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Scaling Inherently Interpretable Language Models
Guide Labs Team, Andreas Madsen, Aya Abdelsalam Ismail, Giang Nguyen, Isaac Plant, Muawiz Chaudhary, Nathaniel Monson, Saqib Azim, Zhichen Guo, Julius Adebayo
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Summary
This paper introduces a method for building inherently interpretable language models by integrating interpretability constraints directly into the training pipeline, rather than relying on post-hoc explanation methods. The authors propose a training recipe that optimizes interpretability alongside the language modeling objective, demonstrating that interpretability scales with capability across different model architectures (autoregressive and diffusion). They instantiate this approach with Steerling-8B, a diffusion-based language model that provides input, concept, and training data attribution. The model enables closed-loop intervention through concept steering without retraining. Experiments show that Steerling-8B remains competitive with opaque models trained on significantly more compute, suggesting that interpretability can be designed into training and improves with scale.
Entities (7)
Relation Signals (6)
Steerling-8B → uses → Diffusion Language Model
confidence 98% · Steerling-8B, a diffusion language model with a causal attention mask.
Steerling-8B → implements → Inherent Interpretability
confidence 95% · We instantiate the training-time recipe with Steerling-8B... suggesting a different scaling paradigm: interpretability can be designed into training
Interpretability → scaleswith → Capability
confidence 93% · interpretability scales with capability rather than against it.
Concept Bottleneck → enables → Logit Decomposition
confidence 92% · we insert an additive concept bottleneck that makes the logit decomposition algebraically exact.
ATLAS → provides → Concept Library
confidence 90% · we build Atlas, a concept annotation pipeline... canonicalizes them into over 33,000 concepts
Steerling-8B → supports → Concept Steering
confidence 90% · correct the behavior through concept steering without retraining.
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Abstract
Abstract:Interpretability is often treated as a tax on capability: language models are trained as opaque systems, then explained after the fact, with methods whose reliability is difficult to establish. In this work, we challenge this premise. Rather than reverse-engineering a model, we make interpretability a constraint of the training pipeline, optimized alongside the language modeling objective. Across three orders of magnitude of compute, on both autoregressive and diffusion language models, interpretability scales with capability rather than against it. Surprisingly, model representations become more disentangled and aligned with human-understandable concepts with scale. We instantiate the training-time recipe with Steerling-8B, a diffusion language model with a causal attention mask. For any group of generated tokens, Steerling-8B attributes the output to relevant input tokens, human-understandable concepts, and training data. This enables closed-loop intervention: diagnose an output through its concept or feature attribution, retrieve similar training data, and correct the behavior through concept steering without retraining. Steerling-8B remains competitive with open peer models trained on substantially 2-16x more compute, suggesting a different scaling paradigm: interpretability can be designed into training, and it improves with scale.
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- Source: https://arxiv.org/abs/2608.07594v1
- Canonical: https://arxiv.org/abs/2608.07594v1
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Scaling Inherently Interpretable Language Models Guide Labs Team Andreas Madsen Aya Abdelsalam Ismail Giang Nguyen Isaac Plant Muawiz Chaudhary Nathaniel Monson Saqib Azim Zhichen Guo Julius Adebayo g Abstract Interpretability is often treated as a tax on capability: language models are trained as opaque systems, then explained after the fact, with methods whose reliability is difficult to establish. In this work, we challenge this premise. Rather than reverse-engineering a model, we make interpretability a constraint of the training pipeline, optimized alongside the language modeling objective. Across three orders of magnitude of compute, on both autoregressive and diffusion language models, interpretability scales with capability rather than against it. Surprisingly, model representations become more disentangled and aligned with human-understandable concepts with scale. We instantiate the training-time recipe with Steerling-8B, a diffusion language model with a causal attention mask. For any group of generated tokens, Steerling-8B attributes the output to relevant input tokens, human-understandable concepts, and training data. This enables closed-loop interven- tion: diagnose an output through its concept or feature attribution, retrieve similar training data, and correct the behavior through concept steering without retraining. Steerling-8B remains competitive with open peer models trained on substantially 2–16× more compute, suggesting a different scaling paradigm: interpretability can be designed into training, and it improves with scale. g Steerling was a team effort; authors other than the last are sorted alphabetically. See author contributionshere. 1 arXiv:2608.07594v1 [cs.CL] 6 Aug 2026 Contents 1 Introduction1 2 Background3 2.1 Transformer notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 2.2 Autoregressive language models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 2.3 Masked diffusion language models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 2.4 Concept bottleneck models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 3 A recipe for building interpretable models5 3.1 Interpretability requirements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 3.2 Why post-hoc methods fail to satisfy the interpretability requirements. . . . . . . . .8 3.3 Faithfulness and inherent interpretability. . . . . . . . . . . . . . . . . . . . . . . . . .10 3.4 The interpretable model-training recipe. . . . . . . . . . . . . . . . . . . . . . . . . . . .12 4 Data15 4.1 Motivation: No human-interpretable concept library exists at scale. . . . . . . . . . .15 4.2 Atlas: Documents to concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 4.3 Concept taxonomy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 4.4 Human interpretability of the concept library. . . . . . . . . . . . . . . . . . . . . . . .27 4.5 Data Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 4.6 Training data indexing for test-time attribution. . . . . . . . . . . . . . . . . . . . . . .32 5 Inherently interpretable architecture33 5.1 Beyond autoregressive models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 5.2 Causal Diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 5.3 Concept module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35 5.4 Model training. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 6 Interpretability capabilities39 6.1 Attribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 6.2 Steering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 7 Interpretability Metrics43 8 Scaling laws45 8.1 Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 8.2 Compute-Optimal scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 8.3 Interpretability scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 9 Steerling-8B: Pretraining55 9.1 Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55 9.2 Pretraining recipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 9.3 Pretraining run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57 9.4 Pretraining lessons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58 10 Steerling-8B: Mid-training60 10.1 Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 10.2 Mid-training recipe changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 10.3 The mid-trained model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 10.4 Evaluation and results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 11 Related work66 11.1 Underspecification and the Rashomon effect. . . . . . . . . . . . . . . . . . . . . . . . .66 11.2 Large language models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 11.3 Scaling laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 11.4 Interpretable-by-design architectures. . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 11.5 Attribution methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 11.6 Model steering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 12 Conclusion70 I Architecture84 A Symbol reference84 I Interpretability capabilities86 B Attribution details86 B.1 Training data attribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86 I Data87 C Atlas: From documents to concepts87 D Additional details on the human interpretability study88 D.1 Sampling and annotation protocol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88 D.2 Phase 1 agreement and coherence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88 D.3 Ordinal mixed-model analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88 D.4 Assumption-light robustness checks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 D.5 Dependence on lifted-word coherence. . . . . . . . . . . . . . . . . . . . . . . . . . . .89 D.6 Power analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 D.7 Limitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90 IV Interpretability metrics91 E Known concept alignment judge91 V Scaling laws92 F Symbol and notations92 G Architectures, IsoFLOP slices, and hyperparameters92 H ELBO estimation for validation loss94 H.1 Estimators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94 H.2 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 H.3 Effect of mask rate on scaling exponents. . . . . . . . . . . . . . . . . . . . . . . . . . .96 H.4 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98 I Annealing each IsoFLOP checkpoint99 I.1 Validation loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 I.2 Interpretability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100 VI Steerling-8B pretraining details102 J Pretraining recipe ablations102 J.1 Diffusion block size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 J.2 Diffusion masking schedule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 J.3 Unknown concept capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 J.4 Unknown embedding factorization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 J.5 Use of the residual term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 J.6 Concept teacher forcing schedule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106 J.7 Unknown concept teacher forcing schedule. . . . . . . . . . . . . . . . . . . . . . . . .107 K Final pretraining Steerling-8B configuration109 L Pretraining diagnostic110 M Steering judge112 VII Steerling-8B mid-training details114 N Mid-training recipe114 N.1 Nemotron: real, synthetic, and mixed. . . . . . . . . . . . . . . . . . . . . . . . . . . . .114 N.2 Final mid-training Steerling-8B configuration. . . . . . . . . . . . . . . . . . . . . . . .114 1Introduction Today’s most capable AI systems are also the least understood. This state of affairs is often treated as the price of progress: if a model is constrained toward human-meaningful structure, the assumption goes, it must have weaker performance. The consequence is a now-pervasive workflow: we train the most capable model we can, then try to reverse-engineer it after the fact, as if training were a law of nature whose results we can only observe, never intervene on, lest we harm performance. In this work, we put this premise to the test. Across three orders of magnitude of compute, on both autore- gressive and diffusion models, we show that building interpretability constraints into the training pipeline introduces a small, fixed scaling offset rather than a growing penalty. More surprisingly, we find that training with interpretability constraints produces models whose representations become moredisentangled and aligned with human-understandable concepts with scale; coupling capability and understanding. Reverse engineering models and post-hoc interpretability.A tremendous amount of scholar- ship has gone into the status-quo: train a model, then inspect it with classifier probes ( Alain and Bengio,2016), feature attributions (Simonyan et al.,2013;Sundararajan et al.,2017b), sparse au- toencoders (Bricken et al.,2023a), perturbation tests (Zeiler and Fergus,2014;Lundberg and Lee, 2017), or chain-of-thought (Wei et al.,2022b;Nye et al.,2021). These tools are useful, but they share a structural limitation: they explain a model that was never trained to make the explana- tion itself a valid interface into the model (Méloux et al.,2025). A probe reports that information is decodable, not that the model uses it. Feature attribution often measures local sensitivity, not necessarily the effect of a human-relevant intervention. A sparse feature may reconstruct an activa- tion without corresponding to a causal unit in the computation. A chain-of-thought may describe a plausible reason without being tied to the computation that produced the answer. The problem is that the standard model training recipe does not create an interface whose variables, interventions, and semantic labels are coupled to the prediction computation. In Section 3.2, we examine these challenges in detail. Inherent interpretability.Instead of asking how to explain an opaque model after training, we ask what conditions must hold for an explanation to be faithful, then build those conditions into the data, architecture, objective, and loss functions. In Section3, we formalize this asinherent interpretability: an attribution is not an auxiliary visualization, but a trained interface satisfying key conditions. Three model understanding axes.Steerling-8B instantiates the inherent interpretability recipe in the diffusion language modeling paradigm. For any output, the model traces its prediction along three axes: 1.to input tokens that affect the output under a trained absence baseline; 2.to human-understandable concepts, in its representations, that contribute to the output; and 3.to training data. From explanation to control.The same interfaces that provide model understanding also support intervention. In the model, each concept is reprsentation as a direction that the model uses in its forward pass to produce an output. Consequently, amplifying or suppressing a concept is a simple edit. Atlas.A central obstacle to our proposed training recipe is that no suitable concept library existed at the scale of modern pretraining corpora. To address this, we build Atlas, a concept annotation 1 pipeline that starts from millions of documents, extracts hundreds of millions of free-form tags, canonicalizes them into over 33,000 concepts, and trains an annotator that labels arbitrary text at chunk level. In total, Atlas annotates over 1 Trillion tokens across web text, code, mathematics, and academic prose. We describe this system in Section 4. Architecture.The architecture makes these concepts native to the model’s computation. Steerling- 8B uses a backbone with block-causal attention that preserves diffusion-style parallelism within blocks while retaining autoregressive-style KV caching across blocks. Between the transformer back- bone and the language-modeling head, we insert an additive concept bottleneck that makes the logit decomposition algebraically exact. The masking objective gives the model a trained representation of “no information at this position,” making feature-removal baselines in-distribution by construction. The overhead is small and decreases with scale: the concept module accounts for 4% of parameters at 8B, and under the same parameterization, would fall below 1% at frontier scales. Section5 describes the architecture and training objective. Scaling.The main empirical question is whether this structure makes the model weaker. We an- swer with IsoFLOP scaling sweeps across three orders of magnitude of compute and four model families: autoregressive, causal diffusion, autoregressive with concepts, and causal diffusion with concepts. Adding the concept module shifts the compute-optimal scaling exponents by a small, fixed per-backbone offset; the cost of interpretability does not grow with scale. Simultaneously, all inter- pretability metrics improve with compute on both backbones: the model predicts concepts more accurately, separates known and unknown representations more cleanly, routes more of its predic- tion through concepts rather than the residual, and aligns its concept embeddings more closely with human-meaningful labels. The validation loss of Steerling-8B is predicted within 0.11 nats from small-scale fits using the joint Chinchilla form, and three of four interpretability metrics are pre- dicted within tight bounds. Under the metrics we measure, the model does not become harder to understand as it scales. It becomes easier. Section 8presents the full analysis. An interpretable model can be competitive with opaque peers trained on far more compute. We train Steerling-8B on 1.2 trillion tokens followed by 150 billion midtraining tokens on a code and math augmented mixture. Compared with open peer models at similar parameter scale, each trained on roughly 2–16more compute, Steerling-8B lands within approximately 10% of their average benchmark performance, despite carrying interpretability constraints throughout training. A model can be both interpretable and competitive. Sections 9and10describe the full training process. Overview.The remainder of the paper is organized as follows. Section2reviews transformers, diffusion language models, and concept bottleneck models. Section 3presents the interpretable training recipe and formalizes faithfulness and inherent interpretability. Section 4describes Atlas, the concept annotation pipeline. Section 5describes the causal-diffusion architecture and concept module. Section6presents attribution and steering capabilities. Section7defines the interpretabil- ity metrics. Section8presents the scaling-law analysis. Sections9and10describe Steerling-8B pretraining and midtraining. Section11discusses related work, and Section12concludes. 2 2Background In this section, we present material that is core to our discussion in the rest of the paper. In addition, we present the notation that we use across the rest of the work. 2.1Transformer notation A Transformer (Vaswani et al.,2017) maps a sequence of input tokens to a sequence of hidden states per position. Each hidden state summarizes the token at that position together with its con- textual information, and the model projects these hidden states into logits over the vocabulary. We summarize the notation in Table1. SymbolMeaning x= (x 1 ,...,x N )Input token sequence of lengthN x i Token at positioni VVocabulary T x i 2R d Learned embedding of tokenx i dHidden dimension LNumber of transformer layers h 2 R d Transformer hidden state at a given position W2R |V|×d Language modeling head W y Row ofWcorresponding to tokeny ℓ y =h ⊤ W y Logit for output tokeny p θ Model with parametersθ x <i Sub-sequence of tokens before positioni L AR Autoregressive training loss x t Corrupted sequence at noise levelt t2[0,1]Noise level M(x t )Set of masked positions inx t L MDM Masked diffusion training loss Table 1.Transformer and language-model notation. 2.2Autoregressive language models Autoregressive language models (Radford et al.,2018) are trained on the task of next-token predic- tion: given an input sequence, the model predicts the next token, one at a time, conditioned on all previous tokens. We use the following notation: •p θ (x i jx <i )is the model’s predicted distribution over the next token at positioni, given all previous tokensx <i = (x 1 ,...,x i−1 ), whereθdenotes the model parameters. •The training loss is the negative log-likelihood of the next token, averaged over all positions in the sequence: L AR = 1 N N X i=1 logp θ (x i jx <i ).(1) 3 2.3Masked diffusion language models Diffusion language models (Austin et al.,2021a;Ou et al.,2024;Sahoo et al.,2024;Shi et al.,2024) are trained by reversing a forward corruption process applied to the input sequence, rather than by predicting the next token. A common variant is the masked diffusion language model (MDM), where the corruption process independently replaces tokens with a special[MASK]token. The model is trained to reconstruct the original tokens from the corrupted sequence. The amount of corruption is controlled by a noise levelt2[0,1], where each token is masked with probabilityt. Thus,t= 0 corresponds to no corruption, whilet= 1corresponds to the fully masked sequence. We use the following notation: •x t is the corrupted sequence at noise levelt. •M(x t )is the set of positions inx t that have been masked. •p θ (x i jx t )is the model’s predicted distribution over the original token at masked positioni, given the corrupted sequence. The training objective is a cross-entropy loss over the masked positions only, averaged over noise levels and sequences: L MDM =E t,x,x t 2 4 1 jM(x t )j X i∈M(x t ) logp θ (x i jx t ) 3 5 .(2) 2.4Concept bottleneck models Concept Bottleneck Models (CBMs) (Koh et al.,2020) add interpretability to a black-box neural network by inserting a layer of human-interpretable concepts between the input and the output. A samplexis first mapped to concept activationsc=φ(x)2R n , where each entry ofccorresponds to a supervised, human-interpretable concept. A second functionψthen predicts the label from the concepts,y=ψ(c). Whenψis linear, as in the original formulation, the final prediction is a weighted sum of interpretable concepts: x φ !c ψ !y.(3) Training a CBM requires ground-truth concept labelscalongside the targety, giving two losses. A concept loss matchesφ(x)toc, and a prediction loss matchesψ(φ(x))toy. Concept Bottleneck Generative Models (CBGMs) (Ismail et al.,2024) extend this idea to generative modeling. Since a fixed set of supervised concepts cannot capture everything in the input, CBGMs add an unsupervised concept channelualongside the known conceptsc, so that generation routes through both an interpretable known part and an unknown part that absorbs the remaining information: x φ !(c,u) ψ !y.(4) CBGMs introduce an additional orthogonality loss, which encourages the unknown embeddings to be orthogonal to the known concept embeddings so that the unknown channel encodes information distinct from the supervised concepts. 4 3A recipe for building interpretable models In this section, we present a general recipe for building interpretable language models. The recipe follows the standard pipeline for training language models (data curation, architecture and loss design, optimization, and evaluation) but modifies each stage to introduce human-interpretability constraints. Roadmap.We proceed in four parts: •Section3.1defines the interpretability requirements that we aim to satisfy in this work. •Section3.2shows why the standard recipe does not provide the required guarantees. •Section3.3formalizes explanation faithfulness and inherent interpretability. •Section3.4presents the interpretable recipe: pipeline modifications traced to specific condi- tions (summarized in Figure1). Figure 1.The recipe for training an interpretable model. 3.1Interpretability requirements Here we formalize our interpretability requirements:input attribution, which measures the effect of removing each input token;concept attribution, which decomposes the model’s output into per- concept contributions; andtraining data (similarity) attribution, which retrieves training examples the model considers similar to its output. Interpretability describes the interface between a model and a human: the structures through which a person can inspect, evaluate, understand, and act on a model’s behavior. We require three specific capabilities of this interface. 5 Informal Interpretability Requirements Given a model,p θ , for any output it produces, we want a human user to be able to answer three questions: 1.“What in the input mattered?”This requirement asks which parts of the input most influence the output? 2.“What topics/ideas are responsible for that?”This requirement asks what topics/ideas in the model’s internal representation drove this output? 3.“Where is the output coming from?”This requirement asks which training data resembles what the model just produced. The literature refers to such artifacts as explanations, interpretations, or attributions. In this work, we useattributionthroughout: each requirement asks the model to attribute to a specific component. We translate each question into a precise operation, which we term anattribution: 1.“What in my input mattered?”!Input attribution. We can measure the change in the model’s output when each input is replaced by a trained absence baseline, and rank tokens by the magnitude of this change. 2.“What topics/ideas are responsible for that?”!Concept attribution. We can decompose the model’s internal representation into contributions from human-understandable topics, and trace each topic’s contribution to the output. 3.“Where is this coming from?”!Training data attribution. We can retrieve the training examples whose representations are most similar to the representation of the model’s output. To move forward, we will now concretize core definitions as we specify the components necessary to build an inherently interpretable language model. Concepts.In this work, and as is common in the literature, we term a human-understandable topic aconcept. More generally, a concept is a coherent unit of meaning that a human recognizes and that we seek to associate with the model’s internal representations. Concepts span multiple levels of granularity: high-level themes like sports , politics , or tourism ; mid-level topics like gradi- ent descent optimization,state elections, orMediterranean cuisine; and fine-grained units likedropout regularization in recurrent networksorranked-choice ballot counting. They may also capture stylistic or functional attributes,formal tone,sarcasm,step-by-step reasoning, rather than topical content. The defining property is not the level of abstraction but human recog- nizability: a domain expert, given sufficient context, can identify whether the concept is present in a piece of text. Definition 3.1formalizes this intuition. Definition 3.1(Concept).Aconceptis a pairc= (z c ,m c )consisting of two components: •z c ,what the model computes with: an activation, embedding, or dimension that participates in producing the model’s output,ℓ y , and can be modified to change the model’s behavior (formalized in Section3.3.2). •m c ,what the human sees: a description that explains, in natural language, whatz c represents. 6 In this work,m c = (l c ,d c ,T c ), wherel c is a short label,d c is a one-sentence description, andT c is a set of characteristic words grounding the concept in observable language. We refer toz c as the concept’smodel variableandm c as itssemantic cardthroughout the paper. The data pipeline (Section4) produces semantic cards before any model exists; the model variablez c is instantiated when the architecture described in Section5is trained. With the definition of a concept in place, we now make each of our three attributions—input, concept, and training data—mathematically precise. We do this for a specific reason: in Section3.2, we will show that post-hoc methods fail to deliver these attributions reliably, and in Section3.3, we will define formal conditions under which theyarereliable. Notation.Letp θ be the model with parametersθ,x= (x 1 ,...,x N )an input sequence ofNtokens, ya predicted output token, andℓ y =h ⊤ W y the output logit for tokeny, whereh2R d is the hidden state andW y is the row of the language modeling head corresponding toy. We writeAfor a generic attribution functional, and use superscripts to distinguish the three types. Input attribution.If an input token were absent, how much would the output change?The input attribution functional maps the model, input, and output to a vector of per-token relevance scores: A input (p θ ,x,y) = α 1 , ..., α N 2R N , where α i =ℓ y (x)ℓ y (x ) measures the effect of replacing token, x i , with a baseline representing absence. This definition has a prerequisite that we formalize later: it is meaningful only ifx is in-distribution. If the model has never seen inputs with positioniabsent, the outputℓ y (x )reflects extrapolation. We return to this in Section3.3.1, where we definevalidity. Concept attribution.Which concepts contributed to the model’s prediction, and by how much?The concept attribution functional maps the model, input, and output to a vector of per-concept relevance scores: A concept (p θ ,x,y) = φ 1 (y), ..., φ n (y), ρ(y) 2R n+1 , whereφ c (y)is the contribution of conceptcto the output logitℓ y andρ(y)is the residual; the portion of the prediction not attributed to any concept. The output logit decomposes as: ℓ y = X c φ c (y) +ρ(y). How the decomposition above is achieved, is an architectural choice described in Section 5. Training data (similarity) attribution.Which training examples does the model consider similar to this output?The similarity attribution functional maps the model output and input to a set of training data, ranked by similarity, under the model’s own encoderr θ : A retrieval (p θ ,x) =TopK D∈D sim r θ (x), r θ (D) . 7 Similarity not causation.Our training data attribution requirement does not seek causation. It only requires that the model represents these training data similarly to its output. We do not claim training influence, which would require approximating counterfactuals over retraining (Koh and Liang ,2017;Bae et al.,2022;Grosse et al.,2023). Retrieval differs from the other two attributions in an important way: it is arepresentational prove- nancefunctional, not a causal attribution. Concept attribution decomposes the output; input attri- bution measures counterfactual effects; retrieval reports similarity in the model’s own geometry. Its criterion for being meaningful is that the similarity scores reflect the model’s actual representation, not an externally imposed metric. This holds by construction when the retrieval encoder is the same encoder that feeds into prediction. 3.2Why post-hoc methods fail to satisfy the interpretability requirements Each of the three attributions defined above has post-hoc analogues: methods that attempt to com- pute input, concept, or training data attributions from an already-trained model, without modifying its training in any way. Each can in principle succeed. Each, in standard practice, does notguarantee the required conditions because the model was not trained to support them. We begin with the root cause: standard training does not create an explanatory interface whose variables are coupled to prediction. We then examine post-hoc attempts at each attribution in turn: concept attribution via probes and sparse autoencoders, input attribution via gradient and pertur- bation methods, and training data attribution via influence functions. We close with the unifying diagnosis and a deeper obstacle, the Rashomon problem, which shows that even faithful post-hoc explanations can be arbitrary. Root cause.The standard model training pipeline optimizes for prediction. The core issue with post-hoc interpretability A standard model is trained for prediction. Its training objective does not ask for representations that decompose into meaningful units, baselines that make “what if this input were absent?” a well-defined question, or reasoning traces that reflect the actual computation. Post-hoc interpretability methods assume one or more of these structures exist and attempt to recover them after training. But they were never created: the model’s representations have no reason to be interpretable, its responses to missing inputs have no reason to be meaningful, and its verbal explanations have no reason to match its internal computations. The failures below are not limitations of specific methods; they are consequences of analyzing a model that was built without specifying formal interpretable conditions. Probes.A common approach to post-hoc concept attribution is to train a linear classifier on the hidden stateshof an already-trained model and interpret high accuracy as evidence that the model “represents” a concept (Alain and Bengio,2016;Belinkov,2022). The fundamental gap is between informationpresenceand informationuse: a probe asks whether conceptcis decodable fromh, not whetherp θ relies oncfor prediction. A perfectly accurate probe is compatible with the model ignoring the probed feature entirely ( Ravichander et al.,2021). Several lines of evidence sharpen this concern: •Linear and structural probes recover information with high accuracy from randomized contex- tualized embeddings ( Conneau et al.,2018;Hewitt and Manning,2019). •Syntactic probes do not generalize across domains (Maudslay and Cotterell,2021). 