CEDAR-GRPO: Process-Aware Reinforcement Learning for General Abductive Reasoning in LLMs

# CEDAR-GRPO: Process-Aware Reinforcement Learning for General Abductive Reasoning in LLMs
Source: [https://arxiv.org/html/2608.14791](https://arxiv.org/html/2608.14791)
Moein Salimi1Danial Parnian1Shaygan Adim1Amirmohammad Ebrahiminasab2,\*Nima Alighardashi1,\*Parsa Gholami1,\*Sahand Akramipour1Mahdi Jafari Siavoshani1,†Mohammad Hossein Rohban1,†1Sharif University of Technology2University of Tehran\{mjafari,rohban\}@sharif\.edu

###### Abstract

Abductive reasoning, often characterized as inference to the best explanation, is central to explanation under uncertainty, from everyday sense\-making and investigation to scientific discovery\. Yet LLM research has mostly studied abduction through narrow, task\-specific benchmarks, making it unclear whether observed gains transfer beyond the benchmark family used for training or evaluation\. We ask whether RL post\-training can improve abduction as a transferable reasoning capability\. We introduce CEDAR\-GRPO, a process\-aware framework that combines final\-answer correctness with abductive rewards for evidence coverage and evidence\-to\-explanation directionality\. Four open\-weight LLMs are post\-trained on a controlled, domain\-neutral mixture of abductive hypothesis\-generation and hypothesis\-selection tasks\. We evaluate them on 11 unseen tasks spanning hypothesis selection, missing\-fact generation, defeasible inference, long\-context investigation, clinical reasoning, code debugging, and non\-abductive controls\. CEDAR\-GRPO improves every model on every held\-out task over both base models and correctness\-only GRPO, with average gains of 7\.4 and 2\.7 points, respectively, and a maximum gain of 30\.8 points\. Ablations confirm that RL, abductive reward design, and task diversity each contribute to transfer\. Process\-level metrics further show stronger abductive behavior, including exploration of alternatives, elimination of rivals, backtracking, and uncertainty marking\.111Code and data are available at[https://github\.com/cedar\-grpo/cedar\-grpo](https://github.com/cedar-grpo/cedar-grpo)\.

11footnotetext:Equal contribution\.22footnotetext:Equal corresponding authors\.## 1Introduction

A clinician interpreting puzzling symptoms, an engineer diagnosing an intermittent failure, and a detective reconstructing a crime from scattered clues all face the same reasoning problem: they must infer a hidden explanation from incomplete evidence\. This explanatory step is abductive reasoning, classically understood as moving from observations to a hypothesis that would make them less surprising, and often described as inference to the best explanation\([26](https://arxiv.org/html/2608.14791#bib.bib16);[24](https://arxiv.org/html/2608.14791#bib.bib35)\)\. For LLMs, abduction is a crucial capability because many real uses of language models—including diagnosis, scientific hypothesis formation, evidence interpretation, debugging, and long\-context investigation—depend on identifying plausible latent causes, missing facts, or underlying mechanisms that best account for uncertain observations\.

ART example\.
Observation 1:Brad was gazing in the fountain downtown\.
Observation 2:Then Brad walked away, whistling happily\.
Hypothesis 1:Brad made a wish on a shooting star\.
Hypothesis 2:Brad made a wish with a penny\.
Gold answer:Hypothesis 2\.Correctness\-only GRPO
Brad was downtown looking into the water of the fountain\.Fountains have still pools of water that act like mirrors\. Since the tall buildings downtown block the view of the sky, he saw a shooting star in the fountain’s reflection\.He made a wish on that star and walked away whistling because he was happy he caught such a rare sight in the city\.Final Answer: Hypothesis 1\.CEDAR\-GRPO
Let us look at the evidence: Brad is “gazing in the fountain” and then walks away “whistling happily\.”Public fountains are strongly associated with throwing in a coin to make a wish\. If we consider Hypothesis 1, Brad would need to be looking upward at the sky\. But he is specifically gazing in the fountain, so Hypothesis 2 better fits the observations\.He likely tossed a penny into the water and walked away happy after making his wish\.Final Answer: Hypothesis 2\.Figure 1:CEDAR\-GRPO grounds its answer in the observed evidence, while correctness\-only GRPO relies on unsupported assumptions\.Recent work has introduced valuable benchmarks for abductive reasoning, including ART for commonsense hypothesis selection\([2](https://arxiv.org/html/2608.14791#bib.bib8)\), e\-CARE for explainable causal reasoning\([6](https://arxiv.org/html/2608.14791#bib.bib19)\), DDXPlus for differential diagnosis\([37](https://arxiv.org/html/2608.14791#bib.bib11)\), ProofWriter and AbductionRules for missing\-fact generation\([35](https://arxiv.org/html/2608.14791#bib.bib20);[40](https://arxiv.org/html/2608.14791#bib.bib10)\), and knowledge\-graph methods for generating hypotheses that explain observed relational structure\([1](https://arxiv.org/html/2608.14791#bib.bib13)\)\. However, progress is often evaluated within a single benchmark family, task format, or structured domain, making it difficult to determine whether a method improves abduction as a reusable reasoning capability or merely adapts a model to a narrow formulation\. This question is especially important because abductive tasks take two principal forms: hypothesis generation, in which a model proposes an explanation, and hypothesis selection, in which it chooses among candidates\([29](https://arxiv.org/html/2608.14791#bib.bib1)\)\. A convincing account of abductive improvement should therefore demonstrate transfer across both forms and beyond the specific formats used for training\.

We investigate whether reinforcement\-learning post\-training can produce such transferable improvements\. We introduceCEDAR\-GRPO\(Correctness, Evidence coverage, and Directionality Abductive Rewards\), a process\-aware GRPO framework that combines deterministic final\-answer correctness—using exact\-match, label\-based, or execution\-based verification—with two abductive rewards: coverage of the observed evidence and preservation of the direction from observations to explanation\. These rewards are designed to encourage explanations that account for the available evidence while maintaining the defining structure of abductive inference\. Figure[1](https://arxiv.org/html/2608.14791#S1.F1)illustrates the resulting behavior on ART, and Figure[2](https://arxiv.org/html/2608.14791#S1.F2)summarizes the training and evaluation pipeline\.

![Refer to caption](https://arxiv.org/html/2608.14791v1/figures/cedar_grpo_overview.png)Figure 2:Overview of CEDAR\-GRPO\. The framework trains LLMs on abductive hypothesis generation and selection tasks using structured outputs and a composite reward for correctness, evidence coverage, and evidence–explanation directionality, then evaluates both task accuracy and reasoning behavior on held\-out datasets\.We post\-train four open\-weight backbones—Qwen3\-4B, Qwen3\-8B, DeepSeek\-R1\-Distill\-Qwen\-7B, and Llama\-3\.1\-8B\-Instruct—on a controlled, domain\-neutral, knowledge\-light mixture spanning both hypothesis generation and hypothesis selection\. The mixture includes formal and commonsense hypothesis selection, missing\-fact generation, and rule\-learning tasks\. This design makes transfer easier to interpret: improvements outside these training formats are less plausibly attributable to narrow domain adaptation\.

We evaluate on 11 unseen tasks covering hypothesis selection, missing\-fact generation, defeasible inference, long\-context investigation, clinical reasoning, code debugging, forward causal reasoning, and general multistep reasoning\. The suite contains direct abductive targets, abduction\-adjacent transfer tasks, and non\-abductive controls\. We also analyze reasoning traces using seven process\-level metrics—branchiness, backtracking, differential elimination, prior invocation, evidence coverage, evidence–explanation directionality, and uncertainty marking—to test whether accuracy gains coincide with changes in abductive behavior\.

Across all four backbones, CEDAR\-GRPO improves performance on every held\-out task relative to both the base models and the correctness\-only GRPO baseline, Cor\-GRPO\. Averaged across settings, it gains 7\.4 points over the base models and 2\.7 points over Cor\-GRPO, with a maximum gain of 30\.8 points over the base model on MuSR\-Murder for DeepSeek\-R1\-Distill\-Qwen\-7B\. As expected, evidence coverage and directionality, both directly optimized during training, improve substantially\. Beyond these reward metrics, CEDAR\-GRPO also improves all five held\-out process metrics, showing greater exploration of alternatives, elimination of competing hypotheses, backtracking, prior invocation, and uncertainty marking\. Ablations indicate that RL, both process\-reward components, and the combination of generation\- and selection\-oriented training data each contribute to transfer\.

Taken together, these results show that abductive reasoning can be strengthened as atransferable capability, rather than through benchmark\-specific optimization alone\. Our main contributions are: \(1\) acontrolled framework for studying cross\-task abductive transfer; \(2\)CEDAR\-GRPO, which augments correctness\-based RL with rewards for evidence coverage and evidence–explanation directionality; and \(3\)systematic evidence of transferacross four backbones and 11 held\-out tasks, supported by task\-level evaluation, held\-out process metrics, and targeted ablations\.

## 2Related Work

### 2\.1Abductive Tasks and Benchmarks

Abductive reasoning in NLP is commonly studied through tasks that ask models to infer plausible explanations, missing facts, or latent causes from incomplete observations\. Existing benchmarks instantiate this ability in several forms, including commonsense hypothesis selection in ART and UNcommonsense, explainable causal reasoning in e\-CARE, clinical differential diagnosis in DDXPlus, abductive missing\-fact generation in ProofWriter and AbductionRules, and structured hypothesis generation over knowledge graphs\([2](https://arxiv.org/html/2608.14791#bib.bib8);[41](https://arxiv.org/html/2608.14791#bib.bib9);[6](https://arxiv.org/html/2608.14791#bib.bib19);[37](https://arxiv.org/html/2608.14791#bib.bib11);[35](https://arxiv.org/html/2608.14791#bib.bib20);[40](https://arxiv.org/html/2608.14791#bib.bib10);[1](https://arxiv.org/html/2608.14791#bib.bib13)\)\. These resources have been essential for measuring abductive behavior, but they often make improvements hard to disentangle from particular domains, output formats, or verification regimes\. Our work therefore treats abduction as a broader capability whose improvement should transfer across both hypothesis\-generation and hypothesis\-selection settings, as well as beyond the tasks used for training\.

### 2\.2Methodologies for Abductive Reasoning

Prior methods for improving abductive reasoning have largely relied on supervised adaptation or inference\-time scaffolding\. Supervised, ranking\-based, and preference\-based objectives train models from labeled abductive data, either by imitating reference explanations or by learning to prefer stronger hypotheses\([43](https://arxiv.org/html/2608.14791#bib.bib18);[13](https://arxiv.org/html/2608.14791#bib.bib15)\)\. In contrast, recent LLM pipelines use prompting, retrieval, multi\-agent decomposition, or symbolic checking to separate observation interpretation, hypothesis generation, and hypothesis evaluation\([25](https://arxiv.org/html/2608.14791#bib.bib2);[23](https://arxiv.org/html/2608.14791#bib.bib4);[14](https://arxiv.org/html/2608.14791#bib.bib5);[21](https://arxiv.org/html/2608.14791#bib.bib7);[16](https://arxiv.org/html/2608.14791#bib.bib6)\)\. Recently, LogiDynamics\([42](https://arxiv.org/html/2608.14791#bib.bib23)\)demonstrated that embedding abductive inference within a broader logical\-reasoning pipeline alongside iterative refinement systematically enhances performance\. While effective for task\-specific adaptation or inference control, these approaches leave open whether post\-training can internalize broadly generalizable abductive behaviors\.

### 2\.3RL Post\-Training and Process\-Aware Rewards

Reinforcement learning offers a natural way to optimize abductive reasoning against task\-level criteria, but its use in abduction remains narrow\. RLF\-KG uses PPO\-style feedback from knowledge graphs to generate logical hypotheses that explain observed facts, while CtrlHGen applies GRPO\-based reinforcement tuning with rewards for semantic alignment and condition satisfaction, and DARK applies a coupled\-GRPO variant to a masked diffusion model for unifying deductive and abductive reasoning over knowledge graphs\([1](https://arxiv.org/html/2608.14791#bib.bib13);[8](https://arxiv.org/html/2608.14791#bib.bib12);[7](https://arxiv.org/html/2608.14791#bib.bib14)\)\. More broadly, GRPO was introduced for mathematical reasoning as a memory\-efficient alternative to PPO\([31](https://arxiv.org/html/2608.14791#bib.bib22)\), and recent RL post\-training work shows that verifiable rewards can elicit behaviors such as verification, reflection, and strategy adaptation\([10](https://arxiv.org/html/2608.14791#bib.bib21)\)\. However, these successes are concentrated in math, code, or structured reasoning\. CEDAR\-GRPO instead studies broad abductive transfer: it combines verifiable correctness with process\-aware rewards for evidence coverage and evidence–explanation directionality, drawing inspiration from process\-supervision work while avoiding reliance on human\-labeled reasoning traces\([22](https://arxiv.org/html/2608.14791#bib.bib24);[38](https://arxiv.org/html/2608.14791#bib.bib25)\)\.

## 3Data Collection

### 3\.1Dataset selection

We construct both the training pool and evaluation suite to probe abductive reasoning in a controlled yet heterogeneous manner\. Following the two\-stage view of abduction adopted in recent survey work, we treat abductive reasoning as comprising*Stage I*hypothesis generation and*Stage II*hypothesis selection\([29](https://arxiv.org/html/2608.14791#bib.bib1)\)\. This distinction is crucial in our setting: a model can improve by proposing better explanatory hypotheses, by evaluating candidate hypotheses more reliably, or by doing both\. To avoid conflating these possibilities, we do not build the training pool around a single benchmark family\.

