Scaling Inherently Interpretable Language Models

Scaling

• Across three orders of magnitude of compute, interpretability constraints add a small, fixed scaling offset rather than a growing penalty, while representations become more disentangled and human-concept aligned.Interpretability metrics improve with compute across autoregressive and diffusion backbones, including concept prediction, representation separation, concept routing, and embedding alignment.Source: §1 · pp. 5–6

Inherent interpretability

• The paper defines inherent interpretability by training explanations as faithful interfaces whose variables, interventions, and semantic labels are coupled to prediction computation.This replaces post-hoc tools that may show decodability, local sensitivity, reconstructive features, or plausible reasoning without establishing causal use.Source: §1 · p. 5

Model understanding

• Steerling-8B traces outputs to affecting input tokens, contributing human-understandable concepts, and training data.These three understanding axes are implemented in a diffusion language model with a causal attention mask.Source: §1 · p. 5

From explanation to control

• The same concept interfaces enable control because each concept is a forward-pass direction that can be amplified or suppressed through a simple edit.This supports closed-loop intervention from explanation to behavior modification without retraining.Source: §1 · p. 5

Atlas

• Atlas canonicalizes hundreds of millions of free-form tags into over 33,000 concepts and annotates over 1 Trillion tokens across web text, code, mathematics, and academic prose.The annotation pipeline addresses the lack of a concept library at modern pretraining scale and labels arbitrary text at chunk level.Source: §1 · pp. 5–6

Architecture

• Steerling-8B makes concepts native to computation through block-causal attention, an additive concept bottleneck, and in-distribution feature-removal baselines.The concept module accounts for 4% of parameters at 8B and would fall below 1% at frontier scales under the same parameterization.Source: §1 · p. 6

Empirical competitiveness

• Steerling-8B lands within approximately 10% of average benchmark performance of similarly sized open peer models trained on roughly 2–16× more compute.It is trained on 1.2 trillion tokens followed by 150 billion midtraining tokens on a code- and math-augmented mixture.Source: §1 · p. 6

2.1 Transformer notation

• Transformers map input tokens to contextual hidden states at each position, then project those states into vocabulary logits.The notation is summarized in Table 1.Source: §2.1 · p. 7, Table 1 · p. 7

2.2 Autoregressive language models

• Autoregressive language models predict each next token conditioned on all preceding tokens and train with averaged next-token negative log-likelihood.The model distribution is pθ(x_i | x_<i).Source: §2.2 · p. 7

2.3 Masked diffusion language models

• Masked diffusion language models reconstruct tokens after independent masking, with noise level t controlling masking from none at t = 0 to complete at t = 1.The model predicts original tokens at masked positions, and training uses cross-entropy over those positions averaged across noise levels and sequences.Source: §2.3 · p. 8

2.4 Concept bottleneck models

• Concept Bottleneck Models route predictions through supervised human-interpretable concepts, using separate concept and prediction losses when concept labels are available.Concept Bottleneck Generative Models add an unsupervised channel for information not captured by fixed concepts, with an orthogonality loss separating the channels.Source: §2.4 · p. 8

3 A recipe for building interpretable models

• The recipe modifies data curation, architecture and loss design, optimization, and evaluation to impose human-interpretability constraints throughout language-model training.The full recipe is summarized in Figure 1.Source: §3 · p. 9, Figure 1 · p. 9

3.1 Interpretability requirements

• The requirements provide input, concept, and training-data similarity attributions answering what mattered, which ideas drove the output, and where the output resembles training data.Concepts are human-recognizable semantic units represented by a modifiable model variable and a natural-language semantic card.Source: §3.1 · pp. 9–11
• Training-data attribution measures representational similarity rather than causal influence, while input attribution requires in-distribution absence interventions and concept attribution decomposes output logits into concept contributions plus residual.Retrieval is meaningful when its encoder is the same representation used for prediction.Source: §3.1 · pp. 11–12

3.2 Why post-hoc methods fail to satisfy the interpretability requirements

• Post-hoc methods lack guarantees because standard training does not create explanatory variables coupled to prediction, valid intervention baselines, informative gradients, or faithful reasoning traces.Probes can show information presence without information use; SAEs can yield seed-dependent features unrelated to the prediction pathway; gradients and perturbations can mismatch valid human interventions; chain-of-thought can diverge from computation.Source: §3.2 · pp. 12–13

3.3.1 Faithfulness

• Faithfulness requires agreement between attribution scores and observed intervention effects, plus validity of interventions within a training-supported family; these conditions diagnose the post-hoc failures.Masking requires masked contexts during training, while concept edits must remain within observed or sampled activation ranges.Source: §3.3.1 · pp. 14–15

3.3.2 Inherent interpretability

• Inherent interpretability additionally requires attributed variables to be native to computation and semantically described, alongside pre-intervention agreement and training-supported validity.The definition includes a semantic map assigning each concept a human-valid semantic card.Source: §3.3.2 · p. 15
• Inherent interpretability guarantees specified attribution queries, not that every neuron, attention head, or model component is human-understandable.The definition targets selected explanations rather than global mechanistic transparency.Source: §3.3.2 · p. 15
• Causal faithfulness concerns whether attributed variables cause outputs as predicted, whereas semantic faithfulness requires human descriptions to represent those variables and explain useful output information.Concept leakage occurs when a variable causally predicts effects but encodes information beyond its semantic description, as in a “sports” concept that also contains hidden gender information.Source: §3.3.2 · p. 16

