# J-space, open models — companion file for language models

This file accompanies the interactive article at https://eliebak.com/viz/jspace-open-v2
(reading modes: `article`, `figures`; figures are numbered Fig. 1–27 and addressable as
`#fig-N`). It is meant to be loaded into a language model's context so the model can
explain the article and its figures to a reader.

**Who wrote this.** This file was compiled by an AI agent (Claude) working with Elie
Bakouch. Quantities marked MEASURED were re-derived by the agent from the committed
result files of the repository below, not copied from its prose. Passages marked
INTERPRETATION are the agent's reading of the data, not established results.

**Source repository.** https://github.com/eliebak/open-jlens-data
(experiments run 2026-07-08 → 2026-07-11 on a Slurm H200 cluster).
The paper being extended: Anthropic's "Verbalizable Workspace",
https://transformer-circuits.pub/2026/workspace.

## Conventions in this file

- Claims carry one of three labels. MEASURED: a number re-derived from the
  repository's committed result files. DESCRIPTION: what a figure shows, with no
  inference. INTERPRETATION: the compiling agent's reading of the data.
- Scope, unless a section states otherwise: 4–6 open models from 1–3 families,
  WikiText as the fitting corpus, context windows of 128 or 4,096 tokens.
- "Undecidable from this data" marks cases where two hypotheses fit the data equally
  well; the data does not distinguish them.

## Revision history

- Earlier materials about this work describe the middle band as "distance-invariant"
  or say its influence "stops fading". The seq-4096 re-measurement (E1 below) found
  that flatness was mostly an artifact of the 128-token measurement window. The
  current window-robust statistics are the decay exponent α and the cliff ratio.
- An earlier report quoted E5 noise margins of 30–500×; the floor's prompts
  overlapped its reference. The corrected margins are 13–93×.

## Where to verify numbers

All paths relative to the source repository's root.

| Quantity | File |
|---|---|
| any per-figure number | `*_analysis.npz`, `summary.json` under the experiment's `results/` |
| E1 metric definitions | `E1_horizon/RESULTS.md` |
| the window-censoring revision | `AMENDMENTS.md` (2026-07-11 entry), `E1_horizon_4k/RESULTS-4k.md` |
| E2 segmentation and trajectories | `E2_training/SYNTHESIS.md` |
| E3 retrieval / injection | `E3_transplant/REPORT.md`, `REPORT-2b.md` |
| E4 bend statistics and seed triples | `E4_delphi/` results |

## Known limitations

- The E4 bend rests on a single 25B fit with a nonstandard 8-shard merge.
- cum90 absolute values are unreliable: the pursuit explains ~1% of activation
  energy at the top of the ladder, and the smallest models press against the
  64-atom cap. The relative 10× fall is the supported statement.
- The "unrelated models score CKA 0.5–0.7" calibration is corpus-dependent
  (0.44–0.70 across E5x pairs and corpora).
- The E1 cliff-on-boundary coincidence rests on two models (qwen3-1.7b, qwen3-4b)
  plus one training trajectory (SmolLM3); other models show related but not
  identical structure (olmo: step up into a flat deep block; gemma-1b: no
  unambiguous k2 deep boundary).
- E6 is descriptive only: two MoE models, one on a reduced prompt budget.
- All E4/E1/E2 headline numbers are WikiText-fit; E5 measures how much the fitting
  corpus moves them.

## The measurement (article section: "The measurement")

MEASURED protocol. For a model reading text, J_ℓ = E[∂h_final,t′ / ∂h_ℓ,t]: the
Jacobian of the final residual state with respect to layer ℓ's residual state,
averaged over source/target positions and prompts. Projected through the unembedding:
V_ℓ = P·J_ℓ, where P holds the unembedding rows of 4,096 frozen token ids
(RandomState(0) over the model's vocab; per-model conventions such as gemma's (1+γ)
gain and tied/untied embeddings are recorded in the repo). V_ℓ is a 4,096 × d
"dictionary": one steering vector per common token. Standard fits use 250 WikiText
prompts (the reference fits in `neuronpedia/jacobian-lens` used n = 460–670 with
stopping rule delta 0.002).

Two summary numbers:

- **CKA** (centered kernel alignment). Center V's rows, form the relation table
  K = V̂V̂ᵀ (4,096 × 4,096), and compare two tables by
  CKA(A,B) = ⟨K_A,K_B⟩ / (‖K_A‖‖K_B‖). Invariant to any rotation of a dictionary;
  sensitive to changes in the *relative* geometry of the 4,096 entries.
  Calibration (MEASURED): two independent fits of the same model ≈ 0.997
  (250-prompt vs 470-prompt endpoint check); two different trained models at matched
  depth ≈ 0.5–0.7 on prose (corpus-dependent, see E5x).
- **PR** (participation ratio). PR = (Σλ_i)² / Σλ_i² over the eigenvalues of the
  dictionary's covariance: the effective number of directions the 4,096 entries span.
  Real layers: roughly 200–600 in the band. PR is quoted in the middle band because
  near the output the measurement collapses into the output matrix.

MEASURED noise scale: a 32-prompt fit already lands within band 1−CKA ≈ 0.005 of the
full 250-prompt fit (from sub-sampled shard lenses of qwen3-1.7b). The repo's
recombined-J convention is a pair-SUM per source position (a constant ×56.0 vs the
pair-mean on the 128-token layout) — the constant cancels in CKA and PR.

INTERPRETATION: the block structure visible in every layer-by-layer CKA map (an
input-side block, a long middle "workspace" block, a small output-side block) is the
page's organizing object. Block edges ("boundaries") are quoted as a fraction of
depth ρ = layer / (n_source_layers).

## E1 · temporal horizon

Question: the paper reads the middle band as a workspace that "holds things in mind";
taken literally a band nudge should out-last nudges elsewhere. Never measured before
because the standard fit averages over all future positions.

MEASURED setup: Δ-resolved fits. effect_ℓ(Δ) = ‖P·J_ℓ(Δ)‖_F on source–target pairs
exactly Δ tokens apart. Six models: gemma3-1b-pt, gemma3-1b-it, qwen3-1.7b, qwen3-4b,
gemma3-27b-it, olmo3-32b. Sequence length 128, skip_first 16, 250 prompts, 9 buckets
with centers Δ = 0, 1, 2, 3.5, 6.5, 12.5, 24.5, 48.5, 96.

Why Δ=1 is the reference: at Δ=0 the measurement is dominated by the position's own
residual/identity path (effect(1) < ½·effect(0) at every layer of every model; the
naive Δ=0 half-life saturates below one token everywhere). Cross-token metrics
(MEASURED definitions):
- r1 = effect(1)/effect(0), "crossing fraction": 0.21–0.36 at early/mid layers,
  0.014–0.06 at the deepest layers, across the six models.
- h1 = first Δ where effect < ½·effect(1) (log2(1+Δ) interpolation), "half-life".
- flatness = effect(96)/effect(12.5); tail_frac = effect(96)/effect(1). Both are
  window-relative (see the 4k re-measurement).

MEASURED results:
- The paper's literal prediction fails in all six models: h1 peaks at ρ = 0.03–0.17
  every time; band half-lives are 2–5 tokens. Sharpest case: gemma3-1b-pt h1 = 20.8
  tokens at L3 (ρ=0.12); instruction tuning cuts that early far tail ~4× (it: 5.1).
- The cliff: cross-token influence collapses within 1–2 layers at ρ ≈ 0.6–0.7.
  qwen3-1.7b r1 steps 0.202→0.158→0.133→0.117→0.077 across L16–L20; cliff at
  ρ = 0.667 = the model's own k2 CKA boundary. qwen3-4b: 0.657 (2.3× scale change).
  olmo3-32b shows a step *up* into a distance-flat deep block instead; its E1
  boundary replicates E2's segmentation digit-for-digit (k2 ρ = 0.4127). gemma-1b has
  no unambiguous frozen k2 deep boundary; the analog is a far-field plateau at
  L15–16 against the exploratory k3 sub-boundary at ρ = 0.68.

MEASURED — the seq-4096 re-measurement (the window correction). Motivation: with
seq 128, Δ≈96 pairs can only start from sources in the first ~40% of the window
(position confound). Re-fit at sequence length 4,096: qwen3-1.7b (112 full-length
prompts, effects accumulated at 160 log-spaced targets per prompt — per-bucket means
unbiased; recombined dictionary matches the standard fit at r = 0.996) and
gemma3-1b-it (200 prompts, r = 0.987). 14 buckets out to Δ ∈ [2049, 4095]. Results:
- The in-band flatness 0.562 (seq-128) collapses to 0.264 in the clean window; the
  seq-128 far buckets were inflated ×1.8 ([33,64]) to ×2.7 ([65,127]).
- Beyond the old window every layer decays as a scale-free power law
  effect ∝ Δ^(−α) with no plateau out to Δ ≈ 1,500. α (qwen, fitted Δ=12.5–1536):
  ≈ 0.59 sensory / 0.72 band / 0.77 motor. gemma-it replicates the collapse and the
  depth ordering (0.55 / 0.56 / 0.67, fitted Δ=12.5–384; its sliding-window
  attention layers add architecture-specific structure noted in the repo).
- The cliff is exactly Δ-independent: the drop ratio at qwen's deep boundary is
  2.21–2.44 at every bucket from Δ=2 to Δ=4095.
- No induction-band bumps: all 13 cross-token buckets decay monotonically at every
  layer (0 upticks).

MEASURED mechanism ablations (gemma-1b): freeze attention patterns → every layer
profile unchanged (r = 0.996–0.9999); ablate the value path → only 5–14% of
cross-token reach remains; ablate calibrated previous-token/copy heads → change
< 10⁻³. INTERPRETATION: the reach is content carried through fixed attention
patterns, and it is not simple copying.

INTERPRETATION (the honest summary): influence shrinks with depth — deeper layers
pass a smaller fraction across tokens, keep it for fewer tokens, and (in the two
long-window fits) decay with a steeper exponent — with one sharp feature on top, the
cliff. Beyond the measured window there is no data, not a null.

## E1×E2 · the horizon through training

MEASURED: the Δ-resolved fit repeated on 12 public SmolLM3-3B checkpoints, 0.095 →
11.19 trillion tokens (tokens = step × 2.37e6). Tracked quantities:
- temporal cliff = depth where r1 first falls below 70% of its mid-band median:
  ρ = 0.6286 at the first checkpoint (0.85% of training) and unchanged for 11 of 12
  checkpoints (final: 0.60).
- CKA boundary (each checkpoint's own k2 segmentation): starts ρ = 0.457, migrates
  onto the cliff (0.629) over ~6T tokens.
- in-band flatness (band fixed to the final checkpoint's): trendless, 0.38–0.42,
  from 0.095T to 11.19T.
- band÷sensory r1 ratio: 0.50 at the first checkpoint → ≈1.0 by ~3T, then locked.

INTERPRETATION: the temporal feature exists before the geometric boundary arrives;
the geometry converges onto it. Training never extends the reach; what matures is
the flat-inside-band shape of the r1 profile.

## E2 · emergence

MEASURED setup: 27 checkpoints across SmolLM3-3B (12), OLMo-32B (11), and an early
OLMo-7B set (random init, 4.2B, 16.8B, 67B tokens). All comparisons use matched-depth
mean CKA (layer ℓ vs layer ℓ, averaged over layers).

Blockiness (MEASURED definition): (mean within-block CKA, off-diagonal) − (mean
between-block CKA), at the dynamic-programming segmentation minimizing within-block
dissimilarity; block count k = argmax_k [blockiness − 0.02(k−1)]. Caveat the article
states and you should repeat: a blockless map whose agreement decays smoothly with
layer distance already scores 0.16–0.27 under this rule (Toeplitz surrogate null).
The excess over that null — the actual block signal — is +0.01 to +0.09 across
trained checkpoints. The random init selects k=2 with raw blockiness 0.127, but its
"blocks" align with nothing: depth-correlation to trained checkpoints ≈ 0, and its
matched-depth CKA to the 32B run starts at 0.31 and falls to 0.17 as that run trains.

MEASURED trajectories:
- Structure is present at the earliest trained checkpoints (4.2B tokens OLMo-7B,
  95B SmolLM3). Both OLMo scales pass through a blockiness *minimum* near ~17B
  tokens (7B: 0.113; 32B: 0.111) — a dissolve-and-reform — before differentiating.
  Blockiness peaks mid-training (32B: 0.363 at 1.09T; 3B: 0.296 at 3.0T) and softens
  toward the end (0.311 / 0.207 final).
- Rate of change, (1 − matched-depth CKA between consecutive checkpoints) / ΔT:
  OLMo-32B falls roughly as 1/t and never reaches zero — its final state scores
  CKA 0.836 against itself 200B tokens earlier. SmolLM3's rate stops falling at
  ~3T and holds constant to the end (consecutive-checkpoint CKA ≈ 0.77–0.79 per
  1.5T interval). One 20–56×-above-trend point in the 32B contains both a data-mix
  change and the LR anneal; cause undecidable.
- Distance to final state: rises from 0.48 (SmolLM3 first) / 0.52 (32B first)
  toward 1.0, with no plateau before the end.
- 7B × 32B cross-grid: at the shared token count (16.8B) the two agree at CKA 0.668
  (depth-corr 0.936) — inside the unrelated-trained-models range. Best matches:
  7B@16.8B ↔ 32B@42B (2.5× more tokens), 7B@67B ↔ 32B@92B (1.4×). The 4.2B row's
  best match is the earliest 32B checkpoint available (censored). FLOPs-matching
  would predict the opposite direction (32B has ~4.6× fewer tokens at equal FLOPs).

INTERPRETATION: what "never settles" means concretely: layer by layer, the relation
table keeps being rewritten — entries change direction relative to one another (a
rigid rotation would leave CKA at 1.0). Token count, not compute, sets a
checkpoint's geometric age, and the bigger model ages more slowly per token.

## E3 · transplant

Question: unrelated models' dictionaries look alike (CKA 0.5–0.7 matched-depth); is
the code causally interchangeable?

MEASURED setup: pairs Llama-3.1-8B ↔ Qwen3-8B (cross-family; 4,096 token-string
matches, 85% hit rate) and gemma-3-4B ↔ 27B (cross-scale; identical tokenizer).
Frozen 3,276 / 820 train/test split. Maps: ridge (inner 90/10 λ selection) and
semi-orthogonal SVD (pure rotation). All fits pre-existed in
`neuronpedia/jacobian-lens`.

MEASURED retrieval (rank the mapped held-out vector against all 4,096 receiver
entries; chance 2.44e-4): best cosine retrieval@1 at matched mid-depth — svd
llama→qwen 0.922 (r@5 0.995), gemma 4b→27b 0.849–0.863. Identity map (possible when
d matches): 0.000–0.001. Shuffled pairing: ≤ 0.002. Note a metric subtlety the agent
found worth flagging: llama→qwen svd scores 0.000 under *euclidean* ranking but 0.922
under cosine — a pure norm-scale effect.

MEASURED injection: 48 hand-vetted single-token concepts from the held-out split × 7
neutral carrier prompts; baseline recall 0/336 (median concept rank ≈ 48,000);
injection adds α·r̄_ℓ·û at a block output, α = 0.5–32 in units of the receiver's
typical residual norm. Best success@1 per direction: 0.940 (svd, L17, α=32, into
qwen), 0.961 (ridge, L20, α=3, into llama), 0.756 (into 27B), 0.560 (into 4B).
All three control arms (random semi-orthogonal rotation, right map + shuffled
concept, norm-matched random vector) score 0.000 over all 104,832 control cells
(48 × 7 × 8 α-values × 3 arms × 13 layer–direction combinations).

