Sparse Readout Prism: Explaining Logit-Lens Scores in Features Instead of Tokens

Motivation

• Token identity is an ambiguous proxy for readout structure because one token can encode multiple senses while distinct tokens can share dominant directions.For example, bug and insect-related readings can obscure whether the decoded structure concerns software defects or animals.Source: §1 · p. 1

Method

• SRP changes lens reports from token identities to readout features without changing state transport, enabling comparisons across tokens, contexts, layers, and lenses.Dictionary learning decomposes readout rows into sparse coefficients over a shared basis; each feature can occur across multiple rows, and each row can contain several features.Source: §1 · p. 2

Applications

• In context, SRP resolves the ambiguity within bug by attributing software-context support to defect features and insect-context support to animal features.This exposes structure that token rankings can obscure and motivates using a reference fixed in the model weights before lens readings are interpreted.Source: §1 · p. 2

Contributions

• SRP reconstructs 8.9–17.3 percentage points more of tested logit differences than the strongest of six row-geometry baselines, while feature ablations shift scores proportionally to their contributions.SRP fits one shared overcomplete basis directly to each model’s readout weights and decomposes token logits or margins into signed feature contributions plus a residual.Source: §1 · p. 2

Corpus conditionality

• Changing a Jacobian lens’s fitting corpus changes the language reported for identical hidden states, but the dominant readout feature remains stable across languages, lens constructions, and shared scripts.This corpus conditionality separates changes in the hidden state from changes in the lens’s decoding instrument.Source: §1 · p. 2

Methods in the lens style

• Logit lens, tuned lens, Future Lens, Patchscopes, and Jacobian lens methods report token identities, while SRP changes the readout unit without changing the lens.This permits differently fitted lenses to be compared on the same hidden state in §5.Source: §2 · p. 2

Output geometry and sparse row dictionaries

• SRP decomposes trained transformer LM-head rows into sparse readout features, unlike neighborhoods, clusterings, and principal directions, which are not constrained to reconstruct the analyzed logit or margin.SRP is evaluated by replacement and reconstruction of selected scores in §4.2.Source: §2 · pp. 2–3

Activation SAEs and decompositions in parameter space

• Activation SAEs decompose hidden states and depend on their source corpus, whereas SRP fits one sparse basis to readout directions that serves every lens and layer.Both factor opposite sides of the lens score ˜h⊤ℓq, but SRP provides a reference independent of the lens, prompt, and corpus.Source: §2 · p. 3

3 Sparse Readout Prism

• SRP expresses token logits and margins as signed readout-feature contributions plus an explicit residual, all in logit units.The decomposition uses the unembedding matrix alone; lenses change only the decoded state presented to that matrix.Source: Figure 2 · p. 3, §3 · p. 3

3.1 Factorizing the LM Head into Sparse Readout Features

• A sparse autoencoder factorizes each unembedding row into a fitted sparse reconstruction and residual, with feature directions defining interpretable readout coordinates.The reported operating points use TopK sparsity k = 256 with 32× expansion, while individual decompositions are sparser.Source: §3.1 · p. 4

3.2 Decomposing Selected Readout Scores

• SRP generalizes from individual token logits to arbitrary linear combinations of readout rows, including token margins and contrasts where shared components cancel.Every margin is a contrast, and other score families replace one side with token-set averages or a ranked competitor.Source: §3.2 · p. 4

3.3 Comparing Decompositions Across Contexts and Lenses

• For a fixed selected direction, only the decoded state changes across contexts, layers, or lenses, while all decompositions share one feature coordinate system.The dominant feature is the feature with the largest absolute signed contribution, allowing feature-level agreement across decompositions.Source: §3 · p. 3, §3.3 · p. 4

3.4 Reporting Fidelity Diagnostics

• SRP reports residuals and three fidelity diagnostics because accurate row reconstruction does not guarantee accurate selected-score reconstruction.The diagnostics cover model-level replacement fidelity, score-level relative reconstruction error, sign agreement, and combined coverage.Source: §3.4 · p. 5

4.1 The Sparse Head Replaces the Readout

• Replacing the readout preserves held-out argmax agreement at 0.89–0.90 for Qwen3.5 and Ministral and 0.75–0.76 for distilled reasoning readouts, while sign agreement reaches at least 0.91.Median KL is 0.09–0.14 bits for Qwen3.5 and Ministral versus 0.43–0.49 bits for distilled reasoning readouts; errors concentrate near ties.Source: §4.1 · p. 6

