# A Primer in Post-Training Reasoning Data: What We Know About How It Works > Reasoning data quality is verifier-relative and base-relative, not intrinsic, so progress attribution requires documenting the full feedback interface, not just prompt-response pairs. - **Source:** [arXiv](https://arxiv.org/abs/2606.02113) - **Published:** 2026-09-05 - **Permalink:** https://picx.dev/p/X7YScR - **Whiteboard:** https://picx.dev/p/X7YScR/image ## Summary # Scaling Laws and the Data Substrate: An Attribution Framework for Post-Training Reasoning Data ## Abstract The rapid advancement of large language models (LLMs) has been substantially driven by post-training on reasoning data. However, the field's understanding of what constitutes a "good" reasoning dataset remains fragmented and often conflates distinct concepts. This primer provides a comprehensive synthesis of the current landscape of post-training reasoning data, moving beyond the dominant "prompt–response" paradigm. We introduce a **verifier-anchored taxonomy** that deconstructs reasoning data across three cross-cutting axes: (1) **verification contracts** (formal, environmental, judgment-based), (2) **supervision granularity** (outcome, step-level, state–action), and (3) **cross-generational lineage** (synthetic, filtering, contamination). We analyze the **quality** of reasoning data not as an intrinsic property but as a **base-relative** and **verifier-relative** attribute, and explore the core question of **attribution** : when a model improves, which component of the data interface changed? We conclude by outlining a **release lineage** framework, synthesizing best practices for documenting the source, construction, and limitations of reasoning data. --- ## 1. Introduction The post-training phase of LLMs has increasingly focused on enhancing reasoning capabilities, with a growing body of work highlighting the critical role of **reasoning data**—the data used to fine-tune models to reason through complex problems. Despite this, the field lacks a unified framework for systematically understanding, comparing, and attributing the gains and failures of these data. This primer addresses this gap by proposing a structured **taxonomy** for reasoning data, moving beyond the simplistic "prompt–response" view. We organize the field around four fundamental questions: 1. **What is reasoning data?** (Data objects, components, and verifiability) 2. **What makes reasoning data useful?** (Quality metrics, difficulty, and base-relative value) 3. **How is reasoning data built?** (Construction pipelines, prompt sourcing, trace generation, and search) 4. **How do we attribute gains?** (Scaling laws, efficiency, and attribution) Our central thesis is that a reasoning data object is not a static artifact but a **verifier-bearing feedback interface**. A data point is useful not merely because it contains a correct answer, but because it provides a verifiable, auditable, and attributable signal. We argue that the field's focus should shift from simply constructing larger datasets to **designing better feedback interfaces**. --- ## 2. What Is Reasoning Data? We define a reasoning data object as a **verifier-bearing feedback interface**, not just a prompt–response pair. This includes: - **Correctness** (formal, environmental, judgment-based) - **Trace quality** (trajectory-relative, chain-relative, process-relative) - **Difficulty** (base-relative, verifier-relative, base-relative) - **Coverage** (lineage-relative, leakage-relative, contamination-relative) A reasoning data object is a **verifier-bearing feedback interface** because it carries within it a verifier (or a verifiable signal) that can be used for training, evaluation, or attribution. --- ## 3. What Makes Reasoning Data Good? Reasoning data quality is a **relational** attribute. We identify three axes: ### 3.1 Correctness Is Verifier-Relative Correctness is a **versioned verifier** contract, not an answer string. A correct answer is only as good as the verifier that validates it. This is particularly critical in domains with **formal verifiers** (e.g., math, code) versus **environmental verifiers** (e.g., tool use, web, code execution). ### 3.2 Good Traces Are Not Just Long Traces A visible trace is **suggestive evidence**, not proof of mechanism. A rationale may be **locally valid**, but it may be **task-relevant** or **grounded** in a verifier. Trace quality is therefore **trajectory-relative** and **process-relative**. ### 3.3 Difficulty Is Base-Relative Difficulty is an **item–model–sampling** relation. A hard item is not necessarily a **useful item**, because it may be **unreachable** for one base, **gradient-producing** for another, and **saturated** for a third. ### 3.4 Coverage, Leakage, and Lineage Coverage is a **recipe**, not a count. A useful release should therefore report the **generator**, **filter**, **split**, **teacher**, **verifier of record**, **decontamination status**, and **known inheritance risks**. --- ## 4. How Is Post-Training Reasoning Data Built? Public reports have made the RL scaffold increasingly visible: **long-CoT SFT**, **distillation**, **RLVR**, **reward modelling**, and **GRPO-style optimization** recur across DeepSeek-R1, Kimi K1.5, Qwen3, Magistral, Phi-4-reasoning, and Llama-Nemotron. But **optimizer visibility** is not causal isolation. **DAPO** changes filtering and loss geometry, **MiniMax-M1** changes importance-weight control, and **Qwen3-Coder-style** systems move signal into long-horizon tool interaction. ### 4.1 Where Do Reasoning Prompts Come From? **Prompt sourcing** is where a recipe first becomes data-specific. **OpenThoughts** makes source, mixture, filtering, answer generation, and teacher choice explicit ablation variables; **math and code corpora** turn olympiad pools, web-mined problems, executable tests, and model-conditioned solve rates into prompt-side fields. Small-set results such as **LIMO** and **s1** provide the counterpoint: few demonstrations can elicit strong behaviour only when the skill already lies within the base policy’s support. Prompt releases should therefore report source, filter, verifier, per-base pass rate, and estimate date, not only domain and count. ### 4.2 Who Writes the Trace? **Trace construction** fixes the author, process field, and grounding of the trajectory. Teacher traces can transfer decomposition style, uncertainty expression, formatting, tool use, and stopping behaviour, not merely answers. Process-supervised variants make hidden structure more explicit: **PRM800K** labels steps, **Math-Shepherd** estimates rollout value, **OmegaPRM** searches for first errors, **PRIME** derives implicit process reward, and **PROF** filters incoherent chains. The dispute is therefore not whether traces matter, but whether **outcome-only RL**, **explicit PRMs**, or **trainable verifiers** expose the relevant process field. ### 4.3 Where Do Search-Time Gains Live? **Agentic reasoning** moves search from decoding into an executable substrate. Repository, web, app, OS, and tool benchmarks define samples through states, actions, observations, files, tests, and terminal predicates. The lesson is not that every task needs an environment, but that **environment data** should be **replayable**. Successful transcripts alone erase failed actions, recoveries, retries, state diffs, and hidden predicates—the branches where credit assignment is visible. ### 4.4 Can Self-Play Eliminate Curation? **Self-play** does not eliminate curation; it moves curation into the **anchor** that turns generated behaviour into trainable feedback. **STaR** anchors rollouts to external answers, **R-Zero** separates Challenger and Solver roles, **Absolute Zero** uses the Python interpreter, **TTRL** turns test-time majority into reward, and **multi-agent** variants distribute these functions across roles. These systems differ less in agent count than in what makes a trajectory admissible: answer availability, executable feedback, majority selection, role-mediated challenge, or archive-based evaluation. ### 4.5 What Does the Policy Train Against? The **reward channel** is also a data object: a “reward” may be a **formal checker**, **process verifier**, **learned reward model**, **rubric judge**, or **closed-loop selection rule**. Master-key attacks, spurious rewards, GSM-Symbolic perturbations, and verifier-gaming tests all show that reward signals can be broad yet brittle. Rubric systems extend the same problem to domains without symbolic checking: **HealthBench**, **RaR**, **OnlineRubrics**, **Omni-RM**, **AutoRubric**, **RubricArm**, and **PoP** make criteria, elicitation, modality, and judge-policy interaction part of the reward-data pipeline. ### 4.6 What Does a Frontier Pipeline Look Like? A **frontier pipeline** is an orchestration of the fields above. **Distill-then-RL** systems buy cold-start speed through teacher traces; **small-warmup multistage RL** systems let RL reshape the policy; **pure-RL** lines reduce teacher inheritance but increase dependence on verifier and prompt support. The optimizer is therefore the wrong final unit of comparison: a reported gain is interpretable only after the prompt support, trace teacher, substrate, anchor, verifier, scaffold, and inference budget are declared. --- ## 5. Scaling: Asymptotes on the Data Substrate, Efficiency on the Optimiser Recent scaling sweeps make reasoning post-training look less like a single law than a ledger of what changed. We use their equations as a reading device rather than a universal law: $$R(C) = R_0 + \frac{A - R_0}{1 + (C_{\mathrm{mid}} / C)^B}, \quad \log L(N,C) = E(N) - k(N) \log C$$ Here A is the reachable ceiling, while B or $k(N)$ captures approach efficiency. A benchmark gain is therefore not self-explanatory: it may move the ceiling, improve approach speed, or change the measurement surface. ### 5.1 Asymptotes and Efficiency The useful commonality between the Khatri and Tan laws is the separation: some choices change the reachable frontier, while others change the path toward it. Data substrate, verifier quality, support coverage, context, architecture, and search topology can move the ceiling; loss design, sampling, rollout budget, curriculum, precision, and warm-start distillation more directly alter efficiency. This distinction also organizes the RLVR