3.4 The interpretable model-training recipe

• The recipe makes interpretability explicit by modifying the standard training pipeline at points tied to the five formal conditions, rather than relying on post-hoc explanations.Its contribution is to make the recipe explicit, connect each element to a formal condition, and instantiate it for language-model pretraining at scale.Source: §3.4 · p. 16
• Concept annotations provide semantic targets and control interfaces, while a concept bottleneck places concept variables on the computational path so their output contributions are directly computed.The bottleneck may use additive, multiplicative, or other decompositions; the paper instantiates an additive concept bottleneck with a linear output head.Source: §3.4 · pp. 16–17
• A trained absence baseline keeps input-attribution interventions in distribution, while interpretability losses align concepts with annotations and prevent residual capacity from making the decomposition vacuous.The losses can penalize residual prediction, reconstruct unexplained hidden state, encourage concept independence, and be combined with task loss using annealed weights.Source: §3.4 · pp. 17–18
• Evaluation must preserve task performance and test the five conditions through causal faithfulness, intervention validity, semantic interpretation, and concept coverage.Coverage reports the fraction of prediction mass carried by attributed concepts alongside the unattributed residual.Source: §3.3.2 · p. 15, §3.4 · p. 18

4 Data

• Atlas annotates a 1.5-trillion-token corpus spanning web text, science, code, and synthetic data with over 33,000 human-understandable concepts.The system supports concept-annotated pretraining, midtraining, and post-training corpora.Source: §4 · p. 19

4.1 Motivation: No human-interpretable concept library exists at scale

• The library is designed to be multi-scale, localizable, stylistically expressive, representative of diverse training data, and human-meaningful, addressing limitations of word-based, activation-derived, and narrow-domain libraries.Human meaningfulness is tested directly because fluent LLM labels or apparently reasonable human ratings do not guarantee that people independently recognize the same concept.Source: §4.1 · pp. 19–20

4.2 Atlas: Documents to concepts

• Atlas proceeds from sampled documents to high-recall tags, from tags to canonical concepts, and from concepts to a model that annotates arbitrary text.Its engineering evaluation scores tag or concept relevance on a 1–5 scale, with scores of 2 or higher counted as successful; human ratings are roughly 0.4 points higher than LLM-judge ratings.Source: §4.2 · pp. 20–21

4.2.1 Stage 1: Corpus sampling, chunking, and documents to tags

• Stage 1 produces nearly 44 million annotated chunks and approximately 500 million intentionally redundant tags, yielding 14 million unique tags before clustering.Across domains, 99.9% of chunks have a per-chunk average relevance score of at least 2, while the raw tag space remains noisy and non-canonical for high recall.Source: §4.2.1 · p. 23

4.2.2 Stage 2: Tags to concepts

• Stage 2 filters, labels, and graph-deduplicates noisy tags into a final canonical library of 33,732 concepts spanning science, technology, philosophy, medicine, law, and other domains.Clustering coherence plateaus around k = 40,000–80,000; starting from k = 80,000, strict filtering and iterative merging produce the final library, while some conceptual overlap remains unavoidable.Source: §4.2.2 · pp. 25–27
• Stage 2 produced concept ratings averaging 3.50, with 98.0% scoring at least 2, while per-concept quality remained uniform at mean 3.51 and standard deviation 0.41.Two-thirds of chunks had averages between 3.2 and 4.0, and 99.7% averaged at least 2.Source: §4.2.2 · p. 28, Figure 9 · p. 28

4.2.3 Stage 3: Concept annotator model

• Stage 3 annotates 33,732 labels, dominated by approximately 33,606 content concepts alongside tone, demographic-reference, and alignment-relevant domains.It uses a long-tail-enriched reservoir, positive-unlabeled supervision, embedding-based content prediction, PU-compatible losses, and rarity-weighted sampling; held-out predictions averaged 2.94, with 95.5% of per-chunk averages at least 2.Source: §4.2.3 · pp. 28–30

4.3 Concept taxonomy

• The concept library is organized using an LCC-derived hierarchy spanning roughly 2,600 occupied nodes, with populated paths reaching depth 9 and all 20 root branches represented.Science accounts for 38% of concepts, followed by Technology and Social Sciences at 15% each and Medicine at 8%.Source: §4.3 · p. 30, Figure 11 · p. 31

4.4 Human interpretability of the concept library

• Atlas labels scored 3.98 versus 3.50 for human labels, received top-two ratings 79% versus 63%, and outscored human labels with probability 0.62.Humans recognized semantic structure in lifted-word evidence, although 17% of responses were incoherent or noisy; Atlas labels also separated from taxonomy and embedding distractors.Source: §4.4 · p. 33, Table 4 · p. 34

4.5 Data Processing

• The Atlas pipeline applies the trained annotator across approximately 11 billion chunks and 1.5 trillion tokens, producing concept annotations, tokenized training data, and cached embeddings for downstream retrieval.Distributed GPU processing preserves source and document metadata while predicting across content, tone, demographic, and alignment domains.Source: §4.5 · p. 34, §4.5.1 · pp. 34–35

4.5.1 Distributed Concept Annotation

• During distributed annotation, each chunk receives global concept IDs and a cached 1024-dimensional FP16 embedding, yielding approximately 11 billion vectors for retrieval and analysis.The embedding is computed in the same forward pass as multi-domain concept prediction.Source: §4.5.1 · p. 35

4.5.2 Tokenization and document structure

• Tokenization preserves document–chunk structure with [BOS], [EOC], and <|endoftext|> markers, while chunk-level concepts align to token spans through [EOC]-delimited OR-aggregation during training.The full vocabulary contains 100,281 tokens, including [MASK] for diffusion training.Source: §4.5.2 · pp. 35–36