Our data collection strategy is intentionally asymmetric\. The training pool is limited to broadly non\-specialized resources and spans both stages of the abductive pipeline, so that gains are less likely to reflect narrow domain adaptation or isolated progress on either generation or selection\. The evaluation suite is broader: alongside classic abductive benchmarks, it includes abstract logical data, neighboring tasks that admit an abductive interpretation, and explicit non\-abductive controls\. This design lets us ask a sharper question: does GRPO learn reusable abductive behavior, or merely adapt to the surface form of a few benchmarks? Throughout, we maintain a strict separation between learning and measurement: evaluation datasets are never used for model optimization or model selection\. Additional dataset\-specific notes are deferred to Appendix[A](https://arxiv.org/html/2608.14791#A1)\.

### 3\.2Training and validation data

Our training pool, detailed in Appendix[A](https://arxiv.org/html/2608.14791#A1)\(Table[5](https://arxiv.org/html/2608.14791#A1.T5)\), is composed of datasets targeting both Stage I and Stage II of abduction\. The Stage II group requires the model to assess the plausibility of an explanation, cause, or evidential relation, or to select among competing hypotheses\. UniADILR\-HGc and Balanced COPA provide compact classic abductive selection signals\. CauseLogics asks whether a candidate premise makes an inference sensible, capturing a core part of hypothesis assessment: checking whether a hypothesis can logically and soundly explain the observations\. CLIMATE\-FEVER similarly provides a Stage II\-adjacent signal, requiring the model to choose among supports, refutes, not enough info, and disputed for a claim–evidence relation\. The Stage I group instead requires constructing missing hypotheses rather than choosing from fixed alternatives\. AbductionRules asks the model to generate a missing explanatory fact from a rule context, while List Function and Crypto require inferring a latent pattern from input\-output examples and generating code that implements it; the code can therefore be interpreted as an explicit candidate hypothesis about the governing rule, making these tasks useful proxies for hypothesis generation\.

This composition is useful for GRPO in two ways\. Selection\-style tasks provide clean signals for plausibility judgments, while generation\-style tasks require the model to construct missing hypotheses rather than only evaluate a closed set of candidates\. Mixing both makes it less likely that post\-training improves only a narrow answer\-selection heuristic\. Just as importantly, the training sources remain non\-specialized, which makes the subsequent evaluation cleaner: if performance improves on downstream tasks from other settings, the gain is less plausibly explained by domain memorization\.

### 3\.3Evaluation data

Our evaluation suite, detailed in Appendix[A](https://arxiv.org/html/2608.14791#A1)\(Table[6](https://arxiv.org/html/2608.14791#A1.T6)\), is deliberately broader than the training pool\. It includes classic abductive benchmarks—ART \(α\\alphaNLI\) for Stage II hypothesis selection and NeuLR’s abductive split for Stage I missing\-fact generation—as well as Defeasible NLI, which tests the Stage II\-adjacent ability to judge whether new evidence strengthens or weakens a hypothesis\.

We also evaluate transfer to neighboring domains that admit abductive interpretations: GoEmotions as latent affect inference from utterances, MuSR: Murder as culprit inference, MedQA as clinical diagnosis, and ML\-debugging as software diagnosis and repair\. Finally, we include non\-abductive controls: Balanced COPA in the effect direction for forward causal reasoning, MuSR: Object and MuSR: Team for long\-context multistep reasoning, and StrategyQA as a broad general\-reasoning control\. Together, these groups help distinguish direct abductive transfer from broader changes in reasoning behavior, while preserving a strict separation between training sources and held\-out evaluation\.

## 4Methodology

### 4\.1Problem Formulation

We operationalize abductive reasoning as selecting or generating the explanation that best accounts for incomplete observations\. Across the training mixture, the target explanation may be a candidate cause, a missing fact, or an inferred transformation rule\. Our experiments use four open\-weight backbones: Qwen3\-4B, Qwen3\-8B, DeepSeek\-R1\-Distill\-Qwen\-7B, and Llama\-3\.1\-8B\-Instruct\. This model suite spans 4B–8B parameters and includes both general instruction\-tuned and reasoning\-oriented backbones, allowing us to test whether our method improves abductive reasoning across model sizes and families\.

### 4\.2Structured CoT Prompting

Models generate structured outputs of the form⟨think⟩​β​⟨/think⟩​⟨answer⟩​α​⟨/answer⟩\\langle\\texttt\{think\}\\rangle\\,\\beta\\,\\langle\\texttt\{/think\}\\rangle\\langle\\texttt\{answer\}\\rangle\\,\\alpha\\,\\langle\\texttt\{/answer\}\\rangle, whereβ\\betais the reasoning trace andα\\alphais the final answer\. Task correctness is computed fromα\\alpha, whereas the process rewards are computed from the user prompt andβ\\beta\. Full generator prompts are provided in Appendix[B](https://arxiv.org/html/2608.14791#A2)\.

### 4\.3Composite Reward Design

For a user promptxx, generated completionyy, ground truthgg, and datasetdd, letβ⁡\(y\)=ExtractThink⁡\(y\)\\beta\(y\)=\\operatorname\{ExtractThink\}\(y\)andα⁡\(y\)=ExtractAnswer⁡\(y\)\\alpha\(y\)=\\operatorname\{ExtractAnswer\}\(y\)\. CEDAR\-GRPO assigns

R⁡\(x,y,g,d\)=rcor\+rcov\+rdir3\.R\(x,y,g,d\)=\\frac\{r\_\{\\mathrm\{cor\}\}\+r\_\{\\mathrm\{cov\}\}\+r\_\{\\mathrm\{dir\}\}\}\{3\}\.\(1\)The correctness reward is

rcor=Vd​\(α⁡\(y\),g\),Vd​\(α⁡\(y\),g\)∈\{0,1\},r\_\{\\mathrm\{cor\}\}=V\_\{d\}\(\\alpha\(y\),g\),\\qquad V\_\{d\}\(\\alpha\(y\),g\)\\in\\\{0,1\\\},\(2\)whereVdV\_\{d\}is the dataset\-specific exact\-match, set\-match, or execution\-based verifier\. For coverage, the judge returnsmmobservation details fromxxand marks each as addressed \(zj=1z\_\{j\}=1\) or unaddressed \(zj=0z\_\{j\}=0\):

rcov=1m​∑j=1mzj\.r\_\{\\mathrm\{cov\}\}=\\frac\{1\}\{m\}\\sum\_\{j=1\}^\{m\}z\_\{j\}\.\(3\)If no valid detail list is returned,rcovr\_\{\\mathrm\{cov\}\}is set to zero\.

The directionality reward is00for traces that assume an explanation and reason back toward the evidence,0\.50\.5for mixed or ambiguous directionality, and11for traces that reason from evidence toward an explanation\. Both process rewards are scored bygpt\-oss\-120bat temperature0\.00\.0from the user promptxxand reasoning spanβ⁡\(y\)\\beta\(y\); correctness is computed only fromα⁡\(y\)\\alpha\(y\)\. Exact verifiers, failure handling, dataset notes, and training\-time judge prompts are given in Appendix[D](https://arxiv.org/html/2608.14791#A4)\.

Figure[3](https://arxiv.org/html/2608.14791#S4.F3)illustrates how three traces with the same correct answer receive different rewards because they differ in evidence coverage and reasoning directionality\. We selected these process terms because they apply across the full training mixture and are less susceptible than alternative trace metrics to simple reward\-hacking strategies such as verbosity and hedging\. The screening analysis is reported in Appendix[D\.5](https://arxiv.org/html/2608.14791#A4.SS5)\.

### 4\.4GRPO Optimization and Compute

For each prompt, we sampleG=4G=4completions and optimize Equation[1](https://arxiv.org/html/2608.14791#S4.E1)with Group Relative Policy Optimization \(GRPO\), which forms group\-relative advantages without a separate value network\. All backbones use NF4 quantization and LoRA fine\-tuning\. All local training and evaluation were conducted on a single NVIDIA GeForce RTX 5090 GPU with 32 GB GDDR7; the reward judge was accessed remotely\. Full hyperparameters are reported in Appendix[C](https://arxiv.org/html/2608.14791#A3)\.

BalancedCOPA example \(cause selection\)\.Premise:“The investigator found inconsistencies in the witness’s story\.”Option 1:The witness fabricated parts of the account\.Option 2:The investigator was reassigned\.Gold: 1\.Coverage judge decomposes the premise into:Ainconsistencies exist;Bthey are in the witness’s story;Cthe investigator detects them\.R=\(rcor\+rcov\+rdir\)/3R=\(r\_\{\\mathrm\{cor\}\}\+r\_\{\\mathrm\{cov\}\}\+r\_\{\\mathrm\{dir\}\}\)/3\.Training progression⟶\\longrightarrowcorrect answer→\\rightarrowgrounded abductive explanationEpoch 1Step 48Generator trace \(condensed\)
<think\>Option 1 says the witness lied; Option 2 is about a reassignment\.If someone lies, their story has problems\.So it is Option 1\.</think\>
<answer\>1</answer\>Judge view\.“Problems” is a generic consequence, not an explicit explanation of any premise detail\. The trace also starts from an assumed lie and predicts the observation\.Explicitly linked details:ABCrcorr\_\{\\mathrm\{cor\}\}1\.00rcovr\_\{\\mathrm\{cov\}\}0/3rdirr\_\{\\mathrm\{dir\}\}0\.00RR0\.33Epoch 2Step 534Generator trace \(condensed\)
<think\>The observation is that the investigator found inconsistencies in the witness’s story\.I need to find what most plausibly caused this\.Option 1: Fabrication means inventing or altering details\. Combining fabricated and truthful sections createsinternal contradictions \(A\)in thewitness’s account \(B\)—exactly what “inconsistencies” describes\.Option 2: Reassignment changes workload, not story content, and gives no mechanism for inconsistencies\. Ruled out\.</think\>
<answer\>1</answer\>Judge view\.The trace begins from the observation and explicitly explains A and B\. It mentions detection, but does not yet explain why the inconsistency is discoverable by the investigator \(C\)\.Explicitly linked details:ABCrcorr\_\{\\mathrm\{cor\}\}1\.00rcovr\_\{\\mathrm\{cov\}\}2/3rdirr\_\{\\mathrm\{dir\}\}1\.00RR0\.89Epoch 3Step 1,018Generator trace \(condensed\)
<think\>Observation: the investigator found inconsistencies in the witness’s story\.I reason abductively—from this observation toward its most plausible cause\.Option 1: Fabricationcreates traceable contradictions \(A\),localizes the error in the witness’s narrative \(B\), andunravels under systematic questioning, explaining why the investigator detects it \(C\)\. All three facts are parsimoniously explained\.Option 2: Reassignment affects caseload, not story content; it has no causal path to the contradictions and would more naturally explain missed, not found, inconsistencies\. Decisively ruled out\.Conclusion: Option 1 uniquely explains the evidence\.</think\>
<answer\>1</answer\>Explicitly linked details:ABCrcorr\_\{\\mathrm\{cor\}\}1\.00rcovr\_\{\\mathrm\{cov\}\}3/3rdirr\_\{\\mathrm\{dir\}\}1\.00RR1\.00Purplecorrect final answerBlueexplicit evidence–hypothesis linkGreenevidence→\\rightarrowexplanationRedexplanation→\\rightarrowevidenceFigure 3:How CEDAR\-GRPO’s composite reward shapes a BalancedCOPA reasoning trace over training\. All three snapshots select the correct answer, sorcorr\_\{\\mathrm\{cor\}\}is already maximal\. At Step 48, the trace reasons from an assumed explanation back to a generic consequence and accounts for none of the three observation details returned by the coverage judge\. By Step 534, it starts from the observation and links two details to the hypothesis\. By Step 1,018, it explains all three details while preserving the evidence\-to\-explanation direction, yielding the maximum reward\.

## 5Experiments

This section evaluates whether post\-training with our composite reward improves general abductive reasoning, rather than simply increasing performance on the training tasks\. We organize the evidence into three parts: \(i\) held\-out task performance on the evaluation suite from Section[3](https://arxiv.org/html/2608.14791#S3); \(ii\) process\-level measurements of the reasoning traces; and \(iii\) ablations that isolate the role of RL, reward design, the two\-stage construction of the training pool, and a matched generic\-reasoning post\-training control\. We use accuracy for closed\-form selection tasks, exact or verifier\-based correctness for formal missing\-fact and rule\-learning tasks, and pass/fix success for ML\-debugging\. All reported scores are computed from the final answer field only; process metrics are computed separately from the reasoning trace\.

### 5\.1Held\-Out Task Performance

Table[1](https://arxiv.org/html/2608.14791#S5.T1)presents the main task\-level comparison\. We compare the original base model, the correctness\-only GRPO checkpoint \(denoted Cor\-GRPO\), and our main composite\-reward checkpoint, CEDAR\-GRPO\.