4.2 SRP Outperforms Every Baseline Built on Row Geometry

• Coverage reaches 0.72–0.81 across six softcap-free readouts, outperforming every row-geometry baseline with non-overlapping intervals and 18 points over capacity-matched k-means on Qwen3.5-2B.SRP also has the lowest mean, 95th-percentile, and maximum absolute error; shuffled codes and random support remain below 0.10 coverage on tested readouts.Source: Table 1 · p. 5, §4.2 · p. 6

4.3 Removing a Feature Moves the Margin as the Decomposition Predicts

• Feature ablations predict margin changes with r2 = 0.83–0.93 across six softcap-free readouts, versus at most 0.06 for random directions.Predicted and measured magnitudes align on Ministral and distilled reasoning readouts with slopes 0.96–1.13, while Qwen3.5 slopes are 1.34–1.69.Source: §4.3 · p. 6, Table 2 · p. 6

4.4 The Decomposition Survives Retraining

• Retrained dictionaries preserve token groupings despite non-repeating feature directions: same-token lists share Jaccard 0.21–0.24 versus 0.014–0.018 for unrelated contrasts.The resulting overlap is 13–15× higher than unrelated overlap, every contrast exceeds the unrelated 90th percentile, and 87–91% of features find matching row-group counterparts.Source: §4.4 · pp. 6–7

4.5 Context Decides Which Features Carry a Token

• The same token’s score is carried by different readout features across contexts: insect-context “bug” uses mosquito, bee, and beetle features, whereas software-context “bug” uses defect, crash, and debug.SRP also decomposes layer-level direct logit attribution into feature terms and supplies candidate readout-edit directions, though the best edit baseline varies by model.Source: §4.5 · pp. 7–8, Figure 3 · p. 7
• For ambiguous words, eight signed feature contributions classify held-out senses with balanced accuracy 0.90, 0.92, and 0.80 across Qwen3.5-2B, Qwen3.5-9B, and R1-Llama-8B.Every word beats its majority baseline, shuffled labels give a null near 0.41, and eight features summarize a weaker but readable signal than full-state probes.Source: §4.5 · pp. 7–8

5 A Fitted Lens Reports the Language of Its Fitting Corpus

• Token readings vary with fitting corpus because the learned map depends on that corpus, so token rankings cannot distinguish hidden-state information from corpus effects.This corpus conditionality applies to fitted lenses; direct unembedding-based latent-language studies use no fitting corpus and are not subject to this dependence.Source: §5 · p. 8

5.1 Design

• The experiment holds hidden states fixed while varying only lens-fitting corpora and uses SRP features learned solely from model weights as a corpus-independent reference.It compares English- and Chinese-fitted Jacobian lenses on Qwen3.5-9B across 80 prompts, four layers, and language-invariant controls.Source: §5.1 · p. 8

5.2 Different Tokens Share a Dominant Feature

• The same SRP feature dominates both lenses on 77 of 80 prompts (96%, 95% CI [0.90, 0.99]), despite different token readings.The lenses reported different scripts on 39 of 80 prompts, including 9 of 10 factual recall prompts; null comparisons produced 0/159 and 0/240 matches.Source: §5.2 · pp. 8–9, Figure 4 · p. 9

5.3 The Pattern Persists Across Languages, Lens Constructions, and Corpus Sizes

• Across extensions, feature agreement remains high despite changing tokens: 67/80 for ridge translators, 84/90 for English–German lenses, and 77/80 after tripling English–Chinese corpora.The same feature also agrees for 76/80 Jacobian-versus-ridge translator comparisons; increasing corpus size changes token disagreement in opposite directions across language pairs, while eleven English–Chinese prompts change language and none changes its dominant feature.Source: §5.3 · pp. 9–10

Safety audits

• SRP can support safety audits by comparing recurring feature contributions for defined contrasts across paraphrases, languages, adversarial prompts, and checkpoints.Recurring features can nominate causal tests or constrained interventions.Source: §6 · p. 10

Evaluation

• Audits should report reconstruction error, transfer, distributional cost, and effects on other outputs so safety claims remain tied to the selected score.Source: §6 · p. 10

Methodological extensions

• Methodological extensions include tracing feature contributions to supplying components, joint activation–readout factorizations, cross-model dictionary alignment, and decompositions covering more than one selected score.Source: §6 · p. 10

• 8.9–17.3 percentage points more of tested logit differences are covered by SRP than by the strongest of six row-geometry baselines.Tokens grouped with a contrast also reproduce across seeds at 13–15 times the overlap of unrelated contrasts.Source: §7 · p. 10
• SRP reveals that different lens-fitting corpora can produce different token readings for the same hidden state while the dominant readout feature remains stable.This provides a reference fixed in the model weights for distinguishing state changes from instrument changes.Source: §7 · p. 10
• Readout features complement sparse dictionaries for hidden states by making output-space organization measurable from weights alone, including shared features across surface forms and scripts.A single row can carry several senses, while one feature can span multiple tokens, forms, and scripts.Source: §7 · p. 10
• Projecting decoded states onto SRP features shows how each state aligns with readout organization layer by layer before token decoding.One dictionary per model serves every lens and layer, so all readings decompose into the same terms without further fitting.Source: §7 · p. 10
• The shared feature terms provide an auditing control for what the fitting corpus contributed to a reading, plus attribution terms and candidate directions for constrained readout edits.Source: §7 · p. 10