8 •Probes extract features merely encoded in token embeddings but unused during infer- ence (Ravichander et al.,2021). •Capacity-controlled, information-theoretic, and amnesic probing mitigate some false discover- ies but do not close the presence-versus-use gap (Voita and Titov,2020;Elazar et al.,2021). Even when probes identify directions useful for steering, one cannot assert that the model uses those directions for prediction. Each new concept requires training a separate probe, with no guarantee of coherence across probes. Sparse autoencoders.Sparse autoencoders (SAEs) decompose a frozen model’s hidden states hinto sparse dictionaries of features (Bricken et al.,2023b), and have surfaced striking results: features corresponding to the Golden Gate Bridge, emotional states, and safety-relevant behav- iors (Templeton,2024). However, the gap between discovering interesting features and guaran- teeing that those features are the variables through whichp θ computes its predictions is substantial. Because the SAE is trained on a frozen model, its features are post-hoc descriptions with no struc- tural relationship to the prediction pathway. More concerning, recent work shows that SAEs trained with different random seeds produce different feature sets, and that random SAE baselines match fully-trained SAEs on sparse probing and causal editing metrics ( Korznikov et al.,2026). The expla- nations may seem convincing, but they are not uniquely grounded in what the model learned. Gradient-based input attribution.Gradient-based attribution methods compute∂ℓ y /∂x i and present the result as a measure of feature importance. Several variants abound including saliency maps (Simonyan et al.,2013), Grad-CAM (Selvaraju et al.,2016), and integrated gradients (Sun- dararajan et al.,2017b), and many others. Gradients measure infinitesimal sensitivity; however, the human typically cares about the effect of removing or substantially changing a feature, a finite but larger perturbation. For gradients to be a useful proxy, the model must be approximately linear over the perturbation budget the human cares about or be off-manifold robust ( Dombrowski et al.,2019;Srinivas et al.,2022,2023). However, unless the training procedure is regularized to produce models that are locally linear, there is no incentive for the model to satisfy this requirement. Handed a model, the user might not have a way to assess how locally linear it is, and therefore no way to know whether the gradient is a valid proxy for the perturbation they care about. Perturbation-based input attribution.Perturbation-based methods like SHAP (Lundberg and Lee, 2017) and occlusion (Zeiler and Fergus,2014) replace input tokens with a reference value and measure the output change. Common reference values (zero vectors, random tokens, arbitrary masks) are out of distribution: the observed effect reflects the model’s extrapolation behavior, not the feature’s actual role (Hooker et al.,2019;Kumar et al.,2020;Jain et al.,2022). The resolution is to choose a null state that is in-distribution by construction for the model at hand. Unfaithful chain of thought.Chain-of-thought reasoning (Nye et al.,2021;Wei et al.,2022b) produces natural-language explanations alongside outputs. The model’s stated reasoning need not reflect its actual computation: a model can produce a correct-sounding chain that arrives at the answer for entirely different internal reasons, or a plausible chain that is confabulated ( Lanham et al.,2023;Madsen et al.,2024a;Turpin et al.,2023;Korbak et al.,2025;Chen et al.,2025). The explanation channel (generated text) and the computation channel (hidden states!output) are not structurally coupled by training. 9 A deeper problem: explanatory multiplicity.Even if post-hoc methods could deliver faithful at- tributions for a given model, the result would be contingent on which model happened to emerge from training. Models with identical performance can differ arbitrarily in their internal mechanisms, and explanatory multiplicity is decoupled from predictive multiplicity: two models with indistin- guishable accuracy can produce contradicting attributions for the same input (D’Amour et al.,2022; Brunet et al.,2022). This instability propagates to any downstream action, e.g. steering, editing, counterfactual recommendations, derived from the explanation (Pawelczyk et al.,2020). The recipe in Section3.4addresses this by anchoring the explanatory interface: the concept library, the absence baseline, and the additive decomposition are shared across all models the training procedure can produce. Different runs yield different parameters, but the interface through which attributions are computed is fixed. Section11discusses the underspecification and Rashomon literature in detail. The common thread Every gap above traces to the same structural absence: the model was not trained to make the attribution valid. The training was indifferent to whether probed features would be causally relevant, perturbation baselines would be in-distribution, gradients would be informative, or stated reasoning would reflect computation. This observation is not new (Méloux et al.,2025): post-hoc attributions are not identi- fiable estimators for the quantities they claim to measure. What is needed is to make the attributions identifiable, and one way to do that is to build the attributions into the training process itself. 3.3Faithfulness and inherent interpretability The gaps identified in Section3.2share a common structure: the attribution’s claims either disagree with what happens when one intervenes on the model, or the intervention itself is invalid. We now formalize these two failure modes as conditions on attributions (Section3.3.1), then define the class of models that satisfy them by construction (Section3.3.2). 3.3.1Faithfulness Prior work on faithfulness.The concept of attribution faithfulness has been widely discussed in the literature (Hooker et al.,2019;DeYoung et al.,2020;Lyu et al.,2024;Atanasova et al., 2023). Most treatments define faithfulness informally as “the explanation/attribution reflects the model’s actual reasoning” (Jacovi and Goldberg,2020) or operationalize it through specific tests, perturbation checks, sufficiency, comprehensiveness (DeYoung et al.,2020), without unifying these into conditions that a training procedure can be designed to satisfy. We distill the literature into two concrete conditions, agreement and validity, that are checkable, traceable to specific design choices, and help to diagnose every post-hoc failure identified in Section 3.2. Definition 3.2(Faithfulness).An attribution functional,A, isfaithfulwith respect to model,p θ , intervention family,I, and toleranceξ, if: 1.Agreement.The output relevance scores, from the attribution, predict the observed effects of interventions inIwithin toleranceξ. 2.Validity.The interventions inIbelong to atraining-supported intervention family: the model has been trained on, or explicitly regularized for, the perturbed states used to define the attri- bution. Two types of interventions.We distinguish input interventionsI x , such as replacing token,x i , with the mask token, from latent interventionsI z , such as modifying a concept variablez c . For masking, 10 validity means the model saw masked contexts during training. For concept interventions, validity means edits are bounded to observed activation ranges or sampled concept values. Mapping to Section3.2.Each post-hoc issue, previously identified, is a violation of one or both conditions. Probes violate agreement: information presence does not imply information use. SAEs violate agreement: reconstruction geometry does not imply causal role. Gradient methods violate agreement: local sensitivity does not match the human’s perturbation budget. Perturbation methods violate validity: the baseline is out of distribution. Chain-of-thought violates agreement: the text channel is not coupled to the computation channel. 3.3.2Inherent interpretability We now have all the relevant definitions and terms to state what we mean by inherent interpretability in this work. Definition 3.3(Inherent interpretability).A model,p θ , isinherently interpretablewith respect to an attribution functional,A, an intervention family,I, a semantic mapM:c7!m c assigning each concept its semantic card (Definition3.1), a toleranceξ, and a coverage thresholdζif its training (or finetuning) procedure produces models for which: Causal faithfulness: 1.Nativeness.The attributed variables are part of the computation that produces the model’s output,ℓ y . 2.Agreement.The attribution predicts the effect of an intervention before it is applied. For an intervention inI, let∆ℓ y be theobservedchange: the difference in the output logit between a forward pass with the intervention applied (e.g., a token replaced by mask, or a concept variable modified) and the unmodified forward pass. Let ˆ ∆ℓ y be thepredictedchange: the change implied by the attribution’s relevance scores alone, computed without running the intervened forward pass. Agreement requires j∆ℓ y ˆ ∆ℓ y jξ. 3.Validity.Interventions inIbelong to a training-supported intervention family. Semantic faithfulness: 4.Interpretation.The semantic mapMgives human-valid descriptions of the attributed vari- ables: each cardm c accurately describes what its variablez c encodes. 5.Coverage.The attributed variables together account for at least a fractionζof the model’s prediction, with the unattributed residualρ(y)explicitly reported. Inherent Interpretability does not mean total mechanistic transparency.Inherent interpretabil- ity is a targeted guarantee for specified attribution queries, not a global claim about every neuron or attention head. When we say inherent interpretability in this work, we do not claim that the entire model is human-understandable. 11 Causal vs. semantic faithfulness Conditions 1–3 establishcausal faithfulness: the attribution correctly describes the model-side variable z c and its role in producingℓ y . Conditions 4–5 establishsemantic faithfulness: the human description m c validly representsz c , and the interpretable component is useful for the output. Concept leakage (Mahinpei et al.,2021) is a failure of semantic faithfulness: the attribution may cor- rectly predict the causal effect ofz c (conditions 1–3 hold) whilem c incompletely describes whatz c encodes (condition 4 fails). For example, ifz c encodes “sports” plus hidden information about “gen- der,” the numeric attributionφ c (y)correctly predicts the effect of modifyingz c , but the semantic card m c =“sports” is incomplete. Examples of inherent interpretability.The definition is agnostic to the form of explanation. We illustrate with examples in the literature: 1.Masking!inherently interpretable w.r.t.A input .Training with a masking objective places the absence baseline in-distribution, satisfying validity.Madsen et al.(2024b) make this move for finetuning; we extend it to pretraining. 2.Concept bottleneck!inherently interpretable w.r.t.A concept .An additive bottleneck gives nativeness and exact agreement (ξ= 0). Koh et al.(2020) introduced this for classification. AndIsmail et al.(2024) extend that approach to generative models. 3.4The interpretable model-training recipe We now have three attribution functionals (Section3.1), a demonstration that post-hoc methods do not guarantee them (Section 3.2), and a formal definition of inherent interpretability specifying what is needed (Definition 3.3). The recipe modifies the standard training pipeline at exactly the points needed to satisfy the five conditions. Every modification traces to a specific condition; Figure 2 summarizes the full pipeline. Why pretraining.Interpretability constraints are most effective when present during representation formation. Finetuning restructures existing representations; pretraining shapes them from the start. Our recipe targets pretraining, though the contracts apply to finetuning with weaker guarantees. The standard recipe.A standard training pipeline trains a modelp θ on input-output pairsf(x i ,y i )g by minimizing a task loss, and evaluates on downstream metrics. This recipe is effective for predic- tion but does not require the model to expose explanations. The interpretable recipe.The interpretable recipe modifies the standard pipeline at five points (right column of Figure2and bottom row of Figure1). Each modification is motivated by a specific condition of Definition3.3. Concept annotations (Interpretation).The training data is augmented with concept annotations: D=f(x i ,y i ,c i )g. Without supervised targets, the model has no signal to organize its representations around human-meaningful categories; it will default to whatever geometry minimizes the task loss (Section3.2). Concept annotations provide the semantic targets that ground the interpretation condition: each concept variable,z c , has a corresponding semantic card,m c , whose meaning is established before training. The annotations also enable a control interface, since concepts with prior semantics can be monitored, amplified, or suppressed.Our instantiation:Section 4describes the Atlas pipeline that produces concept annotations. 12 Standard Training RecipeInterpretable Training Recipe Data D= n Input–Output z| (x i ,y i ) o n i=1 D= n Input–Output z| (x i ,y i ), Explanations z| c i o n i=1 Arch.ℓ y =h |z Opaque ⊤ W y ℓ y = P c φ c (y) |z Bottleneck +ρ(y) |z Residual →concept & retrieval attribution ObjectiveL AR (§2.2) Objective withtrained absence baseline →input attribution Loss P i ℓ (x i ,y i );θ P i ℓ task |z Task + P j λ j ℓ interp,j (c i ) |z Interpretability Evaluatescore | z Task Metric score task |z Task ,score interp |z Interpretability Attribute ˆ A post-hoc (p θ ,x,y) |z Approximate (§3.2) A input |z In-dist. ,A concept |z Exact ,A retrieval |z Native Figure 2.The standard and interpretable recipes, compared stage by stage (Section3.4). The interpretable recipe augments each stage to satisfy the conditions of Definition 3.3. Each attribution functional (bottom row, right) traces to a specific upstream modification (green arrows): the bottleneck architecture enables exact concept attribution and native retrieval in the model’s own concept-aligned geometry; the trained absence baseline in the objective makes feature attribution in-distribution. The standard recipe supports only post- hoc attribution ( orange), which does not guarantee the required conditions (Section3.2). Bottleneck architecture (Nativeness, Agreement).A model architecture that routes the model’s predictions through concepts makes it so that each concept’s contribution to the output can be di- rectly computed from the forward pass. This is a structural requirement: the concept variables,z c , must lie on the computational path from the model’s internal state to the output. The specific mech- anism, additive bottleneck, multiplicative gating, or another decomposition, is an implementation choice. Section 5describes an additive concept bottleneck with a linear output head. Trained absence baseline (Validity).Input attribution requires replacing input features with a baseline representing absence and measuring the change in output. For this to be meaningful, the model must have a learned representation of “no information at this position”. Without this, the per- turbed inputx is out of distribution, and the observed output change reflects extrapolation rather than the feature’s actual role. The contract is that the training objective must include the perturbed states used to defineA input . Standard autoregressive next-token objective does not typically incor- porate this type of masking.Our instantiation:Section 5describes a masked diffusion objective (L MDM ) that trains the model on corrupted sequences, making the mask token an in-distribution absence baseline. Interpretability Losses (Interpretation, Coverage).The architecture provides the slot for con- cepts; the losses ensure the slot is used. Two failure modes must be prevented. First,concept leak- age: the concept variablez c may encode information beyond what the semantic cardm c describes, degrading the interpretation condition. Concept losses that alignz c activations with the annotations 13 c i counteract this by maintaining semantic alignment between the model-side variable and its hu- man description. Second,residual domination: the residualρ(y)may absorb most of the predictive capacity, leaving the concept decomposition algebraically exact but practically vacuous. Losses that penalize the residual, enforce reconstruction of the unexplained hidden state, and encourage inde- pendence between concept components address this. The general form isL=L task + P j λ j L interp,j , where the weightsλ j may be annealed during training to balance task performance and interpretabil- ity.Our instantiation:Section5describes the specific loss components and annealing schedule. Evaluation.Task metrics (perplexity, accuracy) remain necessary to ensure interpretability is not obtained by discarding predictive information. In addition, the model should be evaluated against the five conditions of Definition3.3: 1.Causal faithfulness (Nativeness).Are the attributed variables on the computational path? Agreement: do interventions onz c changeℓ y as predicted byφ c (y)? Validity: are the interven- tions in-distribution? 2.Semantic faithfulness.Interpretation: do humans independently recognize the concepts from the model’s characteristic evidence? Coverage: what fraction of prediction mass is carried by concepts versus the residual? Our instantiation:Section4.4describes the human recoverability study (Condition 4) and the quan- titative evaluation (Conditions 2, 3, 5). Summary.Every modification traces to a condition of Definition3.3; Table2summarizes the con- sequence of removing each one. The recipe itself is not new: its elements are implicit in work on con- cept bottlenecks (Koh et al.,2020), interpretable-by-design architectures (Rudin,2019), masking- based attribution ( Madsen et al.,2024b), and recent theoretical treatments that derive interpretabil- ity constraints from explicit premises (Barbiero et al.,2026). Our contribution is to make the recipe explicit, trace each element to a formal condition, and instantiate it for language models at pretrain- ing scale. Remove...Condition brokenConsequence Concept annotationsInterpretation (4)No semantic targets Bottleneck architecture Nativeness (1), Agreement (2)φ c (y)not computable Trained absence baseline Validity (3)Input-attribution baselines OOD Interpretability lossesCoverage (5), Interpretation (4) Residual absorbs capacity Table 2.Removing any single recipe modification breaks a specific condition of Definition3.3. This completes the recipe. The interpretability requirements (Section3.1) defined what we want; the post-hoc analysis (Section3.2) showed why the standard pipeline cannot deliver it; the defi- nitions of faithfulness and inherent interpretability (Section3.3) formalized what is needed; and the recipe above specified how to modify each stage of the pipeline to satisfy those conditions. The remainder of the paper instantiates this recipe: Section 4instantiates the data contract, building the concept library; Section5instantiates the architecture, objective, loss, and optimization contracts, building the model; and Section8shows that these contracts preserve compute-optimal scaling while the interpretability properties themselves improve with compute. Finally, Sections9and10 carry the recipe to full scale, pretraining and mid-training Steerling-8B. 14 4Data In this section, we describe the process of building large-scale concept-annotated pretraining, mid- training, and post-training corpora. As discussed in Section3, to satisfy the concept interpretability constraints, we need supervision: data annotated with concepts at a level of granularity to capture meaning. Towards this end, we presentAtlas, an automated system for annotating language model- ing corpora with human-understandable concepts at a fine-grained level. Using Atlas, we annotated a 1.5 trillion-token corpus spanning web text, scientific writing, code, and synthetic data with over 33,000concepts across science, technology, philosophy, medicine, law, etc. Overview.First, we discuss why existing concept libraries are inadequate and state the desiderata that a suitable library must satisfy (Section4.1). We then describe the Atlas pipeline: a three- stage process that moves from sampled documents to high-recall tags—short free-form words or phrases associated with local spans of text—(Section4.2.1), from tags to a canonical concept library (Section4.2.2), and from the library to a trained concept annotator that can label arbitrary text (Section4.2.3); each stage is validated under a common evaluation framework introduced at the start of Section4.2. We next describe the hierarchical taxonomy imposed on the library (Section4.3) and a human interpretability study validating that our concepts are human-meaningful rather than merely LLM-fluent (Section4.4). Finally, we describe how the trained annotator is applied to the full pretraining corpus (Section4.5) and how the resulting annotations and embeddings are indexed for test-time training data attribution (Section4.6). 4.1Motivation: No human-interpretable concept library exists at scale Before building our own library, we surveyed existing approaches for large-scale concept extraction. Broadly, existing concept libraries fall into three categories. •Word-based concept dictionaries.Luo et al.(2024) construct a 40,000-item concept dictio- nary by selecting the most frequent words from the Brown Corpus (Francis and Kučera,1967) and prompting GPT-4 ( Achiam et al.,2023) to generate sentences illustrating each word. Sin- gle words, however, cannot capture the higher-level abstractions, multi-sentence topics, or domain-specific ideas that appear throughout pretraining datasets. •Activation-derived, unsupervised concepts. These approaches extract concepts from open- weight model activations, for example, directions discovered via sparse autoencoders (SAEs) ( Bricken et al.,2023b). Activation-derived concepts are not constrained to be human- meaningful; a direction in activation space may be statistically salient without corresponding to anything a person would recognize as a coherent idea. There is also no guarantee that these directions will correspond to topics that are expressed in the corpus that the interpretable lan- guage model will be trained on (Leask et al.,2025;Korznikov et al.,2026). •Narrow domain libraries.Some concept sets target specific tasks such as sentiment analysis or toxicity detection. For instance,Sun et al.(2025) define concepts using ChatGPT for SST2, Yelp Polarity, and AGNews, specifying categories like world, sport, business, and technology news for AGNews. While these libraries offer high-quality labels, they are far too narrow to supervise models trained on diverse corpora spanning web text, code, mathematics, and scientific writing. Desiderata for an ideal concept library.We posit that a concept library suitable for supervising language model pretraining at scale must satisfy five properties. The library must... 15 Figure 3.Overview of the Atlas three-stage annotation pipeline: documents are chunked and tagged (Stage 1), tags are clustered and canonicalized into a concept library (Stage 2), and a concept annotator model is trained for scalable text annotation (Stage 3). 1.Multi-scale: cover both high-level themes (e.g., “machine learning”) and fine-grained units (e.g., “gradient clipping for recurrent networks”). 2.Localizable: be applicable to spans within a document, enabling sub-document-level supervi- sion and control. 3.Stylistically expressive: capture attributes like document tone, formality, or even communica- tive intent, not restricted to semantic categories. 4.Representative: possess the breadth necessary to cover the true distribution of large-scale pre- training corpora across web text, code, mathematics, and scientific writing. 5.Human-meaningful: comprise concepts that people—given expertise appropriate to the topic— can understand and would want to use when assessing a language model’s behavior. The last desideratum deserves emphasis. It is tempting to define “human-meaningful” as “labeled by a human” or “labeled by an LLM in a way that sounds reasonable”. Neither of these is sufficient. It is possible for an LLM to produce a fluent label for even semantically incoherent document groupings. A human can rate a label as “reasonable” without checking whether they would have arrived at the same label independently. In Section 4.4, we describe a direct test that enables us to assess the correspondence between human-generated labels and LLM-generated labels. 4.2Atlas: Documents to concepts Atlas is a three-stage pipeline. Stage 1 samples documents, chunks them into local semantic units, and extracts a high-recall pool of free-form tags. Stage 2 clusters these tags, filters low-quality 16 tag clusters, labels them, and deduplicates them into a canonical library of human-understandable concepts. Stage 3 trains a concept annotator model that can assign concepts from the library to arbitrary text, enabling annotation of the full training corpus. We describe each stage in turn. Evaluation overview.The evaluation of Atlas must answer two questions. The first is an engineering question: does the pipeline produce high-quality annotations at each stage? The second is a scientific question: are the resulting concepts genuinely human-interpretable, or merely LLM-fluent? For the engineering question, we use a single framework throughout the pipeline and report results alongside each stage below. A sampled annotation—a raw tag from Stage 1, a canonical concept from Stage 2, or a predicted label from the Stage 3 annotator—is scored against its text chunk on a 1–5 relevance scale, by an LLM judge at scale and by human annotators on smaller matched samples. A score of 2 or higher counts as successful: the tag or concept is at least minimally present in the chunk. This threshold is intentionally permissive, reflecting the high-recall goal of the pipeline and the fact that minor tags can capture fine-grained contextual details without being the dominant topic. We therefore report both success rates and full score distributions, visualized as histograms of per-chunk and per-tag (or per-concept) average scores. On the matched samples, human ratings track the LLM judge but run roughly 0.4 points higher, so the LLM-judged scores reported below are, if anything, conservative. The scientific question cannot be settled by an LLM judge: an LLM can rate an LLM-generated label as coherent without any guarantee that a human would recognize the same concept. We address it separately with a two-phase human study in Section 4.4, testing whether the concept names are human-interpretable rather than merely fluent. 4.2.1Stage 1: Corpus sampling, chunking, and documents to tags Stage 1 constructs the high-recall tag pool from which the concept library is later derived. We begin with a representative sample of our pretraining and midtraining mixtures, split documents into local semantic chunks, prompt an annotator model to assign structured free-form tags to each chunk, and validate the resulting tags. At this stage, tags are not canonical concepts: they are short words or phrases that may be redundant, overlapping, or overly specific. This is deliberate. Stage 1 is optimized for recall; Stage 2 handles clustering, filtering, labeling, and deduplication. Throughout this section, we distinguish betweentagsandconcepts. Atagis a short free-form string assigned to a specific chunk of text during Stage 1 annotation. Aconceptis a canonical human-interpretable idea produced by clustering, filtering, labeling, and deduplicating many related tags. Each concept has a name, description, and supporting evidence from its associated tags. Tag assignment is an empirical output of a high-recall annotation process: a tag either was or was not assigned to a chunk, but unassigned tags are not treated as negatives. Concept assignment is a calibrated library-level prediction problem: once tags have been canonicalized into concepts, we evaluate whether a named concept is genuinely present in a chunk and whether humans recognize the concept from its supporting evidence. Concepts are necessarily less fine-grained than raw tags, and much of the calibration work in Atlas concerns whether a named concept should be assigned to particular chunks of text. Source mixture.We sample from five major document categories:web text, using a deduplicated version of DCLM from Zyda-2 ( Li et al.,2024;Tokpanov et al.,2024);general academic knowledge, 17 including peS2o (Soldaini and Lo,2023), arXiv (Weber et al.,2024), and Wikipedia and Wikibooks (from Dolma 1.7 (Soldaini et al.,2024));mathematics, including Dolmino-math, GSM8K (Cobbe et al.,2021), OpenWebMath (Paster et al.,2024b), and Algebraic Stack (Azerbayev et al.,2023a); code , using StarCoder (Li et al.,2023); andquestion–answer exchanges, using FLAN v2 (Longpre et al.,2023). In total, the sample contains 6.6 million documents balanced across these categories. This mixture was chosen not as a benchmark distribution, but as a substrate for concept extraction: it spans multiple writing styles, knowledge domains, and levels of conceptual density. Chunking.We annotate at the chunk level rather than the document level. Long documents often contain several unrelated ideas, and whole-document annotation tends to collapse them into coarse summaries. A chunk is a short, semantically coherent span, created by detecting sentence bound- aries with the BlingFire sentence splitter (Microsoft,2019) and concatenating consecutive sentences until a domain-specific token threshold is reached. We use a threshold of 150 tokens for web text and general documents and 256 tokens for mathematics and code, where a single mathematical or algorithmic idea often requires more context. Sentences exceeding the threshold are retained as single chunks; extremely long sentences above 50,000 tokens are dropped as malformed or unin- formative for local annotation. This process yields 44 million chunks from the 6.6 million sampled documents. Chunking at this granularity is essential: it makes tags local to the semantic units they describe, rather than global summaries of entire documents. Domain-aware annotation schemas. Different domains express conceptual structure differently. A Wikipedia article, a mathematical proof, a Stack Exchange question, and a block of Python code call for different annotation fields. We therefore designed domain-specific structured tag schemas, each containing 4–6 fields tailored to the content type. Each field elicits a hierarchy of tags from broad to narrow, and together the fields produce on average 10–15 tags per text chunk. For each domain, the schema separates several complementary views of a chunk: what the chunk is about, what role it plays, and how it is expressed. For web text, fields include topic, communicative purpose, tone or style, and secondary entities or events; Table 3shows an example web text docu- ment split into chunks with its domain-specific structured tags. For mathematics, fields capture the mathematical object or topic, problem type, solution or proof technique, notation conventions, and difficulty. For code, fields capture programming language, design pattern, algorithmic structure, software-engineering role, and documentation style. The schemas are deliberately overcomplete: a single chunk may receive several overlapping tags, because Stage 1 is optimized for recall before Stage 2 clusters the tag space into canonical concepts. Annotator model selection.We evaluated several open-weight models for structured annotation, including Phi-family models ( Abdin et al.,2024),Qwen2.5-7B(Qwen et al.,2025), andMistral- Small-3.1-24B-Instruct(Mistral AI Team,2025). The choice of annotator model is constrained by a practical requirement: reliability at scale matters more than raw capability. Even a 1% schema deviation rate across 44 million chunks produces nearly half a million unusable annotations. We found that Phi-family models could output consistently structured annotations but suffered from repetition collapse and degenerate loops, making them unsuitable for long-running annotation jobs. Qwen2.5-7Bperformed better semantically but less predictably syntactically, often producing the wrong number of fields or responses that were difficult to parse.Mistral-Small-3.1-24B-Instruct consistently adhered to the structured format, avoided repetition collapse, and maintained stable 18 ChunksStructured Tags The game increases the chances of finding a Shiny Pokemon by generating extra personality values in an attempt to find one that results in a Shiny Pokemon, with the number of attempts depending on the size of the current streak. For every Pokemon added to the streak up to 20 Pokemon, the game will make two extra attempts to find a Shiny personality value; i.e., the number of attempts at any given point in the streak is1 + 2∗ streak_size, and caps at a maximum of 41 attempts when the streak is at least 20 Pokemon long. main:gaming, mechanics, video-games, Pokemon, shiny-mechanics purpose:informational-guide, explanatory, how-to tone:neutral, informative, technical minor:gaming-progress, streaks, gaming-algorithms, random-generation In Going for the Gold!, Ash and his friends met a fisherman named Rodman, who was trying to fish up a Shiny Magikarp with a Magikarp-shaped lure. Ash, Serena, and Clemont decided to try fishing too, with Ash giving the inexperienced Serena instructions of how to do it correctly. While fishing, Serena hooked up a Corsola, which she tried to battle with her Fennekin, but it simply hid itself behind Serena when Corsola tried to use Water Gun on it, causing the Coral Pokémon to get away. main:fiction, media-franchise, anime, pokemon, adventure purpose:narrative, storytelling, character-development tone:casual, descriptive, lighthearted minor:fishing-techniques, lure-use, pokemon-battles, strategy-mistakes Table 3.A webtext document aboutPokemonsplit into two chunks, with its domain-specific, hierarchical structured tags. behavior across millions of prompts. Mistral was the smallest model that satisfied our formatting and reliability constraints. Output.Stage 1 produced nearly 44 million annotated chunks, from which we extracted approxi- mately 500 million tags. After tag deduplication, 14 million unique tags remained, of which approx- imately 1 million appeared more than 15 times. The most frequent raw tags, shown in Figure 4, provide a sanity check on this high-recall regime: the tag pool captures broad recurring content and stylistic patterns, but remains redundant and non-canonical before Stage 2 clustering. This raw tag space is intentionally redundant and noisy. At this stage of the pipeline, missing a relevant concept is more damaging than producing overlapping or synonymous tags: synonymy and over-specificity can be removed by clustering, while absent concepts cannot be recovered downstream. The purpose of Stage 1 is therefore to cast a wide semantic net. Stage 2 consolidates this tag space into a large concept library. Using the evaluation framework described at the start of this section, an LLM judge scored 21.3 million (tag, chunk) pairs over a sample of 2.93 million chunks (2.15 million unique tags), with human annotators rating a 50-chunk subsample; the judging prompt is given in Part I. Individual tag ratings average 3.62, and 97.5% score at least 2. Across all domains, nearly every tag is at least minimally present in its chunk: per-chunk average scores center near 3.6, with two-thirds of chunks falling between 3.2 and 4.0 and 99.9% averaging at least 2 (Figure5). 