ModelMethodARTB\-COPADefNLIGoEmo\.MuSR\-MMuSR\-OMuSR\-TNeuLRStratQAMedQAML\-DebugQwen3\-4BBase65\.25%65\.25\\%84\.40%84\.40\\%80\.75%80\.75\\%27\.75%27\.75\\%20\.40%20\.40\\%19\.20%19\.20\\%51\.60%51\.60\\%35\.00%35\.00\\%38\.25%38\.25\\%35\.50%35\.50\\%25\.25%25\.25\\%Cor\-GRPO71\.75%71\.75\\%86\.80%86\.80\\%86\.75%86\.75\\%32\.25%32\.25\\%41\.60%41\.60\\%32\.40%32\.40\\%48\.80%48\.80\\%38\.75%38\.75\\%43\.25%43\.25\\%37\.50%37\.50\\%28\.50%28\.50\\%CEDAR\-GRPO72\.25%\\mathbf\{72\.25\\%\}88\.80%\\mathbf\{88\.80\\%\}88\.50%\\mathbf\{88\.50\\%\}34\.25%\\mathbf\{34\.25\\%\}48\.40%\\mathbf\{48\.40\\%\}33\.20%\\mathbf\{33\.20\\%\}55\.60%\\mathbf\{55\.60\\%\}40\.50%\\mathbf\{40\.50\\%\}44\.75%\\mathbf\{44\.75\\%\}37\.75%\\mathbf\{37\.75\\%\}29\.00%\\mathbf\{29\.00\\%\}Qwen3\-8BBase72\.25%72\.25\\%84\.40%84\.40\\%82\.50%82\.50\\%39\.50%39\.50\\%23\.20%23\.20\\%26\.40%26\.40\\%55\.20%55\.20\\%36\.25%36\.25\\%40\.00%40\.00\\%42\.25%42\.25\\%28\.75%28\.75\\%Cor\-GRPO74\.00%74\.00\\%86\.00%86\.00\\%87\.75%87\.75\\%45\.50%45\.50\\%43\.60%43\.60\\%34\.00%34\.00\\%58\.80%58\.80\\%39\.50%39\.50\\%44\.50%44\.50\\%46\.25%46\.25\\%29\.75%29\.75\\%CEDAR\-GRPO75\.50%\\mathbf\{75\.50\\%\}89\.20%\\mathbf\{89\.20\\%\}90\.50%\\mathbf\{90\.50\\%\}47\.00%\\mathbf\{47\.00\\%\}46\.80%\\mathbf\{46\.80\\%\}36\.80%\\mathbf\{36\.80\\%\}60\.80%\\mathbf\{60\.80\\%\}43\.75%\\mathbf\{43\.75\\%\}50\.50%\\mathbf\{50\.50\\%\}49\.25%\\mathbf\{49\.25\\%\}31\.75%\\mathbf\{31\.75\\%\}DeepSeek\-R1\-Distill\-Qwen\-7BBase70\.25%70\.25\\%85\.20%85\.20\\%81\.50%81\.50\\%30\.00%30\.00\\%26\.40%26\.40\\%38\.40%38\.40\\%49\.60%49\.60\\%31\.50%31\.50\\%42\.50%42\.50\\%35\.75%35\.75\\%23\.75%23\.75\\%Cor\-GRPO73\.50%73\.50\\%88\.00%88\.00\\%82\.25%82\.25\\%31\.50%31\.50\\%56\.40%56\.40\\%40\.40%40\.40\\%50\.00%50\.00\\%33\.00%33\.00\\%42\.25%42\.25\\%36\.25%36\.25\\%22\.75%22\.75\\%CEDAR\-GRPO78\.50%\\mathbf\{78\.50\\%\}89\.60%\\mathbf\{89\.60\\%\}87\.75%\\mathbf\{87\.75\\%\}35\.50%\\mathbf\{35\.50\\%\}57\.20%\\mathbf\{57\.20\\%\}49\.60%\\mathbf\{49\.60\\%\}51\.60%\\mathbf\{51\.60\\%\}35\.50%\\mathbf\{35\.50\\%\}45\.75%\\mathbf\{45\.75\\%\}39\.50%\\mathbf\{39\.50\\%\}24\.75%\\mathbf\{24\.75\\%\}Llama\-3\.1\-8B\-InstructBase73\.50%73\.50\\%86\.40%86\.40\\%86\.00%86\.00\\%33\.75%33\.75\\%24\.80%24\.80\\%35\.20%35\.20\\%48\.80%48\.80\\%29\.00%29\.00\\%42\.50%42\.50\\%33\.50%33\.50\\%19\.00%19\.00\\%Cor\-GRPO78\.75%78\.75\\%89\.20%89\.20\\%87\.75%87\.75\\%35\.50%35\.50\\%52\.00%52\.00\\%36\.00%36\.00\\%49\.20%49\.20\\%30\.25%30\.25\\%42\.25%42\.25\\%34\.25%34\.25\\%19\.75%19\.75\\%CEDAR\-GRPO80\.00%\\mathbf\{80\.00\\%\}90\.40%\\mathbf\{90\.40\\%\}91\.50%\\mathbf\{91\.50\\%\}39\.25%\\mathbf\{39\.25\\%\}53\.20%\\mathbf\{53\.20\\%\}36\.80%\\mathbf\{36\.80\\%\}50\.80%\\mathbf\{50\.80\\%\}32\.25%\\mathbf\{32\.25\\%\}45\.50%\\mathbf\{45\.50\\%\}35\.75%\\mathbf\{35\.75\\%\}21\.25%\\mathbf\{21\.25\\%\}

Table 1:Held\-out task performance across the evaluation suite\. Scores represent task\-native accuracy or exact\-success metrics\. CEDAR\-GRPO is our composite\-reward method; Cor\-GRPO is the correctness\-only baseline\. The best score per backbone is bolded\. Full dataset descriptions and abbreviations are provided in Appendix[A](https://arxiv.org/html/2608.14791#A1)\.This held\-out suite tests transfer beyond the training formats, spanning direct abductive tasks, abduction\-adjacent settings such as clinical diagnosis and ML debugging, and non\-abductive controls\. Comparing against both the base model and Cor\-GRPO separates gains from RL itself from gains due to the composite reward\. The results show that CEDAR\-GRPO consistently improves over Cor\-GRPO, suggesting that the process\-level signals are not merely cosmetic\. Instead, the coverage and directionality rewards are aligned with the underlying reasoning objective and lead to better final\-answer accuracy\.

### 5\.2Reasoning Trace Analysis

MetricDescriptionBranchinessExploring multiple distinct candidate explanations for the same observation\.BacktrackingExplicitly identifying an error or flaw in the reasoning and changing direction\.Differential
EliminationThe active refutation of alternative hypotheses given the specific context\.Prior InvocationIncorporating typicality or prior probability alongside case\-specific evidence\.Evidence
CoverageThe fraction of specific observation details explicitly accounted for by the chosen hypothesis\.Evidence–Expl\.
DirectionalityDemonstrating awareness that reasoning must move from given evidence toward an explanatory conclusion\.Uncertainty
MarkersThe density of probabilistic language and epistemic hedging within the trace\.Table 2:Overview of the process\-level metrics used to evaluate intermediate reasoning behaviors\.While our training improves accuracy on held\-out benchmarks, correct final answers alone do not guarantee that the underlying reasoning process has improved\. Because final\-answer accuracy is a coarse outcome measure, we complement it with process\-level evaluation to examine whether the model’s reasoning chains show the abductive behaviors targeted by our training objective\. To do this, we introduce a novel suite of process\-level metrics, summarized in Table[2](https://arxiv.org/html/2608.14791#S5.T2)and discussed in more detail in Appendix[E](https://arxiv.org/html/2608.14791#A5), that directly evaluate intermediate reasoning chains\. These metrics assess whether the model generates plausible hypotheses, connects them to the available evidence, and uses them to support the final answer\. This gives more concrete evidence that the accuracy gains are accompanied by meaningful improvements in the model’s reasoning behavior\. The reasoning trace analysis in this work is conducted using DeepSeek\-R1\-Distill\-Qwen\-7B\.

Held\-out MetricsReward MetricsMethodBacktrackingBranchinessDiff\. Elim\.PriorUncertaintyCoverageDirection\.Baseline0\.691\.220\.790\.590\.8733\.1%0\.21Cor\-GRPO0\.931\.160\.970\.530\.9239\.1%0\.16CEDAR\-GRPO1\.091\.531\.290\.721\.3752\.9%0\.60Table 3:Mean process\-level metric scores averaged across ten held\-out evaluation datasets for DeepSeek\-R1\-Distill\-Qwen\-7B\. Evidence Coverage and Directionality were explicitly included as rewards in the CEDAR\-GRPO training objective; the other five metrics are fully held\-out metrics used to assess reasoning generalization\. Shading indicates a relative increase \(green\) or decrease \(red\) against the Baseline\. Full per\-dataset breakdowns are provided in Appendix[E](https://arxiv.org/html/2608.14791#A5), Table[10](https://arxiv.org/html/2608.14791#A5.T10)\.As shown in Table[3](https://arxiv.org/html/2608.14791#S5.T3), optimizing with the composite reward \(CEDAR\-GRPO\) understandably leads to substantial increases in the two metrics it directly targets: Evidence Coverage and Evidence–Explanation Directionality\. Evidence Coverage increases substantially from the baseline of 33\.1% to 52\.9%, and Directionality rises from 0\.21 to 0\.60\. Interestingly, correctness\-only optimization \(Cor\-GRPO\) struggles to maintain this structural rigor, with Directionality dropping slightly to 0\.16\. This demonstrates that without explicit grounding, the model may arrive at correct answers without a logically sound evidence\-to\-conclusion flow\.

While the gains in Coverage and Directionality confirm that the optimization successfully induced the specific rewarded behaviors, our central finding is the significant positive transfer to the fully held\-out metrics\. Crucially, we observe consistent improvements in exploratory behaviors that were not explicitly rewarded during training\. Backtracking and Differential Elimination show consistent increases relative to the baseline under both training objectives, indicating that some degree of error correction and active refutation naturally emerges from RL training\. The composite reward, however, amplifies these traits considerably, pushing Differential Elimination from 0\.79 to 1\.29 and Backtracking from 0\.69 to 1\.09\.

A key divergence between the two training regimes appears in how the model generates and explores candidate explanations\. Under correctness\-only optimization, the reasoning traces become narrower: Branchiness decreases from 1\.22 to 1\.16, and Prior Invocation drops from 0\.59 to 0\.53\. In contrast, the composite reward encourages the model to actively hypothesize and explore multiple distinct paths, raising Branchiness to 1\.53 and Prior Invocation to 0\.72\. This suggests that CEDAR\-GRPO induces a broader shift toward exploratory, transferable reasoning rather than narrow optimization for final answers\.

Finally, while the baseline model already exhibits some use of uncertainty markers \(0\.87\), composite training leads to a considerable increase, reaching 1\.37\. This trend suggests that rather than simply generating verbose filler, the model learns to qualify its steps and explicitly acknowledge epistemic uncertainty when evaluating alternative explanations\.

Together, these metrics show that the composite reward improves the model’s abductive reasoning behavior, encouraging it to actively hypothesize and evaluate alternatives rather than passively converging on a correct final answer\.

### 5\.3Ablation Studies

In this section we test whether the observed gains are caused by abductive RL itself, by exposure to the same data under a supervised objective, by the composition of the reward, by the two\-stage structure of the training pool, or by generic reasoning post\-training\. To keep the ablation grid tractable, all ablations are run only on Qwen3\-4B and DeepSeek\-R1\-Distill\-Qwen\-7B\. Table[4](https://arxiv.org/html/2608.14791#S5.T4)summarizes all ablation variants; the following subsections refer to the corresponding row groups\.