Fitting and basis choice

• SRP requires roughly 6–12 GPU-hours per model, and its decoder directions vary with dictionary size, sparsity k, training recipe, and seed.The authors select operating points from fidelity sweeps and report stability across seeds and recipes (§4.4, Appendix I).Source: Limitations · p. 10

Empirical scope

• The empirical scope is limited to Qwen3.5-9B for cross-lens studies and Qwen3.5-2B for readout edits, using selected language pairs, lens constructions, and fitting budgets.Cross-model losses are reported in Appendix N.Source: Limitations · p. 10

Interpretation boundary

• SRP explains selected readout scores and readout organization, not circuits or generation, while feature labels may inherit tokenizer and vocabulary geometry (Press and Wolf, 2017; Inan et al., 2017; Yang et al., 2018a; Bostrom and Durrett, 2020).Quantitative conclusions therefore rely on feature ids, signed contributions, and residual errors, with further discussion in Appendix O.Source: Limitations · p. 10

Ethical Considerations

• The study uses public checkpoints and held-out C4-derived states, avoids human-subject or deployed-system evaluation, does not redistribute checkpoints or raw text, and limits claims to measured quantities.C4 may contain offensive, private, copyrighted, or sensitive text; released SRP dictionaries include artifact-specific licensing and usage notes.Source: Ethical Considerations · pp. 10–11

Ethical Considerations; Appendix Roadmap

• SRP’s inspectable lexical directions and readout edits are dual-use auditing capabilities, so the work reports distribution-shift and off-target probes while leaving system-level safety evaluation beyond scope.The appendix roadmap identifies further protocols for fidelity, transfer, stability, interventions, cross-lens studies, feature audits, direct logit attribution, constrained edits, discussion, and related work.Source: Ethical Considerations · p. 11, Appendix Roadmap · p. 16

A SRP Score Decomposition Identities; A.1 Raw-Space Row Decomposition

• The decomposition evaluates raw readout-space scores by separating an explicit shared offset, sparse feature terms, and residual error, with centering or normalization mapped back before reporting.Keeping the mean outside the dictionary preserves sparse capacity for row deviations and makes the offset separately visible.Source: A SRP Score Decomposition Identities · p. 16, §A.1 · pp. 16–17

A.2 Selected-Score Identity

• For selected readout directions, the grouped identity expresses scores through feature contributions and residuals; the shared offset cancels in zero-sum contrasts such as pairwise logit differences and competitor margins.The selected direction is a linear function of vocabulary rows, and its feature contribution follows from substituting the raw-space row decomposition.Source: §A.2 · p. 17

A.3 Error, Sign, and Display Rules

• Error reporting uses floored relative error ρ0.5 with δ = 0.5 logits for aggregates, unfloored ρ0 for identity and baseline comparisons, and sign agreement or flip rates near decision boundaries.Figures report exact and reconstructed scores, residuals, relative error, and sign status, using full active feature sums even when plots show only largest bars.Source: §A.3 · p. 17

B Model Suite and Selection Rationale; B.1 Selection Rationale

• The evaluation suite spans Qwen, Gemma, Ministral, and distilled DeepSeek-R1 rows to test readout fidelity, scale, portability, tokenizer and geometry differences, softcaps, and tied versus untied readouts.Qwen3.5-9B anchors capacity sweeps, smaller Qwen models test cheaper reruns, Gemma tests distinct geometry, and Ministral and R1-distilled rows broaden transfer comparisons.Source: B Model Suite and Selection Rationale · pp. 17–18, §B.1 · p. 18

B.2 Final Suite; B.3 Interpretation Details

• Qualitative analyses use one low-error display setting, while comparisons account for token frequency, length, special tokens, row norms, tied embeddings, softcaps, and matched-score identity bias.Primary claims concern selected readout scores, especially token logits and contrastive logit differences, with model-specific readout handling before decision-fidelity reporting.Source: §B.2 · p. 18, §B.3 · p. 18

C How the Readout SAEs Were Selected; C.1 Sweep Coverage and Evidence Map; D Qwen Readout SAE Recipe and Capacity Sweeps