MEASURED cost accounting (KL = fluency guard, KL(baseline‖injected) excluding the
concept token): the mapped vector needs 1.0–5.9× the KL budget at which the
receiver's own vector saturates; capped at the own vector's budget, cross-family
transfer is 0.41–0.69; at its own operating point it reaches 94–99% of the
own-vector ceiling at equal KL (gemma pair: 56–76%).

INTERPRETATION: that a pure rotation suffices means the two dictionaries are
congruent shapes, not merely statistically similar. Scope limits to repeat: headline
rates are each direction's best (layer, α) cell of a sweep; all concepts are common
vocabulary; rare words untested.

## E4 · scale

MEASURED setup: the Delphi suite (one architecture/dataset/tokenizer, 447M–25B).
Compute-optimal ladder 3e18 → 1e23 FLOPs, a fixed-compute (isoFLOP) slice of six
sizes, seed triples at 1e21 (3.4B params) and 1e22 (9.7B). "Band PR" here = the
maximum PR over layers with ρ ∈ [0.25, 0.75] (in practice PR at ρ ≈ 0.70–0.75).

MEASURED:
- Ladder band PR: ≈208 at 3e18 rising to ≈444 (1e22), then 379 at 1e23. A single
  power law across the ladder is rejected once seed-measured noise is used
  (heteroscedastic test p ≈ 0.002–0.014; seed log-σ(PR): 0.017 at 3.4B, 0.185 at
  9.7B). Whether the top is flat (break at ≈2.1e20 FLOPs) or slowly saturating is
  undecidable — both fit equally well, and the 25B is a single fit with a
  nonstandard 8-shard merge (per-shard convergence 0.028–0.064 vs ≤0.0036
  elsewhere). Four early-ladder fits stopped early on convergence.
- One surface fits ladder + isoFLOP slice + seeds: PR ∝ d_model^0.58 · (tokens per
  parameter)^(−0.13) — training *compresses* (~26% per 10× tokens/param). An
  undertrained 8.1B measures 520 vs the fully-trained 25B's 379. At fixed compute,
  PR rises with parameter count.
- cum90 (matching pursuit on held-out band activations, 64-atom cap, 50 prompts ×
  111 positions, band median): falls 49 → 5 across the same ladder. Caveats: the
  two smallest models press against the 64-atom cap; at the top of the ladder the
  pursuit explains ~1% of activation energy. The 10× fall is the finding; the
  absolute counts are not reliable.
- Seeds: pairwise band CKA 0.846–0.850 (3.4B) and 0.903–0.914 (9.7B) — between the
  cross-family level (0.76) and the adjacent-layer level (0.995). Seed PR values:
  386/385/375 (3.4B); 444/492/344 (9.7B) — shape agrees, the number does not.

INTERPRETATION: a bigger dictionary, consulted more sparsely; growth in span stops
(or nearly stops) around 2×10²⁰ FLOPs while per-token usage keeps collapsing. All of
E4 is WikiText-measured — see E5 before treating any PR as a model property.

## E5 · corpus dependence (and E5x · cross-model, by corpus)

MEASURED setup: refit the lens on Python code, web math, and PDF-extracted prose for
gemma3-1b-pt, qwen3-1.7b, qwen3-4b, delphi-3.4b; noise floor = a second WikiText fit
on fresh prompts, same budget (band CKA ≈ 0.99).

MEASURED (within-model): input-side layers are rewritten wholesale (qwen3-4b code ×
wikitext at the first layers ≈ 0.26 vs floor 0.99); output side moves least but never
reaches the floor. Block structure survives every corpus. Band PR (qwen3-4b): code
594, math 498, wikitext 329, PDF prose 202 — a swing comparable to E4's whole ladder,
13–93× above the floor across models. (An earlier internal report quoted 30–500×
margins from a floor whose prompts overlapped its reference; 13–93× is the corrected
range.)

MEASURED (E5x, cross-model on the same corpus): qwen3-1.7b × qwen3-4b band means —
code 0.56, wikitext 0.67, math 0.69, PDF prose 0.70. gemma3-1b × qwen3-1.7b — code
0.44, math/wikitext 0.53, PDF prose 0.57. Doubling the prompt count moves these by
< 0.001. Depth-correspondence (correlation between a layer's depth and its best
match's depth) also weakens on code.

INTERPRETATION: code — the corpus that inflates each model's own dictionary the most
— makes two models look *least* alike. The shared, transferable geometry lives in
ordinary prose; code exercises machinery each model built its own way. A dictionary
size is a property of a (model, corpus) pair, not of a model.

## E6 · mixture-of-experts

DESCRIPTION only, by design: layer-agreement maps for Kimi-K2.5 (1T params, 32B
active, 100 prompts) and DeepSeek-V4-Flash (284B, 13B active, 250 prompts), 16
source layers each. Two models, one on a reduced prompt budget, do not support
conclusions; the article deliberately stops at showing the maps. If a reader pushes
for MoE conclusions, decline on those grounds.

## What replicates (the article's closing summary)

Across every model tested: longest-lasting influence from the earliest layers (6/6);
influence shrinks with depth (crossing fraction, half-life, and — in both
long-window fits — decay exponent). The cliff sits on the deep CKA boundary where
that boundary is unambiguous (both qwen models) and precedes it through training
(SmolLM3). Transplant controls at zero in all 104,832 cells. Seeds converge to CKA
0.85–0.91. Corpus effects 13–93× the re-measurement floor. Everything else is
description, not finding.

## Figure index

- Fig. 1 — From a nudge to two numbers
- Fig. 2 — How CKA is computed
- Fig. 3 — A model's layer-by-layer agreement
- Fig. 4 — Why 4,096 entries is not 4,096 directions
- Fig. 5 — Why the clock starts at Δ = 1
- Fig. 6 — How much a nudge still moves the output, as the target gets further away
- Fig. 7 — The long-window re-measurement
- Fig. 8 — The same data as one statistic across depth
- Fig. 9 — Where the cliff and the boundary sit
- Fig. 10 — Does the reach ever grow?
- Fig. 11 — The agreement map, checkpoint by checkpoint
- Fig. 12 — How fast the geometry is still changing
- Fig. 13 — Distance to the final state
- Fig. 14 — Every 7B checkpoint against every 32B checkpoint
- Fig. 15 — Finding the right vector in the other model
- Fig. 16 — Making the receiver say the concept
- Fig. 17 — What the hits and misses look like
- Fig. 18 — Dictionary size across five decades of compute
- Fig. 19 — How many entries a single token engages
- Fig. 20 — Same recipe, different seed
- Fig. 21 — The matrices behind the dots
- Fig. 22 — The same layer, measured through two different texts
- Fig. 23 — Dictionary size depends on the measuring text
- Fig. 24 — The block structure survives every corpus
- Fig. 25 — Two models, measured on the same text
- Fig. 26 — The cross-model maps
- Fig. 27 — Two MoEs under the lens

## Figure data

Generated at build time from the exact arrays the charts render, so these tables
cannot drift from the figures. Values are rounded (2–3 decimals); full precision is
in the repo's result files. Matrices too large to inline (checkpoint atlas maps, seed
matrices, cross-model maps) are summarized here and available via hover on the page
or in the repo.

### Fig. 1 — From a nudge to two numbers

Diagram only (the J-lens pipeline); no data series.

### Fig. 2 — How CKA is computed

Toy relation table K = V̂V̂ᵀ for the six entries (identical for model B after the 35° rotation; CKA(A,B) = 1.000):
```csv
,cat,dog,horse,Paris,London,Rome
cat,0.36,0.41,0.45,-0.47,-0.41,-0.35
dog,0.41,0.49,0.55,-0.55,-0.48,-0.42
horse,0.45,0.55,0.62,-0.6,-0.54,-0.47
Paris,-0.47,-0.55,-0.6,0.62,0.54,0.46
London,-0.41,-0.48,-0.54,0.54,0.48,0.41
Rome,-0.35,-0.42,-0.47,0.46,0.41,0.36
```

### Fig. 3 — A model's layer-by-layer agreement

qwen3-1.7b layer×layer CKA (27 source layers; boundaries ρ = 0.111, 0.667):
```csv
layer,L0,L1,L2,L3,L4,L5,L6,L7,L8,L9,L10,L11,L12,L13,L14,L15,L16,L17,L18,L19,L20,L21,L22,L23,L24,L25,L26
L0,1,1,0.99,0.85,0.83,0.77,0.71,0.68,0.69,0.69,0.68,0.64,0.65,0.64,0.64,0.69,0.66,0.67,0.65,0.57,0.6,0.57,0.53,0.53,0.47,0.44,0.33
L1,1,1,1,0.85,0.82,0.78,0.71,0.69,0.69,0.69,0.68,0.65,0.65,0.64,0.64,0.69,0.66,0.67,0.65,0.56,0.59,0.56,0.52,0.51,0.46,0.43,0.34
L2,0.99,1,1,0.86,0.84,0.81,0.74,0.71,0.72,0.72,0.69,0.67,0.67,0.66,0.66,0.71,0.68,0.69,0.66,0.57,0.6,0.57,0.52,0.52,0.46,0.44,0.35
L3,0.85,0.85,0.86,1,0.99,0.96,0.94,0.91,0.93,0.92,0.89,0.87,0.87,0.85,0.86,0.88,0.82,0.81,0.78,0.71,0.72,0.69,0.65,0.65,0.6,0.56,0.48
L4,0.83,0.82,0.84,0.99,1,0.98,0.97,0.94,0.96,0.95,0.9,0.89,0.89,0.88,0.89,0.9,0.84,0.83,0.78,0.71,0.72,0.69,0.65,0.64,0.6,0.56,0.5
L5,0.77,0.78,0.81,0.96,0.98,1,0.99,0.97,0.98,0.96,0.9,0.91,0.91,0.9,0.9,0.89,0.85,0.82,0.77,0.7,0.7,0.67,0.63,0.62,0.57,0.54,0.51
L6,0.71,0.71,0.74,0.94,0.97,0.99,1,0.99,0.99,0.97,0.9,0.92,0.92,0.9,0.91,0.89,0.84,0.81,0.76,0.7,0.69,0.66,0.62,0.61,0.57,0.54,0.52
L7,0.68,0.69,0.71,0.91,0.94,0.97,0.99,1,0.99,0.96,0.88,0.9,0.9,0.88,0.9,0.88,0.83,0.8,0.75,0.69,0.68,0.65,0.61,0.6,0.56,0.53,0.53
L8,0.69,0.69,0.72,0.93,0.96,0.98,0.99,0.99,1,0.98,0.93,0.94,0.94,0.92,0.94,0.92,0.88,0.85,0.8,0.74,0.73,0.7,0.66,0.65,0.61,0.58,0.55
L9,0.69,0.69,0.72,0.92,0.95,0.96,0.97,0.96,0.98,1,0.96,0.96,0.97,0.95,0.96,0.95,0.91,0.89,0.84,0.79,0.77,0.74,0.7,0.69,0.65,0.62,0.57
L10,0.68,0.68,0.69,0.89,0.9,0.9,0.9,0.88,0.93,0.96,1,0.97,0.97,0.97,0.95,0.96,0.93,0.92,0.88,0.84,0.82,0.79,0.75,0.74,0.69,0.66,0.55
L11,0.64,0.65,0.67,0.87,0.89,0.91,0.92,0.9,0.94,0.96,0.97,1,1,1,0.97,0.96,0.94,0.92,0.87,0.82,0.8,0.77,0.72,0.71,0.67,0.64,0.56
L12,0.65,0.65,0.67,0.87,0.89,0.91,0.92,0.9,0.94,0.97,0.97,1,1,1,0.98,0.97,0.95,0.92,0.88,0.83,0.81,0.78,0.73,0.72,0.68,0.65,0.58
L13,0.64,0.64,0.66,0.85,0.88,0.9,0.9,0.88,0.92,0.95,0.97,1,1,1,0.98,0.97,0.95,0.92,0.88,0.84,0.81,0.78,0.74,0.73,0.68,0.65,0.57
L14,0.64,0.64,0.66,0.86,0.89,0.9,0.91,0.9,0.94,0.96,0.95,0.97,0.98,0.98,1,0.98,0.96,0.93,0.89,0.84,0.82,0.8,0.75,0.74,0.69,0.67,0.61
L15,0.69,0.69,0.71,0.88,0.9,0.89,0.89,0.88,0.92,0.95,0.96,0.96,0.97,0.97,0.98,1,0.98,0.96,0.93,0.89,0.87,0.84,0.8,0.79,0.74,0.71,0.61
L16,0.66,0.66,0.68,0.82,0.84,0.85,0.84,0.83,0.88,0.91,0.93,0.94,0.95,0.95,0.96,0.98,1,0.99,0.95,0.91,0.9,0.88,0.83,0.82,0.77,0.74,0.6
L17,0.67,0.67,0.69,0.81,0.83,0.82,0.81,0.8,0.85,0.89,0.92,0.92,0.92,0.92,0.93,0.96,0.99,1,0.96,0.92,0.92,0.89,0.85,0.84,0.79,0.76,0.59
L18,0.65,0.65,0.66,0.78,0.78,0.77,0.76,0.75,0.8,0.84,0.88,0.87,0.88,0.88,0.89,0.93,0.95,0.96,1,0.97,0.95,0.93,0.9,0.89,0.84,0.81,0.64
L19,0.57,0.56,0.57,0.71,0.71,0.7,0.7,0.69,0.74,0.79,0.84,0.82,0.83,0.84,0.84,0.89,0.91,0.92,0.97,1,0.96,0.94,0.92,0.91,0.88,0.86,0.64
L20,0.6,0.59,0.6,0.72,0.72,0.7,0.69,0.68,0.73,0.77,0.82,0.8,0.81,0.81,0.82,0.87,0.9,0.92,0.95,0.96,1,0.99,0.97,0.96,0.92,0.89,0.66
L21,0.57,0.56,0.57,0.69,0.69,0.67,0.66,0.65,0.7,0.74,0.79,0.77,0.78,0.78,0.8,0.84,0.88,0.89,0.93,0.94,0.99,1,0.99,0.97,0.95,0.92,0.68
L22,0.53,0.52,0.52,0.65,0.65,0.63,0.62,0.61,0.66,0.7,0.75,0.72,0.73,0.74,0.75,0.8,0.83,0.85,0.9,0.92,0.97,0.99,1,0.99,0.97,0.94,0.68
L23,0.53,0.51,0.52,0.65,0.64,0.62,0.61,0.6,0.65,0.69,0.74,0.71,0.72,0.73,0.74,0.79,0.82,0.84,0.89,0.91,0.96,0.97,0.99,1,0.98,0.95,0.67
L24,0.47,0.46,0.46,0.6,0.6,0.57,0.57,0.56,0.61,0.65,0.69,0.67,0.68,0.68,0.69,0.74,0.77,0.79,0.84,0.88,0.92,0.95,0.97,0.98,1,0.97,0.69
L25,0.44,0.43,0.44,0.56,0.56,0.54,0.54,0.53,0.58,0.62,0.66,0.64,0.65,0.65,0.67,0.71,0.74,0.76,0.81,0.86,0.89,0.92,0.94,0.95,0.97,1,0.72
L26,0.33,0.34,0.35,0.48,0.5,0.51,0.52,0.53,0.55,0.57,0.55,0.56,0.58,0.57,0.61,0.61,0.6,0.59,0.64,0.64,0.66,0.68,0.68,0.67,0.69,0.72,1
```

### Fig. 4 — Why 4,096 entries is not 4,096 directions

Toy illustration (6 entries spanning 2 directions, PR ≈ 2); no measured data.