ModelAblationMethodARTB\-COPADefNLIGoEmo\.MuSR\-MMuSR\-OMuSR\-TNeuLRStratQAMedQAML\-DebugAvg\.Δ\\Deltavs BaseQwen3\-4BReferenceBase65\.25%65\.25\\%84\.40%84\.40\\%80\.75%80\.75\\%27\.75%27\.75\\%20\.40%20\.40\\%19\.20%19\.20\\%51\.60%51\.60\\%35\.00%35\.00\\%38\.25%38\.25\\%35\.50%35\.50\\%25\.25%25\.25\\%0\.000\.00ReferenceCor\-GRPO71\.75%71\.75\\%86\.80%86\.80\\%86\.75%86\.75\\%32\.25%32\.25\\%41\.60%41\.60\\%32\.40%32\.40\\%48\.80%48\.80\\%38\.75%38\.75\\%43\.25%43\.25\\%37\.50%37\.50\\%28\.50%28\.50\\%\+5\.91\+5\.91MainCEDAR\-GRPO72\.25%\\mathbf\{72\.25\\%\}88\.80%\\mathbf\{88\.80\\%\}88\.50%\\mathbf\{88\.50\\%\}34\.25%34\.25\\%48\.40%\\mathbf\{48\.40\\%\}33\.20%\\mathbf\{33\.20\\%\}55\.60%\\mathbf\{55\.60\\%\}40\.50%\\mathbf\{40\.50\\%\}44\.75%44\.75\\%37\.75%\\mathbf\{37\.75\\%\}29\.00%\\mathbf\{29\.00\\%\}\+8\.15\+8\.15RL vs SFTSFT66\.75%66\.75\\%86\.00%86\.00\\%80\.75%80\.75\\%28\.50%28\.50\\%22\.80%22\.80\\%17\.60%17\.60\\%48\.40%48\.40\\%33\.25%33\.25\\%37\.25%37\.25\\%33\.25%33\.25\\%24\.25%24\.25\\%−0\.41\-0\.41RewardCor\+Cov\-GRPO69\.50%69\.50\\%88\.00%88\.00\\%87\.00%87\.00\\%32\.00%32\.00\\%43\.20%43\.20\\%33\.20%\\mathbf\{33\.20\\%\}52\.00%52\.00\\%39\.00%39\.00\\%43\.50%43\.50\\%37\.00%37\.00\\%29\.00%29\.00\\%\+6\.37\+6\.37RewardCor\+Dir\-GRPO67\.00%67\.00\\%88\.40%88\.40\\%87\.25%87\.25\\%33\.00%33\.00\\%43\.60%43\.60\\%30\.00%30\.00\\%53\.20%53\.20\\%40\.00%40\.00\\%44\.00%44\.00\\%37\.25%37\.25\\%28\.50%28\.50\\%\+6\.26\+6\.26StageStage\-I CEDAR\-GRPO67\.50%67\.50\\%86\.00%86\.00\\%84\.25%84\.25\\%35\.50%35\.50\\%45\.20%45\.20\\%29\.20%29\.20\\%51\.20%51\.20\\%40\.25%40\.25\\%40\.50%40\.50\\%36\.75%36\.75\\%26\.00%26\.00\\%\+5\.36\+5\.36StageStage\-II CEDAR\-GRPO68\.25%68\.25\\%86\.80%86\.80\\%85\.75%85\.75\\%37\.75%\\mathbf\{37\.75\\%\}34\.40%34\.40\\%26\.80%26\.80\\%51\.20%51\.20\\%35\.00%35\.00\\%46\.25%\\mathbf\{46\.25\\%\}35\.75%35\.75\\%24\.75%24\.75\\%\+4\.49\+4\.49Generic reasoningGeneral Cor\-GRPO64\.50%64\.50\\%86\.40%86\.40\\%83\.50%83\.50\\%26\.50%26\.50\\%19\.20%19\.20\\%17\.60%17\.60\\%41\.60%41\.60\\%37\.75%37\.75\\%37\.25%37\.25\\%33\.75%33\.75\\%26\.50%26\.50\\%−0\.80\-0\.80DeepSeek\-R1\-Distill\-Qwen\-7BReferenceBase70\.25%70\.25\\%85\.20%85\.20\\%81\.50%81\.50\\%30\.00%30\.00\\%26\.40%26\.40\\%38\.40%38\.40\\%49\.60%49\.60\\%31\.50%31\.50\\%42\.50%42\.50\\%35\.75%35\.75\\%23\.75%23\.75\\%0\.000\.00ReferenceCor\-GRPO73\.50%73\.50\\%88\.00%88\.00\\%82\.25%82\.25\\%31\.50%31\.50\\%56\.40%56\.40\\%40\.40%40\.40\\%50\.00%50\.00\\%33\.00%33\.00\\%42\.25%42\.25\\%36\.25%36\.25\\%22\.75%22\.75\\%\+3\.77\+3\.77MainCEDAR\-GRPO78\.50%\\mathbf\{78\.50\\%\}89\.60%\\mathbf\{89\.60\\%\}87\.75%\\mathbf\{87\.75\\%\}35\.50%35\.50\\%57\.20%\\mathbf\{57\.20\\%\}49\.60%\\mathbf\{49\.60\\%\}51\.60%51\.60\\%35\.50%35\.50\\%45\.75%45\.75\\%39\.50%\\mathbf\{39\.50\\%\}24\.75%24\.75\\%\+7\.31\+7\.31RL vs SFTSFT72\.00%72\.00\\%86\.40%86\.40\\%80\.00%80\.00\\%34\.50%34\.50\\%29\.60%29\.60\\%36\.00%36\.00\\%47\.20%47\.20\\%33\.25%33\.25\\%38\.75%38\.75\\%32\.75%32\.75\\%19\.75%19\.75\\%−0\.42\-0\.42RewardCor\+Cov\-GRPO73\.75%73\.75\\%88\.00%88\.00\\%83\.25%83\.25\\%31\.75%31\.75\\%57\.20%\\mathbf\{57\.20\\%\}41\.20%41\.20\\%50\.40%50\.40\\%33\.75%33\.75\\%43\.25%43\.25\\%36\.00%36\.00\\%23\.50%23\.50\\%\+4\.29\+4\.29RewardCor\+Dir\-GRPO72\.75%72\.75\\%87\.60%87\.60\\%82\.75%82\.75\\%31\.25%31\.25\\%56\.80%56\.80\\%40\.80%40\.80\\%49\.60%49\.60\\%33\.75%33\.75\\%43\.50%43\.50\\%36\.75%36\.75\\%22\.50%22\.50\\%\+3\.93\+3\.93StageStage\-I CEDAR\-GRPO69\.50%69\.50\\%87\.20%87\.20\\%82\.75%82\.75\\%33\.25%33\.25\\%42\.00%42\.00\\%27\.60%27\.60\\%55\.20%\\mathbf\{55\.20\\%\}41\.25%\\mathbf\{41\.25\\%\}42\.50%42\.50\\%35\.25%35\.25\\%29\.25%\\mathbf\{29\.25\\%\}\+2\.81\+2\.81StageStage\-II CEDAR\-GRPO72\.75%72\.75\\%88\.80%88\.80\\%86\.50%86\.50\\%40\.25%\\mathbf\{40\.25\\%\}38\.40%38\.40\\%27\.20%27\.20\\%51\.60%51\.60\\%33\.25%33\.25\\%48\.00%\\mathbf\{48\.00\\%\}35\.00%35\.00\\%27\.50%27\.50\\%\+3\.13\+3\.13Generic reasoningGeneral Cor\-GRPO69\.25%69\.25\\%88\.40%88\.40\\%82\.75%82\.75\\%28\.50%28\.50\\%22\.00%22\.00\\%27\.60%27\.60\\%43\.60%43\.60\\%34\.25%34\.25\\%40\.50%40\.50\\%33\.75%33\.75\\%25\.25%25\.25\\%−1\.73\-1\.73

Table 4:Unified ablation results on the two evaluated backbones, isolating the effects of supervised fine\-tuning, reward composition, training stages, and generic reasoning post\-training\. Performance is compared against our main CEDAR\-GRPO method and the Cor\-GRPO baseline\. All columns report task\-native metrics on the same held\-out evaluation datasets as Table[1](https://arxiv.org/html/2608.14791#S5.T1)\.#### 5\.3\.1RL versus supervised fine\-tuning on synthetic rationales

This ablation asks whether the gains come from reinforcement learning or simply from additional exposure to the same abductive tasks and answer formats\. We construct an SFT dataset from the same training pool used for RL\. Because not all original examples include complete reasoning traces, we generate synthetic rationale chains in the same<think\>and<answer\>format used during RL training; details are given in Appendix[F](https://arxiv.org/html/2608.14791#A6)\. We then fine\-tune the same base model with the same PEFT configuration and evaluate it on the identical held\-out suite\.

This creates a matched comparison: SFT sees the same task distribution and response format, but learns by imitating fixed rationales rather than by exploring responses under a reward signal\. As shown in Table[4](https://arxiv.org/html/2608.14791#S5.T4), SFT is weaker than the closest RL counterpart, Cor\-GRPO, which also omits the composite process reward\. On Qwen3\-4B, Cor\-GRPO outperforms SFT on every held\-out task; on DeepSeek\-R1\-Distill\-Qwen\-7B, it is higher on 9 of 11 tasks and better on average\. This suggests that outcome\-driven exploration provides benefits beyond simply imitating synthetic rationales\.

#### 5\.3\.2Reward composition

CEDAR\-GRPO uses a composite reward that combines deterministic final\-answer correctness with evidence coverage and evidence–explanation directionality:

r=λcor​rcor\+λcov​rcov\+λdir​rdir\.r=\\lambda\_\{\\mathrm\{cor\}\}r\_\{\\mathrm\{cor\}\}\+\\lambda\_\{\\mathrm\{cov\}\}r\_\{\\mathrm\{cov\}\}\+\\lambda\_\{\\mathrm\{dir\}\}r\_\{\\mathrm\{dir\}\}\.To isolate the role of the two process rewards, we keep correctness active and remove one process term at a time\. In each two\-term variant, the included rewards receive equal weight \(0\.50\.5each\), while the omitted reward is set to zero\. Thus, Cor\+Cov\-GRPO removes directionality, while Cor\+Dir\-GRPO removes evidence coverage\. The “Reward” rows in Table[4](https://arxiv.org/html/2608.14791#S5.T4)compare these variants with Cor\-GRPO and the full CEDAR\-GRPO objective\.

The results show that both process rewards contribute\. Adding either one to correctness improves average performance over Cor\-GRPO, but neither partial objective matches the full composite reward\. On Qwen3\-4B, CEDAR\-GRPO reaches 52\.1 average accuracy, compared with 50\.3 for Cor\+Cov\-GRPO and 50\.2 for Cor\+Dir\-GRPO; on DeepSeek\-R1\-Distill\-Qwen\-7B, the corresponding scores are 54\.1, 51\.1, and 50\.7\. This suggests that coverage and directionality are complementary: each helps beyond final\-answer correctness, but the strongest transfer comes from optimizing them jointly\.

#### 5\.3\.3Isolating Stage I and Stage II training

Our data collection is built around the claim that abductive reasoning requires both hypothesis generation and hypothesis selection\. To test this directly, we train two restricted variants\. The Stage I\-only model uses only the hypothesis\-generation sources: AbductionRules, Crypto, and List Function\. The Stage II\-only model uses only the hypothesis\-selection or hypothesis\-evaluation sources: UniADILR\-HGc, Balanced COPA cause, CauseLogics, and CLIMATE\-FEVER\. We keep the optimization budget and checkpoint\-selection protocol matched as closely as possible to the full mixture\.

As shown in the “Stage” rows of Table[4](https://arxiv.org/html/2608.14791#S5.T4), the restricted variants retain some task\-specific strengths, but neither matches the breadth of the full training mixture\. On Qwen3\-4B, CEDAR\-GRPO reaches 52\.1 average accuracy, compared with 49\.3 for Stage I\-only and 48\.4 for Stage II\-only training\. The same pattern holds on DeepSeek\-R1\-Distill\-Qwen\-7B, where the full mixture reaches 54\.1, compared with 49\.6 and 49\.9 for the two restricted variants\. These results suggest that generation\- and selection\-oriented data provide complementary signals, and that broad transfer depends on training over both stages\.

#### 5\.3\.4Training on generic reasoning data

Finally, we test whether the observed gains are specific to abductive data or can be obtained by applying GRPO to a generic reasoning mixture\. We create a matched post\-training condition using deductive and general reasoning datasets with the same backbone, similar number of examples, answer format, and RL budget\. The matched general\-reasoning mixture is detailed in Appendix[G](https://arxiv.org/html/2608.14791#A7)\. This controls for improvements from longer reasoning traces, answer\-format practice, or generic reinforcement learning on verifiable problems\.

As shown in the “Generic reasoning” rows of Table[4](https://arxiv.org/html/2608.14791#S5.T4), the generic\-reasoning control underperforms the matched abductive correctness\-only baseline, Cor\-GRPO\. On Qwen3\-4B, Cor\-GRPO is higher on every held\-out task and averages 49\.9 compared with 43\.2 for General Cor\-GRPO\. On DeepSeek\-R1\-Distill\-Qwen\-7B, it also performs better on average, 50\.6 versus 45\.1, with the largest gaps appearing on the MuSR tasks\. This suggests that the gains do not come simply from correctness\-only GRPO on verifiable reasoning data, but from training on abductive data in particular\.

## 6Conclusion

We introduced CEDAR\-GRPO, a composite\-reward post\-training framework that treats abduction as a general reasoning capability rather than a benchmark\-specific skill\. Across four open\-weight backbones and eleven held\-out datasets, CEDAR\-GRPO consistently improved performance over both base models and correctness\-only GRPO, with gains of up to 30\.8 points and transfer to abduction\-adjacent settings such as clinical diagnosis, ML debugging, and long\-context investigation\. Ablations indicate that these gains reflect the abductive training mixture and the composite reward rather than supervised exposure or generic verifiable\-reward post\-training\. Overall, CEDAR\-GRPO is a practical step toward more general and reliable abductive reasoning in language models\.

## 7Limitations

Despite the consistent improvements reported above, several limitations remain\. First, our evaluation is still constrained by benchmark\-style settings and largely closed\-form outcomes, and it remains unclear how far these gains extend to truly open\-ended or interactive explanatory tasks where the answer space is not predefined\. Second, all backbones evaluated are in the 4–8B parameter range, leaving open whether the observed patterns hold at larger scales where base reasoning capabilities and training dynamics may differ substantially\. Third, although we complement final\-answer accuracy with process\-level metrics to validate that the observed improvements reflect genuine changes in reasoning behavior, these metrics are themselves computed by an LLM\-as\-judge, and since the evidence\-coverage and directionality rewards used during training rely on the same kind of judgments, there is a potential circularity between optimization signal and evaluation\. Conclusions about reasoning quality would therefore be strengthened by human evaluation, both as a check on the process\-level scores on our existing evaluation suite and through dedicated human assessment of model outputs on open\-ended generation tasks where automatic metrics are least reliable\. Finally, our training pool is relatively small at 2,400 instances \(Section[5\.3\.1](https://arxiv.org/html/2608.14791#S5.SS3.SSS1)\); while this scale is sufficient to demonstrate the effects studied here, it leaves open how the method behaves under substantially larger or more diverse abductive training data, and we examine the scale question only for the model and not for the data\.

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## Appendix AAdditional Dataset Notes for Data Collection

The notes below distinguish between the original benchmark as introduced in its source paper and the way we operationalize it in our study\. This distinction matters because several datasets are released as task collections or reasoning\-type subsets rather than as fixed train/dev/test corpora, and in several cases our subset choice is narrower than the benchmark as a whole\.