• SAE selection combines rowEV, top-1, KL, selected-logit error, and feature usage, separating native k = 128 strict-budget and native k = 256 high-fidelity operating regimes.The evidence map sequences Qwen3.5-9B recipe selection, capacity comparisons, and transfer to smaller Qwen and Gemma models; Appendix D supplies the Qwen recipe and capacity evidence.Source: C How the Readout SAEs Were Selected · pp. 18–19, §C.1 · p. 19

D.1 Qwen3.5-9B Fixed-k = 128 Recipe Selection

• The selected no-PCA TopK recipe at fixed k = 128 improves reconstruction, top-1 agreement, KL, and feature usage over the previous long-trained matryoshka baseline, with a 20k stop rule.The recipe uses D = 32768, row-seeded initialization, hybrid_50freq_50uniform sampling, warmup-cosine learning rate 10−3 →10−4, and delayed prism ramping; its 15k→20k rowEV gain is +0.0025.Source: §D.1 · pp. 19–20

D.2 Qwen3.5-9B Capacity Sweep

• 0.857 rowEV, 0.900 top-1, and 0.105 KL make Qwen3.5-9B’s 32×, k = 256 run the best capacity-sweep operating point.Its rare feature rate is 0.292, while the compact 16×, k = 256 point reaches 0.761 rowEV, 0.874 top-1, and 0.167 KL.Source: §D.2 · p. 21, §D.4 · p. 22

D.3 Lower Sparsity Budgets

• At fixed width, reducing k strands dictionary capacity: at 16×, k = 64 raises the dead-feature rate to 0.491–0.512 and degrades rowEV from 0.80 to 0.68 and KL from 0.23 to 0.37.At k = 32, dead-feature rates reach 0.694 for Qwen3.5-2B and 0.641 for Qwen3.5-0.8B, while usage is governed by k/D rather than k alone.Source: §D.3 · p. 22

D.4 Operating-Point Claims

• Native k = 128 is the strict interpretability regime and native k = 256 the high-fidelity regime; post-training truncation from k = 256 to k = 128 can raise rowEV while sharply reducing top-1 agreement.Across six softcap-free readouts, a median of 65–107 features carries 80% of a score’s absolute contribution mass, despite dictionaries containing tens of thousands of features.Source: §D.4 · pp. 22–23

E.1 Transfer to Smaller Models

• The transferred recipe scales across smaller Qwen and Gemma models, with 32×, k = 256 producing the largest screening rowEV in every model and Qwen3.5-0.8B 16×, k = 128 selected for the strict-budget point.The screening rowEV values are 0.812, 0.765, 0.711, and 0.742 for Qwen3.5-0.8B, Qwen3.5-2B, Gemma-4-E2B, and Gemma-4-E4B; rowEV, top-1, and KL vary by at most 0.02 across seeds.Source: §E.1 · pp. 23–24, Table 11 · p. 24

E.2 Cross-Family 8B Comparison (Ministral)

• Ministral-3-8B transfers the TopK recipe beyond Qwen: both dictionary settings have dead < 0.03 and KL < 0.14 bits, although rare-feature rates are high.Its matched-width untied, softcap-free readout is geometrically clean, supporting cross-family transfer of the row-factorization recipe.Source: §E.2 · p. 24

E.3 R1-Distilled Comparison Rows

• R1-distilled readouts preserve strong row reconstruction while separating in decision fidelity: R1-Distill-Llama-8B matches Ministral at fidelity rowEV 0.888 but has top-1/KL 0.754/0.434 versus 0.904/0.087.At strict budget, the corresponding rowEV values are 0.796 versus 0.806, with top-1/KL 0.725/0.536 versus 0.885/0.130; the same pattern appears across Qwen2.5-Math-derived (Yang et al., 2024) and Llama-3.1-derived (Llama Team, 2024) lineages.Source: §E.3 · p. 24, Table 14 · p. 25
• The R1 comparisons cannot isolate post-training because the checkpoints also differ in architecture, tokenizer, and pretraining lineage, so a matched base-against-distill ablation remains necessary.The held-out contrastive reconstruction check finds sign agreement near the five-model baseline, with below-half relative-error fractions comparable at fidelity and lower under the strict setting.Source: §E.3 · pp. 24–25