19 Top-50 most frequent tags academic writingresearch reporting science research biology physicsmathematics computer science medicine data analysis molecular biologytechnologyresearch methodology experimental resultsscientific explanationchemistry statisticsreal world context theoretical physics machine learning arithmeticeducational algorithmscomputation education mathematical concepts astrophysics notation analysis fiction programming technical explanation academic discussion genetics informational software developmentequations documentation engineering comparison information sharingmethodology communication narrativematerials sciencefunctions geography comparative analysisresearch methods source codeeconomics study findings material science Figure 4.The 50 most frequent raw tags produced by Stage 1. These tags illustrate the high-recall, redundant, non-canonical tag space before Stage 2 clustering and deduplication. 4.2.2Stage 2: Tags to concepts Stage 2 transforms nearly 14 million noisy, free-form tags into a coherent library of over33,000 human-understandable concepts. The process embeds tags into a semantically meaningful space, clusters them, filters low-coherence clusters, labels each surviving cluster, and merges near-duplicate concepts through graph-based deduplication. We describe each step in turn. Tag normalization and embedding.Raw tags produced by an LLM from Stage 1 are highly variable: minor formatting differences—hyphens, slashes, whitespace, punctuation artifacts—can split seman- tically identical tags into separate strings. We normalize all tags into a standardized form before embedding, e.g.,astronomical-objects!astronomical objects,climate-change-adaptation Figure 5.Stage 1 tag validation scores. Distributions of average tag-relevance ratings on a 1–5 scale: per- chunk averages (left column; each chunk’s tag ratings averaged) and per-tag averages (right column; each unique tag’s ratings averaged over the chunks it appears in), as rated by the LLM judge over 2.93M sampled chunks (top row) and by three human annotators on a 50-chunk sample (bottom row). 20 !climate change adaptation. Each unique tag is then embedded into ann-dimensional (n= 768) space using theall-mpnet-base-v2sentence embedding model (Sentence Transformers Team, 2021). This choice is empirical: in a comparison againstQwen3-Embedding-0.6B(Zhang et al., 2025), the former produced slightly higher-quality clusters, as measured by standard quantitative clustering metrics—Silhouette score (Rousseeuw,1987), Davies–Bouldin index (Davies and Bouldin, 1979)—together with the LLM-judge cluster coherence evaluation described below. In total, this step produces nearly 14 million vectors, one per unique tag. Clustering.With all tags embedded in a common semantic embedding space, we cluster them into groups representing candidate concepts; for instance,missing pet,missing pet incident,missing dog should fall in a single cluster. We usek-means, implemented in the FAISS library (Johnson et al., 2019) for GPU-accelerated efficiency over tens of millions of vectors. The number of underlying semantic clusters is unknown, so we sweep a wide range of cluster counts:k2 f100,500,1k,10k,20k,30k,50k,80k,100kg. Smallkproduces large, diverse clusters that conflate unrelated concepts; largekproduces tight clusters but risks fragmenting genuinely related concepts or introducing clusters based on noise. We evaluate eachkwith the Silhouette score, which rewards clusters that are internally cohesive and well separated, the Davies–Bouldin index, which penalizes clusters whose internal scatter is large relative to their separation, alongside the LLM-based cluster coherence evaluation described next. Figure 6.Comparison ofall-mpnet-base-v2andQwen3-Embedding-0.6Bembeddings for clustering14M LLM-generated tag annotations via thek-means clustering algorithm. We evaluate Silhouette score (left plot), Davies–Bouldin index (middle plot), and coherence score (right plot) on randomly sampled cluster members. Askincreases,all-mpnet-base-v2consistently yields more coherent clusters thanQwen3-Embedding-0.6B. LLM-based cluster coherence evaluation.The two standard metrics capture broad trends across large changes inkbut do not reliably distinguish nearby values, so we complement them with a direct semantic coherence evaluation of every cluster. For each cluster, we sample three strata of tags—core (closest to the centroid), random (uniform), and edge (farthest from the centroid)—group them into sets of ten and query an LLM,Mistral-Small-3.1-24B-Instruct, to score the semantic coherence of each set on a 1–10 scale, yielding a coherence score per stratum. Coherence improves steadily withkand plateaus aroundk= 40,000–80,000as shown in Figure6; beyond this range, clustering computational cost rises while semantic gains diminish, as largerkbegins to split coherent groups across multiple clusters. We therefore set the initial number of clusters tok= 80,000. 21 Quality filtering.Not every cluster corresponds to a coherent concept; some arise from incidental lexical overlap rather than shared meaning. We retain only clusters that clear strict per-stratum coherence thresholds—core9, random8, and edge7. Of the 80,000 clusters, approximately 17,000 fail these criteria and are removed, leaving roughly 63,000 high-quality clusters. Cluster labeling.Each surviving cluster is converted into a human-interpretable concept. We sam- ple 50–100 representative tags per cluster, weighted by tag frequency, and prompt an LLM,Mistral- Small-3.1-24B-Instruct, to generate a concise label (1–6 words) and a one-sentence rich descrip- tion; the prompt is given in PartIII. These labels provide a clean, human-friendly interpretation of what are otherwise dense numerical clusters of tags. The LLMlabelsstructure that emerged from clustering millions of human-comprehensible tags; it does not fabricate concepts from nothing. The tags were produced by annotating real text, the clusters capture statistical regularities in those tags, and the LLM assigns a name to each regularity. Whether the resulting names are genuinely human-interpretable is an empirical question we address in Section4.4. Graph-based cluster deduplication.The labeled concepts, now numbering62,000, still contain substantial redundancy: multiple clusters may express the same underlying idea, differing only in granularity or phrasing. We address this redundancy by iterative graph-based concept merging. Figure 7.Stage-2 transformation of noisy LLM tags into a canonical concept: raw tags are embedded and clustered, incoherent clusters are filtered out, each surviving cluster is labeled into a single concept (name with italic description), and semantically adjacent concepts are merged by cosine similarity into one canonical entry. We embed each concept’s label and description withQwen3-Embedding-0.6Binto a semanticn- dimensional vector space, then connect each concept to itsm-nearest neighbors, retaining edges with cosine similarity above a certain thresholdτ. This produces an undirected similarity graph 22 Figure 8.UMAP projection of a subsample of concept embeddings across five taxonomy groups. The cards showcase representative concepts from distinct regions of the embedding space to illustrate local semantic clusters. whose nodes are concepts and whose edges connect near-duplicates. We partition the graph with Louvain-community detection ( Blondel et al.,2008) to identify groups of related concepts, treat each community as a candidate merge set, and prompt an LLM to regenerate a single unified label and description for each set. A single pass reduces the library to approximately 39,000 concepts. We then re-embed all concepts and repeat—rebuilding the graph and re-running community detection at each iteration—halting after two to three iterations once an iteration merges only a negligible number of concepts. This leaves over33,000concepts. We chosem= 20,τ= 0.95for first iteration, reducedτ= 0.9for second pass, and thenτ= 0.85for third pass based on empirical evidence. Output. The final canonical concept library contains33,732concepts spanning science, technology, philosophy, medicine, law, and other domains. Some conceptual overlap and hierarchy are unavoidable—real-world knowledge is not cleanly partitioned—but the deduplicated library strikes a practical balance between granularity and clar- ity. We visualize the concept embeddings in Figure8using UMAP (McInnes et al.,2018), a scalable nonlinear dimension-reduction method for visualization and manifold learning. Validation.Using the same framework, an LLM judge scored the concepts associated with each of the same 2.93 million chunks via their Stage 1 tags—12.2 million (concept, chunk) pairs covering nearly the entire library—with human annotators again rating a 50-chunk subsample (Figure 9). 23 Individual ratings average 3.50, with 98.0% scoring at least 2; per-chunk averages center near 3.5, with two-thirds of chunks between 3.2 and 4.0 and 99.7% averaging at least 2. Per-concept average scores are notably uniform across the library (mean 3.51, standard de- viation 0.41): concept quality is consistent rather than driven by a head of frequent, well- represented concepts. Figure 9.Stage 2 concept validation scores. Distributions of average concept-relevance ratings on a 1–5 scale: per-chunk averages (left column) and per-concept averages (right column; each concept’s ratings averaged over the chunks it is assigned to), as rated by the LLM judge over 2.93M chunks (top row) and by human annotators on a 50-chunk sample (bottom row). 4.2.3Stage 3: Concept annotator model The outcome of Stages 1 and 2 is a large canonical library of content concepts, defined over the sam- pled corpus. Stage 3 trains a scalable annotator that maps an arbitrary input chunk to a set of target labels, allowing us to annotate the full 1.5 trillion-token pretraining corpus. The annotator predicts over a combined target space consisting of the Atlas-derived content concepts together with several lower-cardinality auxiliary domains: tone, demographic-reference, and alignment-relevant labels. Training targets.The Stage 3 training targets come from two sources. The first source is the content-concept library produced by Stages 1 and 2. For each chunk in the Stage 1 annotated sample, raw tags are mapped through the Stage 2 clustering and deduplication pipeline to the final content- concept IDs. Thus, if a tag assigned to a chunk belongs to a cluster that is ultimately canonicalized as conceptc, thencis inherited as a positive content label for that chunk. These inherited labels provide positive-only supervision: concepts not assigned to a chunk are treated as unlabeled rather than verified negatives. Rather than train directly on all 44 million Stage 1 annotated chunks, we construct a long-tail- enriched training reservoir. The reservoir is selected to improve coverage of rare content concepts by amassing a minimum number of examples per concept. This procedure yields roughly 2.9 million chunks. The resulting reservoir is still highly imbalanced—frequent concepts can appear tens of thousands of times—but it gives substantially better coverage of the tail than uniform sampling from the full Stage 1 sample. We train the annotator on approximately 2.3 million reservoir chunks and reserve approximately 597,000 chunks for held-out evaluation. The second source of supervision consists of fixed candidate vocabularies for auxiliary lower- cardinality domains. These domains are not produced by the Stage 2 clustering procedure. Instead, 24 we define finite candidate lists for tone, demographic references, and alignment-relevant categories, and annotate chunks using the same LLM-based annotation infrastructure with separate lists for each domain. Given a chunk and a candidate list, the annotator is asked which categories are rel- evant to the text. Selected categories are used as positive labels for the corresponding auxiliary heads. The final prediction vocabulary contains33,732labels in total, dominated by approximately 33,606content concepts, together with roughly80tone labels,38demographic-reference labels, and 8alignment-relevant labels. Design constraints.A few key requirements shaped the annotator model design. •First, we want a single model that can handle both the high-cardinality Atlas-derived con- tent concept library and the lower-cardinality auxiliary domains described above, rather than maintaining separate classifiers with separate encoders, thresholds, and inference logic. •Second, in Stage 2, we found that a simplek-nearest neighbor classifier over the concept embeddings, performs surprisingly well on content-label prediction, especially for frequent, well-represented concepts. We want a flexible architecture and loss function family that retains this simplicity, but incorporates learned signals for rare classes. •Third, our training data contains only positive labels for each concept domain since we did not do negative annotation: each chunk is annotated with the concepts ithas, never with the concepts it explicitly does not have. This positive-unlabeled (PU) supervision regime requires careful treatment of loss functions and evaluation metrics, so we will restrict ourselves to settings that make it flexible to incorporate such design constraints (Denis,1998;De Comité et al. ,1999;Denis et al.,2005;Kiryo et al.,2017). Architecture.The annotator uses theQwen3-Embedding-0.6Bmodel as a shared encoder, producing a pooled embedding for each input chunk. This embedding feeds into a lightweight MLP trunk (Lay- erNorm, ReLU, Dropout, projection to 1024 dimensions) shared by the auxiliary prediction heads. The content head follows a different path: it computes dot products between the encoder output and a matrix of content-concept embeddings, one per Atlas-derived content concept. This design preserves the KNN-like signal of direct embedding similarity while allowing learned improvements. The tone, demographic-reference, and alignment-relevant heads use simpler linear classifiers, which suffice for their lower-cardinality fixed vocabularies. Loss.We combine two loss functions. The first is masked binary cross-entropy, applied only to positions where targets are non-zero, which handles the positive-only supervision. The second is a non-negative PU loss, a PU-compatible objective that penalizes overconfident predictions on unla- beled classes and stabilizes learning on long-tail distributions. Rare concepts are further supported by a rarity-weighted sampling scheme that boosts underrepresented labels during training. Evaluation.Standard classification metrics—precision, recall, PR-AUC—are problematic under PU learning, because any unlabeled example predicted as positive is counted as a false positive even if the prediction is correct. This systematically deflates precision. We track these metrics for mon- itoring relative improvement across training runs, but we do not rely on them for absolute quality judgments. Instead, we rely on the LLM and human evaluation framework described at the start of the section. The annotator is trained on 2.3 million of the sampled chunks; we evaluate its predictions on the remaining 597,000 held-out chunks, scoring 4.93 million predicted (concept, chunk) pairs with the LLM judge (Figure 10). Predicted-concept ratings average 2.94, with per-chunk averages clustering 25 Figure 10.Stage 3 annotator validation on held-out chunks. Distributions of average predicted-concept relevance ratings from the LLM judge: per-chunk averages (left) and per-concept averages (right). between 2.5 and 3.5 (median 3.0); 95.5% of per-chunk averages are at least 2, and per-concept averages center near 3.3. The annotator thus assigns acceptable concept sets to the overwhelming majority of held-out chunks, though its predictions score lower and are more dispersed than original tag and con- cept ratings—plausibly reflecting the positive-unlabeled training regime and the per-domain prediction caps. 4.3Concept taxonomy We organized the33,732canonical concepts into a hierarchical taxonomy derived from the Library of Congress Classification (LCC), an established bibliographic classification system maintained by the Library of Congress ( Library of Congress,2023). We used theagentlans/library-classification- systems dataset ( Tseng,2024), which provides a machine-readable outline of LCC entries with parent–child links. The resulting taxonomy maps the concepts onto roughly 2,600 occupied nodes within the full 6,517-node LCC outline, with populated paths reaching depth 9 (Figure11). Concept distribution across taxonomy.Science (Q) dominates with 38% of concepts, followed by Technology (T) at 15%, Social Sciences (H) at 15%, and Medicine (R) at 8%. All 20 root branches are represented to varying degrees. Within the Science branch, mathematics and physics subcategories are particularly prominent: Mathematics (QA) subdivisions such as Analysis, Geometry, and Algebra account for over3,700concepts combined, while Physics (QC) areas such as Atomic and Molecular Physics contribute over 660 concepts. The distribution in Figure 12reflects the composition of the underlying pretraining corpus rather than an editorial choice about which domains matter. The taxonomy provides a structured frame- work for analyzing concept coverage across knowledge domains and, as we describe below, enables stratified evaluation of concept quality. 26 Figure 11.Schematic of the LCC-derived taxonomy used to organize the Atlas concept library. The33,732 canonical concepts are mapped onto approximately 2,600 occupied taxonomy nodes. The figure shows the root structure and expands the Science (Q) branch to illustrate how top-level classes decompose into more specific areas such as Mathematics (QA), Physics (QC), and their subdivisions; other branches are collapsed for readability. 4.4Human interpretability of the concept library The validations above show that Atlas is internally consistent: tags are relevant to chunks, clusters are semantically coherent, and the trained annotator assigns concepts to held-out text that an LLM judge scores as present. This does not, by itself, show that the final concept names are human- interpretable. Since the Stage 2 concepts are labeled by an LLM, there is a specific failure mode we must rule out: the LLM might assign fluent names to statistical patterns that do not correspond to concepts humans recognize. For example, a cluster whose tags co-occur for lexical rather than semantic reasons can still receive a confident, plausible-sounding name—and an LLM judge shown the same evidence may rate that name as coherent—without any person being able to independently arrive at, or even recognize, the underlying concept. We therefore evaluate two claims. First, the evidence associated with a concept must itself contain recoverable semantic structure. Second, the Atlas label—generated by an LLM in Stage 2—must name that structure at least as well as labels generated independently by humans. Lifted-word evidence.For each conceptc, we construct a list oflifted words: lemmatized words appearing in the text chunks assigned conceptc, ranked by how strongly they are associated withc 27 Figure 12.Distribution of over33,000concepts across the top-level LCC classes and notable subclasses. Sci- ence (Q) accounts for the largest share, followed by Technology (T), Social Sciences (H), and Medicine (R); all root classes are represented to varying degrees. The distribution mirrors the composition of the pretraining corpus rather than a curatorial choice. relative to their background frequency across all chunks, lift(w,c) = P(wjc) P(w) , with a minimum-support filter to remove idiosyncratic rare words. Lifted words provide a readable, corpus-level form of concept evidence. They also instantiate the more general evaluation problem used elsewhere in the paper: given only the words or tokens statistically associated with a concept, can a labeler recover a human-meaningful name? Two-phase human study. We ran a two-phase human study. In Phase 1, human annotators saw only the lifted words for a concept. Each annotator wrote a name or short phrase for the lifted-word list and rated, on a1–5coherence scale—from1(the words form no recognizable concept) to5(the words clearly correspond to a single recognizable concept)—whether the words formed a recognizable concept at all. This generation task is deliberately stricter than asking humans to approve a provided label: annotators can mark the evidence as noisy rather than being forced to accept a fluent name. In Phase 2, annotators performed blind comparative scoring. For each concept, we assembled a candidate set containing the Atlas label generated in Stage 2, two human labels written for the same concept by other annotators in Phase 1, ataxonomy distractorfrom a nearby but distinct concept in the taxonomy, and anembedding distractorfrom a different concept with a nearby label embedding. In a small number of cases where a second human label was unavailable, we substituted a deliberately generic filler label as a floor control: a label expected to fit poorly, confirming that raters used the low end of the scale. Annotators rated how well each candidate name fit the same lifted-word list, again on a1–5scale. Candidate order was randomized, annotators were blind to label provenance, and no annotator scored a label they had written. 28 Figure 13.Phase 2 human-evaluation fit scores by candidate type. Human annotators rated, on a1–5scale, how well each candidate name fit the same lifted-word evidence for a concept. Points show mean fit scores and horizontal bars show95% confidence intervals; the dotted vertical line marks the neutral midpoint of the scale. Atlas labels score comparably to or above independently generated human labels, and both are clearly separated from embedding, taxonomy, and filler distractors, indicating that raters were not merely assigning high scores to any plausible label. Do lifted words contain recognizable structure?Yes, although not in every case. We sampled100 concepts stratified by top-level taxonomy branch: ten concepts from each of the nine largest branches and ten from the aggregated remainder. Phase 1 collected303named-concept responses from9 human annotators, with a median of3annotators per concept. Human-written names averaged4.1 words. The mean Phase 1 coherence score was3.52. Annotators judged55% of responses to form a rec- ognizable concept (4), rated27% as borderline (= 3), and flagged17% as incoherent or noisy (2; on this coherence scale, unlike the relevance scale used in the pipeline validations, low scores indicate unrecognizable or noisy evidence). Thus the premise is empirically non-trivial: many lifted-word lists contain semantic structure that humans can independently recognize, while a minority remain ambiguous or noisy. Do Atlas labels name that structure?Yes. Phase 2 collected1,025individual candidate-name ratings:205scoring records over34concepts from8annotators. In blind scoring, annotators sharply separated real labels from distractors, validating the task itself (Table 4). Atlas labels and human-generated labels both scored far above taxonomy and embedding dis- tractors, showing that raters were not merely assigning high scores to any plausible phrase. The strongest comparison is between the Atlas label and independently generated human labels. Atlas labels scored higher on average:3.98versus3.50. They also received a top-two rating (4) 79% of the time, compared with63% for human labels. In paired comparisons, the Atlas label outscores a human label with probability0.62(cluster-bootstrap95% CI[0.58,0.66]). Additional ordinal mixed-model and robustness analyses are reported in Appendix D. Interpretation.These results address the central question of validity: humans can infer meaningful 29 candidate typemean fit (1–5) sdn LLM (Atlas) label3.981.04 205 human label3.501.17 402 embedding distractor2.721.30 205 taxonomy distractor2.081.11 205 filler (floor)1.621.068 Table 4.Phase 2 blind fit scores by candidate type. Human annotators rated how well each candidate name fit the same lifted-word evidence. Counts sum to1,025individual ratings. concepts from lifted-word evidence alone. When asked to judge labels blindly, humans rate the Atlas labels (LLM-generated) at least as highly as independently human-generated labels, and far above nearby distractors. This does not certify every concept in the library individually, but it does show that Atlas is not merely producing LLM-fluent names for arbitrary clusters. On a stratified pilot, the labels are human- recognizable, preferred to strong distractors, and competitive with or better than human-written names. Connection to known concept alignment.The same principle underlies the known concept align- ment metric used in model evaluation (Section7). Known concept alignment asks whether the word/token evidence associated with a concept from the library—for example, lifted words or the tokens most boosted by a steering vector—matches that concept’s name. The human study validates this question as meaningful and supports using an LLM judge as a scalable proxy: humans independently recognize the semantic structure in the evidence and endorse the Atlas labels under blind comparison. The validation chain is therefore: human-recognizable word/token evidence!human-endorsed Atlas labels!scalable LLM-judged concept alignment. 4.5Data Processing The Atlas pipeline described in Section4.2.1–Section4.2.3produces a concept library and a trained annotator model. This subsection describes how we apply the annotator to the full pretraining cor- pus, producing the tokenized, concept-annotated training data that the Steerling model consumes. The process has three stages: distributed concept annotation with embedding extraction, tokeniza- tion with document-structure markup, and retrieval index construction (Section 4.6). 4.5.1Distributed Concept Annotation The Stage 3 annotator (Section4.2.3) must be applied to every chunk in the pretraining corpus— approximately 11 billion text chunks spanning 1.5 trillion tokens. We distribute this workload across GPU nodes using SLURM array jobs, where each task processes a single input parquet file independently. 30 Figure 14.Token stream with chunk-level concept annotations. Each chunk is terminated by an[EOC]token and associated with a set of concept labels. Concept supervision is provided at the chunk level; the model learns to localize concepts to individual tokens via OR-aggregation during training. Input format.Each input parquet file contains pre-chunked text with columns forchunk_text(the main content),data_src,doc_id,source_doc_id,chunk_id,doc_end_flag(marking the final chunk of a document), andmetadata. The chunking follows the same procedure described in Section4.2.1: sentence boundaries are detected with BlingFire sentence splitter (Microsoft,2019), and consecutive sentences are concatenated until a domain-specific token threshold is reached (150 tokens for web text, 256 for mathematics and code). Forward pass.For each chunk, the annotator’sQwen3-Embedding-0.6Bencoder produces a pooled 1024-dimensional embedding, which is then passed to the multi-head classifier. The classifier pre- dicts concept activations across all four domains—content, tone, demographic, and alignment— using domain-specific thresholds and per-domain caps on the maximum number of predicted con- cepts. Each domain’s predictions are local indices into that domain’s concept vocabulary; these are mapped to global concept IDs via fixed offsets, producing a unified list of concept annotations per chunk. Embedding extraction.During the same forward pass, we cache the 1024-dimensional encoder embeddings for every chunk, storing them in Zarr arrays at FP16 precision alongside document metadata (data_src,source_doc_id,doc_id,chunk_id). These embeddings serve two purposes: they form the basis of the retrieval index constructed in Section 4.6, and they are available for downstream analysis without requiring additional encoder inference. At the scale of the full training corpus, this amounts to approximately 11 billion vectors. Output format.Each input parquet file produces a corresponding annotated parquet file with the original columns preserved and new columns added:text(combined chunk text),concepts(list of global concept IDs), and per-domain arrays (content_global_ids,tone_global_ids,demographic_- global_ids,alignment_global_ids). These annotated parquet files are the input to the tokeniza- tion stage. 4.5.2Tokenization and document structure The annotated parquet files are tokenized into the token stream consumed by the model during training. We use thecl100k_baseencoding from tiktoken 2 . We add four special tokens to represent our document and training structure, and reuse the encoding’s native<|endoftext|>token (ID 100,257) as a document-boundary marker. Including the byte-pair tokens, the encoding’s existing special tokens, and our additions, the full vocabulary contains100,281tokens. 