### Fig. 5 — Why the clock starts at Δ = 1

gemma3-1b-pt:
```csv
layer,rho,r1
0,0,0.252
1,0.04,0.265
2,0.08,0.22
3,0.12,0.221
4,0.16,0.162
5,0.2,0.203
6,0.24,0.203
7,0.28,0.198
8,0.32,0.191
9,0.36,0.198
10,0.4,0.173
11,0.44,0.165
12,0.48,0.171
13,0.52,0.17
14,0.56,0.156
15,0.6,0.138
16,0.64,0.091
17,0.68,0.094
18,0.72,0.097
19,0.76,0.078
20,0.8,0.073
21,0.84,0.06
22,0.88,0.044
23,0.92,0.034
24,0.96,0.031
```
gemma3-1b-it:
```csv
layer,rho,r1
0,0,0.291
1,0.04,0.289
2,0.08,0.265
3,0.12,0.258
4,0.16,0.211
5,0.2,0.23
6,0.24,0.228
7,0.28,0.222
8,0.32,0.218
9,0.36,0.217
10,0.4,0.21
11,0.44,0.205
12,0.48,0.202
13,0.52,0.188
14,0.56,0.189
15,0.6,0.148
16,0.64,0.106
17,0.68,0.091
18,0.72,0.086
19,0.76,0.071
20,0.8,0.06
21,0.84,0.051
22,0.88,0.043
23,0.92,0.042
24,0.96,0.035
```
qwen3-1.7b:
```csv
layer,rho,r1
0,0,0.258
1,0.037,0.249
2,0.074,0.24
3,0.111,0.23
4,0.148,0.234
5,0.185,0.238
6,0.222,0.249
7,0.259,0.246
8,0.296,0.249
9,0.333,0.251
10,0.37,0.234
11,0.407,0.243
12,0.444,0.246
13,0.481,0.243
14,0.519,0.24
15,0.556,0.216
16,0.593,0.205
17,0.63,0.161
18,0.667,0.133
19,0.704,0.118
20,0.741,0.078
21,0.778,0.073
22,0.815,0.063
23,0.852,0.053
24,0.889,0.045
25,0.926,0.038
26,0.963,0.03
```
qwen3-4b:
```csv
layer,rho,r1
0,0,0.199
1,0.029,0.183
2,0.057,0.178
3,0.086,0.167
4,0.114,0.171
5,0.143,0.167
6,0.171,0.166
7,0.2,0.158
8,0.229,0.155
9,0.257,0.161
10,0.286,0.17
11,0.314,0.177
12,0.343,0.177
13,0.371,0.182
14,0.4,0.191
15,0.429,0.198
16,0.457,0.216
17,0.486,0.237
18,0.514,0.22
19,0.543,0.184
20,0.571,0.187
21,0.6,0.173
22,0.629,0.149
23,0.657,0.123
24,0.686,0.116
25,0.714,0.104
26,0.743,0.1
27,0.771,0.093
28,0.8,0.08
29,0.829,0.079
30,0.857,0.069
31,0.886,0.064
32,0.914,0.058
33,0.943,0.055
34,0.971,0.04
```
gemma3-27b-it:
```csv
layer,rho,r1
0,0,0.35
1,0.016,0.347
2,0.033,0.338
3,0.049,0.333
4,0.066,0.331
5,0.082,0.332
6,0.098,0.329
7,0.115,0.328
8,0.131,0.353
9,0.148,0.358
10,0.164,0.358
11,0.18,0.344
12,0.197,0.341
13,0.213,0.306
14,0.23,0.277
15,0.246,0.258
16,0.262,0.239
17,0.279,0.232
18,0.295,0.222
19,0.311,0.214
20,0.328,0.21
21,0.344,0.205
22,0.361,0.202
23,0.377,0.201
24,0.393,0.195
25,0.41,0.194
26,0.426,0.201
27,0.443,0.199
28,0.459,0.194
29,0.475,0.193
30,0.492,0.191
31,0.508,0.185
32,0.525,0.182
33,0.541,0.182
34,0.557,0.178
35,0.574,0.178
36,0.59,0.174
37,0.607,0.169
38,0.623,0.163
39,0.639,0.155
40,0.656,0.153
41,0.672,0.158
42,0.689,0.156
43,0.705,0.151
44,0.721,0.147
45,0.738,0.145
46,0.754,0.143
47,0.77,0.142
48,0.787,0.141
49,0.803,0.14
50,0.82,0.138
51,0.836,0.136
52,0.852,0.131
53,0.869,0.129
54,0.885,0.119
55,0.902,0.108
56,0.918,0.088
57,0.934,0.084
58,0.951,0.046
59,0.967,0.036
60,0.984,0.022
```
olmo3-32b:
```csv
layer,rho,r1
0,0,0.243
1,0.016,0.231
2,0.032,0.215
3,0.048,0.205
4,0.063,0.199
5,0.079,0.188
6,0.095,0.177
7,0.111,0.174
8,0.127,0.172
9,0.143,0.173
10,0.159,0.173
11,0.175,0.171
12,0.19,0.171
13,0.206,0.17
14,0.222,0.171
15,0.238,0.169
16,0.254,0.167
17,0.27,0.165
18,0.286,0.169
19,0.302,0.168
20,0.317,0.153
21,0.333,0.152
22,0.349,0.147
23,0.365,0.12
24,0.381,0.117
25,0.397,0.11
26,0.413,0.104
27,0.429,0.1
28,0.444,0.099
29,0.46,0.093
30,0.476,0.089
31,0.492,0.088
32,0.508,0.087
33,0.524,0.084
34,0.54,0.084
35,0.556,0.084
36,0.571,0.082
37,0.587,0.078
38,0.603,0.077
39,0.619,0.077
40,0.635,0.075
41,0.651,0.072
42,0.667,0.07
43,0.683,0.069
44,0.698,0.069
45,0.714,0.066
46,0.73,0.066
47,0.746,0.064
48,0.762,0.064
49,0.778,0.062
50,0.794,0.061
51,0.81,0.061
52,0.825,0.058
53,0.841,0.057
54,0.857,0.056
55,0.873,0.053
56,0.889,0.053
57,0.905,0.052
58,0.921,0.045
59,0.937,0.041
60,0.952,0.038
61,0.968,0.035
62,0.984,0.014
```