### A\.1Training and validation sources

DatasetRelation to abductionSamplesTraining roleUniADILR\-HGc \(abductive\)\([32](https://arxiv.org/html/2608.14791#bib.bib26)\)Stage II400Abstract logical abductive selection with distractor premises; compact signal for evaluating candidate explanations\.Balanced COPA \(cause\)\([19](https://arxiv.org/html/2608.14791#bib.bib27)\)Stage II400Commonsense causal selection while reducing superficial lexical cues\.CauseLogics\([12](https://arxiv.org/html/2608.14791#bib.bib3)\)Stage II\-adjacent400Binary candidate–cause verification with longer reasoning chains; deeper evaluative reasoning\.CLIMATE\-FEVER\([5](https://arxiv.org/html/2608.14791#bib.bib28)\)Stage II\-adjacent400Four\-way claim–evidence plausibility judgment over supports, refutes, not enough info, and disputed; broader verification\-style reasoning\.AbductionRules\([40](https://arxiv.org/html/2608.14791#bib.bib10)\)Stage I400Abductive generation of missing explanatory facts from rule contexts\.Crypto\([20](https://arxiv.org/html/2608.14791#bib.bib34)\)Stage I200Cipher\-rule learning \(Caesar, Atbash\) from examples\.List Function\([28](https://arxiv.org/html/2608.14791#bib.bib29)\)Stage I200Latent list\-transformation inference\.Table 5:Training and validation datasets used for GRPO fine\-tuning\. Each dataset contributes a fixed number of samples \(400 for most datasets, 200 for Crypto and List Function\), split 80/20 into train/validation; Which sums up to a total number of 2400 samples, consisting of 1920 training and 480 validation samples\.##### UniADILR\-HGc \(abductive split\)\.

UniADILR was introduced to study generalization across abductive, deductive, and inductive logical reasoning\([32](https://arxiv.org/html/2608.14791#bib.bib26)\)\. A typical example contains a target statement, a context with both relevant premises and distractors, and a proof\-style justification\. In the source benchmark, the program\-synthesized PSy portion is large, while the human–GPT\-corrected HGc portion is smaller and more realistic\. We use 400 randomly sampled examples from the abductive HGc portion and split them into 320 training and 80 validation samples\. We use this subset because it provides a compact and relatively clean Stage II signal\.

##### Balanced COPA \(cause split\)\.

Balanced COPA keeps the COPA format of a premise plus two alternatives, but mirrors the training data so that lexical cues are balanced between correct and incorrect answers\([19](https://arxiv.org/html/2608.14791#bib.bib27)\)\. The original resource is primarily a rebalanced training set of 1,000 instances, evaluated against the original COPA test set\. In our study, we use the training split only, filter examples withquestion == "cause", randomly sample 400 examples, and split them into 320 training and 80 validation samples\. The cause direction is used because it remains close to canonical commonsense abduction\.

##### CauseLogics\.

CauseLogics is a structured decision benchmark with premises, rules, a phenomenon, a candidate cause, and a binary validity label\([12](https://arxiv.org/html/2608.14791#bib.bib3)\)\. The full benchmark contains four difficulty levels, each corresponding to a different reasoning\-chain length\. We randomly sample 100 examples from each of the four levels and split each level into 80 training and 20 validation examples, producing 320 training and 80 validation samples overall\. This keeps the training pool balanced across reasoning\-chain lengths while still exposing the model to harder multi\-step cases\.

##### CLIMATE\-FEVER\.

CLIMATE\-FEVER is a FEVER\-style claim\-verification dataset built from real\-world climate claims and Wikipedia evidence\([5](https://arxiv.org/html/2608.14791#bib.bib28)\)\. The original release contains 1,535 claims and 7,675 claim–evidence pairs, labeled as supports, refutes, not enough info, or disputed\. In our pipeline, it functions as a Stage II evaluative proxy: the model must assess which evidential relation best characterizes how a piece of evidence bears on a claim\. We randomly sample 400 examples from the test split and divide them into 320 training and 80 validation samples\.

##### AbductionRules\.

AbductionRules is a formal abductive reasoning benchmark in which the model is given a rule context together with an unexpected observation and must generate the missing fact that would explain it\([40](https://arxiv.org/html/2608.14791#bib.bib10)\)\. We construct a balanced subset using the four benchmark variants \(Abduction\-Animal\-Simple,Abduction\-Animal,Abduction\-Person\-Simple, andAbduction\-Person\)\. For training, we sample 80 contexts from each variant usingtrain\.jsonl, producing 320 training examples in total\. For validation, we sample 20 contexts from each variant usingdev\.jsonl, producing 80 validation examples\. One query is sampled per context\. We use this dataset as Stage I training data because the task directly targets hypothesis construction rather than answer selection\.

##### List Function\.

The List Function benchmark is task\-based rather than a single canonical corpus\([28](https://arxiv.org/html/2608.14791#bib.bib29)\)\. Each task provides several input/output demonstrations and asks the model to infer the latent list transformation and apply it to a new input\. Under our formalism, that hidden transformation is the missing explanatory rule; when expressed as code in our setup, the generated function serves as an explicit candidate hypothesis about the governing rule\. We randomly sample 200 tasks from the full dataset and split them into 160 training and 40 validation tasks\. For each task, we use thetrain\["normal"\]demonstrations together with the test examples, while ignoring the out\-of\-distribution and noisy variants\. We use this dataset as Stage I rule\-learning data\.

##### Crypto \(Caesar and Atbash\)\.

We draw Crypto examples from the cryptography tasks of the Robust Rule Induction benchmark introduced by Li et al\.\([20](https://arxiv.org/html/2608.14791#bib.bib34)\)\. In our study, we use only the Caesar and Atbash subsets, sampling 100 examples from each to produce 200 total instances, which are divided into 160 training and 40 validation examples\. Each instance requires inferring the latent cipher rule from input–output demonstrations, so we treat the dataset as Stage I rule\-learning data\.

### A\.2Evaluation sources

DatasetRelationSamplesEvaluation roleClassic abductive benchmarksART \(α\\alphaNLI\)\([2](https://arxiv.org/html/2608.14791#bib.bib8)\)Stage II400Conventional Stage II abductive selection between alternative hypotheses over narrative bridges\.NeuLR\([39](https://arxiv.org/html/2608.14791#bib.bib32)\)Stage I400Conventional Stage I abductive split; missing\-fact generation in abstract logical contexts\.Stage II evidence\-update evaluationDefeasible NLI\([27](https://arxiv.org/html/2608.14791#bib.bib17)\)Stage II\-adjacent400Determines whether an update strengthens or weakens a hypothesis; tests evidence\-based plausibility assessment\.Abductive interpretations in neighboring domainsGoEmotions\([4](https://arxiv.org/html/2608.14791#bib.bib30)\)Domain\-abductive400Infers latent emotion as the explanation for an observed utterance\.MuSR: Murder\([33](https://arxiv.org/html/2608.14791#bib.bib31)\)Domain\-abductive250Detective culprit inference; infers the most plausible explanation for a murder\.MedQA\([18](https://arxiv.org/html/2608.14791#bib.bib37)\)Domain\-abductive400Clinical diagnosis: infers the most plausible illness, problem, or mechanism from patient background and symptoms\.ML\-debugging\([17](https://arxiv.org/html/2608.14791#bib.bib36)\)Domain\-abductive400Debugging as inferring the cause of an observed error or malfunction and producing a fix\.Non\-abductive controlsBalanced COPA \(effect\)\([19](https://arxiv.org/html/2608.14791#bib.bib27)\)Deductive control250Tests forward causal reasoning in the effect direction\.MuSR: Object\([33](https://arxiv.org/html/2608.14791#bib.bib31)\)Multistep control250Tests complex long\-context reasoning over object locations\.MuSR: Team\([33](https://arxiv.org/html/2608.14791#bib.bib31)\)Multistep control250Tests complex long\-context reasoning over team assignments\.StrategyQA\([9](https://arxiv.org/html/2608.14791#bib.bib33)\)General control400Broad reasoning benchmark not explicitly framed as abduction\.Total3,800Table 6:Held\-out evaluation datasets used for model assessment\. These datasets are not used for GRPO training or validation\-based selection\. The Samples column reports the number of held\-out examples evaluated in this study, including full selected subsets for some datasets and fixed\-size samples from larger source splits for others\.##### ART \(α\\alphaNLI\)\.

ART is a canonical abductive commonsense benchmark\([2](https://arxiv.org/html/2608.14791#bib.bib8)\)\. In itsα\\alphaNLI formulation, the model is given two observations and must choose which of two candidate hypotheses best bridges them\. The same paper also introducesα\\alphaNLG for generation, but here we use onlyα\\alphaNLI, since our goal on this benchmark is a clean held\-out test of conventional Stage II hypothesis selection\.

##### NeuLR\.

NeuLR is designed to remove topical and commonsense cues so that logical form is tested in a content\-neutral setting\([39](https://arxiv.org/html/2608.14791#bib.bib32)\)\. The benchmark spans deductive, inductive, and abductive reasoning over abstract facts and rules\. We use 400 examples from its abductive split because this subset provides the clearest held\-out test of conventional Stage I missing\-fact generation rather than surface\-level plausibility scoring\.

##### Defeasible NLI \(DefNLI\)\.

Defeasible NLI extends several existing inference resources with a defeasible\-update formulation\([27](https://arxiv.org/html/2608.14791#bib.bib17)\)\. The classification task takes a premise/hypothesis pair together with an update and asks whether the update strengthens or weakens the hypothesis\. We use this classification view because it directly probes the evaluative side of Stage II reasoning, in a role analogous to CLIMATE\-FEVER in the training pool: the model must judge how new evidence changes the plausibility of a candidate conclusion or explanation\.

##### GoEmotions \(GoEmo\)\.

GoEmotions is a large multi\-label emotion dataset of Reddit comments annotated with 27 emotion categories plus*Neutral*\([4](https://arxiv.org/html/2608.14791#bib.bib30)\)\. The benchmark is not an abductive dataset in the standard sense, but it can still be read as a domain\-specific hidden\-state inference problem: from an observed utterance, the model must infer the latent affective state that best explains it\. We use it precisely for this out\-of\-domain abductive interpretation\.

##### MuSR: Murder \(MuSR\-M\)\.

MuSR is a long\-context multiple\-choice benchmark with three narrative domains: murder mysteries, object placements, and team allocations\([33](https://arxiv.org/html/2608.14791#bib.bib31)\)\. The murder split is the most directly compatible with an abductive interpretation, since the task is to infer the culprit or explanatory sequence of events that best accounts for the available evidence\. We therefore treat MuSR\-M as a domain\-shifted abductive benchmark\.

##### MuSR: Object and MuSR: Team \(MuSR\-O/T\)\.

The object\-placement and team\-allocation splits of MuSR are less cleanly abductive than the murder split\. They still require long\-context, multi\-step reasoning over dispersed constraints, but the target is better understood as recovering an object location or assignment structure rather than explaining an observed outcome\. We therefore use MuSR\-O and MuSR\-T as non\-abductive multistep controls\.

##### MedQA\.

MedQA is a multiple\-choice question\-answering benchmark built around complex multi\-hop medical reasoning\([18](https://arxiv.org/html/2608.14791#bib.bib37)\)\. Each question is accompanied in the original release by a large\-scale collection of medical textbook paragraphs for evidence retrieval, but the standard prediction target is the final multiple\-choice answer\. We use it as a specialized domain\-abductive benchmark because many instances require inferring the most plausible illness, clinical problem, treatment, or underlying mechanism from patient background information, symptoms, and test results\.

##### ML\-debugging\.

ML\-debugging is a code debugging benchmark built around identifying and resolving errors in multi\-library Python scenarios\([17](https://arxiv.org/html/2608.14791#bib.bib36)\)\. Each buggy code snippet is accompanied in the original release by execution feedback and test cases, but the standard prediction target is the final repaired code sequence\. We use it as a specialized domain\-abductive benchmark because debugging requires tracing an observed error, failed test, or malfunction back to its likely cause and then producing a repair\.

##### Balanced COPA \(effect split\)\.

We use the effect direction of Balanced COPA only at evaluation time\([19](https://arxiv.org/html/2608.14791#bib.bib27)\)\. This split keeps the same two\-choice causal format as the cause direction used in training, but reverses the direction of reasoning\. We treat it as a non\-abductive deductive\-style control because the model is asked to reason forward from a premise toward a likely effect, rather than backward from an observation toward an explanation\.

##### StrategyQA \(StratQA\)\.

StrategyQA is a yes/no question\-answering benchmark built around implicit multi\-step reasoning\([9](https://arxiv.org/html/2608.14791#bib.bib33)\)\. Each question is accompanied in the original release by intermediate decompositions and supporting evidence, but the standard prediction target is simply the final yes/no answer\. We do not treat it as an abductive benchmark\. Instead, it functions as a non\-abductive general\-reasoning control that helps us determine whether any observed gains are specific to explanation\-centered reasoning or reflect broader changes in reasoning behavior\.

Taken together, these choices create an intentional gradient: classic abduction in ART and NeuLR, Stage II evidence\-update evaluation in Defeasible NLI, abductive interpretations in neighboring domains through GoEmotions, MuSR\-M, MedQA, and ML\-debugging, and explicit non\-abductive controls through Balanced COPA\-effect, MuSR\-O/T, and StrategyQA\. This structure supports a more precise interpretation of results: improvements on ART, NeuLR, and Defeasible NLI would indicate more direct abductive transfer, improvements on the domain\-abductive group would suggest transfer to explanation\-centered reasoning outside standard abduction benchmarks, and improvements on the controls would indicate broader or less specific changes in the model’s reasoning profile\.