2 https://github.com/openai/tiktoken 31 TokenIDRole <|endoftext|>100257 End of document [PAD]100277 Padding [BOS]100278 Beginning of sequence [EOC]100279 End of chunk (within a document) [MASK]100280 Masked position (for diffusion training) Tokenization procedure.Each chunk’s text is tokenized without special tokens. We prepend a [BOS]token to every training chunk and append anEOCtoken after every chunk, marking the boundary between consecutive chunks within a document. For the final chunk of each document (identified by thedoc_end_flag), an<|endoftext|>token is additionally appended, signaling the document boundary. This structure preserves the document–chunk hierarchy in the token stream: the model can distinguish intra-document chunk boundaries within a document from boundaries between documents. Chunk–concept alignment.The concept annotations produced by the annotator are chunk-level labels: each chunk is associated with a list of concept IDs, with no token-level labels. The[EOC] token serves as the delimiter that aligns concept annotations with their corresponding token spans in the training data. During training, the concept loss (Section5) usesOR-aggregationacross all tokens within a chunk to bridge this chunk-level supervision with the model’s token-level concept activations. Figure 14illustrates this structure. 4.6Training data indexing for test-time attribution The pipeline in Section4.5produces two artifacts: approximately 11 billion text chunks anno- tated with concepts by Stage 3 of the Atlas pipeline (Section4.2.3), and, for each chunk, a 1024- dimensional embedding vector computed during the annotator’s forward pass over the corpus. Be- cause these embeddings are cached at annotation time, index construction reuses them directly and requires no additional annotator forward passes. These vectors form the index that supports test-time training data attribution. Vector database.An exact (flat) index scores a query against every stored vector. While accurate, it imposes two costs that are each prohibitive at our scale: search latency that grows linearly in the number of vectorsN, and memory that grows asNdin full precision. We therefore use an approximate index—an Inverted File with Product Quantization (IVFPQ) index from the FAISS library ( Johnson et al.,2019)—which addresses both costs: an inverted-file structure partitions the space so each query scans only a small fraction of the vectors, and product quantization compresses the stored vectors into compact codes. With our configuration, product quantization reduces the per-vector payload from4,096bytes (a full-precision 1024-dimensional vector) to64bytes, roughly a64compression. Weℓ 2 -normalize all vectors at build and search time, so inner-product retrieval is equivalent to cosine similarity. Even so, the pretraining data index occupies approximately808GB on disk, exceeding available RAM; we therefore store the inverted lists in a memory-mapped on-disk format, so only the lists touched by a query are paged into memory at search time, enabling search over the full corpus without loading the entire index into RAM. Retrieval accuracy.Because IVFPQ is approximate, we assess how faithfully it reproduces exact search. On1,000query chunks drawn from the index itself, we query the index using ann probe = 16, 32 and measure recall@kas the fraction of queries whose own source chunk—the known ground-truth nearest neighbor—appears among the top-k retrieved results. The index achieves arecall@10 of 96.8%, indicating that the quantized search recovers the exact nearest neighbor in the large majority of cases while operating within the memory and latency budget imposed by a corpus of this scale. 5Inherently interpretable architecture In this section, we present the architectural and training choices to build a language model with inherently interpretable outputs, an instantiation of Section3. We depart from the autoregressive paradigm: the masking-based training objective and attention structure of diffusion models are better suited to the interpretability properties we want to enforce (Section5.1), and we adopt a causal block-attention pattern that retains efficient inference while preserving the diffusion training objective (Section5.2). On top of this backbone, we introduce the concept module, a bottleneck inserted between the transformer and the language modeling head that routes every prediction through an explicit concept representation (Section5.3). We then describe the training procedure that ties these components together (Section5.4). 5.1Beyond autoregressive models Autoregressive (AR) models, built on a causal-attention transformer architecture, have been the standard choice for large language models, achieving impressive performance across a wide range of tasks ( Achiam et al.,2023;Anthropic,2024;Team et al.,2023;Liu et al.,2024;Grattafiori et al., 2024b;Bai et al.,2023). Beyond raw capability, the AR paradigm benefits from years of accumu- lated practical knowledge: training procedures, hyperparameter choices, and scaling laws have been studied extensively and documented in detail by the open-source community (OLMo et al.,2025; Grattafiori et al.,2024b;Team et al.,2024;Liu et al.,2024), making AR models a tempting default. However, their inductive biases do not align well with our objectives. We aim to build a model that supports faithful input attribution and concept-level control at inference time, and AR generation makes both properties harder to enforce. AR models predict one token at a time conditioned on a strictly causal context: concepts typically span multiple tokens rather than localizing to one, and attribution lacks a natural absence-of-information baseline. The same left-to-right factorization un- derlies well-documented failure modes, including the reversal curse ( Berglund et al.,2024) and poor performance on tasks requiring graph backtracking (Ye et al.,2025). The limitations above are not incidental to AR models; they follow from the left-to-right factorization itself, so addressing them calls for a different generative paradigm. Diffusion language models offer a more natural fit. The masking objective sidesteps each of the failures identified above and provides three properties we exploit throughout this work. First, it gives the model an explicit, trained representation of “no information at this position,” the absence baseline that faithful input attribution requires and that strictly causal contexts cannot provide. Second, multiple tokens are predicted jointly at each denoising step, giving us a natural interface for concept-level control over phrases rather than individual tokens, directly addressing the fact that concepts span multiple tokens rather than localizing to one. Third, diffusion models generate tokens in any order, so the model can choose where in the sequence to express an intervened concept rather than being forced to commit at the next position. We describe our specific instantiation, Causal Diffusion, in the next section. 33 5.2Causal Diffusion Our objectives place two demands on the architecture. First, we want the model to operate over groups of tokens rather than single positions, since the concepts we interpret and steer (Section5.3) typically span phrases rather than localizing to one token. Second, we want autoregressive-style in- ference efficiency: the KV caching and throughput that make AR models practical at scale. Masked diffusion gives us the first through its joint, any-order denoising, but, as we show below, its stan- dard form sacrifices the second. The remainder of this section develops an attention structure that recovers both. We build on masked diffusion models (MDMs; see Section11and Section2). Standard MDMs use full bidirectional attention: every token attends to every other token in the sequence at each denois- ing step (Figure15, panel b). Bidirectional context is what enables MDMs to denoise multiple tokens jointly in a single forward pass and to predict tokens in any order, since no position is privileged over another. The same property, however, prevents efficient inference. Because tokens that change be- tween denoising steps are attended to by every other token, no representations can be cached across steps (Israel et al.,2026;Arriola et al.,2025). Each step recomputes the full attention over the full sequence, making inference substantially slower than autoregressive models of comparable size. Block diffusion (Arriola et al.,2025) addresses this inference-time cost by modifying the attention pattern. The sequence is partitioned into blocks of fixed lengthb, and the attention mask is bidirectional within each block but causal across blocks (Figure 15, panel c). Generation proceeds one block at a time, and KV caches built from previously generated blocks can be reused across denoising steps. The training algorithm, however, requires concatenating a noisy and a clean copy of the sequence as input: noisy blocks supply the diffusion loss, while clean previous blocks supply context. This roughly doubles the per-step memory and FLOPs relative to standard MDM training. We proposeCausal Diffusion, which retains Block Diffusion’s attention pattern but drops the clean copy of the sequence at training time (Figure 15, panel d). The training objective is the standard masked diffusion loss from Section2.3, applied to a single sequence of tokens with a block-causal attention mask: bidirectional within each block, causal across blocks.Nie et al.(2025) showed that an MDM trained with full bidirectional attention can be sampled block-by-block at inference time with minimal quality degradation, suggesting that the masking objective does not require bidirec- tional context across the entire sequence. If block-causal attention is sufficient at inference, training under the same constraint costs nothing in expressive power: we obtain block-causal structure at half the training cost of Block Diffusion, without the inference-time bottleneck of standard MDMs. At inference, each new block is denoised by masked diffusion while conditioning on the keys and Figure 15.Attention patterns for Autoregressive, Diffusion, Block Diffusion, and Causal Diffusion models. values cached from all previous blocks; once generated, its own keys and values are appended to 34 the cache, exactly as in autoregressive decoding. The result is an MDM that retains diffusion’s par- allelism and any-order flexibility within each block while inheriting autoregressive-style KV caching across blocks. 5.3Concept module One way to build an inherently interpretable language model is to decompose the hidden represen- tation into a set of concepts and use those concepts, combined through a simple and interpretable function, for the model’s predictions. Two properties follow directly from this decomposition: faith- ful concept attribution, since each concept’s contribution to a given output can be computed directly, and concept steering, since we know how each concept is represented and can therefore bias the model towards or away from it. We achieve this with an additive bottleneck: the hidden representa- tion is reconstructed as a sum of concept contributions before being passed to the language modeling head. We refer to this as ourconcept module. Figure 16.The concept module bottleneck. The transformer hidden statehis decomposed into known concept contributions ˆ k, unknown concept contributionsˆu, and a residualε. The concept module sits between the transformer backbone and the language modeling head (Fig- ure16). For each input token, the transformer produces a hidden stateh2R d , which is normally passed straight to the language modeling head. The concept module instead decomposeshinto three additive components: ̄ h= ˆ k+ ˆu+ε,(5) where ˆ kis a weighted sum ofknownconcept embeddings,ˆuis a weighted sum ofunknownconcept embeddings, andε=h ˆ kˆuis the residual term. Only ̄ his passed to the LM head, so every output logit becomes a linear function of concept activations. We apply dropout with ratep ε toε during training, to discourage the model from relying on the residual channel for prediction. Both ˆ kandˆuare computed fromhvia two small heads. The known headfproduces concept activation probabilities for the labeled concept set, and the unknown headgproduces activation 35 probabilities for a set of unknown concepts: k=σ(f(h))2R n , u=σ(g(h))2R m ,(6) wherefandgare small learnable networks,σdenotes the elementwise sigmoid,nis the number of known concepts, andmnis the number of unknown concepts. Each conceptihas a learned embeddingK i 2R d (orU j for unknown), analogous to a token embedding, and the concept-weighted hidden states are ˆ k= n X i=1 k i K i ,ˆu= m X j=1 u j U j .(7) The full set of concept embeddings forms a vocabulary of concepts, in the same way that token embeddings form a vocabulary of tokens. Given the large number of unknown concepts, storing the unknown embedding matrixU2R m×d directly would dominate the parameter count. We therefore factorize it as a low-rank productU=AB, withA2R m×r andB2R r×d for rankrd, reducing the parameter count frommdtor(m+d)while preserving capacity. At inference time, the additive structure of ̄ hmakes every prediction a transparent function of the concept activations. Because the LM head is linear, the logit for any output token decomposes as ℓ y = ˆ k ⊤ W y + ˆu ⊤ W y +ε ⊤ W y ,(8) whereW y is the row of the LM head corresponding to tokeny. Each term is exact: there is no approximation involved in attributing a prediction to known concepts, unknown concepts, or the residual. This decomposition is what allows the model to support faithful concept attribution. Taken together, these components make the concept module a self-contained, modular interface to the model’s predictions: known and unknown concepts, their embeddings, and the residual are separate, inspectable channels, and every output is an explicit, additive function of them. This organization is what makes the attribution and steering described in Section 6possible. 5.4Model training The architecture described in Section5.3forces every prediction to pass through an additive concept representation. In effect, it builds in the linear representation hypothesis (Park et al.,2023), the idea that high-level concepts are encoded as linear directions in representation space. Rather than hoping this property emerges, as it may or may not in a standard language model, our architecture enforces it by construction: every output is an explicit linear function of concept activations. This structural guarantee, however, does not by itself ensure that the concept module learns useful concepts, that the language modeling head produces fluent text, or that the known and unknown concepts encode complementary information. Each of these properties has to be optimized for. We describe the loss objectives that target them in Section 5.4.1, and the training dynamics that control how those losses are applied over time in Section5.4.2. 5.4.1Loss objectives Language modeling loss.The primary objective is the masked diffusion lossL MDM from Sec- tion 2.3, applied to the bottlenecked hidden state ̄ hrather than the raw transformer outputh. Every token prediction is therefore evaluated as a function of the concept module’s output. We denote this lossL LM . 36 Concept loss.The ground-truth concept labels are produced at the chunk level by the annotation pipeline of Section4. A chunk is a contiguous span of tokens terminated by an[EOC]token (Fig- ure14). The labels are positive-only: a label tells us a concept appears somewhere in a chunk, but not at which token. We accommodate this weaker supervision signal with an OR-aggregation across the chunk. Because the diffusion objective only supervises masked positions, the aggregation runs over the masked tokens of the chunk; let chunk M denote this set, whereMis the set of masked positions in the minibatch. For known conceptc, letk c,t denote its predicted activation at token t(thec-th entry ofkfrom Equation (6)). The probability thatcappears at least once among the masked tokens of the chunk is k chunk c = 1 Y t∈chunk M (1k c,t ).(9) Lety c 2 f0,1gdenote the ground-truth chunk-level label for conceptc. The concept loss is the binary cross-entropy between the aggregated probability and the chunk label, averaged over the known concepts and over chunks in the minibatch: L concept = 1 n n X c=1 h y c logk chunk c + (1y c )log(1k chunk c ) i .(10) The OR-aggregation is satisfied as soon as the concept is predicted at any one masked token in the chunk, consistent with our chunk-level supervision. Reconstruction loss.The unknown head is trained to represent the part of the hidden state that is not captured by the known concepts. Given the ground-truth labels for known concepts, the ideal known representation and the corresponding target for the unknown head are ˆ k GT = n X i=1 k GT i K i ,ˆu GT =h ˆ k GT ,(11) whereˆu GT is the residual that remains after subtracting ˆ k GT from the transformer hidden state. The unknown head is trained to match this target under a mean-squared error, averaged over masked positions in the minibatch: L rec = 1 jMj X t∈M kˆu t ˆu GT t k 2 2 ,(12) whereMis the set of masked token positions in the minibatch. Whenˆu= ˆu GT , the residual term in the bottleneck satisfiesε= 0. Independence loss.The reconstruction loss alone does not prevent the unknown head from encod- ing information that is already represented by the known concepts. To discourage such redundancy, we penalize the statistical dependence between the known and unknown representations using a normalized cross-covariance penalty in the spirit of the Hilbert-Schmidt Independence Criterion with a linear kernel ( Mooij et al.,2009;Greenfeld and Shalit,2020), following its use for concept decoupling inAndersson et al.(2026). Over thejMjmasked token positions in a minibatch, let Φ =H k 1μ ⊤ ˆ k ,Ψ =H u 1μ ⊤ ˆu ,(13) whereH k ,H u 2R |M|×d stack the per-token ˆ k,ˆucomponents over the masked positions,μ ˆ k ,μ ˆu 2R d are their column means,12R |M| denotes the all-ones vector, andkk F is the Frobenius norm. The 37 independence loss is L indep = 1 d 2 (jMj1) Ψ ⊤ Φ 2 F .(14) In practice, gradients flow only through the unknown representationΨ; the known representation Φis treated as a fixed input. MinimizingL indep drives the cross-covariance between ˆ kandˆutoward zero, encouraging the two heads to encode complementary rather than redundant information of the hidden state. Final training objective.The four losses are combined linearly with non-negative weights: L=L LM +λ concept L concept +λ rec L rec +λ indep L indep .(15) The backbone and concept module are optimized jointly under this objective, where λ concept ,λ rec ,λ indep are hyperparameters. For both auxiliary lossesL rec andL indep , gradients are detached so that only the unknown head is updated. 5.4.2Training dynamics Per-block masking schedule.Standard masked diffusion models apply a single noise levelt U(0,1)to the entire sequence at each training step. In the block-causal architecture (Section 5.2), we treat each block as an independent unit and sample a separate noise levelt b U(0,1)for each block b, so two blocks within the same context window can be at different stages of the denoising process simultaneously (Figure 17). The model thus observes a richer distribution of partially-denoised contexts at every training step than under a single global noise level. Figure 17.Per-block masking schedule. Each block samples an independent noise levelt b U(0,1), so blocks in the same sequence are denoised to different degrees within a single training step. Concept teacher forcing schedule.The known head’s predictions are unreliable early in train- ing, and even once they stabilize, routing ̄ hthrough the predicted activations ˆ kallows the language modeling loss to push those activations to encode information beyond the labeled concepts, a failure mode known as concept leakage (Mahinpei et al.,2021).Koh et al.(2020) address this by training the concept head independently of the downstream model and feeding ground-truth concepts for- ward at every step. We adopt the same substitution idea but apply it on a schedule: with probability α known (s)at training steps, we replace the predicted known representation ˆ kwith its ground-truth analogue ˆ k GT = P i k GT i K i from Equation (11) when forming ̄ h, which we refer to as teacher forcing. We annealα known from1at the start of training (full teacher forcing) to a smaller steady-state value, so the model relies progressively on its own predictions as the head becomes more accurate. Unknown concept teacher forcing schedule.The unknown head faces the same early-training instability as the known head, but no ground-truth labels exist to substitute for its predicted activa- tions. Instead, we substitute the ideal target itself: from Equation (11), the unknown embedding 38 should reconstructˆu GT =h ˆ k GT , which we can compute directly from the transformer hidden state and the labeled concepts. With probabilityα unknown (s)at training steps, we replace the predicted ˆuwithˆu GT when forming ̄ h. As withα known , we annealα unknown from1to a smaller steady-state value, so the language modeling head is shielded from a poorly-trained unknown head early on and learns to rely on the predicted unknown embedding as it becomes accurate. A consolidated reference for the symbols introduced in this section is provided in AppendixA. 6Interpretability capabilities The architecture of Section5produces models that are bothinterpretableandcontrollable. Inter- pretability answers “why a prediction was produced.”: through attribution, we identify the inputs, concepts, and training data associated with a prediction. Controllability is enabled through steering, which modifies the concept representation at inference time to control how selected concepts affect the model’s behavior. We describe attribution and steering in the following sections. 6.1Attribution Attribution asks why Steerling produced a given output. We approach it from three angles: •Input attribution identifies influential input tokens (Section6.1.1). •Concept attribution traces influential internal concepts (Section6.1.2). •Training data attribution retrieves similar training examples (Section6.1.3). Figure 18shows the three methods applied to a single output chunk. 6.1.1Input attribution Input attribution answers:which input tokens most influenced a given output?We use Integrated Gradients (Equation16) with the[MASK]embedding as the baseline. The baseline determines what the attributionmeasures: how the output changes as the input moves from the baseline to its actual value.[MASK]makes that comparison meaningful for our model. The diffusion objective trains the model to predict masked tokens at every position, so[MASK]becomes a learned representation of “no information at this position,” and the integration runs from this learned “absent” state to the actual token along a path of states the model was trained on. Figure 18A shows the per-token scores for the highlighted chunk in this example. Autoregressive models have no such trained baseline: zero embeddings and padding tokens are out-of-distribution states the model never learned to read as an absence of information. We compute attribution using Integrated Gradients (Sundararajan et al.,2017a) on the token em- beddings. For an input tokenx i with embeddingT x i , we integrate the gradient of the output logit ℓ y along the straight-line path from the baseline embeddingT [MASK] toT x i , giving a score φ(x i ,y) = 1 S S X s=1 (T x i T [MASK] ) ⊤ r T (s) ℓ y ,(16) whereT (s) =T [MASK] + s S (T x i T [MASK] )is thes-th ofSinterpolation points along that path. Replacing a token with[MASK]is an operation the model has performed countless times during 39 Figure 18.Three views of attribution for one output chunk.Input attribution(left): per-token Integrated Gradients scores over the prompt, with tokens that contribute most to the chunk shaded more intensely. Concept attribution(center): top contributing concepts, ranked by their chunk-level contribution.Training data attribution(right): retrieved training chunks, ranked by similarity to the chunk’s representation and identified by chunk ID, which resolves to the source document for verification. training, so the Integrated Gradients path traces a well-defined direction through the model’s learned representation space. 6.1.2Concept attribution Concept attribution answers:which internal concepts most influenced a given output?The additive bottleneck of the concept module already decomposes the logit of any output token exactly into known-concept, unknown-concept, and residual terms (Equation 8). Figure18B ranks the top con- tributing concepts for the highlighted chunk. We report attribution at the chunk level, consistent with how concepts are supervised (Section5.4). The contribution of a concept to a chunk is the sum of its per-token contributions over the tokens in that chunk: Γ known i = X t∈chunk k i,t K ⊤ i W y t ,Γ unknown j = X t∈chunk u j,t U ⊤ j W y t ,(17) wherek i,t andu j,t are the activations of known conceptiand unknown conceptjat the tokeny t produced at positiont, and an analogous sum gives the residual’s contribution. Ranking concepts by these chunk-level contributions identifies which concepts drove the output, while the residual captures the part of the output that the concept inventory does not explain. 6.1.3Training data attribution Training data attribution (TDA) answers:which training examples are most influential to a given output?It enables applications such as alignment fine-tuning, factual provenance tracing, and au- diting whether a model generalizes from proprietary fine-tuning data or falls back on pretraining 40 knowledge. Unlike concept attribution, data attribution does not derive its scores from the model’s own computation. We do not estimate the causal effect of removing a training example, as classical influence-function methods do (Koh and Liang,2017); at our scale the required Hessian inverse is intractable. Instead we frame attribution as approximate semantic-similarity retrieval : given an output, we retrieve the training chunks whose representations are most similar to it. A structural counterpart appears in PRISM (Ley et al.,2026), which implements TDA as a lookup on a learned prototype layer rather than nearest-neighbor search in general representation space. Our work is a scalable proxy for influence rather than a faithful or causal account. The retrieval view suits Steerling in particular: because its representations are aligned to human-interpretable concepts (Section5), semantically related chunks lie close together in representation space, so nearest-neighbor search tends to surface training chunks that share concepts with the output. Because data attribution retrieves rather than computes, its central difficulty is representational. The query vector we extract from Steerling lives in the model’s internal representation space, whereas the index stores corpus embeddings produced by a different model, so an output’s internal repre- sentation cannot be matched against the index directly. We bridge them in two steps: forming a query representation from an output, then mapping it into the index’s space. Given a model out- put, we segment it into chunks at the[EOC]token and forward each chunk through the language model, mean-pooling over its token positions to obtain a single vector that combines the known- head, unknown-head, and residual components of the hidden state. A learnedprojection—a small MLP trained with a cosine-similarity objective—maps this internal vector into the corpus embed- ding space of the index built in Section 4.6, placing an output and its related training chunks close together. The projected query is then matched against the index by approximate nearest-neighbor search, and the nearest chunks are returned as the attributed sources. Figure 18C shows the top- ranked retrieved chunks for the highlighted output chunk, ranked by cosine similarity; their source URLs can be inspected for verification. 6.2Steering Steering is the ability to control model output without using prompts. This is enabled by the concept module and is not possible in other existing models. We steer generation at inference time, without updating any weights, by acting directly on the concept representations the model already uses. Steering can push generation toward a target concept (amplification) or away from it (suppression), and can be applied to both the known concepts supervised during training and the unknown concepts the model discovers on its own. 6.2.1Steering operation Our model learns a direction (or embedding) for each concept and uses it internally to form pre- dictions. We can steer the model toward a given concept by injecting the direction the model has already learned for it into the hidden representation. Injecting a direction into the hidden states to steer generation is a well-established technique (Subramani et al.,2022;Turner et al.,2023;Rimsky et al.,2024;Zou et al.,2023), but these methods must first extract the direction post-hoc from a trained model’s activations, so it is only an estimate of how a concept is represented and varies with the procedure used to recover it. Our architecture is better suited to steering because the direction is not estimated: each concept’s embeddingK c is a model parameter the model itself uses, so we inject a direction the model already relies on rather than one fit after the fact. Concretely, we take the concept embeddingK c and normalize it to unit length, giving a steering directione c =K c /kK c k 2 . To steer toward several concepts at once, we sum their embeddings then 41 normalize. We then add this direction, scaled by a steering strengthγ, to the transformer hidden state at every masked position, at every layer fromL inj onward. The steering signal accumulates as the representation propagates toward the bottleneck: h (l) t h (l) t +γ e c , lL inj .(18) Because each concept aligns with the LM head differently, the effect of a givenγis not comparable across concepts: the same strength shifts the output logits more for some concepts than others. We therefore calibrateγper concept so that its largest effect on any output token equals a fixed targetτ: γ= τ peak(e c ) ,peak(e c ) =max y∈V e ⊤ c W y ,(19) where peak(e c )is the largest logit shift the directione c can induce over the vocabularyV. A single globalτthen gives adaptive steering strength across concepts without per-concept tuning. 6.2.2Steering direction The sign of the steering strengthγsets the direction of the intervention. Amplification pushes gen- eration toward the target concept, while suppression pushes it away. Amplification.Amplification takesγ >0, so the injection of Equation (18) moves the hidden state along e c , toward the concept c . The concept direction is added before the bottleneck, propagating the boosted activation through the concept module then the LM head to increase the logits of concept- expressing tokens. Figure 19.ReLU-gated concept suppression avoids unintended token promotion during negative steering. Naive subtraction suppresses tokens positively aligned with the target concept, but promotes tokens with negative alignment. For anacademic and commercial publishingconcept, this boosts unrelated tokens such as problemandbreakpoints. The ReLU-gated update preserves suppression of publishing-related tokens while leaving anti-aligned tokens unchanged. Suppression.Suppression eliminates text associated with a target conceptc. We combine two mechanisms to ensure the concept is properly suppressed: a negative hidden-state injection (Equa- tion ( 18) withγ <0) that pushes the transformer representation away fromc, and a ReLU-gated 42 logit mask that subtractsc’s contribution from the LM head output. We explain why we use the ReLU gate below. Since the LM head is a linear projection on top of the sum of concept embeddings, the contribution of conceptcto the output is its alignment with each vocabulary token: a c =We c 2R |V| , a c [v] =W ⊤ v e c ,(20) wherea c [v]is the concept’s contribution to the logit of tokenv. Subtracting this contribution directly (ℓ v !ℓ v sa c [v]with suppression strengths >0) will suppress tokens witha c [v]>0as intended, but simultaneouslypromotestokens witha c [v]<0, making unrelated tokens that happen to be anti- aligned with the concept (Figure19) dominate generation. We gate the subtraction with a ReLU so that only positively aligned tokens are affected: ℓ v !ℓ v sReLU a c [v] .(21) This suppresses the tokens aligned with the concept while leaving anti-aligned tokens untouched. 