three measurable pieces, studied under different names:

  1. How related is h_t to h_{t+1}? → "self-similarity / intra-sentence similarity" (Ethayarajh), "direction preservation / trajectory continuity" (Barenholtz), and — as a confound — "anisotropy."
  2. Discontinuities → outlier/sink tokens & "massive activations" (norm jumps); surprisal / garden-path-linked trajectory reversals (direction jumps); structured geometry (belief-state fractals).
  3. How (1) and (2) change across layers → "stages of inference" and the detokenization → sharpening picture.

1. Core empirical picture: token trajectories are locally smooth, and smoothness increases with depth

The most direct numbers: Barenholtz 2026, "Trajectory Dynamics in Language Model Hidden States" (2606.05346). Fit a linear trajectory to the last k=3 hidden states; measure how well the direction of change at one token predicts the direction at the next ("direction preservation" = absolute cosine between the fitted trajectory direction and the actual displacement vector). Natural Stories, GPT-2:

current step+1+2+3random baseline
Layer 6 (mid)0.440.100.080.080.029
Layer 12 (final)0.610.540.540.540.029

This is the answer to the exact question:

  • At a middle layer, the trajectory is "smooth" only over ~1 token. Direction is strongly preserved at the current step (0.44) but collapses almost to chance (0.10) one token later — "direction effectively dies after a single word." Mid-stream is a near-Markov, recompute-from-context regime: locally continuous with the immediate predecessor, essentially no directional memory beyond that.
  • At the final layer, the trajectory has persistent directional structure (0.54 even 3 tokens out). Late-layer residual states drift coherently over many tokens.
  • Surprising tokens are the discontinuities. Direction preservation correlates negatively with surprisal (r = −0.10 at L6); at garden-path disambiguation points ("…the barn fell") extrapolation error spikes — a measurable directional reversal. This is dissociable from raw movement magnitude: one-step displacement ‖h_t − h_{t−1}‖ and trajectory-deviation correlate only r = 0.16 and predict reading time with opposite signs — large moves that continue the trajectory are cheap; moves that break it are costly.
  • Replicates across GPT-2 (absolute pos. emb.) and Pythia (RoPE) → not a coordinate-system artifact.

Foundational older result: Ethayarajh 2019, "How Contextual are Contextualized Word Representations?" (1909.00512). Measures self-similarity (same word, different contexts) and intra-sentence similarity (different tokens, same sentence — the adjacency question in aggregate) layer by layer. Representations get progressively more context-specific (lower self-similarity) in upper layers — a token's identity dissolves into context with depth — but raw similarity is badly contaminated by anisotropy (§2).

2. Mandatory caveat: raw adjacent-token cosine similarity is mostly an artifact

Computing cos(h_t, h_{t+1}) naively gives misleadingly high, smoothly-varying numbers, for two reasons:

Persona-work implication: any "the persona direction is stable across adjacent tokens" claim must be made on standardized activations, or it's measuring anisotropy.

3. Discontinuities come in (at least) four distinct flavors

(a) Norm/outlier discontinuities — specific token positions blow up. Sun et al. 2024, "Massive Activations in LLMs" (2402.17762): a handful of activations are up to ~100,000× larger than the rest, concentrated on special tokens (BOS, delimiters, low-semantic words), roughly input-independent, acting as implicit bias terms and causing attention concentration. The canonical hard discontinuity over the sequence — the norm trajectory across tokens is flat, then spikes massively at sink tokens. Linked to attention sinks ([Xiao et al. StreamingLLM 2023]; anatomy papers 2603.05498, 2603.17771); they emerge predictably during training (2508.03616). Pre-norm + softmax- summing-to-one is the architectural cause; softpick / Softmax1 (2504.20966) removes both.

(b) Direction discontinuities — surprise/syntax forces a turn. Barenholtz (§1). The off-diagonal analysis is telling: low-surprisal/high-deviation tokens are enriched for connectives and clause-openers ("and", "as", "that") — syntactic-boundary discontinuities where the representation pivots into a new constituent. Applied take: Latent Phase-Shift Rollback (2604.18567) monitors a critical layer's residual stream over generation steps and detects "abrupt directional reversals" via a cosine + entropy gate as a reasoning-error signal (single-group 2026 preprint; treat the strong stats with caution).

(c) Structured (non-random) geometry over the sequence. Shai et al. 2024, "Transformers represent belief state geometry in their residual stream" (2405.15943): the sequence of residual states traces the geometry of belief updating over the data-generating process, sometimes with fractal structure. "Discontinuity" isn't always noise — the trajectory over tokens can have lawful self-similar structure tied to what's predictable.