4.6 Training data indexing for test-time attribution

• Cached chunk embeddings support test-time training-data attribution through a FAISS IVFPQ index, which reduces exact-search latency and memory costs by partitioning and compressing vectors.Index construction reuses embeddings from annotation without additional annotator forward passes.Source: §4.6 · p. 36
• Product quantization compresses each vector payload by roughly 64×, while memory-mapped inverted lists enable full-corpus search without loading the approximately 808 GB index into RAM.Vectors are ℓ2-normalized, making inner-product retrieval equivalent to cosine similarity; only query-touched lists are paged into memory.Source: §4.6 · p. 36
• 96.8% recall@10 shows that IVFPQ recovers the exact nearest neighbor in the large majority of evaluated queries.Evaluation used 1,000 query chunks drawn from the index, with nprobe = 16; recall@10 measures whether each query’s known source chunk appears among the top-10 results.Source: §4.6 · pp. 36–37

5.1 Beyond autoregressive models

• The model replaces autoregressive generation with masked diffusion, enabling absence-based attribution, phrase-level concept control, and arbitrary-order expression of intervened concepts.These properties arise because masking provides an explicit no-information state, jointly predicts multiple tokens, and permits any generation order.Source: §5.1 · p. 37

5.2 Causal Diffusion

• Causal Diffusion combines bidirectional attention within blocks with causal attention across blocks, preserving diffusion’s joint denoising while enabling autoregressive-style KV caching.It trains on a single sequence with the standard masked diffusion loss, avoiding Block Diffusion’s clean-copy overhead and caching keys and values across generated blocks.Source: §5.2 · pp. 38–39, Figure 15 · p. 38

5.3 Concept module

• The concept module reconstructs each hidden state from known concepts, unknown concepts, and a residual before the language-model head, making every output an explicit additive function of concept activations.Because the head is linear, each logit’s attribution to known concepts, unknown concepts, and the residual is exact rather than approximate, supporting faithful attribution and concept steering.Source: §5.3 · pp. 39–40, Figure 16 · p. 39
• The module uses labeled and unknown concept heads, while low-rank factorization reduces the unknown embedding parameters from md to r(m + d) for rank r ≪d.The unknown embedding matrix is factorized as U = AB, preserving capacity while avoiding direct storage of the full matrix.Source: §5.3 · p. 40

5.4.1 Loss objectives

• Training jointly optimizes masked language modeling, chunk-level concept prediction, unknown-state reconstruction, and independence losses, with auxiliary gradients updating only the unknown head.The reconstruction loss targets the residual after known concepts are removed, while the independence penalty encourages known and unknown representations to encode complementary information.Source: §5.4 · p. 40, §5.4.1 · pp. 40–42

5.4.2 Training dynamics

• Per-block masking samples an independent noise level for each block, exposing the model to partially denoised contexts at different stages within one training step.This differs from standard masked diffusion, which applies one global noise level to the entire sequence.Source: §5.4.2 · p. 42, Figure 17 · p. 42
• Scheduled teacher forcing stabilizes both concept heads by replacing predicted representations with ground-truth or ideal targets early in training, then annealing toward the model’s own predictions.Known-concept teacher forcing limits concept leakage, while unknown-concept forcing uses u_hat_GT = h−k_hat_GT because unknown labels are unavailable.Source: §5.4.2 · pp. 42–43

6 Interpretability capabilities

• Steerling provides attribution and controllability by linking outputs to influential inputs, concepts, and training data, then modifying concept representations during inference.Attribution explains why a prediction was produced, while steering controls how selected concepts affect behavior.Source: §6 · p. 43

6.1 Attribution

• Three attribution views identify influential input tokens, internal concepts, and similar training examples for a generated output chunk.Figure 18 presents the three methods applied to one output chunk.Source: §6.1 · p. 43, Figure 18 · p. 44

6.1.1 Input attribution

• Input attribution uses Integrated Gradients from the learned [MASK] embedding, which represents no information because diffusion training repeatedly predicts masked tokens.This baseline provides an in-distribution absent state, unlike zero embeddings or padding tokens in autoregressive models.Source: §6.1.1 · pp. 43–44

6.1.2 Concept attribution

• Concept attribution exactly decomposes output logits into known-concept, unknown-concept, and residual terms, then ranks concepts by summed chunk-level contributions.The residual captures output behavior not explained by the concept inventory.Source: §6.1.2 · p. 44

6.1.3 Training data attribution

• Training data attribution retrieves semantically similar training chunks through projected nearest-neighbor search, providing a scalable proxy for influence rather than a causal account.A learned MLP maps pooled output representations into the corpus index’s embedding space; retrieved sources can be inspected for verification.Source: §6.1.3 · pp. 44–45

6.2 Steering

• Steering controls generation without prompts or weight updates by injecting concept directions, supporting amplification or suppression for both supervised known and discovered unknown concepts.The intervention acts directly on the concept representations already used by the model.Source: §6.2 · p. 45

6.2.1 Steering operation

• Each steering direction is a normalized learned concept embedding rather than a post-hoc estimate, and its strength is calibrated per concept to a shared maximum logit-shift target.The direction is injected into hidden states at masked positions from a chosen layer onward.Source: §6.2.1 · pp. 45–46

6.2.2 Steering direction

• Positive steering amplifies a concept, while suppression combines negative hidden-state injection with ReLU-gated logit subtraction to avoid promoting anti-aligned tokens.The gate suppresses positively aligned tokens and leaves anti-aligned tokens unchanged.Source: §6.2.2 · pp. 46–47

Metric Design

• Together, the metrics test whether interpretability is functioning as an architectural design constraint rather than merely a post-hoc analysis.They correspond to concept detection, routing predictions through concepts instead of residual information, disentanglement, and token-level semantic interpretability.Source: §7 · p. 47