## Appendix BPrompts Used in Training and Evaluation

### B\.1Prompts Used in Training

#### B\.1\.1Prompt Template: UniADILR

System Prompt:

Youareanexpertinlogicalreasoningandabductiveinference\.Yourtaskistoidentifywhichsentencesfromagivencontextprovidethenecessaryevidencetosupportorexplainahypothesis\.

Youwillbeprovidedwith:

1\.AContextcontainingmultiplenumberedsentences\(sent1,sent2,sent3,etc\.\)

2\.AHypothesisthatneedstobesupportedorexplained

Yourgoalistoidentifywhichsentence\(s\)fromthecontext,whencombined,providethelogicalfoundationforthehypothesisthroughabductivereasoning\.

\#\#Instructions:

1\.Carefullyreadallsentencesinthecontext

2\.Analyzethehypothesis

3\.Identifywhichsentences,whencombined,bestexplainorsupportthehypothesis

4\.Considerbothdirectevidenceandlogicalconnections

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Sentencenumbersonly,comma\-separated\.Forexample:5,13or2,7,9\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthesentencenumbersseparatedbycommas\.Donotincludetheword"sent"oranyothertext\.

User Prompt:

Context:

\{context\_str\}

Hypothesis:

\{hypothesis\}

Whichsentencenumbersprovidethenecessaryevidenceforthehypothesis?

#### B\.1\.2Prompt Template: Copa \(cause\)

System Prompt:

Youareanexpertinlogicalreasoningandabductiveinference\.Yourtaskistodeterminewhichoftwogivenchoicesrepresentsthemostplausiblecauseforagivenpremise\.

Youwillbeprovidedwith:

1\.APremisedescribingasituationorevent

2\.TwoChoices\(Choice1andChoice2\)

YourgoalistoselectthechoicethatbestexplainsWHYthepremisehappened\-identifyingtherootcausethatledtothedescribedsituation\.

\#\#Instructions:

1\.Carefullyreadthepremise

2\.Thinkstepbysteptoevaluatebothchoicesaspotentialcauses

3\.Considercommonsense,real\-worldknowledge,andtypicalcausalrelationshipswhenmakingyourdecision

4\.Selectthechoicethatrepresentsthemostplausibleanddirectcause

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Either"1"or"2"\-justthenumber,nothingelse\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthenumber1or2\.Donotincludeanyothertext,explanation,orpunctuation\.

User Prompt:

Premise:\{example\[’premise’\]\}

Choice1:\{example\[’choice1’\]\}

Choice2:\{example\[’choice2’\]\}

Whichchoiceisthemostplausiblecauseforthepremise?

#### B\.1\.3Prompt Template: CauseLogics

System Prompt:

Youareanexpertlogicianandcarefulreasoningassistant\.YourtaskistoidentifywhetheragivenPossibleCause,whenaddedtotheprovidedknowledgebase,logicallyentailsanobservedPhenomenon\.

Youwillbeprovidedwith:

1\.AsetofPremises\(facts\)

2\.AsetofRules\(implications\)

3\.AnobservedPhenomenon

4\.APossibleCause\(ahypothesis\)

YourgoalistodeterminewhetherthePhenomenoncanbelogicallyinferredbyforwardreasoningusingONLYthegivenPremises\+Rules\(\+thePossibleCause\)\.

\#\#Instructions:

1\.CarefullyreadallPremisesandRules

2\.AssumethePossibleCauseisaddedasanadditionalpremise

3\.UsingONLYthegivenPremises\+Rules\(\+thePossibleCause\),reasonforward

4\.DecidewhetherthePhenomenoncanbelogicallyinferred

\-IfthePhenomenoncanbeinferred,thePossibleCauseisTRUE

\-IfthePhenomenoncannotbeinferred,thePossibleCauseisFALSE

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Outputexactlyoneofthesetwooptions:TRUE,FALSE\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYoneofthesetwooptions:TRUEorFALSE\.Donotincludeanyothertext\.

User Prompt:

Premises:

\{premises\_text\}

Rules:

\{rules\_text\}

Phenomenon:

\{str\(phenomenon\)\}

PossibleCause:

\{str\(possible\_cause\)\}

IsthePossibleCauselogicallyTRUEorFALSE?

#### B\.1\.4Prompt Template: Climate Fever

System Prompt:

Youareanexpertclimatescientistandprofessionalfact\-checker\.Yourtaskistodeterminewhetherasetofprovidedevidencessupports,refutes,disputedorisinsufficienttoevaluateaspecificclaim\.

Youwillbeprovidedwith:

1\.AspecificClaim

2\.AlistofEvidences

YourgoalistodecidewhethertheEvidenceSUPPORTSorREFUTESorDISPUTEDtheClaim,orifthereisNOTENOUGHINFO,andtojustifythatdecisionbycitingspecificpartsoftheevidence\.

\#\#Instructions:

1\.CarefullyreadtheClaimandallprovidedEvidences

2\.DetermineiftheEvidenceSUPPORTSorREFUTESorDISPUTEDtheClaim,orifthereisNOTENOUGHINFO

3\.Thinkstepbystepabouthowthespecificpartsoftheevidencerelatetotheclaim

4\.Outputthefinallabel

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Outputexactlyoneofthesefouroptions:SUPPORTS,REFUTES,DISPUTED,NOTENOUGHINFO\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYoneofthesefouroptions:SUPPORTS,REFUTES,DISPUTED,NOTENOUGHINFO\.Donotincludeanyothertext\.

User Prompt:

Claim:

\{claim\}

Evidence:

\{evidence\_text\}

DoestheprovidedevidenceSUPPORT,REFUTE,DISPUTEDorprovideNOTENOUGHINFOfortheclaim?

#### B\.1\.5Prompt Template: Abduction Rules

System Prompt:

Youareanexpertinlogicalreasoningandabductiveinference\.Yourtaskistoidentifythesinglemissingfactthat,whenaddedtoagivencontext,makesaquerylogicallydecidable\.

Youwillbeprovidedwith:

1\.AContextcontainingfactsandrules

2\.AQuerythatiscurrentlynotdecidablefromthecontextalone

YourgoalistoinferONEadditionalfactthat,whencombinedwiththecontext,allowsthequerytobeeither:

\-provedtrue,or

\-provedfalse

\#\#Instructions:

1\.Carefullyreadallfactsandrulesinthecontext

2\.Analyzethequery

3\.Identifythesinglemissingfactthatwouldmakethequerydecidable

4\.Preferadirect,minimalexplanation:

\-Outputexactlyonefact

\-Donotoutputarule

\-Donotoutputmultiplefacts

\-Donotparaphrasebeyondthestylealreadyusedinthecontext

5\.Thefactshouldbeonethatworkswiththeexistingrulesandfactstoproveordisprovethequery

6\.Becarefulwithnegation:

\-Sometimestherightmissingfacthelpsprovethequery

\-Sometimesithelpsderivetheoppositeofthequery,therebydisprovingit

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Explainyourthoughtprocess:whichrule\(s\)matter,whichexistingfactsarerelevant,andwhythemissingfactmakesthequeryprovableordisprovable\]

</think\>

<answer\>

\[Outputthesinglemissingfactonly,exactlyasanatural\-languagesentenceendingwithaperiod\]

</answer\>

CRITICAL:

\-TheanswersectionmustcontainONLYonemissingfact\.

\-Donotincludeanyextracommentaryintheanswersection\.

\-Donotoutputmorethanonesentence\.

\-Donotoutputarule;outputafactaboutanentityinthecontext\.

User Prompt:

Context:

\{context\}

Query:

\{query\}

Basedonthecontextandqueryabove,identifythesinglemissingfactthat,whenaddedtothecontext,makesthequerylogicallydecidable\.

#### B\.1\.6Prompt Template: Crypto

System Prompt:

YouareanexpertatinferringexactstringtransformationrulesfromexamplesandexpressingthemascorrectPythonfunctions\.

Youwillbegivenseveraltrainingexamples\.Eachexamplecontains:

\-Input:astring

\-Output:theresultofapplyingthesamehiddentransformationruletotheinput

InferthetransformationrulethatisconsistentwithALLtrainingexamples,thenwriteageneralPythonimplementationofthatrule\.

Beforeanswering,makesurethesameruleexplainsallexamplesexactlyandconsistentlyatthecharacterlevel\.

Thinkabductively:consideralternativehypothesesandchoosetheonethatexplainsallexamplesexactly\.

Outputformat\(MUSTfollowexactly\):

<think\>

\[Explainyourthoughtprocess:reasonstepbystepaboutthepossiblerules,consideralternativehypotheses,andexplainwhytheruleyouchosebestfitsallexamples\.\]

</think\>

<answer\>

deftransform\(s\):

\.\.\.

</answer\>

Coderequirements:

\-DefineEXACTLYonefunctionnamedtransform\.

\-Thefunctiontakesoneargument:s\(astring\)\.

\-ItMUSTreturnastring\.

\-NOIMPORTSallowed\.

\-NOprinting,noinput\(\),norandomness\.

\-Donothardcodespecifictraininginputs/outputs;generalizethelogic\.

\-Preservethebehaviorimpliedbytheexamplesforallcharactersthatappear\.

STRICTFORMATTINGRULES:

\-DoNOTusemarkdowncodeblocks\(like‘‘‘python\)insidethe<answer\>tags\.Justwriterawcode\.

\-DoNOTrepeatthecode\.Writethefunctionexactlyonce\.

\-Ensureyouclosethetagwith</answer\>\.

\-The<answer\>tagmustcontainONLYvalidPythoncode,nocommentsorexplanationsoutsidethefunction\.

\-DoNOTwriteanytextbefore<think\>orafter</answer\>\.

User Prompt:

Trainingexamples:

\{train\_prompt\}

InfertheunderlyingstringtransformationandprovidethePythonfunctionimplementationintherequiredformat\.

#### B\.1\.7Prompt Template: List Function

System Prompt:

YouareanexpertatinferringsimplelisttransformationsfromexamplesandexpressingthemascorrectPythonfunctions\.

Youwillbegivenseveraltrainingexamples\.Eachexamplecontains:

\-Input:alistofintegers

\-Output:theresultofapplyingthesamehiddentransformationruletotheinput

InferthetransformationrulethatisconsistentwithALLtrainingexamples,thenwriteageneralPythonimplementationofthatrule\.

Outputformat\(MUSTfollowexactly\):

<think\>

\[Explainyourthoughtprocess:reasonstepbystepaboutthepossiblerules,consideralternativehypotheses,andexplainwhyyourfinalrulebestfitsALLtrainingexamples\.\]

</think\>

<answer\>

deftransform\(lst\):

\.\.\.

</answer\>

Coderequirements:

\-DefineEXACTLYonefunctionnamedtransform\.

\-Thefunctiontakesoneargument:lst\(alistofintegers\)\.

\-ItMUSTreturnalistofintegers\.Iftheruleresultsinasinglevalue,returnitasasingle\-elementlist\(e\.g\.,\[val\]\)\.

\-NOIMPORTSallowed\.

\-NOprinting,noinput\(\),norandomness\.

\-Donothardcodespecifictraininginputs/outputs;generalizethelogic\.

\-BEROBUST:Handleedgecaseslikeemptylistsorlistswithonly1element\.

STRICTFORMATTINGRULES:

\-DoNOTusemarkdowncodeblocks\(like‘‘‘python\)insidethe<answer\>tags\.Justwriterawcode\.

\-DoNOTrepeatthecode\.Writethefunctionexactlyonce\.

\-Ensureyouclosethetagwith</answer\>\.

\-The<answer\>tagmustcontainONLYvalidPythoncode,nocommentsorexplanationsoutsidethefunction\.

\-DoNOTwriteanytextbefore<think\>orafter</answer\>\.

User Prompt:

Trainingexamples:

\{train\_prompt\}

InfertheunderlyinglisttransformationandprovidethePythonfunctionimplementationintherequiredformat\.

### B\.2Prompts Used in Evaluation

#### B\.2\.1Prompt Template: Art

System Prompt:

Youareanexpertinabductivereasoningandnarrativecomprehension\.Yourtaskistodeterminewhichoftwohypothesesprovidesthemostplausibleexplanationforwhathappenedbetweentwogivenobservations\.

Youwillbeprovidedwith:

1\.Observation1\(theinitialsituationorevent\)

2\.Observation2\(thesubsequentoutcomeorresultingevent\)

3\.TwoHypotheses\(Hypothesis1andHypothesis2\)

YourgoalistoselectthehypothesisthatlogicallyandnarrativelybridgesthegapbetweenObservation1andObservation2,explaininghowthesituationtransitionedfromthefirstobservationtothesecond\.

\#\#Instructions:

1\.CarefullyreadObservation1andObservation2tounderstandthechronologicalandnarrativecontext

2\.EvaluatebothHypothesis1andHypothesis2aspotentialbridgingevents

3\.Considercommonsense,cause\-and\-effectrelationships,andeverydayplausibility

4\.Selectthehypothesisthatbestexplainsthetransition

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Either"1"or"2"\-justthenumber,nothingelse\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthenumber1or2\.Donotincludeanyothertext,explanation,orpunctuation\.

User Prompt:

Observation1:\{obs1\}

Observation2:\{obs2\}

Hypothesis1:\{hyp1\}

Hypothesis2:\{hyp2\}

WhichhypothesisbetterexplainsthetransitionfromObservation1toObservation2?

#### B\.2\.2Prompt Template: Copa \(effect\)

System Prompt:

Youareanexpertinlogicalreasoningandcommon\-sensecausalinference\.Yourtaskistodeterminewhichoftwogivenoptionsrepresentsthemostplausibleeffectforagivencause\.