### Fig. 6 — How much a nudge still moves the output, as the target gets further away

gemma3-1b-pt — effect(Δ)/effect(Δ1), rows = layers:
```csv
layer,rho,d1,d2,d3.5,d6.5,d12.5,d24.5,d48.5,d96
0,0,1,0.676,0.509,0.429,0.387,0.361,0.337,0.345
1,0.04,1,0.657,0.497,0.406,0.357,0.327,0.301,0.3
2,0.08,1,0.743,0.595,0.512,0.465,0.434,0.407,0.413
3,0.12,1,0.753,0.623,0.553,0.516,0.495,0.478,0.491
4,0.16,1,0.764,0.611,0.509,0.431,0.361,0.298,0.266
5,0.2,1,0.76,0.599,0.488,0.407,0.337,0.274,0.239
6,0.24,1,0.748,0.581,0.468,0.385,0.312,0.244,0.199
7,0.28,1,0.753,0.597,0.488,0.404,0.326,0.254,0.207
8,0.32,1,0.723,0.558,0.449,0.37,0.301,0.235,0.19
9,0.36,1,0.699,0.529,0.421,0.347,0.284,0.223,0.182
10,0.4,1,0.736,0.582,0.478,0.408,0.342,0.274,0.224
11,0.44,1,0.685,0.487,0.351,0.27,0.215,0.163,0.128
12,0.48,1,0.66,0.467,0.336,0.258,0.203,0.152,0.119
13,0.52,1,0.657,0.468,0.342,0.264,0.208,0.155,0.12
14,0.56,1,0.605,0.41,0.291,0.231,0.191,0.147,0.117
15,0.6,1,0.529,0.322,0.227,0.207,0.19,0.154,0.125
16,0.64,1,0.558,0.367,0.282,0.27,0.251,0.204,0.164
17,0.68,1,0.557,0.346,0.214,0.157,0.13,0.104,0.086
18,0.72,1,0.549,0.336,0.206,0.133,0.091,0.068,0.058
19,0.76,1,0.505,0.309,0.188,0.124,0.088,0.07,0.061
20,0.8,1,0.481,0.278,0.161,0.11,0.087,0.072,0.065
21,0.84,1,0.52,0.313,0.187,0.134,0.106,0.089,0.081
22,0.88,1,0.555,0.354,0.223,0.169,0.142,0.124,0.113
23,0.92,1,0.515,0.297,0.162,0.1,0.068,0.044,0.032
24,0.96,1,0.518,0.272,0.157,0.102,0.073,0.048,0.035
```
gemma3-1b-it — effect(Δ)/effect(Δ1), rows = layers:
```csv
layer,rho,d1,d2,d3.5,d6.5,d12.5,d24.5,d48.5,d96
0,0,1,0.685,0.486,0.372,0.303,0.26,0.244,0.272
1,0.04,1,0.683,0.507,0.392,0.315,0.262,0.23,0.238
2,0.08,1,0.711,0.54,0.423,0.342,0.285,0.252,0.264
3,0.12,1,0.731,0.571,0.454,0.37,0.311,0.279,0.295
4,0.16,1,0.728,0.56,0.434,0.34,0.276,0.241,0.246
5,0.2,1,0.731,0.561,0.43,0.333,0.268,0.234,0.239
6,0.24,1,0.733,0.563,0.427,0.326,0.258,0.219,0.214
7,0.28,1,0.728,0.561,0.432,0.337,0.267,0.225,0.22
8,0.32,1,0.708,0.527,0.387,0.289,0.223,0.187,0.181
9,0.36,1,0.694,0.506,0.368,0.277,0.219,0.186,0.18
10,0.4,1,0.68,0.499,0.374,0.302,0.257,0.226,0.213
11,0.44,1,0.649,0.441,0.29,0.2,0.148,0.125,0.123
12,0.48,1,0.633,0.428,0.282,0.197,0.148,0.126,0.124
13,0.52,1,0.601,0.392,0.256,0.185,0.146,0.13,0.13
14,0.56,1,0.603,0.39,0.251,0.181,0.141,0.125,0.126
15,0.6,1,0.508,0.305,0.208,0.173,0.155,0.149,0.153
16,0.64,1,0.557,0.355,0.252,0.217,0.196,0.188,0.194
17,0.68,1,0.544,0.329,0.192,0.125,0.093,0.072,0.067
18,0.72,1,0.525,0.312,0.186,0.126,0.097,0.075,0.067
19,0.76,1,0.58,0.364,0.221,0.152,0.117,0.092,0.08
20,0.8,1,0.562,0.356,0.226,0.156,0.123,0.098,0.09
21,0.84,1,0.558,0.355,0.218,0.148,0.111,0.092,0.086
22,0.88,1,0.608,0.394,0.241,0.162,0.121,0.1,0.095
23,0.92,1,0.591,0.366,0.203,0.118,0.076,0.05,0.038
24,0.96,1,0.633,0.387,0.215,0.127,0.076,0.051,0.042
```
qwen3-1.7b — effect(Δ)/effect(Δ1), rows = layers:
```csv
layer,rho,d1,d2,d3.5,d6.5,d12.5,d24.5,d48.5,d96
0,0,1,0.69,0.486,0.393,0.338,0.305,0.298,0.355
1,0.037,1,0.71,0.515,0.418,0.359,0.323,0.316,0.375
2,0.074,1,0.736,0.545,0.442,0.38,0.348,0.346,0.411
3,0.111,1,0.711,0.532,0.415,0.335,0.293,0.274,0.299
4,0.148,1,0.697,0.519,0.413,0.342,0.3,0.285,0.317
5,0.185,1,0.712,0.537,0.424,0.353,0.312,0.299,0.333
6,0.222,1,0.707,0.53,0.415,0.342,0.299,0.283,0.312
7,0.259,1,0.706,0.529,0.414,0.341,0.297,0.282,0.309
8,0.296,1,0.7,0.517,0.399,0.32,0.271,0.249,0.259
9,0.333,1,0.696,0.509,0.38,0.287,0.222,0.183,0.172
10,0.37,1,0.671,0.478,0.343,0.248,0.185,0.145,0.131
11,0.407,1,0.676,0.481,0.345,0.248,0.184,0.145,0.131
12,0.444,1,0.675,0.481,0.343,0.245,0.18,0.141,0.128
13,0.481,1,0.667,0.474,0.339,0.242,0.178,0.141,0.128
14,0.519,1,0.656,0.463,0.332,0.243,0.187,0.155,0.144
15,0.556,1,0.66,0.454,0.306,0.207,0.15,0.113,0.097
16,0.593,1,0.624,0.402,0.255,0.162,0.113,0.085,0.074
17,0.63,1,0.568,0.373,0.25,0.172,0.128,0.099,0.088
18,0.667,1,0.504,0.313,0.198,0.136,0.102,0.08,0.071
19,0.704,1,0.469,0.294,0.188,0.124,0.088,0.065,0.056
20,0.741,1,0.549,0.347,0.216,0.141,0.104,0.081,0.073
21,0.778,1,0.527,0.32,0.189,0.119,0.086,0.067,0.057
22,0.815,1,0.525,0.319,0.181,0.111,0.078,0.057,0.048
23,0.852,1,0.548,0.343,0.202,0.124,0.089,0.064,0.053
24,0.889,1,0.522,0.318,0.184,0.122,0.089,0.067,0.056
25,0.926,1,0.508,0.304,0.177,0.114,0.082,0.061,0.05
26,0.963,1,0.53,0.293,0.138,0.069,0.036,0.02,0.016
```
qwen3-4b — effect(Δ)/effect(Δ1), rows = layers:
```csv
layer,rho,d1,d2,d3.5,d6.5,d12.5,d24.5,d48.5,d96
0,0,1,0.792,0.531,0.364,0.32,0.265,0.256,0.318
1,0.029,1,0.738,0.546,0.378,0.336,0.276,0.264,0.325
2,0.057,1,0.716,0.545,0.38,0.334,0.276,0.266,0.324
3,0.086,1,0.7,0.549,0.373,0.325,0.263,0.248,0.294
4,0.114,1,0.658,0.547,0.36,0.315,0.247,0.226,0.258
5,0.143,1,0.651,0.561,0.366,0.319,0.256,0.241,0.276
6,0.171,1,0.669,0.593,0.388,0.329,0.264,0.253,0.29
7,0.2,1,0.656,0.57,0.373,0.328,0.283,0.275,0.316
8,0.229,1,0.655,0.532,0.364,0.318,0.276,0.266,0.304
9,0.257,1,0.66,0.502,0.365,0.315,0.277,0.267,0.304
10,0.286,1,0.668,0.493,0.373,0.316,0.27,0.255,0.282
11,0.314,1,0.667,0.484,0.369,0.304,0.261,0.245,0.271
12,0.343,1,0.676,0.496,0.375,0.303,0.258,0.239,0.261
13,0.371,1,0.676,0.498,0.379,0.304,0.259,0.241,0.266
14,0.4,1,0.69,0.516,0.396,0.32,0.275,0.257,0.285
15,0.429,1,0.702,0.534,0.412,0.333,0.286,0.271,0.305
16,0.457,1,0.71,0.523,0.384,0.293,0.237,0.207,0.21
17,0.486,1,0.712,0.508,0.345,0.24,0.181,0.153,0.151
18,0.514,1,0.617,0.439,0.312,0.23,0.178,0.15,0.146
19,0.543,1,0.654,0.456,0.313,0.222,0.171,0.146,0.143
20,0.571,1,0.643,0.443,0.301,0.212,0.162,0.138,0.136
21,0.6,1,0.655,0.457,0.312,0.222,0.172,0.15,0.149
22,0.629,1,0.593,0.389,0.247,0.163,0.12,0.097,0.092
23,0.657,1,0.547,0.353,0.233,0.165,0.126,0.105,0.1
24,0.686,1,0.519,0.32,0.2,0.138,0.108,0.092,0.09
25,0.714,1,0.522,0.325,0.205,0.141,0.111,0.095,0.092
26,0.743,1,0.516,0.322,0.202,0.14,0.111,0.094,0.092
27,0.771,1,0.506,0.314,0.195,0.131,0.1,0.082,0.076
28,0.8,1,0.529,0.338,0.213,0.145,0.111,0.091,0.084
29,0.829,1,0.521,0.331,0.208,0.141,0.108,0.089,0.083
30,0.857,1,0.514,0.326,0.206,0.145,0.114,0.095,0.089
31,0.886,1,0.512,0.324,0.208,0.147,0.117,0.099,0.092
32,0.914,1,0.506,0.317,0.198,0.145,0.117,0.102,0.098
33,0.943,1,0.503,0.315,0.197,0.14,0.111,0.091,0.08
34,0.971,1,0.503,0.288,0.167,0.106,0.078,0.058,0.043
```
gemma3-27b-it — effect(Δ)/effect(Δ1), rows = layers:
```csv
layer,rho,d1,d2,d3.5,d6.5,d12.5,d24.5,d48.5,d96
0,0,1,1.214,0.662,0.492,0.297,0.233,0.206,0.236
1,0.016,1,1.226,0.668,0.496,0.299,0.234,0.207,0.237
2,0.033,1,1.248,0.672,0.502,0.303,0.237,0.21,0.241
3,0.049,1,1.081,0.678,0.466,0.304,0.24,0.214,0.244
4,0.066,1,1.067,0.684,0.465,0.306,0.241,0.215,0.246
5,0.082,1,1.052,0.686,0.463,0.305,0.241,0.215,0.244
6,0.098,1,1.038,0.688,0.459,0.305,0.24,0.214,0.241
7,0.115,1,1.025,0.684,0.455,0.305,0.24,0.215,0.242
8,0.131,1,0.97,0.692,0.441,0.301,0.235,0.212,0.239
9,0.148,1,0.921,0.686,0.426,0.296,0.225,0.205,0.226
10,0.164,1,0.899,0.627,0.407,0.284,0.21,0.192,0.206
11,0.18,1,0.855,0.6,0.384,0.282,0.208,0.187,0.197
12,0.197,1,0.755,0.533,0.356,0.269,0.203,0.181,0.19
13,0.213,1,0.754,0.53,0.367,0.278,0.213,0.186,0.197
14,0.23,1,0.77,0.54,0.38,0.294,0.23,0.201,0.21
15,0.246,1,0.77,0.537,0.384,0.303,0.241,0.21,0.216
16,0.262,1,0.758,0.524,0.38,0.302,0.243,0.211,0.215
17,0.279,1,0.745,0.512,0.374,0.297,0.241,0.205,0.2
18,0.295,1,0.719,0.501,0.371,0.298,0.245,0.208,0.2
19,0.311,1,0.695,0.491,0.367,0.298,0.248,0.21,0.2
20,0.328,1,0.679,0.483,0.365,0.297,0.249,0.21,0.198
21,0.344,1,0.667,0.481,0.365,0.298,0.251,0.212,0.197
22,0.361,1,0.659,0.476,0.361,0.297,0.252,0.214,0.198
23,0.377,1,0.65,0.468,0.349,0.279,0.231,0.188,0.166
24,0.393,1,0.647,0.466,0.347,0.278,0.23,0.188,0.165
25,0.41,1,0.64,0.461,0.344,0.275,0.228,0.186,0.162
26,0.426,1,0.63,0.449,0.332,0.265,0.219,0.177,0.152
27,0.443,1,0.613,0.424,0.301,0.233,0.19,0.149,0.122
28,0.459,1,0.605,0.418,0.297,0.231,0.19,0.148,0.121
29,0.475,1,0.596,0.406,0.282,0.218,0.178,0.139,0.112
30,0.492,1,0.585,0.395,0.275,0.213,0.174,0.136,0.109
31,0.508,1,0.576,0.392,0.275,0.217,0.182,0.147,0.124
32,0.525,1,0.57,0.386,0.271,0.215,0.181,0.146,0.123
33,0.541,1,0.565,0.38,0.264,0.21,0.177,0.144,0.121
34,0.557,1,0.56,0.374,0.261,0.208,0.177,0.144,0.121
35,0.574,1,0.556,0.37,0.254,0.202,0.172,0.139,0.117
36,0.59,1,0.555,0.37,0.254,0.202,0.172,0.139,0.117
37,0.607,1,0.544,0.361,0.249,0.2,0.172,0.14,0.119
38,0.623,1,0.545,0.363,0.251,0.202,0.174,0.142,0.12
39,0.639,1,0.54,0.361,0.251,0.204,0.177,0.145,0.123
40,0.656,1,0.54,0.362,0.252,0.204,0.178,0.146,0.124
41,0.672,1,0.551,0.374,0.265,0.22,0.196,0.168,0.148
42,0.689,1,0.549,0.374,0.267,0.224,0.2,0.171,0.152
43,0.705,1,0.537,0.371,0.272,0.232,0.209,0.179,0.159
44,0.721,1,0.538,0.373,0.273,0.235,0.212,0.182,0.162
45,0.738,1,0.538,0.374,0.276,0.237,0.214,0.184,0.163
46,0.754,1,0.539,0.377,0.28,0.242,0.219,0.187,0.166
47,0.77,1,0.536,0.374,0.278,0.239,0.216,0.185,0.164
48,0.787,1,0.53,0.366,0.269,0.233,0.211,0.182,0.162
49,0.803,1,0.532,0.368,0.271,0.234,0.213,0.184,0.163
50,0.82,1,0.536,0.372,0.275,0.238,0.217,0.187,0.166
51,0.836,1,0.534,0.371,0.278,0.244,0.225,0.196,0.175
52,0.852,1,0.541,0.381,0.288,0.255,0.235,0.204,0.183
53,0.869,1,0.526,0.354,0.245,0.204,0.183,0.156,0.137
54,0.885,1,0.527,0.363,0.26,0.22,0.198,0.169,0.148
55,0.902,1,0.509,0.352,0.264,0.229,0.207,0.179,0.158
56,0.918,1,0.523,0.393,0.322,0.298,0.278,0.244,0.217
57,0.934,1,0.534,0.405,0.333,0.309,0.29,0.255,0.227
58,0.951,1,0.62,0.541,0.491,0.478,0.465,0.424,0.397
59,0.967,1,0.433,0.228,0.121,0.07,0.045,0.025,0.017
60,0.984,1,0.342,0.147,0.076,0.044,0.031,0.018,0.013
```
olmo3-32b — effect(Δ)/effect(Δ1), rows = layers:
```csv
layer,rho,d1,d2,d3.5,d6.5,d12.5,d24.5,d48.5,d96
0,0,1,0.68,0.518,0.68,0.31,0.283,0.227,0.271
1,0.016,1,0.684,0.517,0.623,0.308,0.277,0.221,0.256
2,0.032,1,0.692,0.51,0.626,0.31,0.277,0.222,0.252
3,0.048,1,0.693,0.507,0.618,0.312,0.274,0.219,0.238
4,0.063,1,0.662,0.496,0.595,0.308,0.272,0.22,0.235
5,0.079,1,0.671,0.5,0.579,0.314,0.274,0.221,0.228
6,0.095,1,0.683,0.514,0.577,0.329,0.278,0.226,0.23
7,0.111,1,0.698,0.52,0.554,0.335,0.277,0.228,0.232
8,0.127,1,0.701,0.513,0.534,0.334,0.277,0.23,0.231
9,0.143,1,0.708,0.515,0.491,0.335,0.271,0.225,0.223
10,0.159,1,0.703,0.517,0.481,0.334,0.269,0.222,0.216
11,0.175,1,0.701,0.518,0.489,0.343,0.277,0.229,0.223
12,0.19,1,0.69,0.52,0.486,0.347,0.278,0.229,0.22
13,0.206,1,0.689,0.529,0.488,0.356,0.276,0.222,0.202
14,0.222,1,0.684,0.535,0.475,0.346,0.271,0.213,0.186
15,0.238,1,0.69,0.541,0.475,0.35,0.27,0.209,0.177
16,0.254,1,0.692,0.528,0.455,0.328,0.249,0.192,0.167
17,0.27,1,0.666,0.501,0.441,0.309,0.226,0.17,0.145
18,0.286,1,0.638,0.477,0.374,0.288,0.203,0.152,0.128
19,0.302,1,0.612,0.45,0.348,0.266,0.184,0.137,0.113
20,0.317,1,0.619,0.471,0.362,0.276,0.189,0.14,0.118
21,0.333,1,0.608,0.462,0.346,0.256,0.179,0.13,0.106
22,0.349,1,0.577,0.426,0.312,0.228,0.159,0.119,0.1
23,0.365,1,0.58,0.433,0.323,0.231,0.164,0.121,0.102
24,0.381,1,0.566,0.404,0.281,0.204,0.146,0.109,0.094
25,0.397,1,0.54,0.379,0.258,0.187,0.128,0.091,0.085
26,0.413,1,0.513,0.354,0.236,0.17,0.12,0.088,0.088
27,0.429,1,0.504,0.347,0.233,0.167,0.12,0.089,0.09
28,0.444,1,0.492,0.331,0.216,0.155,0.111,0.085,0.091
29,0.46,1,0.474,0.32,0.209,0.154,0.112,0.087,0.095
30,0.476,1,0.455,0.308,0.205,0.152,0.112,0.088,0.099
31,0.492,1,0.444,0.296,0.199,0.148,0.111,0.088,0.099
32,0.508,1,0.441,0.298,0.2,0.149,0.111,0.089,0.1
33,0.524,1,0.438,0.301,0.204,0.153,0.115,0.091,0.105
34,0.54,1,0.436,0.298,0.203,0.151,0.113,0.091,0.105
35,0.556,1,0.431,0.293,0.199,0.15,0.112,0.091,0.105
36,0.571,1,0.43,0.295,0.203,0.153,0.118,0.095,0.108
37,0.587,1,0.446,0.312,0.218,0.168,0.13,0.104,0.113
38,0.603,1,0.446,0.314,0.221,0.171,0.13,0.102,0.111
39,0.619,1,0.44,0.298,0.198,0.144,0.109,0.091,0.111
40,0.635,1,0.426,0.286,0.19,0.145,0.112,0.094,0.113
41,0.651,1,0.423,0.282,0.182,0.132,0.101,0.089,0.115
42,0.667,1,0.425,0.281,0.185,0.134,0.104,0.091,0.116
43,0.683,1,0.423,0.281,0.185,0.134,0.104,0.091,0.117
44,0.698,1,0.427,0.286,0.188,0.137,0.105,0.092,0.12
45,0.714,1,0.399,0.251,0.169,0.13,0.106,0.095,0.122
46,0.73,1,0.395,0.247,0.168,0.128,0.105,0.095,0.122
47,0.746,1,0.399,0.251,0.168,0.129,0.106,0.095,0.121
48,0.762,1,0.398,0.249,0.168,0.129,0.105,0.094,0.121
49,0.778,1,0.405,0.257,0.171,0.131,0.107,0.095,0.12
50,0.794,1,0.406,0.256,0.169,0.129,0.104,0.093,0.117
51,0.81,1,0.403,0.249,0.162,0.119,0.097,0.087,0.111
52,0.825,1,0.399,0.248,0.163,0.121,0.099,0.09,0.114
53,0.841,1,0.395,0.246,0.16,0.118,0.094,0.084,0.106
54,0.857,1,0.395,0.247,0.16,0.119,0.094,0.082,0.103
55,0.873,1,0.368,0.229,0.152,0.117,0.094,0.083,0.102
56,0.889,1,0.364,0.226,0.149,0.113,0.091,0.079,0.098
57,0.905,1,0.362,0.223,0.146,0.112,0.09,0.079,0.094
58,0.921,1,0.408,0.248,0.162,0.122,0.098,0.084,0.098
59,0.937,1,0.399,0.247,0.156,0.112,0.088,0.076,0.093
60,0.952,1,0.377,0.224,0.14,0.098,0.077,0.066,0.082
61,0.968,1,0.374,0.209,0.129,0.086,0.063,0.054,0.06
62,0.984,1,0.464,0.264,0.171,0.121,0.1,0.093,0.121
```