Concept Loss

• Concept loss measures whether the module identifies the correct concepts in unseen text using held-out, pipeline-annotated chunks.It is an OR-aggregated binary cross-entropy between predicted concept presence and ground-truth chunk labels; lower values indicate better detection.Source: §7 · pp. 47–48

Concept Independence Loss

• Concept independence measures linear dependence between known and unknown representations, testing whether their attribution contributions remain additively separable.The linear kernel matches the additive bottleneck and linear LM head, so low dependence indicates that the pathways avoid encoding overlapping information.Source: §7 · p. 48

Concept Contribution

• Concept contribution measures the fraction of output-logit magnitude attributable to known and unknown concepts rather than the residual, directly testing attribution faithfulness.It is computed on held-out predictions; higher values indicate that concepts explain more of each prediction, while reconstruction training pressures the residual toward zero.Source: §7 · p. 48

Known Concept Alignment

• Known concept alignment evaluates whether each concept’s top projected tokens semantically match its human-assigned label and description.An LLM judge rates alignment from 1-5 over sampled concepts; this property is not directly supervised and must emerge from joint concept and language-model optimization.Source: §7 · pp. 48–49

8.1 Setup

• The experiments compare fixed AR and CDLM backbones with and without the same concept module across four IsoFLOP slices and four to six model sizes per slice.All runs use the same data, optimizer, and schedule; the lowest +Concept slice uses slightly higher compute targets because concept-module overhead makes small-scale estimates unreliable.Source: §8.1 · pp. 49–50

8.2.1 Methodology

• Compute-optimal scaling fits per-slice validation-loss parabolas, power laws across compute, and a joint Chinchilla-form fit for irreducible loss and token scaling.The reported 90% confidence intervals use residual bootstrapping over 10,000 iterations.Source: §8.2.1 · pp. 51–52

8.2.2 Results

• Adding the concept module shifts compute-optimal exponents by small fixed per-backbone offsets, while irreducible loss decreases from 1.857 to 1.193 for AR and 2.658 to 1.942 for CDLM.The αP shifts are small, and both αL values become more negative; the authors characterize this as no scaling tax.Source: §8.2.2 · p. 53, Table 6 · p. 54
• Steerling-8B achieves validation loss 2.72 versus 2.61 predicted by the joint Chinchilla fit, a 0.11-nat gap and roughly four times smaller error than log-linear extrapolation.The model was trained at 8B parameters, 1.35T tokens, and 7.6 × 10^22 FLOPs; log-linear extrapolation misses by 0.47 nats.Source: §8.2.2 · p. 54, Figure 23 · p. 55

8.3.1 Methodology

• Interpretability scaling is fit using both compute alone and a joint parameter-token model because fixed-compute checkpoints differ systematically in how their budgets are split.The joint model resolves within-IsoFLOP variation that compute-only fits treat as noise, while bounded metrics constrain extrapolated values to their natural ranges.Source: §8.3.1 · pp. 54–55, §8.3.2 · p. 57

8.3.2 Results

• Across AR+Concept and CDLM+Concept, all four interpretability metrics improve with compute: Concept Loss and Concept Independence Loss decrease, while Concept Contribution and Known Concept Alignment increase.Concept Loss falls toward zero, Concept Independence Loss indicates greater disentanglement, Concept Contribution rises toward one, and Known Concept Alignment approaches the judge ceiling.Source: §8.3.2 · p. 56, Figure 24 · p. 56
• Model size primarily drives Concept Loss and Concept Independence Loss, training data primarily drives Known Concept Alignment, and Concept Contribution lies between these regimes; no single compute split optimizes all four metrics.Source: §8.3.2 · p. 57, Figure 25 · p. 58

8.3.3 Predicting Steerling-8B from small models

• Small-scale interpretability fits predict Steerling-8B within 0.0004 for Concept Loss, 0.04 for Concept Contribution, and 0.10 for Known Concept Alignment, while Concept Independence Loss reaches 1.55 versus 2.16 predicted.The independence result is favorable because the deployed value is lower than the extrapolated value.Source: §8.3.3 · p. 57, Table 7 · p. 57
• Three of four interpretability metrics are predicted within error bounds for Steerling-8B, while the fourth exceeds its prediction in the favorable direction.Predictions use asymptotic fits on small-scale checkpoints; Steerling is evaluated at 8B parameters and 1.35T tokens.Source: §8.3.3 · p. 59, Figure 26 · p. 59

9.1 Data

• The 1.2T-token corpus combined high-quality web, academic, mathematical, and code data, with chunk-level concepts annotated using the Atlas pipeline.Its bulk was Nemotron-CC-HQ, supplemented by peS2o, arXiv, OpenWebMath, Algebraic Stack, StarCoder, Wikipedia, and Wikibooks.Source: §9.1 · pp. 59–60, Table 9 · p. 60

9.2 Pretraining recipe

• The recipe used block size b = 64, moving Gaussian masking, m = 3n unknown concepts, and a rank-R = 256 factorization of U that was roughly 15× smaller.The factorization removed costly per-token matrix multiplication without capability cost, while retaining a residual channel because removing it created a large capability gap.Source: §9.2 · pp. 60–61

9.3 Pretraining run

• Concept contribution rose from 0.62 to 0.85, while concept independence loss eventually peaked near 2.7 and known concept alignment settled below its early peak near 4.Capability metrics diverged: HellaSwag and PIQA held near their peaks, ARC-Challenge peaked late, while MMLU lost roughly a quarter of its peak value.Source: §9.3 · p. 61, Figure 27 · p. 61, Figure 28 · p. 62