E.4 Readout Row-Norm Tail Analysis

• The analysis reports row-norm statistics for base-model and reasoning-distilled checkpoints to characterize the separation between rowEV and fidelity observed in Appendix E.3.Table 16 defines the coefficient of variation as σ/µ and max/median as a summary of the heavy-tail end.Source: §E.4 · p. 25, Table 16 · p. 25
• The upper tail is heavier and the lower tail is compressed on distilled checkpoints, with R1-Distill-Llama-8B showing the most extreme upper-tail behavior.These effects make the distilled comparison rows wider than the base-model rows.Source: §E.4 · pp. 25–26
• Distilled checkpoints show wider readout row-norm distributions, with max/median increasing from 1.5–1.6 to 1.7–2.0 and coefficient of variation from 0.12 to 0.14–0.16.Minimum row norms also rise from 0.25–0.38 on base models to 0.42–0.57 on distilled models.Source: §E.4 · pp. 25–26, Table 16 · p. 25
• Heavier-tailed distilled rows explain why uniform L2 row reconstruction can remain strong while decision fidelity separates, because softmax-sensitive logit errors are dominated by high-norm rows.The analysis motivates norm-aware SAE recipes as a natural extension for reasoning-distilled settings.Source: §E.4 · pp. 25–26

F Readout SAE Diagnostics and Reproducibility

• Appendix F collects secondary diagnostics, alternative-recipe comparisons, denominator controls, and reproducibility details for the selected readout SAEs.It serves as the paper’s repository for supporting evaluations beyond the main SAE results.Source: F Readout SAE Diagnostics and… · p. 26
• All code, per-model training and evaluation recipes, selected k=128 and k=256 SAE checkpoints for eight models, contrast banks, and the blinded label-audit protocol are released.The code is available under the MIT license at github.com/hematteo/sparse-readout-prism, with checkpoints released at hematteo/sparse-readout-prism.Source: F Readout SAE Diagnostics and… · p. 26

F.1 Additional Metric Analyses

• +0.038 rowEV comes from support selection versus +0.006 from coefficient refitting in the 32×, k = 256 run, while sparse reconstruction outperforms dense rank-matched reconstruction.The sparse code is about 5.9× better by unexplained-variance reduction than the dense rank-256 reference; Figure 12 reports the corresponding achieved and ceiling comparisons.Source: §F.1 · p. 26, Figure 12 · p. 27
• Support selection is the dominant remaining bottleneck across four small-model transfers, with signed-OMP gaps of +0.158, +0.099, +0.182, and +0.125.Refitting coefficients on the encoder’s support provides much smaller gains than changing the support.Source: §F.1 · p. 26
• RowEV, top-1, and KL can favor different dictionaries, so selection jointly uses reconstruction, readout, and feature-usage metrics.For D = 65536 truncated to k = 128, rowEV is 0.695 versus 0.621, while top-1/KL are 0.721/0.469 versus 0.846/0.296 for the fixed-k recipe.Source: §F.1 · p. 26

F.2 Alternatives and Negative Controls

• Matched controls favor plain TopK over matryoshka and calibrated BatchTopK recipes, while step- or sparsity-matched TopK controls outperform Qwen JumpReLU.Plain TopK reaches rowEV 0.567 versus 0.559/0.554 for matryoshka and 0.531 for calibrated BatchTopK; matched TopK reaches 0.42–0.44 versus 0.26–0.31.Source: §F.2 · pp. 26–27
• Tokenizer and non-token audits support the denominator choice: structural non-token rows are 5.4% of rows, and excluding them changes rowEV by only +0.0016.These controls accompany the matched recipe comparisons and mixed logit-metric results.Source: §F.2 · p. 27

F.3 Reproducibility and Selection Details

• The artifact bundle provides training and evaluation code, permitted readout-SAE checkpoints or regeneration recipes, prompt materials, and score metadata for recomputing reported metrics.Original model checkpoints and raw C4 text are not redistributed; reproduction uses public identifiers, artifact code, matching revisions, tokenizers, filters, and eligibility rules.Source: §F.3 · pp. 27–28
• Dictionary selection uses row reconstruction, replacement-logit metrics, and feature-usage statistics; TopK readout SAEs are evaluated with rowEV, top-1 agreement, KL in bits, reconstruction error, and sign agreement.Most reported dictionary settings use seed 0, while the Qwen3.5-0.8B 32×, k = 256 three-seed window provides a seed-variation snapshot.Source: §F.3 · pp. 27–28

G Fidelity Checks and Controls

• Fidelity checks cover tokenization and filtering, row-norm and frequency checks, split construction, dictionary settings, and aggregate metrics, with Table 17 organizing local score-decomposition measurements.The decomposition definitions themselves are stated in Appendix A, while quantitative control summaries follow the table’s measurement framework.Source: G Fidelity Checks and Controls · p. 28, §G.1 · p. 28