7Interpretability Metrics This work is built on the claim that interpretability is best treated as a design constraint rather than a post-hoc analysis problem. If that claim is correct, then a model trained with interpretability built into its architecture and objectives should be more interpretable than one trained without, not in some abstract sense, but in ways one can measure directly. This section makes that claim measurable. The metrics defined below test the properties the concept module of Section 5.3was designed to deliver. The concept module was trained to detect a set of concepts in context, via the concept loss. The model was trained to route its predictions through those concepts rather than through the residual, via the reconstruction loss. The known and unknown concepts were trained to be disentangled, so they do not encode the same information, via the independence loss. Each of these properties has a corresponding metric, and each metric tests a necessary condition for the architecture to be working as built. A fourth metric asks whether the learned concept embeddings point at semantically related tokens; this property is not directly supervised by any training objective, and is included to test whether the architecture produces token-level interpretability beyond what was directly optimized for. These metrics are familiar from the broader interpretability literature, where post-hoc methods such as sparse autoencoders, probes, and steering vectors are evaluated along similar axes of concept detection and disentanglement ( Wu et al.,2025;Bhalla et al.,2024). These post-hoc methods exist because representations in standard transformers are in superposition (Elhage et al.,2022;Bricken et al.,2023a): more learned directions than dimensions, packed into overlapping subspaces that have to be recovered after training. The concept module sidesteps superposition by construction: each concept has its own learned embedding, so the metrics do not have to ask whether a recovered direction behaves like a concept, but whether the concepts the model was trained on are doing the work the architecture assigned to them. Concept Loss.The concept loss (Equation10) measures whether the concept module assigns the right concepts to a given chunk. The metric is the OR-aggregated binary cross-entropy between predicted concept presence (Equation 9) and ground-truth chunk-level labels, computed over a held- out validation set of chunks annotated by the pipeline of Section4. A low value indicates that the 43 MetricDirection What it measures Concept Loss#Does the concept module detect the right concepts? Concept Independence Loss#Are known and unknown heads disentangled? Concept Contribution"Do predictions route through the concept module? Known Concept Alignment"Do concept embeddings point at related tokens? Table 5.Interpretability metrics."or#indicates the direction of better performance. concept module identifies concepts in unseen text in the same way it learned to identify them during training. Concept Independence Loss.The independence loss (Equation14) measures the linear statistical dependence between the known and unknown concept representations. The metric is computed over per-token ˆ kandˆuon a held-out validation set. A low value indicates that the two representations carry linearly independent information. A linear kernel was chosen because the bottleneck composes its components additively, ̄ h= ˆ k+ ˆu+ε, and the LM head is linear, so every output logit decomposes linearly into contributions from each pathway (Equation 8). Linear independence between ˆ kandˆu is therefore the exact property concept attribution requires: if the two representations are linearly independent in expectation, their contributions to any logit are additively separable, and the known and unknown pathways do not encode overlapping information that would inflate one attribution at the expense of the other. Concept Contribution.The concept contribution metric measures how much of each output predic- tion is explained by the concept module rather than by the residual. From the logit decomposition in Equation ( 8), the contribution of the concept module to the logit of any tokenyis the sum of the known and unknown terms; the residual termε ⊤ W y denotes what has not been captured by the concept module. The metric is the fraction of total logit magnitude attributable to the concept module: Concept Contribution= j ˆ k ⊤ W y j+jˆu ⊤ W y j j ˆ k ⊤ W y j+jˆu ⊤ W y j+jε ⊤ W y j ,(22) averaged over predictions on a held-out validation set. A high value indicates that predictions can be mostly attributed to the concept module. The reconstruction loss (Equation12) creates training pressure on this property by drivingε!0, but the metric is computed at the logit level rather than onkεkdirectly, so it tests the property that matters for attribution faithfulness: how much of the prediction are the concepts responsible for. Known Concept Alignment.Each known concept has a learned embeddingK c 2R d that the LM head projects into vocabulary space, so the top-ktokens for conceptcareT k (c) =TopK(WK c ). The known concept alignment metric asks whether these top tokens semantically match the human- assigned label and description that the concept was originally annotated with by the pipeline of Section 4. An LLM-judge is given the concept’s label, its description, and the top tokens, and rates the alignment on a 1-5 scale, with 5 indicating that the top tokens unambiguously represent the named concept and 1 indicating no relationship. Details on the judge and the full prompt are given in Appendix E. The metric is the mean rating over a randomly sampled subset of concepts from the library. 44 Known concept alignment is not directly supervised by the training objective. The concept loss (Equation10) supervises chunk-level concept presence using positive-only labels; it does not con- nect a concept embeddingK c to the specific tokens that lexically express conceptc. The language modelling loss supervises next-token prediction; it does not connect token predictions to the concept vocabulary. Any alignment betweenK c and semantically-related tokens therefore arises from the joint optimization of these two objectives, not from a direct training signal. 8Scaling laws A long-standing concern in the interpretability literature is that constraining a model’s representa- tions to be human-understandable necessarily costs capability ( Rudin,2019;Doshi-Velez and Kim, 2017;Koh et al.,2020). A second concern, less studied, is that even when interpretability proper- ties hold at small scale, there is no guarantee they survive at the scales where models actually get deployed (Wei et al.,2022a). This section tests both concerns empirically. Two model families, au- toregressive (AR) and causal diffusion (CDLM), together with their interpretable counterparts, are trained across three orders of magnitude of compute. Scaling laws are fit to the resulting checkpoints for capability (compute-optimal loss) and for the interpretability metrics defined in Section 7, and each fit is then used to extrapolate to Steerling, a CDLM+Concept model trained at 8B parameters and 1.35T tokens. Specifically, this section addresses two research questions: •RQ1.Do inherently interpretable architectures preserve compute-optimal scaling? •RQ2.Do the interpretability properties scale favorably with compute? Across model families and three orders of magnitude of compute, adding the concept module to either backbone shifts the compute-optimal scaling exponents bya small, fixed per-backbone off- set. The interpretability metrics defined in Section 7improve favorably with compute on both backbones, with all four metrics scaling in the expected direction. Extrapolating the small-scale fits to Steerlingpredicts the deployed model’s validation loss within0.11natsand bounds its inter- pretability metrics within their natural ranges. A consolidated reference for the symbols introduced in this section is provided in Appendix F. 8.1Setup Model families.Four model families are compared.ARis a standard autoregressive trans- former;CDLMis the causal block-diffusion model defined in Section 5.2; andAR+Conceptand CDLM+Conceptare their inherently interpretable counterparts, obtained by inserting the concept module of Section5.3between the transformer hidden state and the LM head. The two backbones are held fixed across all experiments; only the presence of the concept module changes. The con- cept library used by the +Concept families is the one constructed in Section 4, fixed across all model sizes. IsoFLOP sweep.Each family is trained across four IsoFLOP slices, with slice targets reported in Table25. Each slice contains four to six model sizes whose token counts are adjusted so that their total compute lands within15% of the slice target. The slice targets differ between base and +Concept families: at the smallest backbones, the concept module’s parameter overhead exceeds the backbone itself, making compute-optimal estimates unreliable in the lowest IsoFLOP slice. Slightly higher targets are therefore used for the +Concept families. Full architecture configurations and slice targets are reported in Appendix G. 45 Training.All runs share the same data, optimizer, and schedule. Pretraining is on the real-data subset of Nemotron-C-HQ (Su et al.,2025) at sequence lengthN= 4096. Optimization uses AdamW under a warmup-stable-decay (WSD) schedule (Hu et al.,2024): a short linear warmup, an extended stable phase at peak learning rate, and a final 20 % linear decay to zero. Optimizer hyperparameters, tokenizer, and per-family configurations are reported in AppendixG. Each IsoFLOP checkpoint is annealed independently rather than sharing a single annealing tail. The 80% stable-phase trajectory is treated as a reservoir of starting points; a20% linear decay is run from each one to its final loss. This is more expensive than estimating annealed losses from non-annealed checkpoints (von Rütte et al.,2025), but it yields a dense compute-loss sweep without a separate full run per slice. The cheaper estimation procedure produced inconsistent results across model families and sizes (AppendixI). Parameters and FLOPs calculation.FollowingBi et al.(2024), per-token forward+backward FLOPs for the AR and CDLM backbones are M base = 6P+ 12LdN,(23) wherePis the non-embedding parameter count,Lthe number of layers,dthe hidden dimension, andNthe sequence length. Total training FLOPs areC=MDforDtraining tokens. The +Concept families add two heads acting on every token (Section5.3). The known head scores allnconcepts through a predictor of sizedn, then composes the result through a separate em- bedding matrix of the same size via top-k known selection. The two matmuls give a per-token cost of2dn. The unknown head is factorized through low-rank embeddings: a down-projectiond!R, a predictor overmunknown concepts, top-k unknown selection, and composition through factorized embeddings, for a per-token cost of2dR+Rm. Combined: M +Concept = 6P+ 12LdN |z backbone + 6 2dn |z known head + 6 2dR+Rm |z unknown head .(24) Concept module hyperparameters and per-backbone parameter counts are reported in Appendix G. The parameter cost of the concept module shrinks rapidly with scale.For a fixed concept library, the concept module adds a parameter overhead ofO(d), while backbone parameters scale asO(d 2 L). The relative overhead therefore decays rapidly: the concept module accounts for89% of the total parameter count at 10M,4% at Steerling, and under1% at frontier scales (Figure20). 8.2Compute-Optimal scaling Compute-optimal scaling is fit per family on the checkpoints from Section8.1, yielding a set of (P i ,D i ,L i )triples per family giving parameter count, training tokens, and validation loss, organized into four IsoFLOP slices. The exponents are then compared pair by pair across each backbone’s inter- pretable and non-interpretable variant, allowing a direct test ofRQ1: do inherently interpretable architectures preserve compute-optimal scaling? 8.2.1Methodology Validation losses.For autoregressive models (AR, AR+Concept),L i is validation cross-entropy on a held-out subset of the Nemotron pretraining corpus. For diffusion models (CDLM, CDLM+Concept), L i is a Monte Carlo estimate of the masked diffusion model (MDLM) ELBO bound on the per-token 46 Figure 20.Concept module overhead as a fraction of total parameters, vs. backbone size. negative log-likelihood (von Rütte et al.,2025;Sahoo et al.,2026;Nie et al.,2024). Per batch, a single noise leveltU(10 −3 ,110 −3 )is sampled and shared across the batch, each token position is independently masked with probabilityt, a forward pass is run, and the mean cross-entropy on the masked positions is computed. The validation loss is the average over batches. Note that this estimator integrates overt2(0,1), while training samplest U(0.05,0.95); the two are therefore unbiased estimators of the same MDLM objective on slightly different intervals and will not match numerically. Details on the ELBO estimator and Monte Carlo sampling are given in AppendixH. Step 1: Per-slice parabola inlogP.Within each IsoFLOP slice (fixed targetC), the per-size(P i ,L i ) pairs are fit to a parabola in logP(Hoffmann et al.,2022): L(logP;C) =a(C) logPlogP ∗ (C) 2 +L ∗ (C),(25) wherea(C), logP ∗ (C), andL ∗ (C)are free parameters, fit by nonlinear least squares (Levenberg- Marquardt). The fitted logP ∗ (C)locates the parameter count that minimizes loss at compute budget C, andL ∗ (C)is the corresponding loss. Doing this once per slice produces four(C i ,P ∗ i ,L ∗ i )triples per family. Step 2: Power laws across compute budgets.With the four per-slice triples, we fit two power laws, P ∗ (C) =a P C α P ,L ∗ (C) =a L C α L ,(26) each by ordinary least squares on log-transformed data:α P and loga P are the slope and intercept of logP ∗ regressed on logC, and analogously forα L , loga L . The exponentsα P andα L characterize how the compute-optimal parameter count and loss scale with available compute; the compute- optimal token exponentα D is estimated from the joint fit of Step 3. Step 3: Joint parametric fit for the irreducible loss.Steps 1 and 2 yieldα P andα L but no irreducible-loss term. FollowingHoffmann et al.(2022);Ni et al.(2025), the joint loss surface L(P,D) =L ∞ + A P P α + A D D β (27) is fit on all(P i ,D i ,L i )observations pooled across the four IsoFLOP slices, not just the slice minima. The five free parameters areL ∞ ,A P ,A D ,α,β. The Huber loss on log-residuals is minimized, L fit (L ∞ ,A P ,A D ,α,β) = X i Huber δ logL(P i ,D i )logL i ,(28) 47 withδ= 10 −3 , using L-BFGS-B with grid-search initialization. The irreducible lossL ∞ and the data exponentα D =α/(α+β)are reported from this fit. The IsoFLOP estimates of Step 2 remain the pri- maryα P andα L values; the joint fit contributes the irreducible-loss term and the data exponentα D . Confidence intervals.Confidence intervals onα P andα L come from bootstrap with residual re- sampling on the per-slice parabola fits of Step 1: residuals are resampled with replacement and added to the fitted values, the parabolas are re-fit on each synthetic dataset, and the power laws are re-fit on the resulting slice minima. Confidence intervals onL ∞ andα D come from bootstrap with log-residual resampling on the joint fit of Step 3, with each iteration re-initialized from grid search. All confidence intervals are reported at the 90% level over 10,000 bootstrap iterations. 8.2.2Results Figure21shows the per-slice parabolic fits for each of the four families. Markers show measured validation losses, andmarkers locate the per-slice minimaP ∗ (C). The fits are well-behaved across all four families: the parabolas tighten with increasing compute, and the per-slice minima move smoothly toward largerPasCgrows. Figure 21.IsoFLOP analysis for each family. Each curve fits a parabola to model sizes within a fixed compute targetC; markers show measured validation losses;markers locate the per-slice minimaP ∗ (C). The resulting power-law fits across compute are shown in Figure22, with exponents and irreducible- loss asymptotes for all four families reported in Table6. The autoregressive baseline givesα P = 48 0.528, in line with prior estimates of0.49,0.464, and0.524reported byHoffmann et al.(2022), Shuai et al.(2024), andBi et al.(2024) respectively. The fit also yieldsL ∞ = 1.857, comparable to Chinchilla’s1.69. The diffusion baseline givesα P = 0.632, on the high end of prior masked-diffusion estimates of 0 . 514 , 0 . 566 , and 0 . 634 reported by Ni et al.(2025),von Rütte et al.(2025), andNie et al. (2024) respectively. The higher value may result from the causal block attention mask (Section5.2). Validation is reported in ELBO to follow prior work, although the choice of validation loss estimator can dramatically affect the fittedL ∗ values (AppendixH).von Rütte et al.(2025) additionally reports α P ranging from0.535to0.589across diffusion families and notes that the validation-loss ranking these exponents induce does not match the downstream performance ranking. The CDLM baseline yieldsL ∞ = 2.658, modestly above Quokka’s2.41but within the variation attributable to the ELBO estimator choice. The goal of this section is not to rank diffusion families against each other, but to measure the effect of adding the concept module on top of a fixed backbone trained on the same data; the comparisons that follow are therefore within-pair rather than cross-family. Figure 22.Compute-optimal scaling laws for each family. Panel (a):L ∗ (C)from the Step 2 power-law fits. Panel (b):P ∗ (C)from the Step 2 power-law fits. Markers show per-slice minima from Figure 21. Adding the concept module shifts the compute-optimal scaling exponents by a small, fixed per- backbone offset. For both pairs,α L becomes more negative when the concept module is added. This direction is consistent with the fixed-library bias: the smallest +Concept models are over- parameterized by the concept library and under-utilize their parameters, biasing their losses upward and steepening the apparent loss-vs-compute slope. The shifts onα P are small in both pairs. The irreducible-loss asymptoteL ∞ shifts downward in both pairs (1.857!1.193for AR,2.658!1.942 for CDLM); within-pairL ∞ comparisons are valid since both members of each pair use the same loss methodology. Interpretability-by-design imposes a small per-backbone offset on compute-optimal scaling, not a scaling tax. Predicting Steerling-8B from small models.The fitted exponents for the +Concept families are likely steeper than the asymptotic scaling rate, because the fixed concept library makes the smallest 49 Modelα P α D α L L ∞ Ours (Autoregressive) AR0.528 [−0.025,+0.023] 0.445 [−0.036,+0.114] 0.060 [−0.002,+0.002] 1.857 [−0.335,+0.071] AR+Concept0.621 [−0.041,+0.077] 0.524 [−0.112,+0.108] 0.070 [−0.002,+0.003] 1.193 [−0.703,+0.394] Ours (Causal Diffusion) CDLM0.632 [−0.091,+0.075] 0.481 [−0.111,+0.162] 0.053 [−0.004,+0.004] 2.658 [−0.708,+0.181] CDLM+Concept0.503 [−0.046,+0.051] 0.374 [−0.170,+0.174] 0.072 [−0.003,+0.003] 1.942 [−1.805,+0.529] Masked diffusion (prior) Ni et al.(2025)0.5140.486—2.41 von Rütte et al.(2025)0.566 [−0.022,+0.019] 0.434 [−0.019,+0.020] 0.0496 [−0.0004,+0.0003] — Autoregressive (prior) Hoffmann et al.(2022)0.4900.510—1.69 Shuai et al.(2024)0.4640.536— Bi et al.(2024)0.5240.476— Table 6.Compute-optimal scaling exponents and irreducible-loss asymptotes. Subscripts are 90% bootstrap confidence intervals. +Concept models artificially under-utilized and biases their losses upward. The joint Chinchilla form (Equation 27) accommodates this curvature by absorbing it into the irreducible lossL ∞ ; a naive log-linear extrapolation does not. Figure 23refits both forms on the small-scale checkpoints only (excluding Steerling), and extrapolates toC= 7.610 22 FLOPs, the compute used to train Steerling at 8B parameters and 1.35T tokens. The Steerling-8B model achievesL ∗ = 2.72, while the joint Chinchilla fit predictsL ∗ = 2.61, a gap of0.11nats. A naive log-linear extrapolation predicts L ∗ 2.25, missing by0.47nats. The joint fit’s prediction error is roughly4smaller than the log-linear extrapolation’s. Inherently interpretable architectures admit scaling-law extrapolation.Steerling-8B’s val- idation loss is predicted from small-scale fits to within0.11nats via the joint Chinchilla form. 8.3Interpretability scaling Section8.2established that adding the concept module preserves compute-optimal scaling: interpretability-by-design imposes at most a fixed per-backbone offset on validation loss. The question that remains is how interpretability itself behaves at scale. The metrics defined in Section 7are fit against training compute on the same checkpoints used in Section8.2, allowing a direct test ofRQ2: does interpretability scale favorably with compute? 8.3.1Methodology Metrics.Four metrics from Section7are fit against training compute: Concept Loss (#), Concept Independence Loss (#), Concept Contribution ("), and Known Concept Alignment ("). Scaling-law fits.Unlike validation loss, interpretability metrics are not analyzed under a compute- optimal frontier. The metrics are modeled in two ways: against training compute alone, and jointly against parameters and tokens. In both cases the fit uses all checkpoints rather than per-slice minima. The metrics are bounded by construction (Concept Loss and Concept Independence Loss below by 50 Figure 23.Extrapolating CDLM+Concept to Steerling. Joint Chinchilla fit (solid,L ∞ dashed) and log-linear fit (dashed) on sliceL ∗ minima. Steerling (filled star) lands within0.11nats of the joint fit. zero, Concept Contribution above by one, Known Concept Alignment above by five), and a log-linear fit ignores these bounds, predicting values outside them when extrapolated past the data range. The first form, following the irreducible-loss scaling laws ofHenighan et al.(2020) as applied to sparse autoencoder scaling byGao et al.(2025), encodes the bound directly as a free parameter: m(C) =eAC −β ,(29) with three free parametersA,β,efit jointly by nonlinear least squares, whereeis the irreducible loss the metric approaches at infinite compute. The sign is positive for#metrics (approachingefrom above) and negative for"metrics (approachingefrom below). Compute alone is an incomplete description of the sweep: within an IsoFLOP slice, compute is fixed while model size and token count vary inversely, so Equation (29) assigns every checkpoint in a slice the same prediction and treats their per-checkpoint variation as noise. The second form resolves each checkpoint at its own parameter countPand token countDby refitting the parametric decomposition of Equation ( 27) to each metric,m(P,D) =e A P P −α +A D D −β , witheplaying the role ofL ∞ and the sign following the convention of Equation (29). The five free parameters are fit by nonlinear least squares. Confidence intervals.Confidence intervals on the fitted parameters come from a residual boot- strap on both fits, matching the procedure of Section 8.2.1. 90% intervals are reported over10,000 iterations, with intervals on the irreducible lossetabulated in Table7, and the joint-fit parameters in Table8. 8.3.2Results Scaling with compute.Figure24shows the fitted curves for the four metrics on both +Concept families, with fitted parameters and bootstrap confidence intervals on the irreducible lossereported 51 in Table7. Every metric improves in the expected direction on both backbones: Concept Loss and Concept Independence Loss decrease with compute, while Concept Contribution and Known Con- cept Alignment increase. The fits explain a modest fraction of the per-checkpoint variance (R 2 between 0 . 49 and 0 . 75 ), reflecting the spread in evaluating interpretability properties on individual checkpoints. For several metrics the fitted asymptote is pinned to its natural bound, with bootstrap intervals concentrated near the boundary; this indicates that the small-scale data does not yet curve enough to identifyefrom the data range alone, and the asymptote estimates should be read as upper or lower bounds on the true plateau rather than precise predictions. Figure 24.Interpretability metrics scaling with compute, fit using the power-law-with-irreducible-loss form (Equation 29). Top row: AR+Concept. Bottom row: CDLM+Concept. Markers colored by IsoFLOP slice; shaded bands are 90% bootstrap confidence intervals on the fitted curve. Across both backbones, all four metrics improve with compute under the asymptotic form. Concept Loss falls toward zero, indicating that the concept module continues to identify concepts more ac- curately as training scales. Concept Independence Loss falls similarly, indicating that the known and unknown heads become more disentangled. Concept Contribution rises toward one, indicating that predictions route through the concept module rather than the residual at increasing fractions. Known Concept Alignment rises toward saturating at the judge ceiling, indicating that concept em- beddings increasingly point at semantically related tokens. Scaling in parameters and tokens.Figure25shows the same metrics under the joint fit of Sec- tion8.3.1, plotted as surfaces over parameters and tokens, with fitted parameters reported in Table8. Contour lines connect(P,D)configurations with equal metric value, and the dashed diagonals mark the IsoFLOP slice budgets. The joint form explains substantially more of the per-checkpoint variance than compute alone: across both families,R 2 rises from0.49to0.75under Equation ( 29) to0.62 52 Familyβe[90% CI]R 2 8B pred 8B actual∆ Concept Loss AR+Concept0.385 0.002 [−0.002,+0.000] 0.704 0.002— CDLM+Concept0.054 0.000 [−0.000,+0.002] 0.648 0.0020.002+0.000 Concept Independence Loss AR+Concept0.279 1.310 [−1.310,+0.000] 0.505 1.550— CDLM+Concept0.217 0.000 [−0.000,+3.687] 0.749 2.1571.5500.611 Concept Contribution AR+Concept0.212 0.937 [−0.047,+0.063] 0.630 0.915— CDLM+Concept0.134 1.000 [−0.149,+0.000] 0.496 0.9150.8760.039 Known Concept Alignment AR+Concept0.099 5.000 [−1.190,+0.000] 0.557 4.200— CDLM+Concept0.349 3.920 [−0.135,+1.080] 0.637 3.8703.7700.100 Table 7.Interpretability scaling fits. Subscripts oneare 90% bootstrap confidence intervals. The 8B columns compare the small-scale extrapolation against the actual Steerling values. to0.94under the joint fit. The variation within an IsoFLOP slice is therefore not evaluation noise but structure: at fixed compute, checkpoints differ systematically in how the budget is split between parameters and tokens, and the joint fit resolves this. The contour orientations separate the metrics into two regimes. Concept Loss and Concept Indepen- dence Loss improve primarily with parameters: their contours run near-vertical over the fitted range, so at fixed compute the larger models in a slice identify concepts more accurately and disentangle the known and unknown heads further. Known Concept Alignment improves primarily with tokens: its contours run near-horizontal, and for AR+Concept the fitted parameter term is negligible over the fitted range, making alignment mostly a function of data. Concept Contribution sits between the two regimes. A consequence is that no single split of a compute budget optimizes all four metrics at once: the token-heavy end of a slice favors alignment while the parameter-heavy end favors dis- entanglement, and the compute-optimal model of Section 8.2is a compromise between them. As with the compute-only fits, several joint asymptotes pin to their natural bounds and should be read as bounds on the plateau rather than point estimates. Interpretability scales favorably with compute across both autoregressive and diffusion, with model size driving the concept and independence losses, and training data driving align- ment. 8.3.3Predicting Steerling-8B from small models The fits of Equation (29) are now refit on the small-scale checkpoints only and extrapolated to C= 7.610 22 FLOPs, the compute used to train Steerling at 8B parameters and 1.35T tokens. Figure 26compares the extrapolations to the deployed model’s measured metrics, and the rightmost columns of Table7report the predicted and actual values per metric. Three of the four metrics land within tight bounds of the small-scale extrapolation: Concept Loss within0.0004on a BCE scale, Concept Contribution within0.04on a[0,1]scale, and Known Concept Alignment within0.10on a1- 5scale. Concept Independence Loss exceeds the small-scale extrapolation in the favorable direction, with the deployed model achieving1.55against a predicted2.16. The joint fit is refit and extrapolated 53 Figure 25.Fitted interpretability surfacesm(P,D)over parameters and tokens; gold indicates better values, and each metric column shares one color scale. White markers: checkpoints at their trained(P,D); dashed diagonals: IsoFLOP slice budgets. Top row: AR+Concept. Bottom row: CDLM+Concept. Near-vertical contours indicate parameter-driven metrics, near-horizontal token-driven ones. Familyαβe[90% CI]R 2 8B pred 8B actual∆ Concept Loss AR+Concept0.093 0.767 0.000 [−0.000,+0.001] 0.945— CDLM+Concept0.427 1.255 0.002 [−0.000,+0.000] 0.893 0.0020.002 +0.000 Concept Independence Loss AR+Concept1.085 0.529 1.347 [−1.347,+0.745] 0.624— CDLM+Concept0.904 0.659 0.000 [−0.000,+1.908] 0.937 0.6241.550 +0.921 Concept Contribution AR+Concept 0 . 541 0 . 739 0 . 903 [−0.028,+0.097] 0.784— CDLM+Concept0.441 2.077 0.864 [−0.012,+0.018] 0.883 0.8440.876 +0.032 Known Concept Alignment AR+Concept2.660 0.154 4.640 [−0.363,+0.360] 0.926— CDLM+Concept 0 . 022 0 . 346 5 . 000 [−1.125,+0.000] 0.933 3.9673.7700.194 Table 8.Joint interpretability scaling fits,m(P,D) =e A P P −α +A D D −β , withαthe parameter exponent andβthe token exponent. Subscripts oneare 90% bootstrap confidence intervals. The 8B columns compare the small-scale extrapolation against the actual Steerling values. 54 the same way, resolving Steerling at its actual parameter and token counts rather than its compute; its predictions are reported in the rightmost columns of Table8. Figure 26.Extrapolating CDLM+Concept interpretability metrics to Steerling. Asymptotic fits on small-scale checkpoints with 90% bootstrap CI bands. Fits use all checkpoints; markers show per-slice means1 SD, with individual checkpoints in the background. Filled stars: Steerling at 8B parameters and 1.35T tokens. Inherently interpretable architectures admit scaling-law extrapolation for interpretabil- ity properties.Three of four metrics are predicted within error bounds from small-scale fits; the fourth exceeds its prediction in the favorable direction. 9Steerling-8B: Pretraining With the recipe established in Section3and its scaling behavior characterized in Section8, this sec- tion describes the full-scale pretraining of Steerling-8B. We cover the concept-annotated 1.2T-token dataset (Section9.1), the architectural and training choices that distinguish Steerling-8B from a standard autoregressive or diffusion backbone (Section9.2), the pretraining run itself (Section9.3), and the lessons we learned (Section9.4). 9.1Data We pretrain Steerling-8B on 1.2T tokens drawn from a mixture of high-quality web, academic, math- ematical, and code corpora. The bulk of the mix is Nemotron-C-HQ (Su et al.,2025), a quality- filtered slice of Nemotron-C composed of both real webtext and synthetic question-and-answer rephrasings generated from the same documents. We choose Nemotron-C-HQ because it yields the strongest downstream performance among public 1T pretraining datasets. The remainder of the 55 corpus consists of peS2o (Soldaini and Lo,2023), arXiv (Weber et al.,2024), OpenWebMath (Paster et al.,2024a), Algebraic Stack (Azerbayev et al.,2023b), StarCoder (Li et al.,2023), and Wikipedia and Wikibooks following the OLMo 2 mixture (Team OLMo et al.,2024). We annotate the corpus with concepts at the chunk level using the Atlas pipeline (Section 4). The full per-source token counts are reported in Table9. SourceDocuments Chunks Tokens Nemotron-C-HQ (real)740M5.1B547B Nemotron-C-HQ (synthetic)971M4.8B498B peS2o38.8M 565.4M59B arXiv3.9M 142.2M 20.4B Wikipedia & Wikibooks6.1M 36.8M3.8B OpenWebMath2.9M 76.8M 12.1B Algebraic Stack2.8M 65.6M 12.1B StarCoder78.6M317M 91.4B Total1.84B 11.1B 1.24T Table 9.Steerling-8B pretraining corpus. Token counts are post-tokenization. 9.2Pretraining recipe Training a standard language model requires choosing a handful of hyperparameters such as se- quence length, batch size, learning rate, optimizer, and weight decay. The community has well- established defaults for each, and we adopt them without modification for Steerling-8B. However, our interpretable causal-diffusion architecture (Section 5) introduces design decisions without prece- dent in the autoregressive or masked-diffusion literature, for which no community defaults yet exist. These fall into two groups: those that govern the diffusion process (Section5.2) and those that govern the concept bottleneck (Section5.3). We describe each choice and the reasoning below; for every choice we ran a small ablation at the 1B scale, reported in full in AppendixJ. Diffusion process.Steerling-8B uses the block-causal attention mask of Section5.2, where tokens attend bidirectionally within a block of sizeband causally across blocks. The block size sets how many tokens are decoded in parallel at inference and the granularity of the key-value cache. We compareb2 f32,64gand find block size leaves the interpretability metrics unchanged while the larger value lowers validation loss, so we setb= 64. Diffusion training must also choose how the noise leveltis sampled at each step. We compare uniform sampling against a moving Gaussian curriculum that shifts from low to high masking over training ( Ni et al.,2025), and adopt the moving Gaussian for a slight edge on validation loss, consistent with prior work. 3 Concept bottleneck sizing.The concept module splits its representation intonknown concepts, fixed by the Atlas library, and m unknown concepts learned during training. Here m is a free hyper- parameter. We find that raising it from3nto5ngives no measurable gain on any metric, so we keep the conservativem= 3n. The unknown embedding matrixU2R m×d is the largest parameter the module adds; we factorize it asU=ABwith rankR= 256, which makes it roughly15smaller and removes a heavy per-token matrix multiplication at no capability cost and only a small drop in concept contribution. 