### Fig. 7 — The long-window re-measurement

qwen3-1.7b (seq 4096) — effect(Δ)/effect(Δ1), rows = layers:
```csv
layer,rho,d1,d2,d3.5,d6.5,d12.5,d24.5,d48.5,d96.5,d192.5,d384.5,d768.5,d1536.5,d3072
0,0,1,0.638,0.436,0.312,0.246,0.187,0.126,0.082,0.052,0.032,0.021,0.015,0.015
1,0.037,1,0.641,0.453,0.325,0.257,0.196,0.13,0.085,0.054,0.033,0.022,0.016,0.016
2,0.074,1,0.654,0.48,0.346,0.276,0.211,0.142,0.094,0.06,0.038,0.025,0.018,0.018
3,0.111,1,0.666,0.482,0.34,0.248,0.183,0.122,0.08,0.052,0.033,0.021,0.015,0.014
4,0.148,1,0.66,0.462,0.331,0.25,0.184,0.125,0.084,0.055,0.035,0.023,0.016,0.015
5,0.185,1,0.669,0.477,0.339,0.255,0.189,0.128,0.086,0.057,0.036,0.024,0.017,0.016
6,0.222,1,0.668,0.474,0.335,0.251,0.184,0.124,0.083,0.055,0.035,0.023,0.016,0.015
7,0.259,1,0.67,0.476,0.338,0.251,0.183,0.124,0.084,0.057,0.037,0.024,0.017,0.016
8,0.296,1,0.667,0.471,0.331,0.241,0.172,0.117,0.079,0.053,0.034,0.022,0.015,0.013
9,0.333,1,0.665,0.467,0.319,0.224,0.151,0.096,0.061,0.039,0.023,0.014,0.008,0.006
10,0.37,1,0.647,0.441,0.292,0.197,0.13,0.081,0.05,0.03,0.018,0.011,0.006,0.005
11,0.407,1,0.654,0.448,0.299,0.2,0.131,0.081,0.05,0.031,0.018,0.011,0.006,0.005
12,0.444,1,0.655,0.448,0.297,0.198,0.129,0.08,0.049,0.03,0.018,0.011,0.006,0.005
13,0.481,1,0.647,0.442,0.294,0.196,0.128,0.079,0.048,0.03,0.018,0.01,0.006,0.005
14,0.519,1,0.635,0.43,0.284,0.192,0.13,0.085,0.055,0.035,0.022,0.013,0.008,0.006
15,0.556,1,0.643,0.428,0.27,0.171,0.11,0.068,0.041,0.024,0.014,0.008,0.005,0.003
16,0.593,1,0.608,0.377,0.219,0.128,0.078,0.046,0.028,0.017,0.01,0.006,0.004,0.002
17,0.63,1,0.538,0.335,0.202,0.129,0.084,0.052,0.033,0.02,0.012,0.007,0.004,0.003
18,0.667,1,0.482,0.286,0.167,0.105,0.068,0.043,0.027,0.017,0.01,0.007,0.004,0.002
19,0.704,1,0.441,0.264,0.158,0.098,0.061,0.036,0.021,0.012,0.007,0.004,0.003,0.002
20,0.741,1,0.521,0.317,0.183,0.111,0.071,0.044,0.027,0.015,0.009,0.006,0.003,0.002
21,0.778,1,0.502,0.296,0.161,0.093,0.059,0.036,0.022,0.013,0.008,0.005,0.003,0.002
22,0.815,1,0.497,0.294,0.156,0.087,0.053,0.032,0.018,0.01,0.006,0.004,0.002,0.002
23,0.852,1,0.519,0.316,0.172,0.099,0.061,0.037,0.021,0.012,0.007,0.004,0.002,0.002
24,0.889,1,0.488,0.291,0.155,0.093,0.059,0.036,0.021,0.012,0.007,0.004,0.002,0.002
25,0.926,1,0.475,0.276,0.149,0.088,0.055,0.033,0.019,0.01,0.006,0.003,0.002,0.002
26,0.963,1,0.501,0.279,0.129,0.061,0.031,0.015,0.008,0.003,0.002,0.001,0.001,0
```
gemma3-1b-it (seq 4096) — effect(Δ)/effect(Δ1), rows = layers:
```csv
layer,rho,d1,d2,d3.5,d6.5,d12.5,d24.5,d48.5,d96.5,d192.5,d384.5,d768.5,d1536.5,d3072
0,0,1,0.652,0.453,0.323,0.244,0.184,0.132,0.096,0.072,0.061,0.05,0.031,0.017
1,0.04,1,0.651,0.471,0.344,0.26,0.193,0.134,0.093,0.067,0.053,0.042,0.027,0.015
2,0.08,1,0.68,0.505,0.376,0.284,0.211,0.145,0.101,0.073,0.058,0.045,0.028,0.016
3,0.12,1,0.697,0.528,0.397,0.302,0.221,0.149,0.102,0.073,0.054,0.039,0.024,0.014
4,0.16,1,0.691,0.519,0.379,0.279,0.199,0.132,0.09,0.062,0.044,0.03,0.018,0.011
5,0.2,1,0.701,0.526,0.383,0.279,0.197,0.13,0.088,0.06,0.042,0.024,0.01,0.005
6,0.24,1,0.708,0.531,0.385,0.275,0.192,0.124,0.081,0.052,0.032,0.018,0.009,0.004
7,0.28,1,0.702,0.529,0.389,0.287,0.202,0.13,0.084,0.054,0.034,0.019,0.01,0.005
8,0.32,1,0.685,0.499,0.35,0.246,0.171,0.11,0.073,0.046,0.028,0.017,0.009,0.005
9,0.36,1,0.675,0.481,0.333,0.234,0.165,0.108,0.072,0.046,0.027,0.017,0.009,0.004
10,0.4,1,0.655,0.464,0.323,0.24,0.181,0.127,0.088,0.057,0.034,0.022,0.011,0.005
11,0.44,1,0.635,0.427,0.27,0.177,0.121,0.08,0.055,0.037,0.022,0.014,0.008,0.004
12,0.48,1,0.618,0.411,0.261,0.174,0.12,0.081,0.056,0.037,0.023,0.014,0.008,0.004
13,0.52,1,0.587,0.376,0.233,0.158,0.114,0.08,0.057,0.038,0.024,0.015,0.008,0.004
14,0.56,1,0.589,0.375,0.229,0.155,0.11,0.077,0.055,0.037,0.023,0.014,0.008,0.004
15,0.6,1,0.492,0.285,0.177,0.137,0.114,0.087,0.065,0.045,0.028,0.018,0.01,0.005
16,0.64,1,0.531,0.33,0.212,0.172,0.144,0.11,0.082,0.057,0.036,0.023,0.013,0.007
17,0.68,1,0.521,0.317,0.173,0.105,0.072,0.045,0.027,0.017,0.011,0.004,0.001,0.001
18,0.72,1,0.498,0.296,0.167,0.107,0.074,0.046,0.027,0.017,0.012,0.004,0.002,0.001
19,0.76,1,0.553,0.349,0.201,0.131,0.091,0.057,0.033,0.021,0.014,0.005,0.002,0.001
20,0.8,1,0.531,0.333,0.2,0.129,0.091,0.058,0.036,0.023,0.016,0.006,0.002,0.001
21,0.84,1,0.523,0.332,0.194,0.123,0.082,0.053,0.033,0.022,0.015,0.007,0.002,0.001
22,0.88,1,0.57,0.371,0.216,0.135,0.088,0.057,0.036,0.023,0.015,0.008,0.003,0.002
23,0.92,1,0.556,0.347,0.186,0.103,0.061,0.033,0.018,0.01,0.007,0,0,0
24,0.96,1,0.597,0.364,0.196,0.11,0.061,0.032,0.018,0.01,0.008,0,0,0
```

### Fig. 8 — The same data as one statistic across depth

gemma3-1b-pt:
```csv
layer,rho,r1,h1,flatness,tail_frac
0,0,0.252,3.76,0.892,0.345
1,0.04,0.265,3.46,0.839,0.3
2,0.08,0.22,7.71,0.888,0.413
3,0.12,0.221,20.77,0.952,0.492
4,0.16,0.162,7.05,0.617,0.266
5,0.2,0.203,6.11,0.585,0.238
6,0.24,0.203,5.5,0.516,0.199
7,0.28,0.198,6.09,0.514,0.207
8,0.32,0.191,4.91,0.515,0.19
9,0.36,0.198,4.17,0.525,0.182
10,0.4,0.173,5.75,0.549,0.224
11,0.44,0.165,3.38,0.476,0.128
12,0.48,0.171,3.2,0.461,0.119
13,0.52,0.17,3.2,0.456,0.12
14,0.56,0.156,2.73,0.506,0.117
15,0.6,0.138,2.18,0.601,0.125
16,0.64,0.091,2.39,0.609,0.165
17,0.68,0.094,2.35,0.547,0.086
18,0.72,0.097,2.29,0.434,0.057
19,0.76,0.078,2.03,0.501,0.062
20,0.8,0.073,1.95,0.584,0.064
21,0.84,0.06,2.12,0.601,0.081
22,0.88,0.044,2.35,0.666,0.113
23,0.92,0.034,2.08,0.315,0.031
24,0.96,0.031,2.09,0.337,0.035
```
gemma3-1b-it:
```csv
layer,rho,r1,h1,flatness,tail_frac
0,0,0.291,3.37,0.898,0.272
1,0.04,0.289,3.64,0.755,0.238
2,0.08,0.265,4.36,0.771,0.264
3,0.12,0.258,5.15,0.796,0.295
4,0.16,0.211,4.74,0.722,0.246
5,0.2,0.23,4.71,0.719,0.239
6,0.24,0.228,4.7,0.655,0.214
7,0.28,0.222,4.73,0.654,0.22
8,0.32,0.218,3.96,0.629,0.181
9,0.36,0.217,3.59,0.647,0.18
10,0.4,0.21,3.49,0.704,0.213
11,0.44,0.205,3.01,0.616,0.123
12,0.48,0.202,2.9,0.628,0.124
13,0.52,0.188,2.65,0.702,0.13
14,0.56,0.189,2.65,0.699,0.126
15,0.6,0.148,2.05,0.885,0.153
16,0.64,0.106,2.37,0.892,0.194
17,0.68,0.091,2.26,0.531,0.067
18,0.72,0.086,2.14,0.523,0.066
19,0.76,0.071,2.48,0.524,0.08
20,0.8,0.06,2.4,0.574,0.09
21,0.84,0.051,2.36,0.58,0.086
22,0.88,0.043,2.68,0.588,0.096
23,0.92,0.042,2.54,0.331,0.039
24,0.96,0.035,2.73,0.345,0.043
```
qwen3-1.7b:
```csv
layer,rho,r1,h1,flatness,tail_frac
0,0,0.258,3.38,1.05,0.355
1,0.037,0.249,3.87,1.045,0.375
2,0.074,0.24,4.62,1.08,0.411
3,0.111,0.23,4.17,0.892,0.299
4,0.148,0.234,3.93,0.926,0.317
5,0.185,0.238,4.31,0.944,0.333
6,0.222,0.249,4.15,0.909,0.311
7,0.259,0.246,4.12,0.908,0.309
8,0.296,0.249,3.85,0.809,0.258
9,0.333,0.251,3.67,0.598,0.172
10,0.37,0.234,3.29,0.529,0.131
11,0.407,0.243,3.33,0.531,0.132
12,0.444,0.246,3.32,0.522,0.128
13,0.481,0.243,3.26,0.529,0.128
14,0.519,0.24,3.17,0.593,0.144
15,0.556,0.216,3.11,0.467,0.097
16,0.593,0.205,2.76,0.458,0.074
17,0.63,0.161,2.45,0.508,0.088
18,0.667,0.133,2.03,0.526,0.071
19,0.704,0.118,1.93,0.454,0.056
20,0.741,0.078,2.31,0.515,0.073
21,0.778,0.073,2.16,0.483,0.058
22,0.815,0.063,2.15,0.427,0.047
23,0.852,0.053,2.3,0.422,0.053
24,0.889,0.045,2.13,0.462,0.056
25,0.926,0.038,2.05,0.45,0.051
26,0.963,0.03,2.15,0.236,0.016
```
qwen3-4b:
```csv
layer,rho,r1,h1,flatness,tail_frac
0,0,0.199,3.95,0.992,0.318
1,0.029,0.183,4.17,0.967,0.325
2,0.057,0.178,4.18,0.971,0.324
3,0.086,0.167,4.19,0.905,0.294
4,0.114,0.171,4.11,0.816,0.257
5,0.143,0.167,4.28,0.865,0.276
6,0.171,0.166,4.67,0.881,0.29
7,0.2,0.158,4.4,0.964,0.316
8,0.229,0.155,3.96,0.956,0.304
9,0.257,0.161,3.54,0.967,0.304
10,0.286,0.17,3.43,0.893,0.282
11,0.314,0.177,3.34,0.893,0.271
12,0.343,0.177,3.46,0.863,0.261
13,0.371,0.182,3.49,0.875,0.266
14,0.4,0.191,3.82,0.89,0.285
15,0.429,0.198,4.18,0.917,0.305
16,0.457,0.216,3.89,0.716,0.21
17,0.486,0.237,3.62,0.629,0.151
18,0.514,0.22,2.92,0.634,0.146
19,0.543,0.184,3.11,0.646,0.143
20,0.571,0.187,3.01,0.642,0.136
21,0.6,0.173,3.12,0.67,0.148
22,0.629,0.149,2.61,0.562,0.092
23,0.657,0.123,2.31,0.611,0.101
24,0.686,0.116,2.11,0.651,0.09
25,0.714,0.104,2.14,0.65,0.092
26,0.743,0.1,2.1,0.661,0.092
27,0.771,0.093,2.04,0.579,0.076
28,0.8,0.08,2.19,0.58,0.084
29,0.829,0.079,2.14,0.59,0.083
30,0.857,0.069,2.09,0.616,0.089
31,0.886,0.064,2.08,0.631,0.093
32,0.914,0.058,2.05,0.675,0.097
33,0.943,0.055,2.02,0.566,0.08
34,0.971,0.04,2.01,0.406,0.043
```
gemma3-27b-it:
```csv
layer,rho,r1,h1,flatness,tail_frac
0,0,0.35,6.33,0.795,0.236
1,0.016,0.347,6.41,0.793,0.237
2,0.033,0.338,6.55,0.793,0.241
3,0.049,0.333,5.92,0.804,0.244
4,0.066,0.331,5.92,0.805,0.246
5,0.082,0.332,5.89,0.8,0.244
6,0.098,0.329,5.84,0.791,0.241
7,0.115,0.328,5.79,0.791,0.242
8,0.131,0.353,5.65,0.793,0.239
9,0.148,0.358,5.48,0.762,0.225
10,0.164,0.358,5.04,0.726,0.206
11,0.18,0.344,4.7,0.699,0.197
12,0.197,0.341,3.95,0.707,0.19
13,0.213,0.306,3.95,0.707,0.197
14,0.23,0.277,4.11,0.716,0.21
15,0.246,0.258,4.09,0.714,0.216
16,0.262,0.239,3.9,0.713,0.215
17,0.279,0.232,3.7,0.673,0.2
18,0.295,0.222,3.52,0.671,0.2
19,0.311,0.214,3.42,0.674,0.201
20,0.328,0.21,3.35,0.667,0.198
21,0.344,0.205,3.32,0.664,0.198
22,0.361,0.202,3.27,0.67,0.199
23,0.377,0.201,3.19,0.594,0.166
24,0.393,0.195,3.17,0.593,0.165
25,0.41,0.194,3.12,0.588,0.162
26,0.426,0.201,3.02,0.573,0.152
27,0.443,0.199,2.82,0.525,0.122
28,0.459,0.194,2.77,0.525,0.122
29,0.475,0.193,2.68,0.516,0.113
30,0.492,0.191,2.6,0.515,0.11
31,0.508,0.185,2.54,0.571,0.124
32,0.525,0.182,2.5,0.575,0.124
33,0.541,0.182,2.46,0.578,0.121
34,0.557,0.178,2.42,0.582,0.121
35,0.574,0.178,2.39,0.581,0.118
36,0.59,0.174,2.39,0.579,0.117
37,0.607,0.169,2.31,0.592,0.118
38,0.623,0.163,2.32,0.594,0.12
39,0.639,0.155,2.29,0.606,0.123
40,0.656,0.153,2.29,0.605,0.124
41,0.672,0.158,2.37,0.673,0.148
42,0.689,0.156,2.36,0.677,0.152
43,0.705,0.151,2.28,0.685,0.159
44,0.721,0.147,2.29,0.689,0.162
45,0.738,0.145,2.3,0.688,0.163
46,0.754,0.143,2.31,0.686,0.166
47,0.77,0.142,2.28,0.687,0.164
48,0.787,0.141,2.23,0.696,0.162
49,0.803,0.14,2.24,0.697,0.163
50,0.82,0.138,2.28,0.698,0.166
51,0.836,0.136,2.26,0.718,0.176
52,0.852,0.131,2.33,0.719,0.183
53,0.869,0.129,2.19,0.669,0.137
54,0.885,0.119,2.21,0.673,0.148
55,0.902,0.108,2.07,0.69,0.158
56,0.918,0.088,2.22,0.729,0.217
57,0.934,0.084,2.34,0.734,0.227
58,0.951,0.046,5.89,0.831,0.397
59,0.967,0.036,1.86,0.228,0.016
60,0.984,0.022,1.72,0.262,0.012
```
olmo3-32b:
```csv
layer,rho,r1,h1,flatness,tail_frac
0,0,0.243,8.99,0.873,0.271
1,0.016,0.231,8.44,0.831,0.256
2,0.032,0.215,8.47,0.816,0.252
3,0.048,0.205,8.41,0.763,0.238
4,0.063,0.199,3.46,0.763,0.235
5,0.079,0.188,7.93,0.728,0.228
6,0.095,0.177,7.99,0.701,0.23
7,0.111,0.174,7.67,0.69,0.232
8,0.127,0.172,7.29,0.692,0.231
9,0.143,0.173,5.22,0.665,0.223
10,0.159,0.173,4.72,0.645,0.216
11,0.175,0.171,5.18,0.65,0.223
12,0.19,0.171,5.1,0.633,0.22
13,0.206,0.17,5.46,0.568,0.202
14,0.222,0.171,5.08,0.537,0.186
15,0.238,0.169,5.18,0.505,0.177
16,0.254,0.167,4.47,0.508,0.167
17,0.27,0.165,3.56,0.468,0.145
18,0.286,0.169,3.25,0.442,0.127
19,0.302,0.168,2.97,0.426,0.113
20,0.317,0.153,3.16,0.426,0.117
21,0.333,0.152,3.05,0.414,0.106
22,0.349,0.147,2.69,0.439,0.1
23,0.365,0.12,2.74,0.442,0.102
24,0.381,0.117,2.54,0.461,0.094
25,0.397,0.11,2.32,0.453,0.085
26,0.413,0.104,2.1,0.518,0.088
27,0.429,0.1,2.03,0.534,0.09
28,0.444,0.099,1.98,0.587,0.091
29,0.46,0.093,1.94,0.619,0.095
30,0.476,0.089,1.9,0.646,0.098
31,0.492,0.088,1.88,0.669,0.099
32,0.508,0.087,1.87,0.676,0.101
33,0.524,0.084,1.87,0.687,0.105
34,0.54,0.084,1.87,0.692,0.105
35,0.556,0.084,1.86,0.706,0.106
36,0.571,0.082,1.85,0.702,0.108
37,0.587,0.078,1.88,0.677,0.113
38,0.603,0.077,1.88,0.654,0.112
39,0.619,0.077,1.87,0.772,0.111
40,0.635,0.075,1.85,0.783,0.113
41,0.651,0.072,1.84,0.868,0.115
42,0.667,0.07,1.84,0.873,0.117
43,0.683,0.069,1.84,0.87,0.117
44,0.698,0.069,1.85,0.872,0.119
45,0.714,0.066,1.8,0.94,0.122
46,0.73,0.066,1.8,0.945,0.122
47,0.746,0.064,1.8,0.931,0.121
48,0.762,0.064,1.8,0.934,0.12
49,0.778,0.062,1.81,0.911,0.12
50,0.794,0.061,1.81,0.917,0.118
51,0.81,0.061,1.81,0.926,0.111
52,0.825,0.058,1.8,0.938,0.115
53,0.841,0.057,1.8,0.889,0.106
54,0.857,0.056,1.8,0.863,0.103
55,0.873,0.053,1.76,0.885,0.102
56,0.889,0.053,1.75,0.867,0.098
57,0.905,0.052,1.75,0.839,0.094
58,0.921,0.045,1.82,0.809,0.098
59,0.937,0.041,1.8,0.82,0.092
60,0.952,0.038,1.77,0.826,0.081
61,0.968,0.035,1.76,0.69,0.059
62,0.984,0.014,1.92,0.972,0.118
```