G.2 Metric Definitions and Fidelity Results

• 0.914–0.958 of selected scores preserve contrast sign across eight 32×-width readout SAEs, with median ρ0.5 ranging from 0.068 to 0.255.The metrics use held-out logit-difference sets at k = 256; top-1 and KL provide additional replacement and distributional checks.Source: §G.2 · p. 28, Table 20 · p. 31
• Under the baseline harness’s unfloored ρ0, SRP reaches 0.774 coverage on Qwen3.5-2B, exceeding nearest-row ridge’s 0.622 and PCA-1024’s 0.539.At matched compactness, PCA-256 drops to 0.272; under the redesigned harness, SRP reaches 0.780, while null controls lower coverage by 0.58–0.71 and sign agreement to 0.38–0.52.Source: §G.2 · p. 29, Table 19 · p. 30

G.3 Direct-Geometry Alternatives Across the Primary Readouts

• Matched-capacity k-means still trails SRP’s coverage on every model, while SRP outperforms direct-geometry alternatives in the primary comparison.The comparison evaluates nearest-row ridge, weighted kNN, k-means, hard clustering, and PCA-256 on identical selected scores and banks.Source: §G.3 · p. 31, Table 21 · p. 32

G.4 Error Tails and Margin Stratification

• 0.996 sign agreement and 0.966 coverage occur for contrasts with |mexact| ≥2, whereas relative-error failures concentrate among near-ties with |mexact| < 0.5.SRP’s absolute error remains nearly flat at 0.64–0.71 logits across margin bins, making the relative tail primarily a denominator effect.Source: Figure 15 · p. 30, §G.4 · pp. 31–32, Table 23 · p. 32

G.5 Nearest-Row Reading of Selected Directions

• Nearest-row readings primarily recover casing, sub-word, morphological, and cross-lingual token families, while sense-bearing structure appears in SRP’s signed feature terms.For wbug−winsect, sense-bearing rows appear only at ranks 13–17; for wbug−werror, the top 30 are entirely variants of the contrast tokens.Source: §G.5 · p. 32, Table 24 · p. 33

G.6 Operating-Regime Cases

• Feature-level readings are used only when reconstruction, sign, target direction, compactness, and tokenization-related checks all pass.If a condition fails, the analysis falls back to the selected score, exact contrast direction, aggregated contribution mass, or a safer token-family contrast; the main Qwen3.5-2B display lies within the regime.Source: §G.6 · pp. 32–33

G.7 Selected-Score Bank Provenance

• The evaluation banks record prompts or hidden-state sources, token information, score families, tokenizer audits, and exact versus reconstructed scores, separating deterministic A/B probes from C4 and benchmark-derived contrast banks.C4 supports model-native replacement and frontier diagnostics, while benchmark formats test fixed readout contrasts without evaluating task accuracy.Source: §G.7 · p. 33

H Sense-Labelled Evaluation

• The classifier framing favors unweighted projections over signed contributions on all three models, showing that signed contributions identify supporting readout directions rather than serving as general sense representations.The reported projection-versus-contribution accuracies are 0.87 versus 0.84, 0.90 versus 0.76, and 0.73 versus 0.67; the evaluation uses one dataset, one sparsity setting per model, and one prompt template.Source: H Sense-Labelled Evaluation · p. 35
• Selected groups recover held-out senses at 0.90, 0.92, and 0.80 balanced accuracy, versus a permutation null near 0.41, while nearly all words exceed the null’s 95th percentile.The groups use eight features per sense selected on training data; a nearest-centroid probe and the full 256-dimensional account score higher, but compressing to eight features costs 5–9 points against the full account and 5–12 points against the full state.Source: H Sense-Labelled Evaluation · pp. 34–35, Table 28 · p. 35

I Stability Across Independent Training Runs; I.1 Explanation-Level Stability

• Across independent training seeds within a fixed recipe, selected-score decompositions reproduce every tested contrast above the null’s 90th percentile at approximately 15× the null level, despite individual decoder directions not recurring.Validation top-1 agreement is 0.805–0.848, so the reported explanation differences are not fidelity artifacts; the decomposition of a selected score is the stable interpretive unit.Source: I Stability Across Independent… · p. 35, §I.1 · pp. 35–36

I.2 Feature-Group Matching

• About 90% of used feature groups have an above-chance counterpart after retraining, roughly 40% are near-exact single matches, and unions of up to three features cover 83–90% of the median group.The remaining instability is characterized as feature splitting, with narrower 16× directions more stable; the thresholds were fixed and the results are reported descriptively.Source: §I.2 · p. 36, Table 30 · p. 36

I.3 Held-Out Recovery of the Stable Grouping Core

• Held-out SRP recovers 72–75% of the stable grouping core, compared with 43–49% for row kNN and 21–25% for clustering.All methods are evaluated on identical leave-one-out cores, and SRP/kNN ratio confidence intervals exclude parity.Source: §I.3 · p. 36, Table 31 · p. 36