3 This schedule proved too aggressive over the full pretraining run (see Section9.4.1). 56 Concept bottleneck training dynamics.Both concept heads are trained from scratch, so their early predictions are unreliable. We therefore route ground-truth concepts to the LM head early and anneal toward the model’s own predictions, holding the teacher forcing floor at0.5for both heads, since decaying further hurts known concept alignment for no capability gain. The model can optionally carry a residualε=h ˆ kˆuin the bottleneck, an uninterpreted channel that absorbs whatever the two heads fail to reconstruct. Dropping it forces every dimension through the known and unknown heads, raising concept contribution to1.0by construction, but in our comparison the capability gap is large enough that we keepε. 9.3Pretraining run Pretraining used the configuration of AppendixKon320A100 GPUs (40nodes, 8 GPUs per node) for approximately21days, totaling161K GPU-hours. The run completed its full 1.2T-token budget without divergence or manual intervention to the loss curve. We monitored various metrics during the pretraining run. Validation loss was logged continuously, computed as a Monte Carlo estimate of the MDLM ELBO (Section 8.2.1). At every50B-token interval, we additionally saved a checkpoint and ran a broader evaluation suite of five downstream bench- marks from the language modelling harness: ARC-Challenge (Clark et al.,2018), HellaSwag (Zellers et al.,2019), PIQA (Bisk et al.,2020), MMLU (Hendrycks et al.,2020), and WinoGrande (Sakaguchi et al.,2021); together with the four interpretability metrics introduced in Section7: concept loss, concept contribution, concept independence loss, and known concept alignment. Figure27shows the capability metrics across the run. Validation loss descends rapidly through the first half and then plateaus, oscillating around a stable value. The harness benchmarks behave differently from one another. HellaSwag and PIQA rise smoothly and hold near their peaks to the end, and ARC-Challenge peaks late with only a small dip at the very end. MMLU and WinoGrande, by contrast, peak around the midpoint and then decline through the final third, with MMLU losing roughly a quarter of its peak value. Figure 27.Validation loss and LM-Harness metrics for Steerling-8B across the pretraining run. Figure28shows the interpretability metrics across the run. Concept loss drops early and then climbs steadily, though the climb is small in absolute terms. Concept independence loss stays low through most of training, then rises sharply in the final third, peaking around2.7. Concept contribution dips slightly early before climbing from0.62to0.85. Known concept alignment moves modestly on its [1,5]scale, rising to a peak near4in the first third before settling back. 57 Figure 28.Interpretability metrics for Steerling-8B across the pretraining run. 9.4Pretraining lessons During the run, most metrics looked healthy. Validation loss kept descending or held steady, three of the four interpretability metrics drifted only modestly, and HellaSwag, PIQA, and ARC-Challenge plateaued near their peaks. However, four signals were less encouraging: •MMLU and WinoGrande began declining around the midpoint of training. •Concept independence loss spiked in the final third. •A late checkpoint showed weak math and code performance. •The same checkpoint did not steer reliably through the concept module. At the time we expected some of these to recover as the schedules stabilized, so we continued the run to its planned 1.2T token budget. Once the run completed, we examined the issues closely and traced them to four causes: an over-aggressive masking curriculum (Section 9.4.1), late-training entanglement of the concept heads (Section9.4.2), insufficient math and code in the corpus (Sec- tion 9.4.3), and the absence of steering operations during training (Section9.4.4). 9.4.1Over-aggressive masking MMLU and WinoGrande peak around the midpoint of training and decline thereafter. Overlaying the masking curriculum on the capability metrics (Figure49) makes the cause visible: both inflection points coincide with the curriculum reaching0.5on its climb to its endpoint of0.8. The moving Gaussian curriculum was validated at small scale (Appendix J.2), where it held a slight edge on validation loss. At pretraining scale, however, the curriculum spends a substantial fraction of training above50% masking, where most of the input is replaced with[MASK]tokens and the model has little context to leverage. 9.4.2Late-training entanglement of the concept heads Concept independence loss stays low through most of the run and rises sharply in the final third. Overlaying the three schedules on the interpretability metrics shows why (Figure50). Two of them move into adverse territory over the same window. The masking curriculum reaches its hard regime, so most input tokens are masked. At the same time the teacher forcing floors have been reached, withα known at its floor andα unknown decayed substantially, so the LM head leans increasingly on 58 predicted concepts rather than ground-truth ones. With less supervision and more reliance on pre- dicted contributions, the two heads come to rest on overlapping information, and disentangling their representations becomes harder. At ablation scale this effect was not visible. 9.4.3Limited math and code in the corpus Our corpus is dominated by Nemotron-C-HQ, which carries little math or code. Together with the smaller dedicated sources in the mix (Table9), the run saw slightly over100B math and code tokens out of1.2T total.von Rütte et al.(2025) report a similar shortfall on the same corpus. We caught this only at the end of pretraining, when GSM8K and HumanEval both came back low. With more math and code in the mixture, and these benchmarks included in the in-run evaluation sweep, the issue would have surfaced earlier and been addressable on the data side. 9.4.4The model does not respond to steering A model built around an explicit concept module should be steerable through it, since injecting a concept’s own direction (Section6.2) should bias generation toward that concept. We tested this on a late pretraining checkpoint following the evaluation protocol ofWu et al.(2025). We elicit 128- token generations from a fixed prompt, sampling a continuation under the method being evaluated, and score each continuation with an LLM-judge on two axes. Aconceptscore (0–2) measures how strongly the target concept is present in the generated text, and aqualityscore (0–2) measures the fluency and coherence of the output. Details on the judge and the full prompts are given in AppendixM. Finally, we summarize the two scores by their harmonic mean: Harmonic= 2 1 concept + 1 quality ,(30) which weights both axes equally and strongly penalizes a method that sacrifices one for the other. We report the mean over 72 randomly sampled concepts. We study steering on three conditions:unsteered,prompting(via prepending the concept label and description to the prompt), andsteeredgeneration (layer injection in Section6.2). In Table10, unsteered generation produces fluent text that does not surface the target concept, which is expected. Steering achieves the highest concept score of 1.072, showing that intervening through the model’s own concept direction induces the target concept more strongly than though a text prompt. Yet, the cost of steering is quality. As a result, prompting yields the best harmonic mean of 1.156. MethodConcept"Quality"Harmonic" Unsteered0.0331.1080.065 Prompting0.9081.5881.156 Steered1.0720.9721.020 Table 10.Steering results on random concepts of the pretrained Steerling-8B.Steereddenotes layer injection. From this experiment, we identify two limitations for steering. First, the quality drop under steering is substantial, suggesting Steerling-8B does not gracefully integrate the injected direction into its forward pass. Second, layer injection does not generalize on less-frequent concepts, as roughly one third never activate at any injection strengthγ, with the bottleneck activationk c staying near zero throughout generation. Both trace to the same cause, that Steerling-8B never encounters concept injection during pretraining, so addingγK c at inference is an out-of-distribution perturbation the model has no mechanism to respond to. 59 10Steerling-8B: Mid-training Mid-training is a short 150B-token run initialized from the final pretraining checkpoint. It has two aims. First, address the four weaknesses identified in pretraining (Section9.4): aggressive masking, thin math and code coverage, unstable independence loss, and no exposure to steering during train- ing. Second, tighten the model for downstream use, by reducing its reliance on teacher forcing and increasing its reliance on the concept heads. Here we describe the data mixture (Section10.1), the recipe changes from pretraining including a dedicated steering phase (Section10.2, Section10.2.4), the resulting mid-trained model (Section10.3), and its benchmark performance against open base models of comparable size (Section10.4). 10.1Data Our goal in mid-training is to improve Steerling-8B on math and code, where the pretrained model lagged furthest behind, and to recover the reasoning and knowledge capabilities degraded by heavy masking late in pretraining. We follow the OLMo 2 mid-training recipe (Team OLMo et al.,2024): start from a high-quality data mixture that lifts performance across the benchmark suite, then patch the specific capabilities the pretrained model is weakest on. For the natural-language portion of the mixture we again use Nemotron-C-HQ, but restrict to real tokens; we benchmarked real, synthetic, and mixed against each other (AppendixN.1), and real tokens won. SourceMath-heavy Balanced Code-augmented Code-only Nemotron (real)5.0B4.7B5.0B– Dolmino Math5.0B2.1B2.0B– StarCoder–1.0B10.0B FLAN–1.7B1.0B– peS2o–0.6B0.4B– Wikipedia & Wikibooks–0.7B0.5B– Stack Exchange–0.2B0.2B– Total10.0B10.0B10.0B10.0B Table 11.Data compositions compared in the mid-training ablation. Each arm is a 10B-token run from the final pretraining checkpoint of 1.2T tokens; entries are token counts in billions. To find the best midtraining mixture, we run a data ablation: starting from the final pretraining checkpoint of 1.2T tokens, we midtrain on 10B tokens for each of four candidate compositions and compare their downstream performance. The compositions, listed in Table 11, are a math-heavy mixture, a balanced mixture following OLMo 2’s Dolmino Mix, a code-augmented mixture that adds StarCoder to the balanced mixture, and a code-only mixture. The first three hold the Nemotron-real share roughly fixed and vary the math, code, and high-quality reference sources around it; the code-only mixture is an extreme that drops Nemotron entirely, isolating the effect of training on code alone. The results are shown in Table12. Every mixture except code-onlyimproves substantiallyover the pretrained model, but each isolates a different lesson: •The math-heavy mixture delivers the largest math gain, lifting GSM8K from 0.140 to 0.441, and posts the best overall average, but this is driven by large gains on only two benchmarks while it lags on the rest. 60 CompositionMMLU GSM8K ARC-C HSwag HEval MBPP WinoG Avg. Pretrained model 0.298 0.140 0.484 0.673 0.049 0.004 0.596 0.321 Math-heavy0.3760.4410.492 0.681 0.0370.0120.6160.379 Balanced0.4160.328 0.4970.6930.037 0.006 0.616 0.371 Code-augmented0.4160.3280.498 0.6930.0550.012 0.6280.376 Code-only0.303 0.086 0.434 0.6300.061 0.0120.583 0.301 Table 12.Downstream performance of the four mid-training compositions, each a 10B-token run from the final pretraining checkpoint, against the base model. Best in each column in bold. HSwag: HellaSwag; HEval: HumanEval; WinoG: WinoGrande. •The balanced mixture drives the largest knowledge gain, recovering MMLU from 0.298 to 0.416, but with no dedicated code source, HumanEval again fails to improve. •The code-only mixture pushes HumanEval the highest, as expected, but lags well behind every other composition elsewhere. •The code-augmented mixture strikes a balance across reasoning, math, and code, and is the only composition to improve on every benchmark over the pretrained model, so we adopt it as the mid-training mixture. The final midtraining mixture applies the code-augmented recipe over 150B tokens, with proportions given in Table 13. SourceNumber of Tokens Ratio (%) Nemotron (real)72.79B48.5 StarCoder30.75B20.5 peS2o21.98B14.7 Dolmino Math(2)16.05B10.7 FLAN6.38B4.3 Wikipedia & Wikibooks1.50B1.0 Stack Exchange0.56B0.4 Total150B100.0 Table 13.The final Steerling-8B midtraining mixture of 150B tokens.Dolmino Mathis upsampled roughly twofold. Proportions follow the code-augmented composition of Table11. 10.2Mid-training recipe changes 10.2.1Schedules Masking schedule.The first and most consequential change is to revert the masking schedule. Pre- training used a moving Gaussian curriculum whose center rose from 0.2 to 0.8, and Section9.4 traced Steerling-8B’s declining knowledge and reasoning scores to the high-masking regime this curriculum entered late in the run. Midtraining instead samples the masking rate uniformly, fol- lowing MDLM training ( Nie et al.,2025), so the model sees a balanced spread of masking levels rather than a curriculum that drifts toward heavy masking. To confirm that uniform sampling is the 61 right choice at this stage, we compare it against an 80% Gaussian schedule in a 10B-token ablation from the final pretraining checkpoint (Table14). The two schedules are close on most benchmarks, and the 80% schedule is even slightly stronger on the math tasks. On MMLU, however, the heavy schedule falls to 0.280, below the pretrained model’s 0.298, while uniform sampling lifts it to 0.416; the heavy-masking regime that hurt knowledge during pretraining hurts it again here, confirming our initial suspicion. We adopt a uniform schedule during midtraining. Masking MMLU GSM8K ARC-C HSwag HEval MBPP WinoG Avg. Pretrained model 0.298 0.140 0.484 0.673 0.049 0.004 0.596 0.321 50% uniform0.4160.328 0.4980.693 0.055 0.012 0.628 0.376 80% Gaussian0.2800.355 0.5000.6860.0550.0080.6280.359 Table 14.Masking schedule ablation, each a 10B-token run from the final pretraining checkpoint. Best in each column in bold. HSwag: HellaSwag; HEval: HumanEval; WinoG: WinoGrande. Teacher forcing.During pretraining the model’s predicted concepts were partly replaced with ground-truth ones: with probabilityα known the known head’s predicted activations were replaced by their labeled values, and the unknown head was also mixed with its residual at rateα unknown . Mid-training annealsα known from its end-of-pretraining value of0.5to0, and holdsα unknown at0 throughout. By the end of mid-training, every concept entering the bottleneck is model-predicted, matching the inference regime where labeled concepts are unavailable. Learning rate.Following standard mid-training practice (Team OLMo et al.,2024;Grattafiori et al., 2024a;Liu et al.,2024), the learning rate is decayed linearly to zero. 10.2.2Losses The second pretraining weakness was the independence loss (Equation14), which destabilized late in the run as the model came to rely on its predicted concepts (Section 9.4). Midtraining makes the model rely on the predicted concepts entirely, which exposes a limitation of the single-term loss. The pretraining loss penalized the dependence between the known representation ˆ kand the unknown representation routed to the bottleneck, but with the unknown head now always supplying its own prediction, detached before the bottleneck (gradient detachment), the term applies no pressure on the transformer residual. We therefore replace it with two terms,L indep (ˆu, ˆ k)+L indep (ε, ˆ k): the first keeps the predicted unknown independent of the known concepts, the second keeps the residual εindependent of them. Each term carries half the original weight, leaving the total magnitude unchanged. 10.2.3Architecture Two architectural changes tighten interpretability. First, the residual dropout ratep ε is raised from 0.1to0.3, applying more pressure onεto vanish and forcing the concept heads to carry a larger share of the prediction. Second, the concept heads are sparsified: the known head composes the bottleneck from its top 32 concepts, and the unknown head from its top 128 . Sparsity makes attribution more interpretable, since each logit decomposes into at most160concepts rather than the full bottleneck of thousands. 62 10.2.4Steering training Pretraining labels mark that a concept appears in a chunk but not which tokens actually express it. Steering supervision needs that detail because injection happens at the token level: at each generation step, we add a concept embedding into the transformer’s hidden state at each token position, so the loss must know which positions should actually respond. We therefore mid-train on a token-level dataset (Figure29) of roughly400M tokens, where each token is tagged with its attributed concepts. Figure 29.Token-level concept annotations used for steering training. Each token is tagged only with the concepts attributed to it, rather than with all chunk-level concepts in pretraining. Steering losses.During a steering phase the target concept’s directionK c , scaled byγ, is injected at the masked positions attributed toc, exactly as steering is applied at inference. LetIdenote the set of these injected positions. Two losses are added on top of the masked diffusion lossL MDM , one for each objective: •Respond objective.At every injected position, the concept bottleneck should activate the injected concept. Its activationk c,t is driven toward1by minimizing its negative log-likelihood: L respond = 1 jIj X t∈I logk c,t .(31) •Express objective.A high bottleneck activation does not by itself produce concept-expressing output: the concept direction may still place weight on unrelated tokens such as fillers. We therefore push the concept’s output distribution at attributed positions onto the tokens that expressc. These are the lifted setT c : vocabulary tokens with the highestlift, defined as the ratio of token frequency in concept-tagged chunks to token frequency in the corpus. We maximize the total probability assigned toT c , i.e., minimize its negative log: L express = 1 jIj X t∈I log P y∈T c e ℓ t,y P y∈V e ℓ t,y ,(32) 63 whereℓ t,y is the logit of tokenyat positiont. The masked diffusion lossL MDM is kept active throughout so that generation stays coherent, while the pretraining interpretability losses (the concept, reconstruction, and independence terms) are disabled during steering phases. The steering-phase objective is L=L MDM +λ respond L respond +λ express L express ,(33) whereλ respond andλ express are both set to1. Figure 30.Interleaved mid-training schedule for Steerling-8B. Four steering phases (90 steps each, shaded) are inserted within the standard run; the first normal block is a 100-step warmup. Normal phases use chunk- level concept data, steering phases use token-level concept data. Steering training schedule.Steering is interleaved into the main midtraining run rather than added as a separate stage (Figure30). We front-load it: four short steering phases come imme- diately after a 100-step warmup, and the remainder of mid-training is normal capability training. This placement keeps the steering objectives from interfering with the capabilities the rest of the run builds. Each phase is one pass over the token-level dataset (400M tokens); we use four phases because bothL respond andL express converge by the fourth epoch. Normal phases use chunk-level data and the standard objective; steering phases use token-level data and the steering losses above. Steering training works and does not cost capability.We evaluate steering as an ablation on the mid-training recipe: starting from the math-heavy composition tested in Section10.1, the 10B-token run is repeated with the steering phases interleaved, so that the only difference from the recipe in Table 12is steering itself. Adding steering leaves LM Harness performance essentially unchanged across all five benchmarks (Table15), so teaching the model to respond to injection does not trade off against capability. It also achieves its intended effect: the steering benchmark scores all improve, and the model activates the target concept in a single injection step atγ= 1. LM Harness"Steering benchmark" MMLU GSM8K ARC-C HSwag WinoG Concept Quality Harmonic Math-heavy 0.3700.4310.4990.682 0.6181.208 0.9891.088 + steering0.3840.4150.5050.681 0.6111.244 1.139 1.189 Table 15.Steering ablation: Adding the steering phases improves every metric on the steering bench- mark (Wu et al.,2025), while LM Harness performance stays mostly unchanged. 64 10.3The mid-trained model The final mid-training run is 150B tokens on the code-augmented mixture (Section10.1), starting from the final pretraining checkpoint, with the recipe changes of Section10.2: uniform masking, teacher forcing annealed to zero, the two-term independence loss, residual dropout raised to0.3, sparsified concept heads, and a learning rate decayed to zero. Steering is interleaved into this run rather than added as a separate stage, as four short phases over a token-level dataset of roughly 400M tokens (Section10.2.4). The complete hyperparameter list is given in AppendixN.2. Mid-training rescues the capabilities lost in pretraining.Mid-training addresses the capability weaknesses of pretraining at once (Table16). The knowledge lost under heavy masking comes back, with MMLU up17 percentage points. The two capabilities the pretraining corpus starved improve the most: math rises 30 percentage points and code around 7 percentage points on average (HEval & MBPP). The benchmarks that were already healthy stay healthy, so nothing is traded away.Every benchmark improves, lifting the overall average by10 percentage points. MMLU GSM8K ARC-C HSwag HEval MBPP WinoG Avg. Pretrained29.814.048.467.34.90.459.6 32.1 Mid-trained46.444.452.370.38.5 11.0 64.2 42.4 Table 16.Steerling-8B before and after mid-training across the LM Harness suite (accuracy, %). HSwag: HellaSwag; HEval: HumanEval; WinoG: WinoGrande. Pretrained Mid-trained Concept Loss#0.0020.002 Concept Independence Loss#1.9071.546 Concept Contribution"0.8510.876 Known Concept Alignment"3.7303.770 Table 17.Interpretability metrics for Steerling-8B before and after mid-training. Mid-training improves interpretability.The effects of midtraining on interpretability metrics are shown in Table17. The late-pretraining rise in concept independence loss, the third weakness of Section 9.4, is mitigated after mid-training: the two-term penalty of Section10.2brings concept independence loss down by19%, so the known and unknown representations are more cleanly separated than at the end of pretraining. Concept contribution rises, meaning the concept module accounts for a larger share of each prediction, and known concept alignment improves slightly. Mid-training improves steering.We evaluate the midtrained Steerling-8B using the steering benchmark used for the pretraining checkpoint (Wu et al.,2025). Midtraining improves every met- ric in Table 18: mean concept rises from 1.072 to 1.247, mean quality from 0.972 to 1.064, and their harmonic mean from 1.020 to 1.148. We additionally reportmean sample harmonic(the harmonic mean computed per sample, then averaged across samples). This rises from 0.843 to 0.963. 10.4Evaluation and results Table19compares Steerling-8B against open base models of comparable size on seven benchmarks, and Figure 31places those scores on a compute axis. Every peer was trained on2 to 16more 65 CheckpointConcept"Quality"Harmonic"Sample harmonic" Pretrained (1.2T)1.0720.9721.0200.843 Mid-trained (1.35T)1.2471.0641.1480.963 Table 18.Steering benchmark scores for the pretrained and mid-trained Steerling-8B checkpoints. Mid- training improves every steering metric. compute than Steerling-8B, yet it lands within10%of them on average and ahead of the models trained at comparable budgets. Although Steerling-8B is an inherently-interpretable architecture, we find this does not come at the cost of capability. A model can be both interpretable and competitively performant. ModelHSwag WinoG PIQA MMLU ARC-CGSM8K MathAvg. Steerling-8B70.364.275.9 46.452.344.48.051.6 LLaMA2 7B76.072.579.1 45.946.313.14.348.2 DeepSeek 7B 75.470.579.2 48.248.1 17.46.049.3 Gemma 1 7B 81.272.381.2 64.353.246.424.360.4 LLaDA 8B70.574.873.6 65.945.9 70.331.461.8 LLaMA3 8B79.177.380.6 65.453.148.716.060.0 OLMo 2 7B83.877.2 80.1 ∗ 63.779.867.5 19.1 ∗ 67.3 Table 19.Steerling-8B base model against open base models of comparable size. HSwag: HellaSwag; WinoG: WinoGrande. Values marked ∗ are taken from the OLMo 3 report (OLMo et al.,2025); all other peer numbers are from the respective model reports. 11Related work 11.1Underspecification and the Rashomon effect The phenomenon.For overparameterized neural networks, many distinct models achieve equiva- lent performance on the training objective while differing in their internal mechanisms ( D’Amour et al.,2022;Breiman,2001;Black et al.,2022;Rudin et al.,2024).Fisher et al.(2018) formalized this as theRashomon set, the collection of models within a small tolerance of the optimum, and showed that feature importance varies substantially across its members. Semenova et al.(2022) demonstrated that these sets are large in practice, with structurally diverse models routinely coex- isting at equivalent performance. The problem for post-hoc interpretability.Even a perfectly faithful explanation is an account of one arbitrary member of the Rashomon set; a different training run could yield an equally valid model with an entirely different internal decomposition. Worse, Brunet et al.(2022) show that models with nearly identical accuracy can produce contradicting explanations, e.g. opposite-sign attributions for the same feature, with no diagnostic to predict when this occurs.Pawelczyk et al. 66 Figure 31.Average performance across the benchmark suite against approximate training FLOPs, with verti- cal lines at multiples of Steerling-8B’s compute. (2020) show that counterfactual recommendations derived from one model’s decision boundary can be invalid for an equally valid alternative. How inherent interpretability addresses this?Our approach constrains the Rashomon set during training. The concept library is fixed before training begins, the concept module forces every trained model to decompose its output through the same concept variables, and the masking objective en- sures a shared absence baseline. Different runs still yield different parameters, but the attribution interface: which concepts exist, how attribution is computed, and what “absent” means, is fixed. 11.2Large language models Most large language models are autoregressive (AR): they generate a sequence left to right, one token at a time, each conditioned on all preceding tokens (Radford et al.,2018). Trained at scale with a next-token prediction objective, these models attain strong performance across a wide range of tasks, with training recipes and open model weights now widely documented by the community (Grattafiori et al.,2024b;OLMo et al.,2025;Liu et al.,2024). Most deployed production models belong to this family ( Team et al.,2023;Anthropic,2024;Achiam et al.,2023), making the AR pipeline a mature and well-understood baseline. Diffusion language models generate by iteratively denoising a corrupted sequence, producing many tokens in parallel and in arbitrary order. Discrete diffusion comes in several families that trade off quality and efficiency: masked diffusion attains the strongest perplexity ( Nie et al.,2024,2025), uniform-state diffusion yields higher-quality samples in the few-step regime and is well suited to guidance (Austin et al.,2021b;Schiff et al.,2025;Sahoo et al.,2025a), and interpolating diffu- sion supports KV caching for faster decoding (Sahoo et al.,2025b). Block Diffusion bridges the two paradigms by factorizing autoregressively over blocks while denoising within each block, re- covering KV caching at the cost of full block-wise attention ( Arriola et al.,2025). Recent work has 67 further shown that masked diffusion models scale to billions of parameters (Nie et al.,2025) and now serve production workloads (Song et al.,2025;Labs et al.,2025). Masked diffusion models are also a natural substrate for interpretability: the[MASK]token gives a learned baseline for attri- bution, parallel generation supports concept-level control, and any-order generation enables clean interventions. Steerling builds on this with Causal Diffusion, a block-causal formulation that attains the benefits of Block Diffusion at roughly half the cost, and adds the concept module on top. 11.3Scaling laws Scaling laws characterize how model performance improves with compute, parameters, and data, and have become the standard tool for planning large training runs. For autoregressive language models, this line of work established power-law relationships between loss and scale and the compute-optimal allocation of parameters and tokens (Kaplan et al.,2020;Hoffmann et al.,2022; Bi et al.,2024). More recently, the same methodology has been extended to diffusion language models, both for masked diffusion (Nie et al.,2024;von Rütte et al.,2025) and through compute- optimal studies tailored to the diffusion objective (Ni et al.,2025;Sahoo et al.,2026). We follow this methodology and fit compute-optimal scaling laws to inherently interpretable models, measuring the effect of the concept module on both the autoregressive and diffusion families. A separate line of work asks whether interpretability itself scales. Sparse autoencoders trained on frozen model activations recover more and finer features as the autoencoder grows, with reconstruc- tion quality and feature-quality metrics following clean power laws (Gao et al.,2025;Templeton, 2024), and recurring neuron populations become more selective and monosemantic as the base model grows ( Dravid et al.,2026). These works ask whether a post-hoc probe gets better as the probe is scaled, on a fixed, uninterpretable model. They do not ask how a model’s own interpretability scales with its training compute. Adapting the irreducible-loss scaling form of Gao et al.(2025), we test whether interpretability-by-design preserves compute-optimal scaling and whether the model’s interpretability metrics improve predictably with compute across several distinct measures. 11.4Interpretable-by-design architectures A range of architectures build interpretability into the model rather than recovering it post hoc. Among these, Concept Bottleneck Models (CBMs) (Koh et al.,2020) route predictions through a layer of human-understandable concepts. Concept Embedding Models relax the bottleneck to recover accuracy ( Espinosa Zarlenga et al.,2022). Originally developed for classification, the approach was later extended to generative models, where intervening on the concept layer enables interpretable and controllable generation (Ismail et al.,2024). It has since reached protein language modeling ( Ismail et al.,2025) and single-cell counterfactuals generation (Andersson et al.,2026). Backpack language models pursue a similar goal through a different mechanism, attaching interpretable sense vectors to each token in place of named concepts (Hewitt et al.,2023). A parallel line pursues interpretability through prototypes rather than concepts. PRISM (Ley et al.,2026) forms each prediction from a sparse, non-negative mixture of learned prototypes, where each prototype anchors to a coherent neighborhood of training examples. This yields structural training-data attribution at scale without post-hoc estimation. All of these operate at a substantially smaller scale, and with far fewer concepts, than Steerling. We use an additive CBM to scale the concept vocabulary to over a hundred thousand concepts, with the goal of reaching millions. 