9.4 Pretraining lessons

• MMLU and WinoGrande declined after midpoint, concept independence loss spiked late, math and code performance was weak, and steering was unreliable, prompting four diagnosed causes.The causes were over-aggressive masking, late-training concept-head entanglement, insufficient math and code, and no steering operations during training.Source: §9.4 · p. 62

9.4.1 Over-aggressive masking

• At pretraining scale, moving Gaussian masking passed 50% and coincided with MMLU and WinoGrande inflections, leaving the model with little input context.The curriculum had shown a slight validation-loss advantage at small scale but became over-aggressive during full pretraining.Source: §9.2 · p. 60, §9.4.1 · p. 62

9.4.2 Late-training entanglement of the concept heads

• Late masking and reduced teacher forcing made the concept heads rely on overlapping predicted information, causing concept independence loss to rise and representations to entangle.This effect was not visible at ablation scale.Source: §9.4.2 · pp. 62–63

9.4.3 Limited math and code in the corpus

• The corpus contained slightly over 100B math and code tokens out of 1.2T, and low GSM8K and HumanEval scores exposed the imbalance only after pretraining.The dominant Nemotron-CC-HQ source carried little math or code, so earlier in-run evaluation could have enabled data correction.Source: §9.4.3 · p. 63

9.4.4 The model does not respond to steering

• 1.156 was the best harmonic mean for prompting, while steering achieved the highest concept score at 1.072 but substantially reduced output quality.Steering also failed to activate roughly one third of less-frequent concepts, because concept injection was never encountered during pretraining and was therefore out of distribution.Source: Table 10 · p. 63, §9.4.4 · p. 63

10 Steerling-8B: Mid-training

• The 150B-token mid-training run starts from the final pretraining checkpoint and targets math, code, knowledge recovery, reduced teacher forcing, and greater reliance on concept heads.Its data mixture uses real Nemotron-CC-HQ tokens, with the code-augmented recipe selected after candidate-mixture ablations.Source: §10 · p. 64, §10.1 · p. 64

10.1 Data

• The adopted code-augmented mixture is the only candidate improving every benchmark, balancing reasoning, math, and code across the 10B-token ablation.Math-heavy training gives the largest math gain and best average, balanced training gives the largest knowledge gain, and code-only training leads on HumanEval but lags elsewhere.Source: §10.1 · pp. 64–65, Table 12 · p. 65

10.2.1 Schedules

• Uniform masking replaces heavy-masking curriculum, lifting MMLU to 0.416 versus 0.280 under the 80% schedule and 0.298 for the pretrained model.Teacher forcing is also annealed to zero so every bottleneck concept is model-predicted during inference-like training.Source: §10.2.1 · pp. 65–66, Table 14 · p. 66

10.2.2 Losses

• The independence loss is replaced by Lindep(ˆu, ˆk)+Lindep(ε, ˆk), preserving total loss weight while separately discouraging dependence in predicted unknown and residual representations.This change addresses the limitation exposed when predicted concepts fully supply the bottleneck and are detached before it.Source: §10.2.2 · p. 66

10.2.3 Architecture

• Residual dropout rises from 0.1 to 0.3, while sparsified concept heads use the top 32 known and 128 unknown concepts, limiting each logit to at most 160 concepts.These changes force more prediction through the concept heads and make attribution more interpretable.Source: §10.2.3 · p. 66

10.2.4 Steering training

• Interleaved steering phases leave LM Harness performance essentially unchanged while improving every steering metric and activating the target concept after one injection at γ = 1.The phases use roughly 400M token-level annotated tokens and are front-loaded after a 100-step warmup rather than added as a separate stage.Source: Table 15 · p. 68, §10.2.4 · p. 68, Figure 30 · p. 68

10.3 The mid-trained model

• Mid-training improves every benchmark, raising MMLU by 17 percentage points, math by 30 points, code by around 7 points, and the overall average by 10 points.Benchmarks that were already healthy remain healthy, so the capability recovery does not trade away prior strengths.Source: Table 16 · p. 69, §10.3 · p. 69

10.4 Evaluation and results

• Steerling-8B lands within ∼10% of open peers trained with 2–16× more compute and ahead of peers trained at comparable budgets.The comparison covers seven benchmarks and shows that inherent interpretability does not require sacrificing capability.Source: §10.4 · pp. 69–70, Table 19 · p. 70

11.1 Underspecification and the Rashomon effect

• In the Rashomon setting, equally performant models can use structurally different mechanisms, making post-hoc explanations and counterfactuals unstable across valid training outcomes.Feature importance can vary substantially across equivalent models, and nearly identical-accuracy models may produce contradictory attributions or invalid counterfactual recommendations.Source: §11.1 · pp. 70–71
• Steerling constrains the Rashomon set by fixing the concept library, additive attribution interface, and shared [MASK]-based absence baseline during training.Different runs may still learn different parameters, but the concepts, attribution computation, and meaning of absence remain fixed.Source: §11.1 · p. 71

11.2 Large language models

• Diffusion language models generate tokens by iterative denoising, and Steerling’s Causal Diffusion combines block-causal structure with the concept module at roughly half the cost of Block Diffusion.Masked diffusion also provides a learned [MASK] baseline, parallel generation, and arbitrary-order generation that support attribution and interventions.Source: §11.2 · pp. 71–72

11.3 Scaling laws

• Steerling extends scaling-law methodology to inherently interpretable autoregressive and diffusion models, testing whether concept modules preserve compute-optimal scaling.This differs from prior work that studies scaling of post-hoc probes on fixed, uninterpretable models, while Steerling measures interpretability changes with training compute.Source: §11.3 · p. 72