I.4 Cross-Recipe Disclosure

• The stability claims are recipe-specific: the earlier paper dictionary shares only 36% of the stable core and recovers 0.444 of leave-one-out cores, below row kNN in that comparison.Variation across preprocessing and training recipes exceeds seed variation within a recipe, and dictionary families are not mixed across comparisons.Source: §I.4 · p. 37

J Predicted Versus Realized Local Readout-Side Changes

• r2 = 0.83–0.93 versus random-control r2 ≤0.06 across six readouts, supporting SRP contributions as predictors of local readout-side score changes.Slopes preserve sign and rank, but exceed 1 on Qwen3.5-0.8B/2B/9B; the validation concerns readout-side ablations of realized forward passes, not circuit-level or generation-level effects.Source: J Predicted Versus Realized Local… · p. 37

K Cross-Lens Study Protocol, Controls, and Per-Family Results

• 77/80 prompts (96%, CI [0.90, 0.99]) share the dominant SRP feature across English- and Chinese-fitted lenses, versus empty unrelated-token and shuffled-pairing null floors.The same feature carries both surface forms within a lens on 61/80 English-lens and 62/80 Chinese-lens prompts.Source: K Cross-Lens Study Protocol, Controls,… · p. 38, Table 32 · p. 38
• The shared feature f112 carries both 大 and big across English- and Chinese-fitted lenses, while German groß and English large use distinct features f6764 and f12474.On the German antonym example, three fitted lenses shift surface tokens by language, but the decomposition separates the language pairs using LM-head row codes alone.Source: K Cross-Lens Study Protocol, Controls,… · pp. 38–39, Table 33 · p. 38, Table 34 · p. 39
• 84/90 English–German prompts (0.93, CI [0.86, 0.97]) retain the same dominant feature across lenses despite top-token divergence on 73/90 prompts.Agreement remains above null floors of 1/204 for unrelated tokens and 0/270 for shuffled pairings, with family counts reported across antonym, cloze, exemplar, recall, translation, and controls.Source: K Cross-Lens Study Protocol, Controls,… · pp. 38–39
• 67/80 prompts preserve the dominant feature across English- and Chinese-fitted ridge translators, while holding English fitting fixed yields 76/80 agreement across Jacobian and ridge constructions.The ridge translators reproduce corpus-conditioned surface divergence, and both comparisons have the stated null floors.Source: K Cross-Lens Study Protocol, Controls,… · p. 39
• Tripling the fitting budget leaves English–Chinese feature agreement at 77/80 with identical passing prompts, while English–German agreement rises from 84/90 to 85/90.Surface language reports change with fitting scale, but no English–Chinese prompt changes its dominant feature; the study covers one model, two language pairs, two lens constructions, and two fitting scales.Source: K Cross-Lens Study Protocol, Controls,… · pp. 39–40, Table 35 · p. 40

L Additional Qwen Displays and Feature Audits

• Qwen companion analyses include all-layer logit-lens displays, curated pairwise logit differences, selected-score and group-contrast decompositions, fixed-token context shifts, projection profiles, and token-row audits.All figures use the same Qwen3.5-2B 32×, native-k = 256 SAE trained on unembedding rows.Source: L Additional Qwen Displays and Feature… · p. 40, §L.1 · p. 40, §L.2 · p. 40

L.5 Fixed-Token Context Comparisons

• Fixed-token profiles distinguish context-specific feature support for bug, bridge, ring, graph, memory, virus, and pipe, while preserving shared positive support in companion views.The technical comparisons separate chart/diagram from graph-structure, human from computer memory, and biological virus from malware features; pipe remains mixed.Source: §L.5 · p. 43, Figure 22 · p. 43, Figure 23 · p. 43

L.6 Readout-Feature Profiles

• Readout-feature profiles rank SAE decoder directions directly from the decoded state before token or logit-difference selection, with vocabulary rows labeling the selected directions.Figures 26 and 27 apply this projection view to shell, Java, port, and cell, complementing score-specific local contribution plots.Source: §L.6 · pp. 43–44

L.7 What SRP Adds Beyond the Logit Lens

• SRP feature profiles distinguish shell prompts’ shell, command, parser, and editor directions from Java prompts’ beverage, framework, and language directions.Each feature label lists its top associated unembedding rows, linking feature profiles to token-level readout structure.Source: §L.7 · p. 45, Figure 26 · p. 45
• SRP separates the ordinary lens’s local token ranking into signed feature terms, showing deletion-related features opposing summary and communication features in the prompt-injection case.The displayed contributions include -2.938, -1.700, and -1.606 for delete-related terms, opposed by +0.969, +0.933, and +0.922 for summarize-related terms.Source: §L.7 · p. 45