68 11.5Attribution methods Input attribution.We use Integrated Gradients (Sundararajan et al.,2017a), but the core of our input attribution is not the algorithm: it is the choice of baseline. Prior Integrated Gradient imple- mentations use a zero or padding embedding as the baseline (Kokhlikyan et al.,2020;Nguyen et al., 2021); the model never learned to interpret either as absence of information. We use[MASK]in- stead. The diffusion training objective makes[MASK]a learned representation of “no information at this position,” so the integration path stays inside the model’s training distribution. Autoregressive models offer no comparable learned baseline through their training objective. Concept attribution.Methods that attribute predictions to human-interpretable concepts divide into post-hoc and architectural approaches. Post-hoc methods include linear probes (Alain and Ben- gio,2016) and TCAV (Kim et al.,2018), which detect concepts in activations or test their influence on predictions, and sparse autoencoders ( Huben et al.,2024;Gao et al.,2025), which factorize acti- vations into interpretable features. All are approximations of an internal representation that was not designed to be decomposed. Architectural methods embed concept supervision in the model itself, and we cover this family in Section11.4. Steerling sits in the architectural family and produces an exact additive decomposition of every output logit, so concept attribution reads off the forward pass rather than estimating it. Training data attribution.Influence functions estimate the effect of perturbing a training point on a model’s predictions (Koh and Liang,2017). Scaling them to large language models requires approximations to the Hessian inverse ( Grosse et al.,2023), and the resulting estimates can be brittle in non-convex regimes (Bae et al.,2022). TracIn takes a different approach, tracing influence along the optimization trajectory (Pruthi et al.,2020). A complementary line avoids estimation altogether: OLMoTrace (Liu et al.,2025) retrieves verbatim substring matches between an output and the training corpus, surfacing documents the model could have seen. A separate line makes TDA a first-class output of the architecture rather than a post-hoc estimate, achieving retrieval roughly 500× faster than influence-function baselines at matched memory ( Ley et al.,2026). We adopt the same retrieval framing but match in latent representation space rather than by surface form: given an output, we return the training chunks most similar to it in that space. We do not claim our method estimates causal influence at trillion-token scale. 11.6Model steering Methods for steering model behavior at inference time fall into two families. Both add a learned direction to hidden activations, with the sign of the injection determining whether the target behav- ior is amplified or suppressed. They differ in how the direction is obtained. The first family derives the direction from positive and negative examples of the target behavior, generating a new direction per task: representation engineering (Zou et al.,2023), steering vectors (Turner et al.,2023), and contrastive activation addition (Rimsky et al.,2024). The second family selects directions from a pre-built feature dictionary that is independent of any specific behavior we desire to steer. Parsimo- nious concept engineering ( Luo et al.,2024) and sparse autoencoder features (Huben et al.,2024) decompose activations into a fixed set of interpretable features, any of which can serve as a steer- ing direction. Both families obtain the directionpost-hoc, from activations the model has already produced. Our model instead has its steering directions built into the architecture: each concept con- tributes an exact additive term to every output logit, so positive or negative steering is a closed-form edit on that term, requiring neither contrastive derivation nor post-hoc dictionary construction. 69 12Conclusion Interpretability is often treated as a cost paid against capability. In this work, we tested whether that cost, indeed, grows with scale. However, we demonstrate concrete settings where this is not the case. Concretely, across three orders of magnitude of compute, on both autoregressive and causal-diffusion language models, training a model with interpretable structure shifts compute-optimal scaling by a small, fixed offset rather than a penalty that compounds with scale. More surprisingly, we find that all the interpretability constraints we train for improve with compute: larger models have more independent, and more semantically aligned use of human-understandable concepts. Under the metrics we measure, the model does not become harder to understand as it becomes more capable; it becomes easier. The central change is where interpretability enters the modeling process. Standard pipelines train opaque predictors and then ask whether post-hoc methods can recover faithful explanations. In- stead, we ask which conditions training must enforce for explanations to be faithful, and build those conditions into the data, architecture, objective, and losses. The resulting recipe is not a collection of interpretability add-ons: each component exists because removing it breaks a specific condition required for faithful attribution. Because the same concept variables support both attribution and intervention, a user can decompose an output, inspect the relevant concepts and similar training data, edit the responsible concept direction, and verify the immediate logit-level effect, all without retraining. The choices made in this work are first instantiations, not settled directions. We fixed the concept library before training, to topics mostly describing the content of training document segments. We chose one bottleneck design, the additive module, among many possible alternatives. We supervised concepts at the chunk level, and our training stops at supervised finetuning. Each of these choices can be improved upon, and we expect future work to do so substantially. A few prospects seem tantalizing. The first is post-training: reward objectives can be expressed over concepts, so that training targets not only what a model says, but which concepts it uses to decide. In agentic settings, the same structure lets an agent’s decisions be decomposed into inspectable concepts that are monitored and corrected mid-trajectory. Concept libraries can grow to match, becoming hierarchical and adaptive rather than fixed. Furthermore, individuals should be able to interactively define, audit, and extend the vocabularies through which models explain themselves. Second, the attribution interfaces can become more legible in turn: today they return structured artifacts, ranked concept contributions, token-level scores, and retrieved training chunks; future work can translate these into natural-language explanations that remain grounded in the underlying decomposition. Third, our scaling laws suggest that a frontier-grade interpretable model is feasible, including agentic systems whose every action can be decomposed, audited, and steered. We therefore view Steerling- 8B less as a finished system, but as evidence that this research program is viable. Taken together, the results suggest a different scaling paradigm for capable AI systems. Interpretabil- ity, steerability, and other reliability requirements need not be retrofitted after training, nor treated as a tax against capability. They can be specified as a contract, optimized as part of the training pro- cess, and measured as models scale. Our results point to another possibility: if interpretability can be specified, trained, and scaled like any other capability, then the opacity of today’s most capable systems is not a law of nature. 70 Authorship and Credit Attribution The results outlined in this manuscript was a collaborative effort that spanned data annotation, architecture, training infrastructure, post-training, and product. No component was built isolation; each underwent iterative feedback from all members of the team. The statements of work below describe key primary responsibilities rather than any exclusive ownership. Writing.All members of the team contributed to writing the manuscript. Data.Nathaniel Monson, Saqib Azim, and Julius Adebayo built the Atlas data annotation system. Nathaniel Monson led the work on the LLM annotation infrastructure, designed the concept library, and the human validation study. Saqib Azim led and implemented the pipeline for Stage 2 of the Atlas system, covering clustering, de-duplication, and labeling clusters of tags as concepts; he also built the baseline k-nearest-neighbor embedding annotators and led the implementation of the dis- tributed index of the training data. Julius Adebayo designed and trained the multi-head concept annotator, and then used it to annotate the entire pre-training corpus. Model Architecture, Training Recipe, Scaling Laws, and Training.Aya Abdelsalam Ismail led the design of the model architecture, training recipe, and concept module. She devised, along with Andreas Madsen, the causal diffusion attention formulation. Giang Nguyen devised the mid-training steering formulation and led its design, empirical validation, and production implementation. He also contributed to training recipe design, and implementation of both input feature and concept attribution. Aya Abdelsalam Ismail designed and executed the scaling law formulation. Infrastructure.Andreas Madsen developed the new dataloader and distributed multi-node training framework used throughout model training. Using the infrastructure, Julius Adebayo then wrote the pre-training package for model training. Andreas Madsen and Julius Adebayo managed the internal SLURM cluster used for research and production. Open Source.Aya Abdelsalam Ismail and Giang Nguyen led the release of Steerling base and in- struct models along with the necessary artifacts to enable attribution and steering. Model Post-Training.Andreas Madsen led post-training for Steerling. He reproduced the LLaDA- 8B instruction supervised fine-tuning setup, adapted the recipe to block-diffusion and interpretabil- ity training, trained the supervised fine-tuned model, and identified and fixed correctness issues across the training and evaluation stack. Model Post-Processing.Nathaniel Monson led the work on concept-naming for the trained model with feedback from Giang Nguyen and Aya Abdelsalam Ismail. Clarity Platform.Zhichen Guo and Isaac Plant designed the Clarity product that serves Steerling- 8B to users. Andreas Madsen led platform engineering: he designed and implemented the mi- croservice architecture and gRPC interfaces on which the serving, attribution, and product backend systems are built. Zhichen Guo implemented the frontend and Clarity API. Muawiz Chaudhary and Andreas Madsen designed the model serving engine for the new causal diffusion model. Muawiz Chaudhary led the implementation of the serving engine with feedback from Andreas Madsen. 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Representation engineering: A top-down approach to ai transparency.arXiv preprint arXiv:2310.01405, 2023. 83 Part I Architecture ASymbol reference SymbolTypeMeaning Hidden states hvectorTransformer hidden state ̄ hvectorBottlenecked state passed to the LM head ̄ h= ˆ k+ ˆu+εequation Concept module decomposition Concept module heads ffunction Known head gfunction Unknown head k=σ(f(h))vectorPer-concept activation probabilities, known u=σ(g(h))vectorPer-concept activation probabilities, unknown k known scalarTop-kcount for the known head k unknown scalarTop-kcount for the unknown head Concept embeddings KmatrixKnown concept embedding matrix UmatrixUnknown concept embedding matrix K i vectorEmbedding of known concepti U j vectorEmbedding of unknown conceptj nscalarNumber of known concepts mscalarNumber of unknown concepts (mn) RscalarFactorization rank of unknown embedding matrix Concept contributions ˆ k= P i k i K i vectorKnown concept contribution ˆu= P j u j U j vectorUnknown concept contribution ε=h ˆ kˆuvectorResidual term Logit decomposition W y vectorRow of the LM head for output tokeny ℓ y scalarOutput logit for tokeny Table 20.Concept module notation, grouped by role. 84 SymbolTypeMeaning Losses L LM lossLanguage modeling loss (L MDM on ̄ h) L concept lossConcept loss (chunk-level BCE) L rec lossReconstruction loss for the unknown head L indep lossIndependence loss between ˆ kandˆu LlossCombined training objective λ concept ,λ rec ,λ indep scalarsLoss weights Supervision and targets y c scalarGround-truth chunk label, known conceptc k chunk c scalarOR-aggregated chunk-level activation k GT i scalarGround-truth activation of known concepti ˆ k GT vectorGround-truth known concept contribution ˆu GT =h ˆ k GT vectorTarget for the unknown head Independence loss H k ,H u matrices Stacked per-token ˆ k,ˆuover a minibatch μ ˆ k ,μ ˆu vectors Column means ofH k ,H u ΦmatrixCentered known features ΨmatrixCentered unknown features Training dynamics MsetMasked token positions in the minibatch t b scalarPer-block noise level (blockb) α known (s)scalarTeacher forcing prob., known, steps α unknown (s)scalarTeacher forcing prob., unknown, steps p cfg scalarDropout rate for the known head p ε scalarDropout rate for the residualε BscalarMinibatch size bscalarBlock size in causal block-diffusion Table 21.Concept module notation, grouped by role. 85 Part I Interpretability capabilities BAttribution details B.1Training data attribution The training data attribution pipeline of Section6.1.3indexes the training corpus offline and re- trieves against it at inference time. We describe each component here. Corpus index.We index the full training corpus, approximately 11 billion chunks, with a fine-tuned embedding model,FT-Qwen3-Embedding-0.6B, derived fromQwen3-Embedding-0.6B(the annotator model of section4), encoding each chunk into a 1024-dimensional vector. To make nearest-neighbor search tractable at this scale, we build a FAISS index with an inverted-file structure and product quan- tization (IVFPQ): the inverted file partitions the corpus into coarse clusters so a query is compared only against the closest few, and product quantization compresses each stored vector to keep the index in memory. Retrieval performs approximate nearest-neighbor search over the top-ncoarse clusters. Chunk representation.At inference time, each chunk of the model response is tokenized and forwarded through the language model to obtain per-token hidden states. These are mean-pooled over the real (non-padding) positions into a single 4096-dimensional representation, formed from the known-head, unknown-head, and residual components of the hidden state, so it encodes the meaning of the chunk as the model represents it. Transducer.The chunk representation is 4096-dimensional while the corpus index is 1024- dimensional, so we bridge the two spaces with the transducer: a two-hidden-layer MLP with roughly 15 million parameters that mapsR 4096 !R 1024 , trained to preserve semantic content under a cosine- similarity objective, L trans = 1cos( ˆ e,e ∗ ),(34) where ˆ eis the transducer’s predicted embedding ande ∗ is the target embedding fromFT-Qwen3- Embedding-0.6Bfor the same chunk. Pipeline.End to end, each response chunk is encoded and mean-pooled into a single representation, transduced into the corpus embedding space, and used to query the IVFPQ index by approximate nearest-neighbor search over the top-ncoarse clusters, returning the most similar training chunks as the attributed sources. 86 Part I Data CAtlas: From documents to concepts Text-assigned concept evaluation.For the tag, concept, and predicted-concept relevance evalua- tions described in Section4.2, we use an LLM judge to rate whether candidate concepts are present in a text chunk. The judge isMistral-Small-3.1-24B-Instructrun at temperature 0. It receives a text chunk together with one or more candidate concepts, each represented by a label and one- sentence description, and returns a JSON object assigning each candidate concept a relevance score from 1 to 5. The full prompt is shown in Figure32. SYSTEM: You are an expert at evaluating whether concepts are genuinely present in a passage of text. For each concept , you need to score based on how well the concept is exhibited (either explicitly or in a subtle manner) by the text passage. Use the following 1-5 scoring criteria: - 5: Very strongly present/central to the text - concept is explicitly discussed and /or forms a core theme - 4: Strongly present/well -demonstrated - concept is clearly evident with substantial supporting content - 3: Moderately present/clearly related - concept is reasonably connected with some supporting evidence - 2: Weakly present/tangentially related - minimal connection or only surface -level mention - 1: Not present/completely inappropriate - concept is absent , irrelevant , or text merely uses related vocabulary without actually being about that concept Be precise in your evaluation - consider both explicit mentions and substantive demonstration of the concept. Respond with a single -line JSON object where keys are the concept letters (A, B, C, etc.) and values are scores (1-5). **DO NOT include any explanations or additional text .** USER: Evaluate how well each concept below is exhibited in the given text chunk using the 1-5 scale. === TEXT CHUNK === text_chunk === CONCEPTS TO EVALUATE === concepts_with_descriptions Figure 32.Prompt for text-assigned concepts evaluation 87 DAdditional details on the human interpretability study This appendix gives additional methodological and statistical details for the human interpretability study in Section4.4. The main text reports the study design and headline results; here we report agreement statistics, robustness checks, and model-based analyses. D.1Sampling and annotation protocol We sampled100concepts stratified by top-level taxonomy branch: ten concepts from each of the nine largest branches and ten from the aggregated remainder. For each sampled concept, we constructed a lifted-word list by ranking lemmatized words according to lift(w,c) = P(wjc) P(w) , subject to a minimum-support filter. Annotators saw only these lifted words, not the pipeline concept name, concept description, taxonomy position, or source documents. In Phase 1, annotators wrote a name or short phrase for the concept and rated whether the lifted words formed a recognizable concept on a1–5scale. Phase 1 collected303named-concept responses from9annotators, with a median of3annotators per concept. Human-written names averaged4.1 words. In Phase 2, annotators rated candidate names for the same lifted-word lists. Each candidate set con- tained the pipeline’s LLM-generated label, two human labels from Phase 1, an embedding-neighbor distractor, and a taxonomy-neighbor distractor. A small number of cases also included a low-effort filler label as a floor control when one of the other candidate types was unavailable. Candidate order was randomized, annotators were blind to candidate provenance, and no annotator scored a label they had written. Phase 2 collected205scoring records over34concepts from8annotators, for1,025individual candidate-name ratings. The20most-rated concepts were scored by all eight Phase 2 annotators. D.2Phase 1 agreement and coherence Phase 1 was designed to test whether lifted-word evidence contains recoverable semantic structure before any pipeline label is shown. The mean coherence score was3.52. Annotators judged55% of responses to form a recognizable concept (4), rated27% as borderline (= 3), and flagged17% as incoherent or noisy (2). Agreement was moderate rather than perfect, as expected for a task involving short word lists and concepts drawn from many technical domains. ICC(1)was0.43. The within-concept standard devi- ation was0.65, compared with total scale standard deviation1.05, and53% of concepts were unan- imous on the coherent-vs-not split. These agreement statistics support the main-text conclusion: lifted-word evidence is often meaningful, but not uniformly so. D.3Ordinal mixed-model analysis The primary Phase 2 outcome is an ordinal fit ratingy i 2f1,2,3,4,5gfor a candidate label. Because ratings are ordinal and clustered by rater, concept, label, and scoring context, we fit a Bayesian cumulative-link mixed model with crossed random effects. Human labels are the reference category. The model has the form Pr(y i k) =logit −1 (τ k η i ), 88 AnalysisResult Mean Phase 2 fit scoreLLM3.98vs. human3.50 Top-two rating rate (4)LLM79% vs. human63% Bayesian cumulative-link modelOR2.38,95% CrI[1.23,4.01] Posterior probability of LLM advantage0.99 Paired comparison probability0.62, bootstrap95% CI[0.58,0.66] Gaussian mixed model+0.47points,95% CI[0.28,0.65] Table 22.Robustness checks for the Phase 2 comparison between pipeline labels and independently generated human labels. wherek2f1,2,3,4gindexes the ordinal thresholds and η i =β LLM 1 LLM,i +β emb 1 embedding,i +β tax 1 taxonomy,i +β fill 1 filler,i +u rater r[i] +u concept c[i] +u context s[i] +u label ℓ[i] . Positive coefficients indicate higher expected fit scores. The main contrast isβ LLM , comparing the pipeline label to human-written labels. The LLM–human proportional-odds ratio was2.38(95% CrI[1.23,4.01]), with posterior probability 0.99that the pipeline label receives higher fit ratings than a human label. On the response scale, pipeline labels received a top-two rating (4)79% of the time, compared with63% for human labels. D.4Assumption-light robustness checks We also ran simpler checks that make fewer modeling assumptions. In paired comparisons, the pipeline label outscores a human label with probability0.62(cluster-bootstrap95% CI[0.58,0.66], 402pairs). A Gaussian mixed model on the raw1–5ratings estimates a+0.47point advantage for pipeline labels over human labels (95% CI[0.28,0.65]). These checks agree with the ordinal mixed model: pipeline labels are not merely competitive with human labels, but are rated higher on average under blind evaluation. D.5Dependence on lifted-word coherence If the LLM were hallucinating plausible names uniformly, its advantage should not depend strongly on whether the lifted-word evidence itself is coherent. We therefore examined how Phase 2 label fit varies with Phase 1 coherence. Concepts with higher Phase 1 coherence receive higher Phase 2 fit scores for both human and pipeline labels. The correlation between Phase 1 coherence and Phase 2 fit is+0.42for human labels and+0.46for pipeline labels. The pipeline advantage is positive across coherence bins, but larger when the underlying word evidence is clearer: the LLM–human gap is+0.22for low-coherence concepts,+0.54for mid-coherence concepts, and+0.52for high-coherence concepts. This pattern is consistent with the LLM naming real statistical structure rather than assigning plausible labels independently of the evidence. D.6Power analysis Finally, we checked whether the realized Phase 2 sample size was sufficient for the observed LLM– human contrast. A simulation under the fitted clustered generative model gives power above0.99 89 for the observed contrast at roughly20fully-rated concepts and essentially1.00at the realized sample size. This analysis should not be interpreted as certifying every individual concept in the full library. Rather, it shows that the stratified pilot is well-powered for the aggregate comparison between pipeline labels and human-written labels. D.7Limitations The human study validates the labeling operation used by Atlas, not every concept individually. A minority of lifted-word lists are ambiguous or noisy, and some concepts require domain expertise that may not be uniformly available across annotators. The Phase 2 comparison also evaluates names relative to lifted-word evidence, rather than full source-document evidence. These limitations make the result conservative in one respect and incomplete in another: humans often recover and endorse the same semantic structure from sparse evidence alone, but the study does not eliminate the need for additional per-domain or per-concept audits in downstream use. 90 Part IV Interpretability metrics EKnown concept alignment judge The Known Concept Alignment metric from section7relies on an LLM-judge to rate concepts. The judge usesMistral-Small-3.1-24B-Instructat temperature 0. It receives a concept’s human-assigned label, its one-sentence description, and the top-Ktokens scored by the concept embedding through the LM head. It returns a single integer rating from 1 to 5. The full prompt is shown in Figure33. SYSTEM: You are an expert in neural network interpretability. You are evaluating whether a concept head in a language model has learned to represent a specific named concept. You will be given: 1. The concept 's human -assigned label and description 2. The top activated tokens from the concept head Rate how well the top tokens represent the named concept on a 1-5 scale: - 5: Tokens strongly and clearly represent the concept. Most tokens are directly related. - 4: Tokens mostly represent the concept with minor noise or tangential items. - 3: Tokens partially represent the concept. Some relevant tokens but also significant off -topic items. - 2: Tokens weakly relate to the concept. Only a few tokens connect; most are unrelated. - 1: Tokens do not represent the concept at all. IMPORTANT: A token does not need to be an exact word from the label or description to count as relevant. Proper nouns , abbreviations , sub -words , and semantically related terms all count. For example , if the concept is 'academic publishers ', then 'Penguin ', 'Wiley ', 'ISBN ', 'paperback ', and 'imprint ' are all relevant even though none appear in the label. Respond with ONLY one line: ALIGNMENT: <integer 1-5> USER: Concept label: concept_label Concept description: concept_description === TOP ACTIVATED TOKENS === top_tokens Figure 33.Prompt for the Known Concept Alignment judge. 91 Part V Scaling laws FSymbol and notations Symbol TypeMeaning Compute and resources CscalarTotal training FLOPs MscalarPer-token FLOPs (forward + backward) PscalarNon-embedding parameter count DscalarNumber of training tokens (total) D i scalarTraining tokens for thei-th checkpoint Validation losses L i scalarMeasured validation loss for checkpointi L ∗ (C)function Compute-optimal validation loss at budgetC L ∞ scalarIrreducible validation loss L(P,D)function Joint Chinchilla loss surface L fit function Huber loss minimized in the joint fit Power-law parameters P ∗ (C)function Compute-optimal parameter count at budgetC a P ,a L scalars Power-law coefficients (parameter and loss) α P scalarExponent on compute for parameter scaling α L scalarExponent on compute for loss scaling α D scalarExponent on compute for training-token scaling A P ,A D scalars Chinchilla coefficients (parameter and data sides) α,βscalars Chinchilla exponents (parameter and data sides) Interpretability scaling m(C)function A metric as a function of compute escalarIrreducible value of a metric A,βscalars Coefficient and exponent for metric scaling Table 23.Notation introduced in the scaling-law analysis, grouped by role: compute and resources, validation losses, power-law parameters, and interpretability scaling. GArchitectures, IsoFLOP slices, and hyperparameters Backbones.Table24lists the configuration of each backbone size used in the scaling sweep. All four families share the same backbone architecture; +Concept variants add the concept module on top, configured per Table 26and Table27. IsoFLOP slices.Table25reports the four IsoFLOP target compute budgets per family. The +Con- cept families start at higher targets because the concept module’s per-token FLOPs are non-negligible at small backbone sizes. Each slice contains four to six model sizes whose per-checkpoint compute lands within15% of the target. 92 Size LayersLHiddendBackbone params +Concept total params 10M63209.2M82.8M 25M651223.6M110.2M 85M1076886.5M190.5M 200M131024197.7M319.1M 400M171280401.1M540.0M 800M171792779.9M953.6M 1.5B2023041,510M1,718M 3B2430723,228M3,489M 5B2438405,694M6,007M Table 24.Backbone architectures used across all four families. Backbone parameter counts exclude embed- dings; +Concept totals include the concept module heads (concept classifier, and factorized unknown head concept embeddings are excluded). Sequence length is 4096 throughout. FamilySlice 1 Slice 2 Slice 3Slice 4 AR610 18 10 19 310 19 10 20 CDLM610 18 10 19 310 19 10 20 AR+Concept10 19 310 19 1.110 20 3.0910 20 CDLM+Concept10 19 310 19 1.110 20 3.0910 20 Table 25.IsoFLOP target compute budgets per family. Shared hyperparameters.All scaling-law runs share optimizer, learning rate, batch size, warmup, and architectural defaults. The optimizer is AdamW withβ 1 = 0.9,β 2 = 0.95,ε= 10 −8 , weight decay0.1(excluding embeddings), and gradient clipping at1.0. Peak learning rate is fixed at410 −4 across all model sizes and all families. The schedule is warmup-stable-decay (WSD) with an80/20 stable/decay split and decay to zero. Warmup is the minimum of2000steps or2% of total training steps, to accommodate runs with smaller token budgets. The total batch size is524,288tokens (128 sequences4096 tokens) across every run. Backbone architectural defaults (post-norm RMSNorm, QK-norm, RoPE base510 5 , SwiGLU MLP with ratio4, no biases, clip_qkv=10, dropout0) are held fixed across all backbones. Per-(size, slice) token counts are determined by the slice targetC and the FLOP equation (Equation ( 23)); intermediate stable-phase checkpoints are also included in IsoFLOP fits where they fall within15% of a slice target. CDLM-specific settings.The two diffusion families (CDLM and CDLM+Concept) share an addi- tional set of settings governing the masking process: causal block size64, with the noise level sam- pled uniformly at training time,t U(0.05,0.95). These are held fixed across all CDLM model sizes. Concept module settings.The two +Concept families (AR+Concept, CDLM+Concept) share the concept module configuration. Settings differ between the stable phase (Table26) and the anneal phase (Table27), with the anneal phase additionally applying top-ksparsification to both heads. The parametersα known (t)andα unknown (t)are the probabilities of using ground-truth components in place of predicted ones during training; see Section 5.4.2for the motivation and definition. 93 SettingValueNotes n(known concepts)33,732shared across all sizes Unknown ratio3)m= 101,196 Factorization rankR256for unknown head Top-k known 16predictor + compose α known (t)1.0!0.5by step0.10T max cosine α unknown (t)0.0!0.5by step0.10T max linear λ concept 1.0concept loss weight λ rec 1.0reconstruction loss weight λ indep 1.0independence loss weight p cfg 0.1known head dropout rate p ε 0.3residual dropout rate Table 26.Concept module settings during the stable phase. Bothαschedules ramp from initial value to0.5 by step0.10T max , then hold constant for the remainder of the stable phase.T max denotesmax_steps. SettingValueNotes α known (t)0.5!0.0over anneal linear α unknown (t)constant at1.0 Top-k known 32topk_known Top-k unknown 128factorized compose,apply_topk_to_unknown Table 27.Concept module settings during the anneal phase. The anneal phase resumes from the final stable- phase checkpoint and runs a linear LR decay to zero. Inherited settings (n, unknown ratio, factorization rank, loss weights, dropouts) match Table26. HELBO estimation for validation loss Autoregressive models report exact negative log-likelihood (cross-entropy on the next token); how- ever, in diffusion models, the per-token NLL cannot be computed exactly and must instead be esti- mated via Monte Carlo on an Evidence Lower Bound (ELBO). Different MC schemes give materially different absolute loss values, which propagates into reportedL ∞ values and, when the bias is non- uniform across model sizes, into reported scaling exponents. Here we compare four estimators on our CDLM checkpoints. We find that the compute-optimal parameter countP ∗ (C)and the exponent α P are robust to estimator choice, while absolute loss valuesL ∗ and the irreducible-loss asymptote L ∞ vary by up to a nat between estimators. H.1Estimators Letx 0 = (x 1 0 ,...,x N 0 )denote a sequence of lengthNdrawn from the validation distributionD. At noise levelt2[0,1], the forward process replaces each token independently with[MASK]with probabilitytand leaves it unchanged with probability1t: x i t = ( [MASK]with probabilityt, x i 0 with probability1t. (35) We denote byM t =fi:x i t =[MASK]gthe set of masked positions inx t , and byp θ (x i 0 jx t )the model’s predicted distribution over the token at positionigiven the corrupted sequencex t . All estimators below average over100M tokens of validation data. 94 Fixed-rate mask loss.The first estimator computes per-token cross-entropy at a single fixed mask rateτ, with no integration over the noise schedule: ˆ L fixed (τ) =E x 0 ,x τ " 1 jM τ j X i∈M τ logp θ (x i 0 jx τ ) # .