debate. Some read current RLVR as sharpening trajectories already accessible to the base policy. The counter-literature relaxes different closures: **ProRL** extends horizon length; **RL-PLUS** injects external rollouts; **CoT-Pass@K** changes the success metric; and **PASS@(k, T)** adds interaction depth. Thus “RL expands capability” is not a scalar claim; it specifies which closure was broken. ### 5.2 Data Uniqueness and Stage Composition Tan et al.’s decomposition $D_{\mathrm{total}} = D_{\mathrm{unique}} \times \tau$ makes reuse part of the scaling surface rather than a preprocessing detail. Small-pool releases such as **LIMO** and **s1** show hundreds or thousands of examples can elicit strong behaviour when the base already supports the skill. Cross-base studies sharpen the point: cognitive behaviours and even spurious rewards are base-conditional rather than generic. Distillation and stage composition decide what enters RL before scaling is measured, since teacher traces, grounded thoughts, mid-training, SFT, and RL prefer different quality–diversity trade-offs. Scaling is therefore parameter count under a base prior, uniqueness budget, teacher lineage, and stage order. ### 5.3 Inference-Time Compute and Trace Topology Inference-time scaling changes the observable, not merely the decoding setting. Repeated sampling can scale coverage when candidates are verifiable; **s1** makes budget forcing an explicit intervention on thinking length; and **parallel-distill-refine**, **TTRL**, and **MiGrATe** make the deployment-time selector, verifier, or mixed policy part of the reported data object. Long-context reasoning adds another topology: more context can raise the ceiling while reducing efficiency, whereas **Markovian Thinking** changes the environment so reasoning proceeds through fixed-size chunks and learned state. A pass@N score under serial extension, repeated sampling, parallel refinement, test-time training, or Markovian chunking is therefore not the same observable. ### 5.4 Verifier Scaling and Release Lineage The verifier scales with the policy, but not necessarily in the right direction. Master-key attacks, GSM-Symbolic perturbations, and verifier-robustness studies show that reward signals can be broad yet brittle. Process-verifier corpora follow the same logic: **PRM800K**, **Math-Shepherd**, **OmegaPRM**, and **Qwen2.5-Math-PRM** differ in whether supervision comes from human labels, Monte-Carlo scores, first-error localization, or judge consensus. CoVerRL and DeepSeekMath-V2 suggest co-evolving generators and verifiers, but co-evolution increases the need for versioning rather than removing it. --- ## 6. Conclusion Post-training reasoning data have become the hidden substrate of reasoning-model progress, and the stream of releases is accelerating. The evidence reviewed here suggests that the reusable unit is not a prompt–response pair, but a **verifier-bearing feedback interface** whose value depends on the verifier, base model, lineage, optimizer, scaffold, and inference budget. The central unresolved question is therefore **attribution** : when a model improves, which part of that interface changed? This primer helps the community move from reporting gains to making them **inspectable**, **comparable**, and **testable**. --- ## Limitations This primer is limited by the public evidence available for post-training reasoning data. Closed pipelines, proprietary data mixtures, and undocumented release practices are necessarily absent, and many public reports omit lineage cards, verifier versions, compute and inference budgets, and contamination audits. Our synthesis is also question-driven rather than a formal meta-analysis: we include work that exposes reasoning-data objects, feedback interfaces, construction recipes, or scaling surfaces, which may underrepresent work that reports only aggregate benchmark gains or uses non-public artifacts. In addition, the evidence base is heterogeneous, spanning peer-reviewed papers, arXiv preprints, technical reports, benchmarks, and model cards; we use these sources to document public recipes and release practices, not to establish causal rankings. Finally, we do not independently re-run training recipes, audit contamination, or validate every verifier. The taxonomy should therefore be read as an **attribution and reporting framework** whose fields will need updating as reasoning post-training expands to new multimodal, multilingual, agentic, and co-evolving verifier-generator settings. --- ## Acknowledgments We thank the authors of the works discussed in this primer for their contributions to the field, and we refer readers to the original papers for full details. We also thank the authors of the **DeepSeek-R1**, **Kimi K1.5**, **Qwen3**, **Magistral**, **Phi-4-reasoning**, and **Llama-Nemotron** teams for their public releases and technical reports, which have made this synthesis possible. We also thank the broader community for the open-source releases of data, models, and benchmarks that have shaped the field. --- _Markdown view of https://picx.dev/p/X7YScR, served by PicX — AI-generated visual whiteboard summaries of research papers._