11.4 Interpretable-by-design architectures

• Prior interpretable-by-design models use concept bottlenecks, concept embeddings, token sense vectors, or prototypes, whereas Steerling scales an additive concept vocabulary beyond a hundred thousand concepts.Earlier approaches support interpretable generation, protein modeling, counterfactuals, or training-data attribution, but operate at substantially smaller scale and with fewer concepts.Source: §11.4 · p. 72

11.5 Attribution methods

• Steerling produces exact architectural attributions: Integrated Gradients uses learned [MASK] absence, concept contributions are read directly from the forward pass, and training-data retrieval matches latent representations.This contrasts with zero or padding baselines, post-hoc concept probes, approximate influence estimates, and surface-form substring retrieval.Source: §11.5 · p. 73

11.6 Model steering

• Unlike post-hoc activation steering, Steerling embeds each concept’s steering direction in the architecture, enabling closed-form positive or negative edits to its additive logit contribution.Existing steering methods derive directions from contrastive examples or feature dictionaries after the model has produced activations.Source: §11.6 · p. 73

• Across three orders of magnitude of compute, interpretability constraints improved with compute, with larger models using human-understandable concepts more independently and semantically aligned.Under the measured metrics, models became easier rather than harder to understand as they became more capable.Source: §12 · p. 74
• The recipe builds faithful interpretability into data, architecture, objectives, and losses, enabling concept-based attribution, training-data inspection, intervention, and logit-level verification without retraining.Its components are justified by specific conditions for faithful attribution rather than functioning as post-hoc add-ons.Source: §12 · p. 74
• The presented design is a first instantiation limited by a fixed concept library, one additive bottleneck, chunk-level supervision, and supervised-finetuning-only training.Each choice could be substantially improved in future work.Source: §12 · p. 74
• Future directions include concept-based post-training rewards, inspectable agent trajectories, hierarchical adaptive vocabularies, interactive vocabulary design, and grounded natural-language explanations.Current interfaces provide structured artifacts, ranked concept contributions, token-level scores, and retrieved training chunks.Source: §12 · p. 74
• Steerling-8B is evidence that frontier-grade interpretable and agentic systems may be feasible, but it remains a research-program demonstration rather than a finished system.The scaling results motivate systems whose actions can be decomposed, audited, and steered.Source: §12 · p. 74
• Interpretability, steerability, and reliability requirements can be specified as a training contract and scaled like capabilities rather than retrofitted or treated as taxes.This challenges the idea that opacity in highly capable systems is inevitable.Source: §12 · p. 74

Collaboration

• Contributions spanned data annotation, architecture, training, infrastructure, post-training, and product, with no component built in isolation or exclusively owned.The manuscript describes iterative feedback across the team and frames listed responsibilities as primary rather than exclusive.Source: Authorship and Credit Attribution · p. 75

Data

• Nathaniel Monson, Saqib Azim, and Julius Adebayo built the Atlas annotation system, covering infrastructure, concept design, clustering, labeling, annotators, indexing, and corpus annotation.Monson led the annotation infrastructure, concept library, and validation study; Azim led Stage 2 and the distributed training-data index; Adebayo designed the multi-head annotator and annotated the pre-training corpus.Source: Authorship and Credit Attribution · p. 75

Model Architecture, Training Recipe, Scaling Laws, and Training

• Aya Abdelsalam Ismail led model architecture, training recipe, concept modules, causal diffusion attention, and scaling laws, while Giang Nguyen led steering design, validation, implementation, and attribution.Ismail devised causal diffusion attention with Andreas Madsen; Nguyen also contributed to training-recipe design and input-feature attribution.Source: Authorship and Credit Attribution · p. 75

Infrastructure

• Andreas Madsen developed the dataloader and distributed multi-node framework, Julius Adebayo wrote the pre-training package, and both managed the internal SLURM cluster.Madsen and Adebayo jointly supported the infrastructure used for research and production training.Source: Authorship and Credit Attribution · p. 75

Model Post-Training, Model Post-Processing, and Open Source

• Andreas Madsen led Steerling post-training, while Nathaniel Monson led concept naming and Aya Abdelsalam Ismail and Giang Nguyen led release of the base and instruct models.The release included artifacts enabling attribution and steering, and post-training adapted supervised fine-tuning to block diffusion and interpretability training.Source: Authorship and Credit Attribution · p. 75

Clarity Platform and Supervision

• Zhichen Guo and Isaac Plant designed Clarity, with engineering, serving, attribution, frontend, API, and production data-attribution responsibilities distributed across the team; Julius Adebayo supervised the project.Andreas Madsen led platform engineering, Muawiz Chaudhary led serving-engine implementation, and Saqib Azim implemented production training-data attribution.Source: Authorship and Credit Attribution · p. 75

A Symbol reference

• The appendix organizes concept-module notation, atlas evaluation, and additional human-interpretability analyses alongside the attribution and study protocols.These materials are presented as symbol reference, prompt-based concept evaluation, and methodological/statistical details supporting the human study.Source: Table 20 · p. 88, Table 21 · p. 89, Figure 32 · p. 91

B.1 Training data attribution

• The training-data attribution pipeline indexes approximately 11 billion chunks as 1024-dimensional vectors in a FAISS IVFPQ index for approximate nearest-neighbor retrieval.The inverted file restricts comparisons to nearby coarse clusters, while product quantization compresses stored vectors for in-memory indexing.Source: §B.1 · p. 90
• Response chunks are mean-pooled from per-token hidden states into 4096-dimensional representations combining known-head, unknown-head, and residual components.A roughly 15-million-parameter, two-hidden-layer MLP maps these representations to the corpus index space while preserving semantic content under cosine similarity.Source: §B.1 · p. 90
• End to end, each response chunk is encoded, transduced into corpus-embedding space, and used to retrieve the most similar training chunks as attributed sources.The retrieval searches the top-n coarse clusters of the IVFPQ index.Source: §B.1 · p. 90