(d) Token-level compute demand varies. FlexiDepth (2503.23798): repetitive/fixed-phrase tokens need few layers; high-uncertainty/computational tokens need many — indirect evidence that some positions require much larger representational updates than their neighbors.

4. How the layer axis modulates everything

Organizing frame: Lad, Gurnee & Tegmark 2024, "The Remarkable Robustness of LLMs: Stages of Inference?" (2406.19384) — four stages; early/late layers far more intervention-sensitive than the middle:

  1. Detokenization (early): local context fused into token-spanning units — adjacent-token mixing is most local here. Confirmed weight-only by Kamoda et al. 2025 (2501.15754): GPT-2's first-layer attention is biased toward nearby tokens by construction.
  2. Feature engineering (middle): the near-Markov regime — strong 1-step continuity, no longer-range directional memory (Barenholtz L6: 0.44 → 0.10).
  3. Prediction ensembling (mid-late).
  4. Residual sharpening (final): irrelevant features suppressed — both high persistent directional drift (Barenholtz L12: 0.54) and the possibility of a sharp final-layer change as the distribution collapses onto the unembedding.

Two more depth facts relevant to adjacency:

  • Adjacent layers' residual streams become more similar in larger models (MLSAE 2409.04185; Group-SAE's "Average Maximum Angular Distance" 2410.21508) — so late-layer token trajectories are also smoother across depth, compounding sequence-axis smoothness.
  • A 2026 geometric-phases paper finds three depth phases (seeding → hoisting → focal convergence) in the predictive subspace (2605.09011) — consistent picture, newer/less-vetted.

5. The depth-axis cousin (one paragraph)

The residual stream is "iterative inference" (logit/tuned lens — Belrose et al.); adjacent layers are highly redundant, so deep layers can be pruned by angular distance (Gromov et al. "Unreasonable Ineffectiveness of the Deeper Layers" [2403.17887]; ShortGPT); "your transformer is secretly linear" between consecutive layers (Razzhigaev et al.). Different axis, but the method (angular distance / linear-fit between adjacent states) transfers directly to the token axis.

What's missing — and where it touches persona work

  • No standard "consecutive-token similarity vs. layer" curve exists the way the depth-redundancy curve does. Barenholtz is closest, on GPT-2/Pythia only, on English narratives. A clean Qwen-2.5-7B sweep of standardized cos(h_t, h_{t+1}) and direction-preservation across all layers would be a novel, cheap (~0 GPU-h, analysis-only) measurement.
  • Persona connection: the two regimes (mid-layer near-Markov vs. late-layer persistent drift) predict where a persona signal behaves smoothly over a generation vs. where it can flip. A persona that "turns on" mid-generation should show a directional discontinuity (trajectory-extrapolation spike), and it should look different at mid vs. late layers — a measurable hypothesis that ties to trait-direction and marker-leakage "where does the behavior live" questions.

Ranked reading list

  1. Barenholtz 2026 — Trajectory Dynamics (2606.05346) — most directly the question; the direction-preservation table is the key result.
  2. Ethayarajh 2019 — How Contextual… (1909.00512) — foundational self-/intra-sentence similarity across layers.
  3. Timkey & van Schijndel 2021 — Rogue Dimensions (2109.04404) — the methodological landmine; read before measuring anything.
  4. Lad, Gurnee, Tegmark 2024 — Stages of Inference (2406.19384) — the depth-modulation frame.
  5. Sun et al. 2024 — Massive Activations (2402.17762) — the canonical sequence-axis discontinuity.
  6. Shai et al. 2024 — Belief State Geometry (2405.15943) — structured trajectory geometry over tokens.

Provenance caution: items 1–6 are solid (1 and 6 are single-/small-group but methodologically careful). Several other 2026 hits cited in passing (Latent Phase-Shift Rollback, the geometric-phases and GEM papers) are recent single-group preprints with unusually clean stats — fine for ideas, verify before citing.

All sources

Barenholtz 2606.05346 · Ethayarajh 1909.00512 · Timkey & van Schijndel 2109.04404 · Godey 2306.07656 · Rudman 2310.17715 · Sun 2402.17762 · attention-sink anatomy 2603.05498 · gradient sinks 2603.17771 · softpick 2504.20966 · massive-activation training dynamics 2508.03616 · Latent Phase-Shift Rollback 2604.18567 · Lad/Gurnee/Tegmark 2406.19384 · Kamoda 2501.15754 · Shai 2405.15943 · MLSAE 2409.04185 · Group-SAE 2410.21508 · geometric phases 2605.09011 · FlexiDepth 2503.23798 · mikexcohen LLM breakdown