### Fig. 9 — Where the cliff and the boundary sit

SmolLM3-3B through training (shared by Fig. 9 and Fig. 10):
```csv
tokens_T,cliff_rho,boundary_rho,in_band_flatness,band_over_sensory_r1
0.095,0.629,0.457,0.394,0.495
0.19,0.629,0.457,0.408,0.57
0.379,0.629,0.486,0.391,0.666
0.758,0.629,0.543,0.387,0.792
1.517,0.629,0.571,0.39,0.86
3.034,0.629,0.6,0.382,0.999
4.55,0.629,0.6,0.399,1.014
6.067,0.629,0.6,0.395,1.032
7.584,0.629,0.6,0.402,1.073
9.101,0.629,0.6,0.407,1.059
10.428,0.629,0.6,0.384,1.047
11.186,0.6,0.629,0.417,1.026
```

### Fig. 10 — Does the reach ever grow?

Same table as Fig. 9.

### Fig. 11 — The agreement map, checkpoint by checkpoint

smollm3:
```csv
branch,tokens_T,k3_band_lo,k3_band_hi,blockiness,decay_only_null,block_signal
stage1-step-40000,0.095,0.4,0.629,0.222,0.207,0.015
stage1-step-80000,0.19,0.4,0.629,0.278,0.26,0.018
stage1-step-160000,0.379,0.4,0.629,0.253,0.241,0.012
stage1-step-320000,0.758,0.4,0.629,0.265,0.236,0.029
stage1-step-640000,1.517,0.2,0.6,0.285,0.258,0.028
stage1-step-1280000,3.034,0.114,0.6,0.296,0.259,0.037
stage1-step-1920000,4.55,0.4,0.657,0.255,0.215,0.04
stage1-step-2560000,6.067,0.114,0.6,0.257,0.217,0.04
stage1-step-3200000,7.584,0.114,0.6,0.239,0.21,0.029
stage2-step-3840000,9.101,0.114,0.6,0.204,0.189,0.015
stage3-step-4400000,10.428,0.4,0.657,0.204,0.179,0.025
stage3-step-4720000,11.186,0.4,0.686,0.207,0.175,0.032
```
olmo32b:
```csv
branch,tokens_T,k3_band_lo,k3_band_hi,blockiness,decay_only_null,block_signal
stage1-step1000,0.008,0.27,0.635,0.174,0.164,0.01
stage1-step2000,0.017,0.222,0.54,0.111,0.091,0.02
stage1-step5000,0.042,0.191,0.476,0.147,0.104,0.043
stage1-step11000,0.092,0.222,0.46,0.243,0.158,0.085
stage1-step26000,0.218,0.222,0.444,0.3,0.218,0.082
stage1-step58000,0.487,0.27,0.476,0.321,0.251,0.069
stage1-step130000,1.091,0.286,0.444,0.363,0.269,0.094
stage1-step292000,2.449,0.27,0.444,0.304,0.245,0.059
stage1-step656000,5.503,0.365,0.651,0.297,0.247,0.05
stage3-step1000,5.611,0.381,0.651,0.323,0.274,0.05
stage3-step11921,5.703,0.381,0.651,0.311,0.271,0.039
```

### Fig. 12 — How fast the geometry is still changing

smollm3:
```csv
branch,tokens_T,cka_vs_prev,rate_per_T
stage1-step-80000,0.19,0.847,1.614
stage1-step-160000,0.379,0.841,0.839
stage1-step-320000,0.758,0.813,0.493
stage1-step-640000,1.517,0.779,0.291
stage1-step-1280000,3.034,0.759,0.159
stage1-step-1920000,4.55,0.783,0.143
stage1-step-2560000,6.067,0.792,0.137
stage1-step-3200000,7.584,0.77,0.152
stage2-step-3840000,9.101,0.767,0.154
stage3-step-4400000,10.428,0.786,0.161
stage3-step-4720000,11.186,0.89,0.145
```
olmo32b:
```csv
branch,tokens_T,cka_vs_prev,rate_per_T
stage1-step2000,0.017,0.777,26.584
stage1-step5000,0.042,0.802,7.868
stage1-step11000,0.092,0.879,2.404
stage1-step26000,0.218,0.88,0.954
stage1-step58000,0.487,0.893,0.399
stage1-step130000,1.091,0.891,0.18
stage1-step292000,2.449,0.875,0.092
stage1-step656000,5.503,0.863,0.045
stage3-step1000,5.611,0.844,1.439
stage3-step11921,5.703,0.953,0.513
```

### Fig. 13 — Distance to the final state

smollm3:
```csv
branch,tokens_T,cka_vs_final
stage1-step-40000,0.095,0.481
stage1-step-80000,0.19,0.507
stage1-step-160000,0.379,0.544
stage1-step-320000,0.758,0.585
stage1-step-640000,1.517,0.594
stage1-step-1280000,3.034,0.634
stage1-step-1920000,4.55,0.669
stage1-step-2560000,6.067,0.683
stage1-step-3200000,7.584,0.686
stage2-step-3840000,9.101,0.767
stage3-step-4400000,10.428,0.89
stage3-step-4720000,11.186,1
```
olmo32b:
```csv
branch,tokens_T,cka_vs_final
stage1-step1000,0.008,0.525
stage1-step2000,0.017,0.498
stage1-step5000,0.042,0.555
stage1-step11000,0.092,0.61
stage1-step26000,0.218,0.705
stage1-step58000,0.487,0.762
stage1-step130000,1.091,0.774
stage1-step292000,2.449,0.795
stage1-step656000,5.503,0.836
stage3-step1000,5.611,0.953
stage3-step11921,5.703,1
```

### Fig. 14 — Every 7B checkpoint against every 32B checkpoint

Matched-depth CKA, rows = OLMo-7B checkpoints, columns = OLMo-32B checkpoints:
```csv
,8.4B,16.8B,41.9B,92.3B,218.1B,486.5B,1.09T,2.45T,5.50T,5.61T,5.70T
7B@init,0.308,0.301,0.278,0.268,0.252,0.239,0.23,0.206,0.194,0.19,0.171
7B@4.2B,0.593,0.538,0.523,0.522,0.519,0.516,0.507,0.513,0.492,0.454,0.45
7B@16.8B,0.58,0.668,0.694,0.671,0.643,0.619,0.604,0.607,0.583,0.523,0.524
7B@67B,0.552,0.636,0.742,0.763,0.742,0.716,0.7,0.707,0.688,0.594,0.592
```

### Fig. 15 — Finding the right vector in the other model

gemma4b-27b__a2b (cosine retrieval@1):
```csv
rho,svd,ridge,identity,shuffle
0,0.167,0.245,,0.002
0.03,0.165,0.259,,0.002
0.061,0.161,0.256,,0
0.091,0.171,0.26,,0.001
0.121,0.167,0.272,,0.001
0.152,0.199,0.273,,0
0.182,0.41,0.371,,0
0.212,0.533,0.472,,0
0.242,0.589,0.524,,0
0.273,0.678,0.641,,0.001
0.303,0.765,0.624,,0.001
0.333,0.771,0.729,,0.001
0.364,0.769,0.676,,0
0.394,0.813,0.738,,0
0.424,0.815,0.742,,0
0.455,0.837,0.772,,0
0.485,0.846,0.842,,0
0.515,0.845,0.755,,0
0.545,0.861,0.774,,0
0.576,0.863,0.776,,0
0.606,0.855,0.784,,0
0.636,0.852,0.763,,0.001
0.667,0.837,0.774,,0
0.697,0.857,0.776,,0
0.727,0.856,0.763,,0
0.758,0.87,0.79,,0
0.788,0.851,0.766,,0
0.818,0.848,0.766,,0
0.848,0.816,0.748,,0.001
0.879,0.874,0.739,,0.001
0.909,0.862,0.683,,0.001
0.939,0.828,0.693,,0.001
0.97,0.913,0.828,,0.001
```
gemma4b-27b__b2a (cosine retrieval@1):
```csv
rho,svd,ridge,identity,shuffle
0,0.129,0.483,,0
0.016,0.135,0.528,,0
0.033,0.135,0.534,,0
0.049,0.137,0.513,,0
0.066,0.137,0.511,,0
0.082,0.139,0.502,,0
0.098,0.151,0.502,,0
0.115,0.151,0.505,,0
0.131,0.129,0.508,,0
0.148,0.16,0.538,,0
0.164,0.182,0.539,,0
0.18,0.291,0.715,,0
0.197,0.374,0.716,,0
0.213,0.452,0.734,,0
0.23,0.472,0.738,,0
0.246,0.491,0.74,,0
0.262,0.55,0.751,,0
0.279,0.641,0.751,,0
0.295,0.701,0.801,,0
0.311,0.716,0.798,,0
0.328,0.72,0.795,,0
0.344,0.722,0.798,,0
0.361,0.742,0.794,,0
0.377,0.794,0.793,,0
0.393,0.783,0.796,,0
0.41,0.784,0.788,,0
0.426,0.802,0.751,,0
0.443,0.838,0.85,,0
0.459,0.846,0.784,,0
0.475,0.838,0.795,,0
0.492,0.85,0.802,,0
0.508,0.827,0.771,,0
0.525,0.839,0.782,,0
0.541,0.856,0.793,,0
0.557,0.865,0.796,,0
0.574,0.871,0.808,,0
0.59,0.884,0.813,,0
0.607,0.874,0.802,,0
0.623,0.879,0.79,,0
0.639,0.876,0.796,,0
0.656,0.877,0.776,,0
0.672,0.879,0.771,,0
0.689,0.871,0.783,,0
0.705,0.873,0.785,,0
0.721,0.86,0.766,,0
0.738,0.865,0.772,,0
0.754,0.862,0.766,,0
0.77,0.859,0.768,,0
0.787,0.828,0.726,,0
0.803,0.818,0.702,,0
0.82,0.816,0.7,,0
0.836,0.798,0.658,,0
0.852,0.8,0.662,,0
0.869,0.828,0.74,,0
0.885,0.828,0.732,,0
0.902,0.815,0.704,,0
0.918,0.79,0.689,,0
0.934,0.776,0.679,,0
0.951,0.8,0.692,,0
0.967,0.896,0.782,,0
0.984,0.895,0.776,,0
```
llama8b-qwen8b__a2b (cosine retrieval@1):
```csv
rho,svd,ridge,identity,shuffle
0,0.123,0.205,0,0
0.032,0.127,0.205,0,0
0.065,0.133,0.2,0.001,0
0.097,0.173,0.205,0,0
0.129,0.25,0.277,0,0
0.161,0.273,0.307,0.001,0
0.194,0.285,0.331,0,0
0.226,0.318,0.381,0,0
0.258,0.339,0.406,0,0
0.29,0.444,0.439,0.001,0
0.323,0.454,0.344,0,0
0.355,0.523,0.363,0.001,0
0.387,0.551,0.394,0.001,0
0.419,0.618,0.413,0.001,0
0.452,0.694,0.433,0,0
0.484,0.777,0.446,0.001,0
0.516,0.845,0.461,0,0
0.548,0.907,0.496,0,0
0.581,0.913,0.482,0,0
0.613,0.917,0.478,0,0
0.645,0.922,0.502,0.001,0
0.677,0.904,0.487,0.001,0
0.71,0.899,0.489,0,0
0.742,0.893,0.487,0,0
0.774,0.877,0.459,0,0
0.806,0.867,0.44,0,0
0.839,0.872,0.422,0,0
0.871,0.857,0.398,0.001,0
0.903,0.84,0.35,0.001,0
0.935,0.791,0.3,0.001,0
0.968,0.739,0.129,0.001,0
```
llama8b-qwen8b__b2a (cosine retrieval@1):
```csv
rho,svd,ridge,identity,shuffle
0,0.027,0.305,0,0
0.029,0.026,0.32,0,0.001
0.057,0.032,0.367,0,0
0.086,0.039,0.402,0,0.001
0.114,0.044,0.504,0,0
0.143,0.048,0.507,0,0
0.171,0.046,0.543,0,0
0.2,0.048,0.552,0,0
0.229,0.048,0.557,0,0
0.257,0.052,0.568,0,0
0.286,0.087,0.658,0,0
0.314,0.078,0.677,0,0
0.343,0.098,0.705,0.001,0.001
0.371,0.112,0.721,0,0
0.4,0.126,0.732,0.001,0
0.429,0.148,0.752,0,0
0.457,0.217,0.791,0.001,0
0.486,0.299,0.813,0,0
0.514,0.373,0.829,0,0
0.543,0.701,0.696,0,0
0.571,0.69,0.694,0,0
0.6,0.696,0.706,0.001,0
0.629,0.796,0.756,0,0
0.657,0.802,0.768,0,0
0.686,0.79,0.761,0,0
0.714,0.769,0.761,0,0
0.743,0.758,0.756,0,0
0.771,0.763,0.76,0,0
0.8,0.76,0.754,0,0
0.829,0.752,0.743,0,0
0.857,0.709,0.723,0,0
0.886,0.699,0.726,0,0
0.914,0.706,0.716,0,0
0.943,0.674,0.643,0,0
0.971,0.633,0.589,0.001,0
```