(36) We setτ= 0.5. This estimator computes per-token cross-entropy at one fixed corruption level rather than an ELBO bound onlogp θ (x 0 ), but is the closest validation analogue of the training objective, which integrates over[0.05,0.95]. Fixed-grid discretized ELBO.The second estimator approximates the ELBO integral overtviaK fixed binsft 1 ,...,t K gaveraged uniformly: ˆ L grid = 1 K K X k=1 E x 0 ,x t k 2 4 1 jM t k j X i∈M t k logp θ (x i 0 jx t k ) 3 5 .(37) We useK= 9witht k =k/10fork= 1,...,9. With uniform weights this is a midpoint-rule approximation to the ELBO integral overt2[0.05,0.95], matching the interval sampled during training. MDLM ELBO.The third estimator follows the low-variance form of the MDLM ELBO from equa- tion2. Under the linear scheduleα t = 1t, the canonical NELBO bound onlogp θ (x 0 )is logp θ (x 0 )E t∼U(0,1) " 1 t X i∈M t logp θ (x i 0 jx t ) # .(38) The1/tprefactor cancels in expectation withE[jM t j/Njt] =twhent U(0,1), giving the per-token form ˆ L MDLM =E t∼U(0,1) E x 0 ,x t " 1 jM t j X i∈M t logp θ (x i 0 jx t ) # .(39) We sample onetper batch and average over batches. This estimator is unbiased for the per-token NELBO bound and matches the formulations used in Sahoo et al.(2024);Nie et al.(2024);Ni et al. (2025);von Rütte et al.(2025);Sahoo et al.(2026); it is also what we report in section8.2. Note that the CDLM training loss is itself an unbiased estimator of the same quantity, except that training clips tto[β,ω] = [0.05,0.95]to avoid degenerate extremes. The two estimators are therefore numerically close but not identical: ˆ L MDLM integrates over the full[0,1]interval, while the training loss integrates over[0.05,0.95]. Per-block ELBO.The fourth estimator matches the training distribution of our causal block- diffusion model ( Section5.2), where each block has its own independently sampled noise level. We partition each sequence intoN/bblocks of sizeb. For each block, we sample an independent noise levelt b U(0,1)and mask positions within that block at ratet b . We then compute mean cross-entropy on all masked positions: ˆ L block =E t b iid ∼U(0,1) E x 0 ,x t " 1 jM t j X i∈M t logp θ (x i 0 jx t ) # .(40) 95 We useb= 64, matching the training configuration, so each sequence contributes 64 independent noise levels per validation step. The cancellation argument from Equation (38) applies per-block, so this estimator is also unbiased for the per-token NELBO bound, with lower Monte Carlo variance than ˆ L MDLM because each batch covers a wider distribution of t values. H.2Results IsoFLOP shapes and absolute losses.Figure34compares the four estimators across the four CDLM IsoFLOP slices. The per-slice parabolas (top row) have similar shapes and per-slice minima at com- parable parameter countsP ∗ (C)across all estimators, indicating that the location of the compute- optimal model size is robust to estimator choice. The absolute loss values, however, differ sub- stantially: at any fixed slice, the four estimators are vertically offset by up to a nat, with uniform mask consistently lowest and per-block ELBO highest. The bottom row shows each estimator’s full IsoFLOP grid; the parabolas tighten with compute under all four estimators, and the slice minima move smoothly toward largerPasCgrows. Figure 34.IsoFLOP comparison of the four ELBO estimators on CDLM. Top row: per-slice parabolic fits with all four estimators overlaid. Bottom row: per-estimator IsoFLOP grids with all four slices overlaid. Effect on power laws.Figure35plots the compute-optimal scaling lawsL ∗ (C)andP ∗ (C)for each estimator. TheP ∗ (C)fits (panel b) are nearly indistinguishable across estimators, with all four lines overlapping within marker width andα P values clustering tightly between0.602and0.632. The L ∗ (C)fits (panel a) show parallel-ish slopes shifted vertically by an estimator-dependent offset; the loss exponentα L varies modestly between0.071(uniform mask) and0.039(per-block ELBO). Table 28summarizes the exponents with90% bootstrap confidence intervals. H.3Effect of mask rate on scaling exponents The four estimators above differ in how they aggregate across the noise schedule, mixing the effect ofwhichmask rates are sampled with the effect ofhowthey are integrated. To isolate the role of the mask rate alone, the IsoFLOP fit is re-run at nine fixed mask ratest2 f0.1,0.2,...,0.9gon the same CDLM checkpoints. Figure 36shows the per-slice parabolas at each mask rate, Figure37 96 Figure 35.Compute-optimal scaling laws under the four ELBO estimators. (a)L ∗ (C)versus compute. (b) P ∗ (C)versus compute. Figure 36.IsoFLOP analysis at nine fixed mask ratest2 f0.1,0.2,...,0.9gon CDLM.markers locate the per-slice minima P ∗ ( C ) . 97 Estimatorα P α L α D L ∞ Fixed-rate mask0.612 [0.564,0.661]0.071 [0.075,0.067] 0.374 [0.317,0.525] 1.914 [1.263,2.070] Fixed-grid ELBO0.611 [0.568,0.658]0.054 [0.057,0.051] 0.368 [0.309,0.555] 2.416 [1.786,2.560] MDLM ELBO0.632 [0.541,0.707]0.053 [0.058,0.049] 0.481 [0.370,0.643] 2.658 [1.950,2.839] Per-block ELBO0.615 [0.572,0.660]0.039 [0.041,0.036] 0.340 [0.281,0.558] 3.353 [2.685,3.526] Table 28.IsoFLOP power-law exponents and irreducible-loss asymptotes under four ELBO estimators on CDLM. Subscripts are 90% bootstrap confidence intervals. shows the resultingP ∗ (C)andL ∗ (C)fits, and Table29reports the exponents.s The parameter exponentα P separates into two groups:α P 0.60fort0.7andα P 0.62fort0.8. The loss exponentα L flattens monotonically and substantially, from0.120att= 0.1to0.013att= 0.9: at low mask rates the loss has substantial room to fall with compute, while at high mask rates it approaches the random-token floor where additional compute buys little. This decoupling explains the four-estimator pattern documented above: any aggregation acrosstinherits a similarα P from the per-rate fits but produces anα L determined by which mask rates carry weight. Figure 37.Compute-optimal scaling laws at nine fixed mask rates on CDLM. (a)L ∗ (C)versus compute. (b) P ∗ (C)versus compute. TheL ∗ (C)slopes flatten monotonically astgrows;α P falls into two groups,0.60 fort0.7and0.62fort0.8. H.4Discussion Implications forL ∞ .The IsoFLOP exponentsα P andα L are robust to estimator choice, but the irreducible-loss asymptoteL ∞ from the joint Chinchilla fit is not: across the four estimators,L ∞ spans roughly1.9to3.4, a1.5-nat range that exceeds the entireL ∞ spread reported across diffusion scaling papers. This indicates that absoluteL ∞ values are not directly comparable across diffu- sion methodologies that use different ELBO estimators, even on the same model and data. Within- methodology comparisons remain valid, as both members of eachConcept pair in the main results use the same estimator. The CDLM model is trained withtU(0.05,0.95). The fixed-rate estimator atτ= 0.5matches the middle of this training interval and yields the lowestL ∞ across the four estimators, withα P in line 98 Mask ratet α P α L R 2 (P)R 2 (L) 0.10.6120.120 0.9910.992 0.20.6120.110 0.9910.994 0.30.6120.099 0.9920.995 0.40.6090.086 0.9920.996 0.50.6090.071 0.9930.997 0.60.6050.057 0.9930.997 0.70.6040.042 0.9930.998 0.80.6240.026 0.9960.998 0.90.6230.013 0.9950.997 Table 29.IsoFLOP power-law exponents at each fixed mask ratet(CDLM family). with prior masked-diffusion estimates (Ni et al.,2025;von Rütte et al.,2025;Nie et al.,2024). The other three estimators integrate over a widertrange, including the boundary regionst2[0,0.05] andt2[0.95,1]that the model was not directly trained on. The main paper reports MDLM ELBO to follow the convention adopted by prior diffusion scaling papers (Nie et al.,2024;Ni et al.,2025; von Rütte et al.,2025;Sahoo et al.,2026), which makes the reportedα P directly comparable to theirs; the higher absoluteL ∞ relative to e.g. Quokka’s falls inside the estimator-induced spread documented here. IAnnealing each IsoFLOP checkpoint The IsoFLOP analysis of Section8relies on annealed checkpoints, since the warmup-stable-decay schedule’s final20% decay reduces validation loss by a non-trivial amount. Annealing each check- point independently is more expensive than estimating annealed losses from raw stable-phase check- points with a constant correction, as proposed by von Rütte et al.(2025). For the +Concept families, annealing is not solely an LR-decay procedure: the interpretable- component schedules also shift during the anneal phase (AppendixG, Table26and Table27).α known ramps from0.5to0.0, transitioning the model from teacher-forced ground-truth concept represen- tations to its own learned known-head predictions.α unknown shifts to1.0, transitioning the model to fully rely on the learned unknown head. Top-ksparsification activates on both heads, restricting predictions to a small subset of the135K concepts so that attribution remains interpretable. The anneal phase therefore transitions the model from its training-time configuration to the deployed inference-time configuration. CDLM+Concept checkpoints in both states are compared on validation loss and the four inter- pretability metrics of section 7. The gap is substantial in two places: compute-optimal allocation (α P shifts by0.11) and Concept Independence Loss (which decreases3-10under annealed evaluation). I.1Validation loss Figure38shows IsoFLOP fits to CDLM+Concept checkpoints evaluated in both states, with the re- sulting power-law exponents reported in Table30. The parameter exponentα P shifts from0.574 pre-anneal to0.686post-anneal, a0.11absolute increase. The loss exponentα L is essentially un- changed, and the slice-wiseL ∗ values shift by at most0.11nats. Theα P shift is not a uniform ”more tokens” effect of the anneal phase: a uniform downward shift in L ∗ across slices would change the intercept of logL ∗ versus logCbut leave the per-slice logP ∗ min- 99 ima fixed, preservingα P exactly (the slice-wise parabolas would shift vertically without translating along the parameter axis). The observed shift inα P requires that annealing improves lossasymmet- ricallyacross model sizes within each slice, moving the parabola minimum along logP. The result is that the compute-optimal model size P ∗ ( C ) is systematically larger under annealed evaluation than under pre-anneal evaluation. Figure 38.CDLM+Concept validation loss, pre-anneal versus annealed. Top row: per-slice parabolas in each state. Bottom row:L ∗ (C)andP ∗ (C)power-law fits. Conditionα P α L R 2 (P)R 2 (L) Pre-anneal0.5740.073 0.9590.956 Annealed0.5030.072 0.9830.976 Table 30.Compute-optimal scaling-law exponents for CDLM+Concept, evaluated pre- and post-anneal. I.2Interpretability Figure39shows the four interpretability metrics evaluated in both states across the same CDLM+Concept checkpoints, with the resulting power-law fits in Table 31. Three of the four metrics are robust to annealing: Concept Loss and Concept Contribution shift by at most0.02 on their respective scales, with near-identical slopes; Known Concept Alignment is essentially unchanged (β= 0.447pre-anneal vs0.437annealed,R 2 within0.01). Concept Independence Loss is the outlier: pre-anneal HSIC sits in the30-80range with a noisy trend (R 2 = 0.14), while annealed HSIC drops to the5-20range with a clean trend (R 2 = 0.79), a3-10reduction across checkpoints. The pattern is consistent with which inference-time quantities each metric depends on. Known Concept Alignment is computed from the concept embeddingsK c projected to vocabulary space and is invariant to the inference-time mixing of the known and unknown heads, so the schedule changes during anneal do not affect it. Concept Loss and Concept Contribution depend on the relative contributions of known, unknown, and residual pathways at inference; these shift modestly withα known andα unknown . Concept Independence Loss is much more sensitive: multiple schedule 100 changes (top-ksparsification, the shift to fully model-predicted heads) act on the representations ˆ k andˆuduring anneal, and the metric drops3-10. Figure 39.CDLM+Concept interpretability metrics, pre-anneal versus annealed. Three metrics are robust; Concept Independence Loss decreases3-10under annealed evaluation. MetricConditionβR 2 Concept LossPre-anneal0.000 0.014 Annealed0.000 0.648 Concept Independence Loss Pre-anneal17.425 0.141 Annealed7.497 0.801 Concept ContributionPre-anneal+0.057 0.350 Annealed+0.073 0.536 Known Concept AlignmentPre-anneal+0.447 0.609 Annealed+0.449 0.645 Table 31.Interpretability metric trends versus compute for CDLM+Concept, evaluated pre- and post-anneal. βis the log-linear slope of the metric against log 10 (C). 101 Part VI Steerling-8B pretraining details JPretraining recipe ablations The interpretable causal-diffusion architecture introduces design choices with no established de- faults in the autoregressive or masked diffusion literature. For each choice in the recipe (Section9.2) we run a small ablation that sweeps one variable from a fixed baseline, a 1B model trained on 30B sampled from Nemotron-C-HQ, with the full default configuration in Table32. SettingValue Architecture Backbone size1.5B (non-embedding) Hidden dimensiond2304 LayersL20 Attention heads18 Sequence lengthN4096 Block sizeb64 Unknown concept capacitym= 3n Unknown factorization rankR256 Unknown decompositionMLP Gradient flow to unknown head detached Optimization OptimizerAdamW (β 1 = 0.9,β 2 = 0.95) Weight decay0.1 (excluding embeddings) Peak learning rate410 −4 LR scheduleconstant at peak, 2% warmup, no decay Batch size524,288tokens Training tokens20B Concept module schedules MaskingGaussian curriculum, center0.2!0.8,σ= 0.3 α known (s)1.0!0.5, cosine α unknown (s)1.0!0.5, linear p ε 0.3 p cfg 0.1 Loss weights λ concept 1.0 λ rec 1.0 λ indep 1.0 Data SourceNemotron-C-HQ (Su et al.,2025) Table 32.Default configuration for pretraining ablations. Each ablation swaps one setting from this baseline. We track five metrics at every saved checkpoint. Three measure capability: validation loss; MMLU soft score, the log-probability the model assigns to the correct answer letter on MMLU ( Hendrycks et al.,2020), taken relative to the four choices (Held,2026), that is 102 logP(correct)log P c∈A,B,C,D P(c), which is bounded above by0and varies continuously with capability while hard accuracy is still pinned near chance; and HellaSwag accuracy (Zellers et al.,2019). Validation loss and MMLU soft score are continuous and retain signal at the 1B-parameter scale, where standard accuracy is near random and unable to discriminate between configurations (OLMo et al.,2025;Held,2026). The remaining two measure interpretability: concept contribution (Equation22) and known concept alignment (Section7). J.1Diffusion block size Steerling is a causal diffusion model with the block-causal attention mask of Figure15: tokens within a block attend to each other and to all previous blocks, so within a single block the attention is effectively bidirectional. The block sizebcontrols how many tokens see each other bidirectionally, the maximum number of tokens that can be decoded in parallel at inference, and the granularity at which keys and values are cached across blocks. We compareb2 f32,64gon the five metrics introduced above, with results shown in Figure40. Figure 40.Block size ablation,b2f32,64g. The two settings are indistinguishable on both interpretability metrics: concept contribution and known concept alignment track each other within the noise band over the whole run. On capability the picture tilts toward the larger block. MMLU soft score is noisy and overlapping, with neither value holding a consistent lead, but validation loss separates cleanly, withb= 64sitting below b= 32for essentially the entire run, and HellaSwag shows the same tilt, withb= 64pulling slightly ahead over the back half. Since the larger block is at least as good everywhere, better on validation loss and HellaSwag, and decodes more tokens in parallel at inference, we adopt it. Block size leaves interpretability unchanged;b= 64 improves validation loss and HellaSwag, so we adopt it. J.2Diffusion masking schedule Unlike autoregressive models, where the training objective is fixed at next-token cross-entropy, diffu- sion language models must additionally choose how the noise leveltis drawn at each step.Nie et al. (2025) sampletU(0,1)uniformly.Ni et al.(2025) report a small but consistent gain from a mov- ing Gaussian curriculum, in whichtis drawn from a Gaussian window whose center shifts from low to high noise over training, exposing the model to easier (less masked) sequences early and harder ones late. The block-causal attention of Steerling differs from the full-attention setting in which both were measured, so we re-examine the choice here. We compare uniformtU(0.05,0.95)(Nie et al. ,2024,2025;Sahoo et al.,2024) against a moving Gaussian curriculum with center increasing linearly from0.2to0.8andσ= 0.3; the schedules are shown in Figure41and results in Figure42. 103 Figure 41.Mean mask rate per training step under the two sampling schedules. Figure 42.Masking schedule ablation, uniform vs. moving Gaussian. At this scale the two schedules are very close. The moving Gaussian holds a very slight edge on validation loss over the back half of the run, in line with whatNi et al.(2025) report, while MMLU soft score and HellaSwag overlap throughout, with uniform finishing marginally ahead on HellaSwag. The interpretability metrics track each other after an early transient on concept contribution. With no setting clearly ahead, we adopt the moving Gaussian on the validation-loss edge and consistency with prior work. 4 The two masking schedules are very close at ablation scale, with a slight edge to the moving Gaussian on validation loss; we adopt it. J.3Unknown concept capacity Steerling’s concept module splits its representation into known concepts, given by the data pipeline, and unknown concepts, learned during training. The number of known conceptsnis fixed by the Atlas concept library, but the number of unknown conceptsmis free. A largermgives the model more room to discover recurring patterns the known library does not cover, at the cost of parameters, compute, inference latency, and memory that could otherwise serve the language modeling objective. We comparem= 5nagainstm= 3n; results are shown in Figure43. The two settings are indistinguishable on the interpretability metrics. On capability they are also close, and where they separate it slightly favours the smaller setting:m= 3nholds a mild edge on MMLU soft score over the back half of the run, while validation loss and HellaSwag overlap throughout. The extra capacity ofm= 5nbuys no measurable improvement, so we adopt the more conservative setting. 4 This schedule proved too aggressive over the full pretraining run; see Section9.4.1. 104 Figure 43.Unknown concept capacity ablation,m2f3n,5ng. Raising unknown capacity from3nto5nyields no measurable gain; we keep the conservativem= 3n. J.4Unknown embedding factorization The unknown head’s embedding matrixU2R m×d is the largest single parameter the concept module adds: atmnit dominates the parameters Steerling carries over a standard backbone of the same size, and computingˆu=u ⊤ Uat every token is a heavy matmul in the forward pass. We therefore factorizeU=ABwithA2R m×R ,B2R R×d , and rankR= 256d, cutting both the parameter count and the per-step compute of the unknown pathway. We compare the denseU against the factorized form; results are in Figure44. Figure 44.Unknown embedding factorization ablation: dense vs. factorized (R= 256). Factorization is close to free on capability: validation loss, MMLU soft score, and HellaSwag all overlap the dense baseline throughout, and known concept alignment is identical. The one cost is concept contribution, where the dense head holds a steady0.90against0.80for the factorized head, as the low-rank bottleneck forces the model to rely more on the residual. However, the saving is large: at the Steerling-8B scale the denseUis roughly15larger than its factorized form, and factorization also removes the corresponding matrix multiplication from the per-token forward pass. We judge the small contribution cost well worth this and adoptR= 256. FactorizingUatR= 256makes the unknown embedding roughly15smaller with no capability cost and a small drop in concept contribution; we adopt it. J.5Use of the residual term The bottlenecked hidden state passed to the LM head, ̄ h= ˆ k+ ˆu+ε, carries a residualε=h ˆ kˆu that absorbs whatever the two heads fail to reconstruct, so that ̄ h=hidentically. Dropping it 105 sets ̄ h= ˆ k+ ˆuand forces every dimension of the bottleneck to be the sum of a known and an unknown concept contribution, leaving the model no uninterpreted channel. We compare the two formulations; results are in Figure45. Figure 45.Residual term ablation: with vs. withoutε. Removing the residual sharply harms the capability metrics: validation loss separates from the first tokens and never recovers, ending well above the residual baseline; HellaSwag plateaus around0.31 against0.36for the baseline; and MMLU soft score is noisier and weaker over the back half. The perfect concept contribution here is true by construction, not a sign of a better decomposition: with noεchannel the hidden state has nowhere else to fall. It does not justify the capability degradation, and given the uncertainty over how that cost would compound at larger scale, we keep the residual. Droppingεforces concept contribution to1.0by construction but inflicts a large, persistent capability penalty; we keep the residual. Figure 46.Teacher forcing schedules for the two ablations. (a)α known (s)with cosine decay. (b)α unknown (s) with linear decay. Defaults in bold. J.6Concept teacher forcing schedule The known head is trained from scratch alongside the transformer, so early on its predicted acti- vations ˆ kare unreliable, and routing ̄ hthrough them can drive concept leakage into the language modeling loss ( Mahinpei et al.,2021). The standard mitigation is to feed the ground-truth ˆ k GT to the LM head instead, known as independent training (Koh et al.,2020), but at our scale a model that never sees its own predictions would be unprepared for inference, where no ground truth exists. We therefore use the schedule of Section 5.4.1, whereα known (s)starts at full teacher forcing, decays via 106 cosine annealing during the early phase, and holds at a floor afterwards. Shape, warmup, and start- ing value are fixed at the defaults of Table32, and we ablate the floor overα known,min 2 f0.5,0.0g. The schedules are shown in Figure46(a) and results in Figure47. Figure 47.Concept teacher forcing floor ablation,α known,min 2f0.5,0.0g. Capability is unaffected by the floor, with validation loss, MMLU soft score, and HellaSwag overlap- ping across the run. The difference is on the interpretability side, and it is large. Holdingα known at0.5keeps known concept alignment climbing to3.7, whereas decaying to0.0stalls it near2.6 from early in training. We therefore hold the floor at0.5. Decayingα known to0.0during pretraining leaves capability intact but collapses known concept alignment; we hold the floor at0.5. J.7Unknown concept teacher forcing schedule The unknown head faces the same early-training instability as the known head, but no ground-truth labels exist to substitute for its predicted activationsˆu. The natural target is instead the analyti- cal residualˆu GT =h ˆ k GT from Equation11, computed from the transformer hidden state and the labeled known concepts. We use the schedule of Section5.4.1, whereα unknown (s)starts at full teacher forcing, decays linearly, and holds at a floor afterwards. As with the known head, shape, warmup, and starting value are fixed at the defaults of Table 32, and we ablate the floor over α unknown,min 2f0.5,0.0g. The schedules are shown in Figure46(b) and results in Figure48. Figure 48.Unknown concept teacher forcing floor ablation,α unknown,min 2f0.5,0.0g. Unlike the known head, the unknown floor has only modest effects, but where the metrics separate they favour the higher floor: validation loss is slightly lower and HellaSwag consistently higher at α unknown,min = 0.5, while concept contribution is slightly higher atα unknown,min = 0.0, and the rest overlap. We adopt0.5. The unknown teacher forcing floor has modest effects that favour0.5on capability; we adopt it. 107 ChoiceValuesDefault Finding Block attention sizebf32,64g64No interp. effect; lower loss Masking scheduleuniform vs. Gaussian Gaussian Very close; slight edge Unknown capacitymf3n,5ng3nNo gain from more capacity Unknown factorization rankRdense vs.25625615smaller, no cost Residual termεwith vs. withoutwithRemoving it costs capability α known floorf0.5,0.0g0.50.0collapses alignment α unknown floorf0.5,0.0g0.5Little effect Table 33.Summary of the architecture and training ablations in Section9.2. 108 KFinal pretraining Steerling-8B configuration SettingValueNotes Backbone LayersL32 Hidden dimensiond4096 Attention heads32GQA with4KV heads Sequence lengthN4096 MLPSwiGLUratio4, no biases NormalisationRMSNormpost-norm, QK-norm Position encodingRoPEbase510 5 Weight tyingyes Diffusion Block sizeb64 Masking schedulemoving Gaussiancenter0.2!0.8,σ= 0.3 trange[0.05,0.95]clipped at sampling Concept module Known conceptsn33,732from Atlas Unknown conceptsm101,196m= 3n Unknown factorization rankR256 Top-k known 16 Unknown decompositionMLP α known (s)1.0!0.5, cosinewarmup15%, decay until50% α unknown (s)1.0!0.5, linearwarmup25%, decay until100% p cfg 0.1known head dropout p ε 0.1residual dropout λ concept 1.0 λ rec 1.0 λ indep 1.0 Gradient flow to unknown head detached Optimizer OptimizerAdamWβ 1 = 0.9,β 2 = 0.95,ε= 10 −8 Peak learning rate410 −4 ScheduleWSD100% stable; decay deferred to mid-training Warmup2000steps Weight decay0.1excluding embeddings Gradient clipping1.0 Run Batch size5,242,880tokens/step1280sequences4096tokens Token budget1.2T Hardware40nodesA100s Precisionbf16 Table 34.Final pretraining configuration of Steerling-8B. 109 LPretraining diagnostic Here we look into the issues raised in the pretraining lessons (Section9.4). We plot each metric alongside the training schedule that most plausibly drives it, on a secondary axis, allowing the align- ment between schedule transitions and metric inflections to be read directly. Figure49overlays the masking curriculum on validation loss and the five downstream benchmarks. The schedule begins at center0.2and rises linearly to0.8over the course of pretraining, passing 0.5around the midpoint. MMLU and WinoGrande peak shortly before the schedule crosses0.5 and decline as it continues to steepen. ARC-Challenge peaks slightly later with a smaller decline; HellaSwag and PIQA hold near their peak values; validation loss plateaus rather than continuing to descend. The decline on the harder reasoning benchmarks tracks the masking curriculum directly. Figure 49.Capability metrics during Steerling-8B pretraining with the masking curriculum overlaid (right axis, blue dashed). The curriculum passes center0.5around the midpoint of training. Figure50overlays the masking, concept teacher forcing, and unknown concept teacher forcing schedules on the four interpretability metrics. The teacher forcing schedules decay from1.0to their floor of0.5:α known via cosine annealing over the first50% of training,α unknown linearly across the full run. Concept independence loss is near zero through the first two thirds of training and rises sharply in the final third, coinciding with the masking curriculum reaching its hard regime and the two teacher forcing schedules having moved substantially away from their starting values. Con- cept contribution climbs over the same window as the LM head depends increasingly on predicted concepts. Known concept alignment and concept loss change only modestly throughout, on small absolute scales. 110 Figure 50.Interpretability metrics during Steerling-8B pretraining with the three training schedules overlaid: α known ,α unknown , and masking curriculum. 111 MSteering judge The steering evaluation in Table10relies on an LLM-judge to score generated continuations. We use Mistral-Small-3.1-24B-Instruct at temperature 0. For each continuation we call the judge twice: once with the prompt in Figure51to produce aconcept scorefrom 0 to 2 measuring how strongly the target concept appears in the continuation, and once with the prompt in Figure52to produce aquality scorefrom 0 to 2 measuring fluency and coherence. Each prompt is sent as a single user message through the model’s chat template, with no separate system instruction. Both prompts instruct the judge to score only the continuation, not the prefix it was generated from. You are an impartial evaluator. Task: Rate how strongly the ANSWER expresses the TARGET CONCEPT , based on meaning. - Score the ANSWER text only. - The concept may be expressed without using the exact label words; judge by meaning. Examples: Rating: [[2]] - if the answer clearly matches the concept description with multiple explicit supporting details. Rating: [[1]] - if the answer is loosely related but provides only minimal or indirect evidence. Rating: [[0]] - if the answer is unrelated to the concept. Score: 2 = concept clearly present with strong evidence 1 = concept present but weak/indirect/minimal 0 = concept absent Output ONLY one line in exactly this format: Rating: [[0]] or Rating: [[1]] or Rating: [[2]] [TARGET CONCEPT LABEL] concept_label [TARGET CONCEPT DESCRIPTION] concept_desc [ANSWER] answer Now score the ANSWER 's concept expression. Answer with 'Rating: [[0]] ' or 'Rating: [[1]] ' or 'Rating: [[2]] ' only. Figure 51.Prompt for the steering concept-score judge. 112 You are an impartial evaluator. Task: Rate the text quality of the CONTINUATION text ONLY , considering fluency , coherence , and readability. IMPORTANT: - Do NOT score the PREFIX quality; score ONLY the continuation 's text quality. Examples: Rating: [[2]] - The continuation is fluent and coherent , and it follows naturally from the prefix. Rating: [[1]] - The continuation is mostly understandable but has noticeable issues (awkwardness , jumps , mild repetition). Rating: [[0]] - The continuation is hard to read (incoherent , severe repetition) or does not connect to the prefix. Score: 0 = very poor (incoherent , severe repetition , hard to read) 1 = understandable but with issues (awkward phrasing , jumps , mild repetition) 2 = fluent , coherent , easy to read Output ONLY one line in exactly this format: Rating: [[0]] or Rating: [[1]] or Rating: [[2]] [PREFIX - for context only , do not score] prompt_text [CONTINUATION - SCORE THIS ONLY] answer Now score the CONTINUATION 's text quality. Answer with 'Rating: [[0]] ' or 'Rating: [[1]] ' or 'Rating: [[2]] ' only. Figure 52.Prompt for the steering quality-score judge. 113 Part VII Steerling-8B mid-training details NMid-training recipe N.1Nemotron: real, synthetic, and mixed Nemotron-C-HQ contains two kinds of natural-language data: real webtext, and synthetic question- and-answer rephrasings generated from the same documents. To decide which to use for the mid- training mixture, we compare three natural-language sources, each a 10B-token run from the final pretraining checkpoint: real only, synthetic only, and a mixture of the two (Table35). SourceMMLU GSM8K ARC-C HSwag HEval WinoG Avg. Base model 0.298 0.140 0.484 0.673 0.049 0.596 0.373 Real0.376 0.4410.492 0.681 0.0370.616 0.440 Synthetic0.370 0.4260.5030.6770.0550.615 0.441 Mixed0.370 0.431 0.4990.6820.0310.6180.439 Table 35.Nemotron natural-language source comparison, each a 10B-token run on a fixed math base from the final pretraining checkpoint. HSwag: HellaSwag; HEval: HumanEval; WinoG: WinoGrande. All three improve substantially over the base model and track closely on most benchmarks, with near-identical averages. Real tokens give the strongest knowledge and math scores (MMLU and GSM8K), the capabilities mid-training most needs to recover, so we use real tokens for the natural-language portion. N.2Final mid-training Steerling-8B configuration Table36lists the midtraining configuration alongside the pretraining values for comparison. Hyper- parameters not listed are unchanged from pretraining (AppendixK). 114 PretrainingMid-training Data Tokens1.2T150B MixtureNemotron-C-HQreasoning + Code Natural-language sourcereal + syntheticreal Masking Schedulemoving Gaussianuniform Range / center0.2!0.8U(0.05,0.95) Concept module α known floor0.50(annealed) α unknown floor0.50(fixed to prediction) Independence losssingle termtwo terms Residual dropoutp ε 0.10.3 Known head top-kdense32 Unknown head top-kdense128 Optimization Learning rateconstantdecayed to0 Steering phases Phases—4 (interleaved) Steering data—token-level,400M tokens Injection strengthγ—1.0 Injection layers—ℓL inj λ respond ,λ express —1.0, 1.0 Table 36.Mid-training configuration for Steerling-8B, with pretraining values for comparison. 115