L.8 Token-Row Audit for Main Case-Study Features

• The token-row audit distinguishes readable semantic families from surface-form artifacts and tests whether each coherent family matches the panel sense and signed contribution direction.Labels are drafted from highest-scoring unembedding rows and verified against those same rows.Source: §L.8 · pp. 45–46
• Three blinded rater runs agreed unanimously on 93.4% of items, with Fleiss’ κ = 0.87 and 81.8% agreement with the author audit.Consensus over 99 display occurrences classified 76 as coherent, 4 as mixed or ambiguous, and 19 as token form under a fixed protocol.Source: §L.8 · p. 46
• 85 of 99 displayed claim-bearing labels name coherent lexical families, including 66 on-target and side-consistent families, while 14 are token-form artifacts.The remaining 19 coherent families are off-target, naming another sense or being dominated by the panel’s target token.Source: §L.8 · p. 46, Table 38 · p. 47

M Feature-Resolved Direct Logit Attribution

• The SRP–DLA identity applies to pairwise token-token and vocabulary-mean contrasts, decomposing a realized forward pass across residual-stream components, SAE features, and residual terms.For zero-sum contrasts, the vocabulary offset cancels, while separate DLA additivity and sparse reconstruction gaps are reported.Source: M Feature-Resolved Direct Logit… · pp. 47–48
• Feature-resolved attribution groups the verify-minus-assume margin by semantics and components, exposing verification/checking and assumption-family contributions across residual-stream stages.The component view stacks each component’s readout-feature terms against ordinary direct logit attribution.Source: M Feature-Resolved Direct Logit… · p. 48, Figure 28 · p. 48

N Constrained Readout-Side Edit Test

• At scale 16, the readout SAE direction edit flips the same 23 held-out candidates as oracle token bias, reaching 23/64 = 0.359 while discovery-token bias has zero held-out effect.The edit selects directions from five discovery terms and transfers to held-out lexical variants; Qwen3.5-2B shows specificity 1.0 for 12 of the top 13 profanity-associated features and 0.988 for the remaining feature.Source: N Constrained Readout-Side Edit Test · pp. 49, 51, Table 41 · p. 51

N.1 Matched-KL Frontier and Cross-Model Outcome

• At matched KL, SRP outperforms every label-free baseline on Qwen3.5-2B, but this advantage is model-specific across the broader model comparison.Against the mean-row direction, SRP’s paired ∆pbad advantage is +0.032 at KL 0.02, +0.087 at 0.05, +0.085 at 0.10, and +0.074 at 0.20; it loses to the mean row on Qwen3.5-0.8B and Qwen3.5-9B.Source: §N.1 · pp. 52–53, Figure 33 · p. 53

O Discussion and Future Work; O.1 What Sparse Readout Prism Decomposes

• SRP decomposes a selected vocabulary logit, pairwise difference, group contrast, vocabulary-mean contrast, or top-competitor margin into sparse feature terms plus preprocessing and reconstruction errors.The decomposition is local to the selected score, so reconstruction quality and the chosen contrast determine what the reported feature contributions mean.Source: §O.1 · pp. 53–54

O.2 Relation to Activation SAEs

• Unlike activation SAEs, SRP fits sparse codes to unembedding rows, and a feature’s contribution depends jointly on fixed row coefficients and the decoded state’s projection.Thus, the same token row can have different dominant feature contributions across contexts.Source: §O.2 · p. 54

O.3 Model-Family Results

• Across model families, the paper reports reconstruction, LM-head replacement, and selected-score metrics alongside feature contributions, with Qwen rows supporting the main local decompositions.The additional families vary in tokenizer, readout, and post-training conditions.Source: §O.3 · p. 54

O.4 Feature-Resolved DLA

• Feature-resolved DLA combines readout-side SRP terms with residual-stream component attribution, reporting both DLA additivity error and readout-SAE reconstruction error.The two views distinguish what reads the state from which components produced the state’s projection onto the selected readout direction.Source: Figure 30 · p. 49, §O.4 · p. 54

O.5 Future Work

• Proposed extensions include decoder-direction interventions, cross-model alignment, distributional top-k decompositions, broader model-family sweeps, and joint activation–readout feature factorizations.Future analyses could also census projection mass used, competing, or unused by a selected score and test these categories across layers and generations.Source: §O.5 · p. 54

P Extended Related Work

• Related work positions SRP as complementary to contextual lenses, activation and weight-based sparse dictionaries, latent-language readings, output-space edits, and component-level attribution methods.SRP decomposes LM-head weight rows rather than hidden states or embedding outputs, while its readout-side edits compose with component-level logit prisms.Source: P Extended Related Work · pp. 54–55