### Fig. 16 — Making the receiver say the concept

gemma4b-27b__a2b · receiver layer L30 (ρ=0.49) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0.018,0.021,0.711,0,0,0
0.906,0.134,0.039,0.821,0,0,0
1.641,0.223,0.042,0.839,0,0,0
2.972,0.295,0.042,0.804,0,0,0
5.384,0.271,0.039,0.807,0,0,0
9.752,0.25,0.024,0.804,0,0,0
17.665,0.253,0.021,0.795,0,0,0
32,0.25,0.021,0.792,0,0,0
```
gemma4b-27b__a2b · receiver layer L33 (ρ=0.54) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0.036,0.021,0.857,0,0,0
0.906,0.277,0.039,0.917,0,0,0
1.641,0.292,0.027,0.935,0,0,0
2.972,0.333,0.048,0.914,0,0,0
5.384,0.333,0.042,0.878,0,0,0
9.752,0.333,0.042,0.881,0,0,0
17.665,0.333,0.042,0.893,0,0,0
32,0.333,0.042,0.896,0,0,0
```
gemma4b-27b__a2b · receiver layer L36 (ρ=0.59) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0.155,0.027,0.938,0,0,0
0.906,0.482,0.083,0.982,0,0,0
1.641,0.548,0.101,0.958,0,0,0
2.972,0.574,0.107,0.958,0,0,0
5.384,0.592,0.149,0.958,0,0,0
9.752,0.583,0.188,0.958,0,0,0
17.665,0.58,0.188,0.958,0,0,0
32,0.598,0.188,0.958,0,0,0
```
gemma4b-27b__a2b · receiver layer L39 (ρ=0.64) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0.39,0.042,0.958,0,0,0
0.906,0.601,0.057,1,0,0,0
1.641,0.714,0.086,1,0,0,0
2.972,0.756,0.104,1,0,0,0
5.384,0.75,0.11,1,0,0,0
9.752,0.75,0.125,1,0,0,0
17.665,0.75,0.125,1,0,0,0
32,0.75,0.146,1,0,0,0
```
gemma4b-27b__b2a · receiver layer L16 (ρ=0.48) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0.125,0.033,0.836,0,0,0
0.906,0.146,0.033,0.899,0,0,0
1.641,0.161,0.036,0.97,0,0,0
2.972,0.152,0.06,0.979,0,0,0
5.384,0.158,0.062,0.979,0,0,0
9.752,0.167,0.062,0.97,0,0,0
17.665,0.167,0.062,0.958,0,0,0
32,0.167,0.074,0.958,0,0,0
```
gemma4b-27b__b2a · receiver layer L18 (ρ=0.55) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0.214,0.119,0.961,0,0,0
0.906,0.262,0.143,0.979,0,0,0
1.641,0.31,0.146,0.979,0,0,0
2.972,0.315,0.146,0.979,0,0,0
5.384,0.298,0.125,0.979,0,0,0
9.752,0.292,0.125,0.979,0,0,0
17.665,0.292,0.125,0.979,0,0,0
32,0.271,0.125,0.979,0,0,0
```
gemma4b-27b__b2a · receiver layer L21 (ρ=0.64) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0.506,0.205,1,0,0,0
0.906,0.542,0.188,1,0,0,0
1.641,0.56,0.196,1,0,0,0
2.972,0.56,0.223,1,0,0,0
5.384,0.56,0.229,1,0,0,0
9.752,0.56,0.229,1,0,0,0
17.665,0.542,0.229,1,0,0,0
32,0.542,0.229,1,0,0,0
```
llama8b-qwen8b__a2b · receiver layer L17 (ρ=0.49) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0,0,0,0,0,0
0.906,0,0,0,0,0,0
1.641,0.051,0,0,0,0,0
2.972,0.411,0,0,0,0,0
5.384,0.622,0,0,0,0,0
9.752,0.821,0,0.051,0,0,0
17.665,0.872,0,0.61,0,0,0
32,0.94,0.104,1,0,0,0
```
llama8b-qwen8b__a2b · receiver layer L20 (ρ=0.57) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0,0,0,0,0,0
0.906,0.003,0,0,0,0,0
1.641,0.128,0,0,0,0,0
2.972,0.601,0,0,0,0,0
5.384,0.842,0,0.057,0,0,0
9.752,0.878,0,0.818,0,0,0
17.665,0.881,0.039,1,0,0,0
32,0.884,0.589,1,0,0,0
```
llama8b-qwen8b__a2b · receiver layer L23 (ρ=0.66) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0,0,0.003,0,0,0
0.906,0.071,0,0.104,0,0,0
1.641,0.372,0,0.616,0,0,0
2.972,0.667,0.006,0.967,0,0,0
5.384,0.78,0.039,1,0,0,0
9.752,0.804,0.211,1,0,0,0
17.665,0.804,0.75,1,0,0,0
32,0.795,0.818,1,0,0,0
```
llama8b-qwen8b__b2a · receiver layer L15 (ρ=0.48) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0,0,0.003,0,0,0
0.906,0.003,0,0.205,0,0,0
1.641,0.018,0.006,0.97,0,0,0
2.972,0.003,0.083,1,0,0,0
5.384,0,0.146,0.911,0,0,0
9.752,0,0,0.625,0,0,0
17.665,0,0,0.988,0,0,0
32,0.104,0.033,1,0,0,0
```
llama8b-qwen8b__b2a · receiver layer L17 (ρ=0.55) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0,0,0.182,0,0,0
0.906,0.009,0.021,0.932,0,0,0
1.641,0.077,0.211,0.997,0,0,0
2.972,0.11,0.747,0.994,0,0,0
5.384,0.045,0.101,0.693,0,0,0
9.752,0.018,0,0.905,0,0,0
17.665,0.211,0.051,1,0,0,0
32,0.185,0.116,1,0,0,0
```
llama8b-qwen8b__b2a · receiver layer L20 (ρ=0.65) — success@1:
```csv
alpha,mapped_svd,mapped_ridge,own,rand_orth,shuffled,rand_vec
0.5,0,0.006,0.655,0,0,0
0.906,0.11,0.161,0.967,0,0,0
1.641,0.711,0.804,0.973,0,0,0
2.972,0.619,0.961,0.952,0,0,0
5.384,0.223,0.244,0.923,0,0,0
9.752,0.345,0.188,1,0,0,0
17.665,0.503,0.268,1,0,0,0
32,0.542,0.375,1,0,0,0
```

### Fig. 17 — What the hits and misses look like

```csv
direction,kind,concept_sent,receiver_said,concept_rank
gemma4b-27b__a2b,hit,sapphire,sapphire,1
gemma4b-27b__a2b,hit,gypsum,gypsum,1
gemma4b-27b__a2b,hit,emblem,emblem,1
gemma4b-27b__a2b,hit,beaker,beaker,1
gemma4b-27b__a2b,hit,auditorium,auditorium,1
gemma4b-27b__a2b,hit,parlour,parlour,1
gemma4b-27b__a2b,hit,pillows,pillows,1
gemma4b-27b__a2b,hit,halls,halls,1
gemma4b-27b__a2b,near,rose,k,2
gemma4b-27b__a2b,near,chainsaw,shrubs,2
gemma4b-27b__a2b,near,priests,preachers,2
gemma4b-27b__a2b,near,spain,Spain,2
gemma4b-27b__a2b,near,Hernandez,Garcia,2
gemma4b-27b__a2b,near,soundtrack,soundtracks,3
gemma4b-27b__a2b,near,genie,to,4
gemma4b-27b__a2b,near,haunted,Haunted,4
gemma4b-27b__b2a,hit,sapphire,sapphire,1
gemma4b-27b__b2a,hit,emblem,emblem,1
gemma4b-27b__b2a,hit,auditorium,auditorium,1
gemma4b-27b__b2a,hit,pillows,pillows,1
gemma4b-27b__b2a,hit,halls,halls,1
gemma4b-27b__b2a,hit,sewing,sewing,1
gemma4b-27b__b2a,hit,priests,priests,1
gemma4b-27b__b2a,hit,uncles,uncles,1
gemma4b-27b__b2a,near,soundtrack,soundtracks,2
gemma4b-27b__b2a,near,giants,titans,2
gemma4b-27b__b2a,near,endurance,dürü,2
gemma4b-27b__b2a,near,seven,nine,2
gemma4b-27b__b2a,near,road,roads,2
gemma4b-27b__b2a,near,Buckingham,shire,2
gemma4b-27b__b2a,near,Salisbury,Harcourt,2
gemma4b-27b__b2a,near,handwriting,handwritten,3
llama8b-qwen8b__a2b,hit,grape,grape,1
llama8b-qwen8b__a2b,hit,needle,needle,1
llama8b-qwen8b__a2b,hit,compass,compass,1
llama8b-qwen8b__a2b,hit,ceiling,ceiling,1
llama8b-qwen8b__a2b,hit,feathers,feathers,1
llama8b-qwen8b__a2b,hit,pottery,pottery,1
llama8b-qwen8b__a2b,hit,pudding,pudding,1
llama8b-qwen8b__a2b,hit,restaurant,restaurant,1
llama8b-qwen8b__a2b,near,doorway,door,2
llama8b-qwen8b__a2b,near,academy,acad,2
llama8b-qwen8b__a2b,near,Chester,Club,7
llama8b-qwen8b__b2a,hit,grape,grape,1
llama8b-qwen8b__b2a,hit,needle,needle,1
llama8b-qwen8b__b2a,hit,compass,compass,1
llama8b-qwen8b__b2a,hit,ceiling,ceiling,1
llama8b-qwen8b__b2a,hit,doorway,doorway,1
llama8b-qwen8b__b2a,hit,feathers,feathers,1
llama8b-qwen8b__b2a,hit,pottery,pottery,1
llama8b-qwen8b__b2a,hit,pudding,pudding,1
llama8b-qwen8b__b2a,near,warfare,war,2
llama8b-qwen8b__b2a,near,intimacy,intim,2
```

### Fig. 18 — Dictionary size across five decades of compute

Ladder (band PR = max PR over ρ∈[0.25,0.75]):
```csv
model,params,flops,band_pr,cum90,seed_prs
delphi-3e18-447Mparams-1.2Btokens,447244032,3000000000000000000,205.9,49,
delphi-9e18-550Mparams-2.9Btokens,550337664,9000000000000000000,230.2,49,
delphi-2e19-837Mparams-3.6Btokens,837007744,20000000000000000000,269,47,
delphi-3e19-998Mparams-5Btokens,998036992,30000000000000000000,297.3,47,
delphi-9e19-1.4Bparams-10.6Btokens,1384584448,90000000000000000000,320.9,40,
delphi-2e20-1.9Bparams-14.8Btokens,1934716160,200000000000000000000,397.9,38,
delphi-3e20-2.5Bparams-18.6Btokens,2544614912,300000000000000000000,389.1,33,
delphi-1e21-3.4Bparams-46.3Btokens,3383110656,1000000000000000000000,386.2,13,386 / 385 / 375
delphi-1e22-9.7Bparams-160Btokens,9714698752,10000000000000000000000,443.9,5,444 / 492 / 344
delphi-1e23-25Bparams-628Btokens,24963098112,99999999999999991611392,378.7,5,
```
Fitted laws (predicted band PR):
```csv
flops,single_law,broken_at_2.1e20,smooth_saturation
3,245,202,196
3579,248,207,204
427,251,213,211
5094,254,219,218
6078,257,225,226
7251,26,231,233
8651,263,237,241
1032,266,244,248
1231,269,25,256
1469,272,257,264
1753,275,264,271
2091,279,272,279
2495,282,279,287
2976,285,287,294
3551,288,294,302
4236,292,302,309
5054,295,311,316
603,299,319,323
7194,302,328,33
8583,306,337,337
1024,309,346,343
1222,313,356,349
1458,316,366,355
1739,32,376,361
2075,324,386,366
2475,328,396,371
2953,331,396,376
3523,335,397,38
4203,339,397,384
5015,343,397,388
5983,347,397,391
7138,351,397,394
8515,355,397,397
1016,359,398,399
1212,364,398,402
1446,368,398,403
1725,372,398,405
2058,376,398,406
2456,381,398,408
293,385,398,409
3495000000000000262144,39,398,41
417,394,399,41
4975000000000000262144,399,399,411
5935000000000000262144,403,399,411
7080999999999999737856,408,399,412
8448,413,399,412
1008,418,399,412
12030000000000000524288,422,4,412
14350000000000000524288,427,4,412
1712,432,4,412
20419999999999998951424,437,4,412
24359999999999997902848,442,4,412
29070000000000000524288,448,4,412
34680000000000002097152,453,4,412
4136999999999999737856,458,4,413
49359999999999995805696,463,401,413
58890000000000001572864,469,401,413
70260000000000003145728,474,401,413
83819999999999996854272,48,401,413
99999999999999991611392,485,401,413
```

### Fig. 19 — How many entries a single token engages

cum90 column in the Fig. 18 table.

### Fig. 20 — Same recipe, different seed

```csv
scale,seed_pair_ckas,band_mean,anchor_cross_family,anchor_adjacent_layer
3.4B@1e21,0.846 / 0.844 / 0.85,0.847,0.76,0.995
9.7B@1e22,0.904 / 0.914 / 0.903,0.907,,
```

### Fig. 21 — The matrices behind the dots

```csv
scale,seed_pair,band_mean_cka
3.4B · 1e21,base × s42,0.846
3.4B · 1e21,base × s62746,0.844
3.4B · 1e21,s42 × s62746,0.85
9.7B · 1e22,base × s42,0.904
9.7B · 1e22,base × s62746,0.914
9.7B · 1e22,s42 × s62746,0.903
```
Full matrices: page Fig. 21 hover, or repo.