Youwillbeprovidedwith:

1\.ACausedescribingasituationorevent

2\.TwoOptions\(Option1andOption2\)

Yourgoalistoselecttheoptionthatbestdescribesthedirecteffect,logicalconsequence,ormostlikelyresultingactionofthegivencause\.

\#\#Instructions:

1\.Carefullyreadtheprovidedcause

2\.EvaluatebothOption1andOption2aspotentialeffectsorconsequences

3\.Considercommonsense,real\-worldknowledge,andtypicalcause\-and\-effectrelationships

4\.Selecttheoptionthatrepresentsthemostplausibledirecteffect

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Either"1"or"2"\-justthenumber,nothingelse\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthenumber1or2\.Donotincludeanyothertext,explanation,orpunctuation\.

User Prompt:

Cause:\{premise\}

Option1:\{choice1\}

Option2:\{choice2\}

WhichofthefollowingisthemostplausibleEFFECTofthiscause?

#### B\.2\.3Prompt Template: DefasibleNLI

System Prompt:

Youareanexpertindefeasiblereasoningandlogicalanalysis\.Yourtaskistodeterminehownewinformationaffectsthelikelihoodofagivenhypothesis\.

Youwillbeprovidedwith:

1\.AHypothesis\(atentativeconclusion\)

2\.AnUpdate\(newinformation\)

3\.APremise\(optionalcontextualbackground\)

YourgoalistoanalyzethecontextanddecideifthenewUpdatemakestheHypothesismorelikelyorlesslikelytobetrue\.

\#\#Instructions:

1\.ReadtheHypothesisandthePremise\(ifprovided\)tounderstandtheinitialsituation

2\.CarefullyevaluatethenewUpdate

3\.DetermineiftheUpdateprovidesevidencethatsupportstheHypothesis\(strengthensit\)orcontradictsit\(weakensit\)

4\.ClassifytheeffectaseitherSTRENGTHENSorWEAKENS

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[STRENGTHENSorWEAKENS\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYtheexactwordSTRENGTHENSorWEAKENS\.Donotincludeanyothertext,explanation,orpunctuation\.

User Prompt:

Premise:

\{premise\}

Hypothesis:

\{hypothesis\}

Update:

\{update\}

DoesthisUpdateSTRENGTHENorWEAKENtheHypothesis?

#### B\.2\.4Prompt Template: GoEmotion

System Prompt:

Youareanexperttextanalystandemotionclassifier\.Yourtaskistoidentifyallemotionsexpressedinagiventext\.

Youwillbeprovidedwith:

1\.AshortTexttoanalyze

Yourgoalistodetectthepresenceofspecificemotionsfromthefollowingpredefinedlist:

\[\{\_emotions\_list\_str\}\]

\#\#Instructions:

1\.Carefullyreadtheprovidedtext

2\.Analyzethecontext,tone,andnuancetounderstandtheunderlyingfeelings

3\.Matchtheexpressedfeelingsstrictlyagainstthepredefinedlistofavailableemotions

4\.Identifyallapplicableemotions\(use"neutral"ifnospecificemotionisstronglyexpressed\)

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Comma\-separatedlistofapplicableemotions\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYtheexactemotionnamesfromtheavailablelist,separatedbycommasiftherearemultiple\(e\.g\.,joy,surprise\)\.Donotincludeanyothertext,explanation,orcapitalization\.

User Prompt:

Text:"\{text\}"

Whatemotion\(s\)areexpressedinthistext?

#### B\.2\.5Prompt Template: MedQA

System Prompt:

Youareanexpertmedicalcliniciananddiagnostician\.Yourtaskistosolvecomplexmedicalmultiple\-choicequestionsaccurately\.

Youwillbeprovidedwith:

1\.AmedicalProblem,whichtypicallyincludesaclinicalvignetteormedicalquestionalongwithfourcandidatechoices\(A,B,C,D\)

Yourgoalistoevaluatetheclinicalpresentationandselectthesinglemostaccurateanswer\.

\#\#Instructions:

1\.Carefullyreadthemedicalproblem,notingkeypatientdemographics,symptoms,physicalexamfindings,andlabvalueswhereapplicable

2\.Identifythecoremedicalquestionbeingasked\(e\.g\.,nextbeststepinmanagement,mostlikelydiagnosis,underlyingmechanism\)

3\.Evaluateallfourcandidateoptions\(A,B,C,D\)usingevidence\-basedclinicalreasoning

4\.Selectthelettercorrespondingtothecorrectmedicalanswer

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Exactlyoneletter:A,B,C,orD\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthesingleuppercaseletterofthecorrectchoice\(A,B,C,orD\)\.Donotincludeparentheses,periods,oranytextualexplanation\.

User Prompt:

Problem:\{problem\}

Whichoptionisthecorrectanswer?

#### B\.2\.6Prompt Template: Musr Murder

System Prompt:

Youareabrilliantdetectiveandanexpertindeductivereasoning\.Yourtaskistoanalyzecluestosolvecomplexmysteries\.

Youwillbeprovidedwith:

1\.Context:Adetaileddetectivestorycontaininginformationaboutacrime,suspects,alibis,andclues

2\.Problem:Aquestionaboutthemystery,followedbyalistofnumberedmultiple\-choiceoptions

Yourgoalistologicallydeducethetruthfromthecontextandidentifythecorrectchoicebyitsindexnumber\.

\#\#Instructions:

1\.CarefullyreadtheContexttoidentifytimelines,motives,means,andlogicalinconsistenciesamongthesuspects’statements

2\.EvaluatetheProblemandalltheprovidedchoices

3\.Usedeductivereasoningtoeliminateimpossiblescenariosandidentifytheonlylogicallysoundanswer

4\.Notetheindexnumber\(e\.g\.,0,1,2,\.\.\.\)ofthecorrectchoice

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Exactlyoneintegerrepresentingtheindexofthecorrectchoice\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthenumericindexnumberofthecorrectchoice\.Donotincludethetextofthechoice,punctuation,oranyotherexplanationsinsidetheanswertags\.

User Prompt:

Context:

\{context\}

Problem:

\{problem\}

Whatistheindexnumberofthecorrectchoice?

#### B\.2\.7Prompt Template: Musr Team

System Prompt:

Youareanexpertlogicalreasonerspecializinginevaluatingteamskillsandassigningpeopletotasksoptimally\.Yourtaskistoanalyzeastorydescribingpeople,theirabilities,andtheirteamworkdynamicsinordertodeterminethebestassignmentofpeopletotasks\.

Youwillbeprovidedwith:

1\.Context:Astorydescribingseveralpeople,theirabilitiesatdifferenttasks,andhowwelltheyworkwithothers

2\.Problem:Aquestionaskingwhichassignmentofpeopletotasksresultsinthemosteffectivecompletionofthetasks,alongwithmultiple\-choiceoptionsindexedas0,1,2,\.\.\.

Yourgoalistodeterminewhichassignmentbestutilizeseachperson’sskillswhilealsoconsideringteamworkeffectivenesswhentwopeoplemustworktogetheronatask\.

\#\#Instructions:

1\.CarefullyreadtheContextandidentifyeachperson’sskilllevelfortherelevanttasks\(e\.g\.,great,acceptable,orbad\)

2\.Determinehowwelldifferentpairsofpeopleworktogetherwhenassignedtothesametask

3\.Rememberthatonetaskwillrequiretwopeopleworkingtogether

4\.Considerthatifonepersonisbadatatask,theotherperson’sskillmaynotfullycompensateunlesstheyworkwelltogether

5\.Evaluatetheoveralleffectivenessofeachassignmentoption

6\.Selecttheoptionthatresultsinthemosteffectiveoverallcompletionofalltasks

7\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Exactlyoneintegerrepresentingtheindexofthecorrectchoice\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthenumericindexnumberofthecorrectchoice\.Donotincludethetextofthechoice,punctuation,oranyadditionalexplanation\.

User Prompt:

Context:

\{context\}

Problem:

\{problem\}

Whatistheindexnumberofthecorrectchoice?

#### B\.2\.8Prompt Template: Musr Object

System Prompt:

Youareanexpertlogicalreasonerspecializingintrackingbeliefsandobjectlocationsinnarrativestories\.Yourtaskistoanalyzeastoryanddeterminewhereacharacterbelievesanobjectislocated\.

Youwillbeprovidedwith:

1\.Context:Astorydescribingcharacters,theiractions,andmovementsofobjects

2\.Problem:Aquestionaboutwhereaspecificcharacterbelievesanobjectislocated,alongwithmultiple\-choiceoptionsindexedas0,1,2,\.\.\.

Yourgoalistodeterminethecorrectanswerbyreasoningaboutwhatthecharacterobservedandthereforebelievesabouttheobject’slocation\.

\#\#Instructions:

1\.CarefullyreadtheContextandtracktheobject’slocationthroughoutthestory

2\.Trackwhateachcharacterobserveswhentheobjectismoved

3\.Ifacharacterobservestheobjectmoving,theyupdatetheirbeliefabouttheobject’slocation

4\.IfacharacterdoesNOTobservetheobjectmoving\(e\.g\.,theyareabsentordistracted\),theywillcontinuetobelievetheobjectremainsinthelastlocationwheretheysawit

5\.AnalyzetheProblemandevaluateallprovidedchoices

6\.Determinewhichoptioncorrectlyrepresentsthecharacter’sbeliefabouttheobject’slocation

7\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Exactlyoneintegerrepresentingtheindexofthecorrectchoice\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthenumericindexnumberofthecorrectchoice\.Donotincludethetextofthechoice,punctuation,oranyadditionalexplanation\.

User Prompt:

Context:

\{context\}

Problem:

\{problem\}

Whatistheindexnumberofthecorrectchoice?

#### B\.2\.9Prompt Template: NeuLR Abductive

System Prompt:

YouareanexpertForensicLogicAnalystanddeductivereasoningspecialist\.Yourtaskistoperformabductivereasoningtoidentifyamissinglogicalpremise\.

Youwillbeprovidedwith:

1\.LogicalRulesandKnownFacts:Asetofestablishedrules\(If/Thenstatements\)andgivenbasefacts\.

2\.TargetConclusion:Anobservedfactoroutcomethatcurrentlycannotbeprovenusingonlytheprovidedfactsandrules\.

YourgoalistoidentifythesingleMISSINGFACT\(premise\)that,whenaddedtotheknownfacts,makestheTargetConclusionlogicallytruebasedontheRules\.

\#\#Instructions:

1\.CarefullyreadtheLogicalRulesandKnownFactstounderstandtheestablishedlogicaluniverse\.

2\.AnalyzetheTargetConclusionthatneedstobeproven\.

3\.WorkbackwardfromtheTargetConclusiontoidentifywhichrule\(s\)couldproduceit\.

4\.Checktheconditionsforthoserule\(s\)againsttheKnownFacts\.

5\.Identifytheexactmissingcondition\(fact\)requiredtocompletethelogicalchainandtriggertheruletoprovetheTargetConclusion\.

6\.Formulatethismissingfactasacompletesentence,matchingtheexactsyntax,terminology,andstyleoftheprovidedcontext\.

7\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[Theexactmissingfactwrittenasacompletesentence\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthemissingfactasasinglecompletesentence\(e\.g\.,"NPsw0v0kisADP37scy8\."\)\.Donotincludequotationmarks,introductorytext,oranyadditionalexplanationswithintheanswertags\.

User Prompt:

LogicalRulesandKnownFacts:

\{rules\_block\}

TargetConclusion:

\{target\_fact\}

WhatmissingfactisrequiredtoconcludetheTargetConclusion?

#### B\.2\.10Prompt Template: StrategyQA

System Prompt:

Youareanexpertdeductivereasonerandfact\-checker\.Yourtaskistoanswerayes/noquestionusingtheprovidedevidence\.

Youwillbeprovidedwith:

1\.AQuestion:AspecificqueryrequiringaYESorNOanswer\.

2\.Evidence:Alistoffactsorparagraphscontainingrelevantinformation\.

Yourgoalistodeducethecorrectanswerbasedonthelogicalimplicationsoftheprovidedevidence\.

\#\#Instructions:

1\.CarefullyreadtheQuestiontounderstandwhatisbeingasked\.

2\.AnalyzetheprovidedEvidenceparagraphs,identifyingfactsrelevanttothequestion\.

3\.SynthesizethefactstologicallyformulateadefinitiveYESorNOconclusion\.

4\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

\[OutputexactlyYESorNO\]

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthewordYESorthewordNO\.Donotincludeanyothertext,punctuation,orexplanations\.

User Prompt:

Question:

\{question\}

Evidence:

\{evidence\_text\}

IstheanswertothequestionYESorNO?

#### B\.2\.11Prompt Template: MLDebugging

System Prompt:

YouareanexpertPythondeveloperanddebugger\.YourtaskistoidentifyandfixerrorsinPythoncodesnippets\.

Youwillbeprovidedwith:

1\.TaskInstructions:Theintendedbehaviorandrequirementsforthecode

2\.BuggyCode:TheincorrectPythoncodesnippetthatisfailingitstests

3\.RuntimeError/TestFeedback:Theexecutionlogs,tracebacks,orfailingtestresults

Yourgoalistoanalyzethefailure,correctthebug,andprovidethecomplete,workingPythoncode\.

\#\#Instructions:

1\.CarefullyreadtheTaskInstructionstounderstandthedesiredfunctionality

2\.AnalyzetheBuggyCodetogetherwiththeRuntimeError/TestFeedbacktoidentifytherootcause

3\.Determinetheminimalnecessaryfixthatpreservescorrectexistingbehavior

4\.Outputafullyself\-containedcorrectedPythonsolution\(noplaceholders,noomissions\)

5\.Thinkstepbystep\.