D.1 Sampling and annotation protocol

• Phase 1 sampled 100 concepts and collected 303 named-concept responses from 9 annotators, with a median of 3 annotators per concept.Annotators saw only lifted-word lists, wrote names or short phrases, and rated recognizability on a 1–5 scale.Source: §D.1 · p. 92
• Phase 2 collected 1,025 candidate-name ratings across 34 concepts from 8 annotators using randomized, provenance-blinded comparisons.Candidate sets included the pipeline label, human labels, embedding- and taxonomy-neighbor distractors, and occasional filler controls.Source: §D.1 · p. 92

D.2 Phase 1 agreement and coherence

• 55% of Phase 1 responses formed recognizable concepts, with a mean coherence score of 3.52 and 17% judged incoherent or noisy.The remaining 27% were borderline, rated 3 on the 1–5 scale.Source: §D.2 · p. 92
• Agreement was moderate: ICC(1) was 0.43, within-concept standard deviation was 0.65, and 53% of concepts were unanimous on coherence.Total scale standard deviation was 1.05, indicating that lifted-word evidence was meaningful but not uniformly so.Source: §D.2 · p. 92

D.3 Ordinal mixed-model analysis

• 2.38 odds ratio: pipeline labels received higher fit ratings than human labels, with 79% versus 63% top-two ratings and posterior probability 0.99.The LLM–human proportional-odds ratio was 2.38 (95% CrI [1.23, 4.01]).Source: §D.3 · p. 93

D.4 Assumption-light robustness checks

• Robustness checks found pipeline labels outscored human labels with probability 0.62 and by 0.47 raw-rating points, confirming higher average blind-evaluation ratings.The paired comparison used 402 pairs; its cluster-bootstrap 95% CI was [0.58, 0.66], and the Gaussian mixed-model 95% CI was [0.28, 0.65].Source: §D.4 · p. 93

D.5 Dependence on lifted-word coherence

• Pipeline labels outscored human labels across coherence bins, with the LLM–human gap increasing from +0.22 for low-coherence to +0.54 mid-coherence concepts.Phase 1 coherence correlated with Phase 2 fit at +0.42 for human labels and +0.46 for pipeline labels; the high-coherence gap was +0.52.Source: §D.5 · p. 93

D.6 Power analysis

• Power exceeded 0.99 at roughly 20 fully rated concepts and was essentially 1.00 at the realized sample size, supporting the aggregate pipeline–human comparison.The analysis does not certify every individual concept in the full library.Source: §D.6 · p. 94

D.7 Limitations

• The human study validates Atlas’s labeling operation, not every concept, because lifted-word evidence can be ambiguous or noisy and domain expertise may vary.Phase 2 compares names with lifted-word evidence rather than full source-document evidence, so downstream per-domain or per-concept audits remain necessary.Source: §D.7 · p. 94

E Known concept alignment judge

• Known Concept Alignment uses a temperature-0 Mistral-Small-3.1-24B-Instruct judge that rates each concept from 1 to 5 using its label, description, and top-K tokens.The full judging prompt is shown in Figure 33.Source: E Known concept alignment judge · p. 95, Figure 33 · p. 95

F Symbol and notations

• The scaling-law analysis introduces notation grouped around compute and resources, validation losses, power-law parameters, and interpretability scaling.Table 23 organizes the notation by these four roles.Source: Table 23 · p. 96

G Architectures, IsoFLOP slices, and hyperparameters

• All four backbone families share one architecture, while +Concept variants add concept-module heads; sequence length is fixed at 4096.Backbone parameter counts exclude embeddings, whereas +Concept totals include concept-module heads.Source: G Architectures, IsoFLOP slices, and… · p. 96, Table 24 · p. 97
• Each family is evaluated in four IsoFLOP slices containing four to six model sizes within ±15% of each target compute budget.+Concept families use higher initial compute targets because their concept modules add non-negligible per-token FLOPs at small backbone sizes.Source: G Architectures, IsoFLOP slices, and… · p. 96, Table 25 · p. 97
• Scaling runs hold optimizer, learning rate, batch size, warmup, decay schedule, and backbone architectural defaults fixed across model sizes and families.Training uses AdamW, peak learning rate 4×10^-4, WSD scheduling, and a total batch size of 524,288 tokens.Source: G Architectures, IsoFLOP slices, and… · p. 97
• CDLM and CDLM+Concept models use a causal block size of 64 and uniformly sampled noise levels t ∼ U(0.05, 0.95) across sizes.The shared masking settings are specific to the two diffusion families.Source: G Architectures, IsoFLOP slices, and… · p. 97
• The concept module is shared by both +Concept families, with distinct stable and anneal configurations and top-k sparsification applied during annealing.The αknown(t) and αunknown(t) schedules control replacement of predicted components with ground-truth components during training.Source: G Architectures, IsoFLOP slices, and… · p. 97, Table 26 · p. 98, Table 27 · p. 98

H ELBO estimation for validation loss

• Compute-optimal parameter counts and scaling exponents are robust across four CDLM validation-loss estimators, while absolute loss and irreducible-loss estimates differ by up to one nat.The estimator-dependent bias can propagate into reported L∞ values and scaling exponents when it varies across model sizes.Source: H ELBO estimation for validation loss · p. 98