### Fig. 22 — The same layer, measured through two different texts

gemma3-1b-pt (band = ρ∈[0.25, 0.75]):
```csv
rho,code_x_wikitext,math_x_wikitext,finepdfs_x_wikitext,floor_wikitext_x_orig
0,0.318,0.464,0.177,0.989
0.04,0.281,0.424,0.139,0.987
0.08,0.232,0.377,0.1,0.986
0.12,0.222,0.383,0.099,0.987
0.16,0.463,0.64,0.284,0.994
0.2,0.372,0.598,0.27,0.992
0.24,0.413,0.655,0.385,0.992
0.28,0.433,0.667,0.419,0.993
0.32,0.488,0.698,0.49,0.994
0.36,0.498,0.715,0.507,0.994
0.4,0.642,0.851,0.654,0.997
0.44,0.704,0.888,0.75,0.999
0.48,0.701,0.885,0.755,0.998
0.52,0.701,0.879,0.744,0.998
0.56,0.757,0.916,0.765,0.999
0.6,0.742,0.898,0.787,0.999
0.64,0.763,0.904,0.81,0.999
0.68,0.793,0.923,0.852,0.999
0.72,0.824,0.932,0.859,0.999
0.76,0.874,0.959,0.895,0.999
0.8,0.886,0.965,0.904,1
0.84,0.895,0.968,0.912,1
0.88,0.892,0.974,0.906,1
0.92,0.915,0.985,0.925,1
0.96,0.925,0.99,0.952,1
```
qwen3-1.7b (band = ρ∈[0.25, 0.75]):
```csv
rho,code_x_wikitext,math_x_wikitext,finepdfs_x_wikitext,floor_wikitext_x_orig
0,0.297,0.607,0.415,0.998
0.037,0.283,0.585,0.384,0.998
0.074,0.283,0.593,0.392,0.997
0.111,0.358,0.773,0.573,0.997
0.148,0.356,0.777,0.54,0.997
0.185,0.318,0.752,0.505,0.996
0.222,0.337,0.747,0.498,0.996
0.259,0.322,0.719,0.482,0.996
0.296,0.395,0.759,0.583,0.996
0.333,0.478,0.819,0.634,0.997
0.37,0.529,0.842,0.685,0.998
0.407,0.495,0.83,0.646,0.997
0.444,0.508,0.839,0.651,0.998
0.481,0.52,0.849,0.651,0.998
0.519,0.489,0.848,0.654,0.998
0.556,0.552,0.87,0.698,0.998
0.593,0.607,0.878,0.696,0.999
0.63,0.645,0.886,0.694,0.999
0.667,0.681,0.899,0.761,0.999
0.704,0.74,0.921,0.803,0.999
0.741,0.793,0.937,0.835,0.999
0.778,0.833,0.951,0.856,1
0.815,0.875,0.966,0.888,1
0.852,0.882,0.969,0.92,1
0.889,0.905,0.972,0.929,1
0.926,0.932,0.982,0.952,1
0.963,0.858,0.98,0.909,1
```
qwen3-4b (band = ρ∈[0.25, 0.75]):
```csv
rho,code_x_wikitext,math_x_wikitext,finepdfs_x_wikitext,floor_wikitext_x_orig
0,0.274,0.526,0.314,0.993
0.029,0.265,0.518,0.317,0.994
0.057,0.264,0.523,0.323,0.995
0.086,0.307,0.622,0.385,0.996
0.114,0.358,0.71,0.424,0.998
0.143,0.361,0.71,0.408,0.998
0.171,0.334,0.707,0.369,0.997
0.2,0.305,0.698,0.382,0.997
0.229,0.351,0.72,0.442,0.996
0.257,0.361,0.718,0.456,0.997
0.286,0.352,0.695,0.433,0.998
0.314,0.338,0.662,0.444,0.998
0.343,0.335,0.687,0.496,0.998
0.371,0.337,0.679,0.497,0.998
0.4,0.316,0.654,0.467,0.998
0.429,0.314,0.634,0.466,0.998
0.457,0.347,0.655,0.507,0.998
0.486,0.394,0.715,0.542,0.998
0.514,0.408,0.742,0.559,0.998
0.543,0.434,0.765,0.592,0.998
0.571,0.451,0.773,0.59,0.998
0.6,0.458,0.772,0.594,0.999
0.629,0.557,0.835,0.681,0.998
0.657,0.611,0.847,0.688,0.999
0.686,0.678,0.883,0.753,0.999
0.714,0.7,0.891,0.77,0.999
0.743,0.708,0.886,0.774,0.999
0.771,0.784,0.927,0.818,0.999
0.8,0.807,0.934,0.842,0.999
0.829,0.821,0.939,0.847,0.999
0.857,0.85,0.949,0.866,0.999
0.886,0.856,0.952,0.881,1
0.914,0.882,0.957,0.909,1
0.943,0.912,0.973,0.922,1
0.971,0.968,0.992,0.963,1
```
delphi-3.4b (band = ρ∈[0.25, 0.75]):
```csv
rho,code_x_wikitext,math_x_wikitext,finepdfs_x_wikitext,floor_wikitext_x_orig
0,0.454,0.71,0.584,0.996
0.04,0.497,0.748,0.599,0.996
0.08,0.531,0.775,0.611,0.996
0.12,0.567,0.821,0.661,0.998
0.16,0.582,0.838,0.691,0.998
0.2,0.597,0.848,0.701,0.998
0.24,0.6,0.858,0.712,0.998
0.28,0.612,0.859,0.727,0.998
0.32,0.649,0.888,0.768,0.999
0.36,0.667,0.896,0.782,0.999
0.4,0.684,0.898,0.799,0.999
0.44,0.734,0.918,0.823,0.999
0.48,0.778,0.932,0.834,0.999
0.52,0.786,0.94,0.84,0.999
0.56,0.813,0.948,0.849,0.999
0.6,0.834,0.95,0.863,0.999
0.64,0.856,0.959,0.877,1
0.68,0.88,0.966,0.886,1
0.72,0.888,0.969,0.892,1
0.76,0.898,0.971,0.894,1
0.8,0.908,0.974,0.9,1
0.84,0.926,0.98,0.905,1
0.88,0.942,0.987,0.917,1
0.92,0.919,0.99,0.923,1
0.96,0.958,0.993,0.961,1
```

### Fig. 23 — Dictionary size depends on the measuring text

gemma3-1b-pt:
```csv
rho,pr_code,pr_math,pr_finepdfs,pr_wikitext
0,44.4,47.3,10.9,17.1
0,36.1,39.7,9.8,11
0.1,26.6,28.3,7.7,8.1
0.1,22.7,20.7,7.4,6.4
0.2,30.8,33.8,15.6,16.7
0.2,36.4,39.9,19,21.4
0.2,43,38.2,26.1,25.6
0.3,48,39.8,28.3,27.7
0.3,55.7,42.7,29.1,30.9
0.4,55.6,42.9,26.8,30.8
0.4,39.6,30.8,13.7,22.6
0.4,90.5,82,48.6,60.1
0.5,93.9,86.6,52.6,66.8
0.5,91.9,87.4,52.4,72
0.6,86.4,77.2,36.9,58.2
0.6,105.5,96.6,63.1,76.4
0.6,111.8,105.6,75.8,87.7
0.7,114.5,106.8,82.1,89.6
0.7,124.3,125.6,96.5,113.7
0.8,116.6,120.1,93.6,109.2
0.8,113.5,118.8,93.6,107.8
0.8,114.6,120.2,99.5,108.3
0.9,105.7,104.4,92.1,90.8
0.9,95.4,100.1,93.2,94.5
1,98.7,117.6,105.3,117.3
```
qwen3-1.7b:
```csv
rho,pr_code,pr_math,pr_finepdfs,pr_wikitext
0,35.8,43.2,19,17.3
0,32.7,37.6,17.6,14.2
0.1,26.6,31.8,16.8,13.7
0.1,37.7,41.8,20.6,25.2
0.1,38.4,42.8,19.8,24.9
0.2,31.8,39.5,17.4,21.7
0.2,36.3,41.4,17.4,22.5
0.3,35.9,45.7,19.6,22.9
0.3,61.4,49.2,24.3,29.5
0.3,81.9,60.2,24.2,38.4
0.4,102,80.1,31.9,56.9
0.4,98.7,80.7,28.1,50.9
0.4,104.3,85.3,29.2,54.4
0.5,108.6,84.6,29,56
0.5,104.9,87.8,29.8,55
0.6,132.8,115.2,33.3,73.3
0.6,156.4,129.6,28.9,86.5
0.6,166.6,130,22.7,87.3
0.7,264.4,205.9,68.4,150.4
0.7,320.9,275,104.1,228.4
0.7,364,292.4,110.3,239.6
0.8,418.8,322.8,131.7,270.1
0.8,467.6,360.9,164.9,323
0.9,493.1,384.2,210.2,343.8
0.9,544.3,463.6,274.9,441
0.9,658.7,584.3,419.4,554.1
1,422,295.2,299.7,250.8
```
qwen3-4b:
```csv
rho,pr_code,pr_math,pr_finepdfs,pr_wikitext
0,49.2,41.8,21.2,14
0,46.1,40,21.7,13.4
0.1,45.9,40.2,22.4,13.6
0.1,47.5,40.4,21.9,19.6
0.1,56.4,40.8,19.3,26
0.1,60.3,42.4,18.2,26
0.2,52.2,39.5,15.6,23
0.2,47.2,41.1,16.4,23.2
0.2,56.6,42.1,18.8,27.4
0.3,62.1,45.1,19.9,26.1
0.3,80,52.3,19.9,25.6
0.3,86.9,60.2,24.3,24.2
0.3,92.4,70.5,28.1,28.6
0.4,96.5,74.4,28.9,27.7
0.4,93.7,72.7,28.8,22.2
0.4,92.1,67.7,30.6,18.3
0.5,100.4,69.4,33.5,22.5
0.5,111.9,78.1,35.7,31.2
0.5,124.5,90.6,37.8,38.7
0.5,150.3,118.9,41.3,56.7
0.6,167.1,131.5,42.4,61.7
0.6,187.7,140,44.7,61.4
0.6,284.6,240.5,84.6,148.9
0.7,368.6,316,104.4,206.2
0.7,474.3,394.1,148.4,266.6
0.7,537.8,449.6,179.1,307.3
0.7,594.4,498,201.7,328.5
0.8,645.1,547.9,238.8,445.8
0.8,699.4,595.1,288.1,496.6
0.8,736.2,624.6,300.9,523.4
0.9,784.9,688.7,345.7,596.4
0.9,813.1,705.7,372.8,605.8
0.9,856.1,779.1,498.1,696.5
0.9,886.1,800,541.1,749.6
1,996,982.3,836.5,985
```
delphi-3.4b:
```csv
rho,pr_code,pr_math,pr_finepdfs,pr_wikitext
0,86.3,110.9,40.6,39
0,104.1,124.6,42,42.6
0.1,118.6,129.9,45.2,44.1
0.1,135.5,137.8,50.9,50.4
0.2,149,142.4,55,54.5
0.2,154.6,145.2,60.1,56.6
0.2,172.2,149.9,67.1,57.3
0.3,184.8,159.9,77.1,67.2
0.3,204.6,184.7,82.6,85.2
0.4,212.5,190,89.6,98
0.4,239.3,217.6,110.4,116
0.4,290.9,280,130.7,165.1
0.5,330.9,312.2,143,195.6
0.5,377.1,344.4,169.8,234.9
0.6,424.7,375.4,189.1,259.7
0.6,433.2,405.1,200.5,310.1
0.6,468.3,430.7,214.8,332.5
0.7,509.4,472.2,235.5,377.1
0.7,530.6,480.6,251.9,387.7
0.8,551.4,491,263.2,397.2
0.8,565.9,506.3,278.7,421
0.8,588,531.1,294.7,453.2
0.9,596.5,531.7,311.1,479.7
0.9,624.5,467.9,348.8,404.4
1,700.1,634.3,512,583.4
```

### Fig. 24 — The block structure survives every corpus

Maps shown on the page; summary (band-peak PR per fitting corpus):
```csv
model,code,math,finepdfs,wikitext
gemma3-1b-pt,124,126,96,114
qwen3-1.7b,364,292,11,24
qwen3-4b,594,498,202,328
delphi-3.4b,531,481,252,388
```

### Fig. 25 — Two models, measured on the same text

qwen3-1.7b__qwen3-4b (matched-depth CKA):
```csv
rho,code,math,finepdfs,wikitext,wikitext_orig
0,0.273,0.535,0.58,0.502,0.505
0.037,0.281,0.546,0.581,0.496,0.499
0.074,0.273,0.555,0.593,0.534,0.535
0.111,0.341,0.62,0.63,0.547,0.554
0.148,0.356,0.614,0.641,0.547,0.553
0.185,0.368,0.644,0.634,0.559,0.565
0.222,0.392,0.636,0.65,0.609,0.612
0.259,0.406,0.635,0.658,0.599,0.601
0.296,0.43,0.644,0.678,0.601,0.599
0.333,0.493,0.67,0.656,0.653,0.656
0.37,0.516,0.683,0.682,0.631,0.633
0.407,0.546,0.697,0.669,0.64,0.642
0.444,0.571,0.687,0.701,0.652,0.654
0.481,0.583,0.701,0.714,0.664,0.666
0.518,0.576,0.693,0.71,0.69,0.693
0.556,0.571,0.713,0.716,0.707,0.708
0.593,0.626,0.722,0.728,0.71,0.709
0.63,0.603,0.696,0.661,0.702,0.701
0.667,0.621,0.717,0.749,0.716,0.715
0.704,0.626,0.713,0.741,0.699,0.698
0.741,0.641,0.719,0.74,0.713,0.711
0.778,0.649,0.714,0.74,0.713,0.711
0.815,0.656,0.712,0.738,0.709,0.708
0.852,0.656,0.702,0.749,0.7,0.699
0.889,0.661,0.702,0.742,0.69,0.688
0.926,0.658,0.695,0.715,0.689,0.688
0.963,0.543,0.493,0.532,0.485,0.486
```
gemma3-1b-pt__qwen3-1.7b (matched-depth CKA):
```csv
rho,code,math,finepdfs,wikitext,wikitext_orig
0,0.235,0.424,0.351,0.345,0.333
0.04,0.297,0.479,0.348,0.335,0.32
0.08,0.29,0.456,0.333,0.316,0.301
0.12,0.312,0.444,0.346,0.316,0.306
0.16,0.336,0.546,0.446,0.438,0.435
0.2,0.353,0.521,0.455,0.436,0.428
0.24,0.349,0.506,0.526,0.442,0.436
0.28,0.391,0.512,0.547,0.448,0.442
0.32,0.44,0.511,0.58,0.463,0.463
0.36,0.444,0.524,0.605,0.485,0.487
0.4,0.417,0.524,0.489,0.499,0.5
0.44,0.426,0.53,0.548,0.535,0.534
0.48,0.434,0.531,0.563,0.536,0.535
0.52,0.412,0.497,0.549,0.503,0.499
0.56,0.431,0.558,0.593,0.563,0.563
0.6,0.426,0.507,0.548,0.541,0.542
0.64,0.425,0.493,0.507,0.542,0.542
0.68,0.493,0.599,0.644,0.613,0.613
0.72,0.487,0.574,0.639,0.579,0.58
0.76,0.57,0.632,0.662,0.626,0.625
0.8,0.575,0.635,0.667,0.627,0.626
0.84,0.571,0.624,0.667,0.616,0.614
0.88,0.551,0.612,0.675,0.595,0.593
0.92,0.557,0.609,0.66,0.595,0.595
0.96,0.515,0.463,0.51,0.436,0.436
```

### Fig. 26 — The cross-model maps

```csv
pair,corpus,band_mean_cka,depth_corr
qwen3-1.7b__qwen3-4b,code,0.558,0.914
qwen3-1.7b__qwen3-4b,math,0.692,0.978
qwen3-1.7b__qwen3-4b,finepdfs,0.7,0.949
qwen3-1.7b__qwen3-4b,wikitext,0.67,0.956
gemma3-1b-pt__qwen3-1.7b,code,0.435,0.788
gemma3-1b-pt__qwen3-1.7b,math,0.53,0.875
gemma3-1b-pt__qwen3-1.7b,finepdfs,0.568,0.84
gemma3-1b-pt__qwen3-1.7b,wikitext,0.526,0.876
```
Full cross matrices: page Fig. 26 hover, or repo.

### Fig. 27 — Two MoEs under the lens

Descriptive only (see E6 section). Fit summaries:
```csv
model,n_source_layers,boundary_rho,blockiness,band_peak_pr,pr_over_d
Kimi-K2.5 (100 prompts),16,[0.617],0.415,553.4,[0.00027, 0.00028, 0.00028, 0.00028, 0.0003, 0.00035, 0.00042, 0.00059, 0.00075, 0.00127, 0.00203, 0.00444, 0.01568, 0.07721, 0.13716, 0.12038]
DeepSeek-V4-Flash (250 prompts),16,[0.977],0.499,15.5,[0.00132, 0.00125, 0.00127, 0.00116, 0.00106, 0.00097, 0.00101, 0.00097, 0.00097, 0.0009, 0.00086, 0.001, 0.00213, 0.00378, 0.008, 0.19437]
```

## Figure notes

Figure-by-figure notes an explainer may need beyond the titles: Fig. 5's step down
at ρ≈0.6–0.7 is the cliff; Fig. 6's slider default is the band's flattest layer;
Fig. 7's dashed curve is the same layer's seq-128 measurement and the vertical line
the old window edge; Fig. 11's readout shows blockiness against its decay-only null;
Fig. 14's ● marks each row's best-matching 32B checkpoint and ▢ the equal-token one;
Fig. 16's grey curve is the receiver steering itself (ceiling) and the dashed line
the best of three controls; Fig. 18's open circles are seed re-runs; Fig. 20's
anchors are 0.76 (unrelated models) and 0.995 (adjacent layers); Figs. 25–26 share
the corpus color coding with Figs. 22–24.