H.1 Estimators

• Validation compares fixed-rate mask loss, a nine-bin discretized ELBO, the MDLM ELBO, and a per-block ELBO over approximately 100M validation tokens.The fixed-rate estimator uses τ = 0.5, while the discretized estimator uses K = 9 bins spanning the training interval.Source: §H.1 · pp. 98–99
• The MDLM estimator is unbiased for the per-token NELBO bound, whereas the per-block estimator matches blockwise training noise and achieves lower Monte Carlo variance.The per-block method uses b = 64 and independently samples one noise level per block; the MDLM estimator integrates over [0, 1].Source: §H.1 · pp. 99–100

H.2 Results

• αP fits overlap across estimators at 0.602–0.632, while αL ranges from −0.071 to −0.039 and estimator offsets shift absolute losses.The four IsoFLOP parabolas have comparable minima, tighten with compute, and move toward larger P as C grows.Source: §H.2 · p. 100, Figure 35 · p. 101

H.3 Effect of mask rate on scaling exponents

• At fixed mask rates, αP is ≈0.60 for t ≤0.7 and ≈0.62 for t ≥0.8, while αL flattens from −0.120 at t = 0.1 to −0.013 at t = 0.9.Thus, aggregation preserves similar parameter scaling but weights mask rates differently when determining loss scaling.Source: §H.3 · pp. 100, 102, Figure 37 · p. 102

H.4 Discussion

• L∞ spans roughly 1.9 to 3.4 across estimators, so absolute asymptotes are not directly comparable across diffusion methodologies, although within-methodology comparisons remain valid.The CDLM is trained with t ∼U(0.05, 0.95), and τ = 0.5 gives the lowest L∞ among the four estimators.Source: §H.4 · pp. 102–103

I Annealing each IsoFLOP checkpoint

• Annealing CDLM+Concept checkpoints changes both learning-rate and interpretable-component schedules, transitioning from teacher-forced concept representations toward learned heads with top-k sparsification.The annealed configuration is intended to match deployed inference, rather than merely applying learning-rate decay.Source: I Annealing each IsoFLOP checkpoint · p. 103

I.1 Validation loss

• αP increases from 0.574 pre-anneal to 0.686 post-anneal, while αL is essentially unchanged and slice-wise L∗ shifts by at most 0.11 nats.The shift requires asymmetric loss improvements across model sizes, moving compute-optimal P∗ systematically larger under annealed evaluation.Source: §I.1 · pp. 103–104, Figure 38 · p. 104, Table 30 · p. 104

I.2 Interpretability

• Concept Independence Loss decreases 3–10× under annealed evaluation, while Concept Loss, Concept Contribution, and Known Concept Alignment remain broadly stable.Known Concept Alignment changes from β = 0.447 pre-anneal to 0.437 annealed, whereas annealed HSIC falls from 30–80 to 5–20 with R2 = 0.79.Source: §I.2 · pp. 104–105, Figure 39 · p. 105, Table 31 · p. 105

J Pretraining recipe ablations; K Final pretraining Steerling-8B configuration

• The ablation baseline fixes a 1B model trained on 30B Nemotron-CC-HQ tokens and tracks validation loss, MMLU soft score, HellaSwag, concept contribution, and known concept alignment.The selected architecture and training settings are consolidated in the final pretraining configuration.Source: J Pretraining recipe ablations · pp. 106–107, Table 32 · p. 106, Table 34 · p. 113

J.1 Diffusion block size; J.2 Diffusion masking schedule; J.3 Unknown concept capacity

• The recipe adopts b = 64, moving-Gaussian masking, and m = 3n because larger blocks improve capability, the moving schedule slightly improves validation loss, and extra unknown capacity yields no gain.Interpretability is unchanged across block sizes and unknown capacities, while the two masking schedules are otherwise closely matched.Source: §J.1 · p. 107, §J.2 · p. 108, §J.3 · pp. 108–109

J.4 Unknown embedding factorization

• Factorizing the unknown embedding at R = 256 is roughly 15× smaller with no capability cost, trading a small concept-contribution drop for lower compute and memory.Concept contribution is approximately 0.90 for the dense head versus 0.80 for the factorized head.Source: §J.4 · p. 109

J.5 Use of the residual term

• Removing the residual forces concept contribution to 1.0 by construction but causes a persistent capability penalty, so the residual is retained.HellaSwag ends around 0.31 without ε versus 0.36 with it.Source: §J.5 · p. 110

J.6 Concept teacher forcing schedule

• Keeping the known teacher-forcing floor at 0.5 preserves capability and raises known concept alignment to ∼3.7, whereas decaying it to 0.0 stalls alignment near 2.6.The schedule begins with full teacher forcing, then cosine-decays to its floor.Source: §J.6 · pp. 110–111

J.7 Unknown concept teacher forcing schedule

• The unknown teacher-forcing floor has modest effects favoring αunknown,min = 0.5 on validation loss and HellaSwag, so the higher floor is adopted.Concept contribution is slightly higher at a floor of 0.0, while other metrics overlap.Source: §J.7 · p. 111

L Pretraining diagnostic

• During pretraining, harder reasoning benchmarks decline as the masking curriculum crosses its midpoint, while concept independence loss rises in the final third as schedules enter harder regimes.Concept contribution rises as the LM head increasingly depends on predicted concepts; known alignment changes only modestly.Source: L Pretraining diagnostic · p. 114, Figure 49 · p. 114, Figure 50 · p. 115

M Steering judge; N.1 Nemotron: real, synthetic, and mixed; N.2 Final mid-training Steerling-8B configuration

• Real Nemotron tokens are selected for natural-language mid-training because real, synthetic, and mixed sources improve substantially over the base, while real data gives the strongest knowledge and math scores.The final mid-training hyperparameters are listed alongside pretraining values, and steering-judge prompts are provided separately.Source: Figure 51 · p. 116, Figure 52 · p. 117, §N.1 · p. 118