\#\#OutputFormat:

YouMUSTprovideyouranswerinthefollowingformat:

<think\>

\[Thinkstepbystephere\]

</think\>

<answer\>

‘‘‘python

\[Yourfull,correctedPythoncodehere\]

‘‘‘

</answer\>

CRITICAL:TheanswersectionmustcontainONLYthefullcorrectedPythoncodeblock\.Noexplanations,noextratext,andnoadditionalformattingoutsidethecodeblock\.

User Prompt:

TaskInstructions:

\{instruct\_prompt\}

BuggyCode:

‘‘‘python

\{bug\_code\}

‘‘‘

RuntimeError/TestFeedback:

\{runtime\_feedback\}

WhatisthefullycorrectedPythoncode?

## Appendix CTraining and Implementation Details

### C\.1Hyperparameters and Model Specifications

The training pipeline uses Unsloth for memory\-efficient post\-training ofDeepSeek\-R1\-Distill\-Qwen\-7B,Qwen3\-4B,Qwen3\-8B, andLlama\-3\.1\-8B\-Instruct\. All models use 4\-bit NormalFloat \(NF4\) quantization and Low\-Rank Adaptation \(LoRA\)\. Table[7](https://arxiv.org/html/2608.14791#A3.T7)reports the shared configuration\.

HyperparameterValueHyperparameterValueBase ModelsDeepSeek\-R1\-Distill\-Qwen\-7B, Qwen3\-4B, Qwen3\-8B, Llama\-3\.1\-8B\-InstructLearning Rate1×10−51\\times 10^\{\-5\}QuantizationNF4 \(4\-bit\)LR ScheduleCosineLoRA Rank \(rr\)64Warmup Steps2LoRA Alpha \(α\\alpha\)64Weight Decay0\.1GRPO Group Size \(GG\)4OptimizerAdamW \(Torch\)Max Sequence Length4096Adamβ1,β2\\beta\_\{1\},\\beta\_\{2\}0\.9, 0\.99Max Prompt Length2048Max Completion Length2048Per\-device Batch Size4Gradient Accumulation1Rollout Temperature0\.7KL Penalty \(β\\beta\)0\.01Training Epochs5Clip Epsilon \(ε\\varepsilon\)0\.2Max Grad Norm0\.1Evaluation / Save Interval128 / 128 stepsTable 7:Training and generation hyperparameters across model architectures\.
### C\.2Compute and Reproducibility

All local training and evaluation were conducted on one NVIDIA GeForce RTX 5090 GPU with 32 GB GDDR7; no multi\-GPU parallelism was used\. Thegpt\-oss\-120breward judge was accessed remotely\. We fix the pipeline\-level random state to 3407 and the PyTorch and NumPy seeds to 42\.

## Appendix DReward Function Formulation

### D\.1Composite Objective

For promptxx, completionyy, ground truthgg, and datasetdd, letβ⁡\(y\)\\beta\(y\)andα⁡\(y\)\\alpha\(y\)denote the text extracted from the<think\>and<answer\>spans\. The reward passed to GRPO is

R=rcor\+rcov\+rdir3\.R=\\frac\{r\_\{\\mathrm\{cor\}\}\+r\_\{\\mathrm\{cov\}\}\+r\_\{\\mathrm\{dir\}\}\}\{3\}\.\(4\)No additional transformation is applied before GRPO\. Cor\-GRPO uses onlyrcorr\_\{\\mathrm\{cor\}\}\.

### D\.2Correctness Reward

rcor\(x,y,g,d\)=𝕀\[Vd\(α\(y\),g\)=1\]\.r\_\{\\mathrm\{cor\}\}\(x,y,g,d\)=\\mathbb\{I\}\\\!\\left\[V\_\{d\}\(\\alpha\(y\),g\)=1\\right\]\.\(5\)If the<answer\>span is absent,rcor=0r\_\{\\mathrm\{cor\}\}=0\. Table[8](https://arxiv.org/html/2608.14791#A4.T8)summarizes the dataset\-specific verifiers\.

DatasetVerifierBalancedCOPAParse an integer and compare it with the one\-based gold label\.CauseLogicsUppercase prediction and gold label and require exact equality\.ClimateFeverUppercase prediction and gold label and require exact equality among the four labels\.AbductionRulesCollapse whitespace, strip, lowercase, and require exact equality\.UniADILRExtract standalone integers and compare their set with the gold proof\-antecedent sentence identifiers; order and duplicates are ignored\.CryptoExecutetransform\(s\)on every held\-out test input and require exact string outputs for all tests\.ListFunctionExecutetransform\(lst\)on ten held\-out test pairs and require matching integer\-list outputs for all tests\.Table 8:Dataset\-specific correctness verifiers used during training\.For executable tasks, code is extracted from the answer span, optional Markdown fences are removed, and each test runs in a separate process with a five\-second timeout\.

### D\.3Process Rewards

The coverage judge receivesuu,β⁡\(y\)\\beta\(y\), and a dataset\-specific note and returnsmmdetail records with addressed indicatorszjz\_\{j\}\. The implementation computes

rcov​\(x,y,d\)=\{1m​∑j=1mzj,m\>0,0,otherwise\.r\_\{\\mathrm\{cov\}\}\(x,y,d\)=\\begin\{cases\}\\frac\{1\}\{m\}\\sum\_\{j=1\}^\{m\}z\_\{j\},&m\>0,\\\\ 0,&\\text\{otherwise\}\.\\end\{cases\}\(6\)The denominator is the number of details returned by the judge; the code applies no additional alias merging or deduplication\.

The directionality judge receives the same inputs and returns

rdir​\(x,y,d\)∈\{0,0\.5,1\},r\_\{\\mathrm\{dir\}\}\(x,y,d\)\\in\\\{0,0\.5,1\\\},\(7\)for explanation\-to\-evidence, mixed, and evidence\-to\-explanation reasoning, respectively\. Invalid or unparseable judge outputs receive zero\. If the<think\>span is absent, the full completion is passed to the process judges\. Both judges useopenai/gpt\-oss\-120bat temperature0\.00\.0, and both process rewards are active for all seven training datasets\.

### D\.4Training\-Time Judge Prompts

The following prompts are used online to compute the process rewards\. They are distinct from the generator prompts in Appendix[B](https://arxiv.org/html/2608.14791#A2)and the post\-hoc evaluation prompts in Appendix[E\.4](https://arxiv.org/html/2608.14791#A5.SS4)\.

#### D\.4\.1Evidence Coverage

##### System message\.

\#\# Training System ContextYou are an automated reward judge embedded in a Group Relative Policy Optimization \(GRPO\) reinforcement\-learning training loop\. A language model \(the \*generator\*\) is being fine\-tuned on abductive reasoning datasets\. At each training step the generator produces a reasoning trace in response to an abductive problem; your structured score is used directly as a reward signal to update the generator’s weights\. Evaluate precisely and strictly from what is written in the trace\.The generator is learning abductive reasoning: given observations, identify the most plausible hypothesis and reason transparently from evidence to conclusion\.\-\-\-You are an expert evaluator of abductive reasoning traces\.\#\# Your TaskGiven a reasoning\_trace in which a model selects one hypothesis to explain an observation, you must:1\. \*\*Extract every specific detail\*\* that appears in the observation \(or the model’s description of the observation\) \-\- not just the main event, but also peripheral facts, contextual clues, timing details, quantities, locations, named entities, and any other particulars mentioned\.2\. \*\*For each detail\*\*, decide whether the reasoning\_trace \*explicitly\* connects that detail to the chosen hypothesis:\- A detail is "addressed" only if there is a direct explanation of \*why or how\* the hypothesis connects to that detail\.\- Set addressed = false, evidence = an empty string if the detail is not explicitly connected\.3\. \*\*Write a brief overall\_analysis\*\* summarising the observation coverage\.\#\# Important rules\- Be exhaustive across main and peripheral details\.\#\# Output formatYou may first think freely — identify each detail, deliberate on whether the reasoning trace addresses it, and work through any ambiguous cases\. Once you have finished your analysis, place a single valid JSON block at the very end of your response with this structure:\{"overall\_analysis": "Brief analysis of observation coverage in this reasoning\_trace","observation\_details": \[\{"detail": "Specific atomic detail from the observation","addressed": true,"evidence": "Quote from reasoning\_trace showing how this detail is explained"\}\]\}

##### User\-message template\.

Dataset: \{dataset\_name\}Dataset context: \{coverage\_dataset\_note\}Analyse the following observation and reasoning\_trace and produce the structured observation\-coverage evaluation\.<observation\>\{policy\_user\_message\}</observation\><reasoning\_trace\>\{extracted\_think\_span\}</reasoning\_trace\>

#### D\.4\.2Evidence–Explanation Directionality

##### System message\.

\#\# Training System ContextYou are an automated reward judge embedded in a Group Relative Policy Optimization \(GRPO\) reinforcement\-learning training loop\. A language model \(the \*generator\*\) is being fine\-tuned on abductive reasoning datasets\. At each training step the generator produces a reasoning trace; your score is used directly as a reward signal to update the generator’s weights\.The generator is learning that abductive reasoning runs strictly from evidence to explanation — not the reverse\. Your score tells the training loop how well the current trace respects that directional constraint\.\-\-\-You are an expert evaluator assessing the logical directionality of reasoning chains generated by AI models\.\#\# Metric Definition\*\*Evidence\-Explanation Directionality:\*\* The model demonstrates awareness that abduction runs strictly from evidence to explanation \(not explanation to evidence / not deduction\)\.In proper abductive reasoning, the model MUST start from the known observations \(the evidence\) and logically reason forward to determine the best hypothesis \(the explanation\)\. Assuming an explanation is true and then checking whether it predicts or aligns with the given observations is a core logical flaw for this metric\.\#\# Scoring Rubric\*\*Score 1\.0 – Proper Abductive Direction\*\*\- The model reasons from observations/evidence as the foundation\.\- It builds its argument forward from the observations toward the hypothesis\.\- Language cue: "Given these observations\.\.\."\*\*Score 0\.5 – Ambiguous or Mixed Directionality\*\*\- The model shows partial abduction but also some backwards reasoning\.\- The logical flow is not clearly unidirectional\.\*\*Score 0\.0 – Explanation\-Evidence Direction \(deduction\)\*\*\- The model presupposes an explanation and checks if it predicts the observations\.\- Reasons backwards from assumed conclusions to evidence\.\#\# Output FormatYou may first think freely — examine directional cues in the language, deliberate on which score best fits, and explain your reasoning\. Once you have finished, place a single valid JSON block at the very end of your response:\{"directionality\_score": 0\.0,"explanation": "Brief explanation of the logical direction observed"\}Note: directionality\_score must strictly be 0\.0, 0\.5, or 1\.0\. Do not wrap the JSON in markdown code blocks\.

##### User\-message template\.

Dataset: \{dataset\_name\}Dataset context: \{directionality\_dataset\_note\}Evaluate the reasoning chain below for Evidence\-Explanation Directionality and assign a score of 0\.0, 0\.5, or 1\.0\.\*\*Observations / Evidence:\*\*<observations\>\{policy\_user\_message\}</observations\>\*\*Model’s Reasoning Chain:\*\*<reasoning\_chain\>\{extracted\_think\_span\}</reasoning\_chain\>Assign a directionality score \(0\.0 / 0\.5 / 1\.0\) and briefly explain your assessment in the \`explanation\` field\.

#### D\.4\.3Dataset\-Specific Notes

The strings below replacecoverage\_dataset\_noteanddirectionality\_dataset\_notein the templates above\.

##### UniADILR\.

Coverage:The text passages and numbered sentences are the observation\. Extract all factual claims together with their sentence identifiers\.Directionality:The numbered Context sentences are the evidence, and the stated Hypothesis is the candidate explanation\. Assess whether the trace reasons from the context evidence toward support for that hypothesis\.

##### BalancedCOPA\.

Coverage:The ’Premise:’ field is the observation\. Extract all stated facts and contextual details from the premise\.Directionality:The Premise is the observed event, and Choice 1 and Choice 2 are candidate causes\. Assess whether the trace reasons from the premise toward selecting the more plausible cause\.

##### CauseLogics\.

Coverage:The Premises, Rules, and Phenomenon form the observation, while the Possible Cause is the candidate hypothesis\. Extract all stated facts, rule conditions and conclusions, and phenomenon details\.Directionality:The Premises and Rules are background evidence, the Phenomenon is the observation, and the Possible Cause is the candidate explanation\. Assess whether the trace reasons from the stated evidence toward evaluating that explanation\.

##### ClimateFever\.

Coverage:The Claim and provided Evidence sentences together form the observation\. Extract all factual statements and their individual components\.Directionality:Treat the provided Evidence sentences as the observations and the selected claim label as the conclusion\. Assess whether the reasoning moves from the evidence toward that label\.

##### AbductionRules\.

Coverage:The Context facts and rules together with the Query form the observation\. Extract all relevant conditions, entity properties, stated facts, and query details\.Directionality:Treat the Context facts, rules, and Query as the observations and the proposed missing fact as the hypothesis\. Assess whether the reasoning moves from the given context and query toward that fact\.

##### ListFunction\.

Coverage:The ten training input\-output pairs are the observation\. Treat each demonstrated list transformation as a detail and assess whether the reasoning explains the pattern across all examples\.Directionality:The observed list input\-output pairs are the evidence, and the inferred list transformation is the explanation\. Assess whether the reasoning moves from the examples to the rule\.

##### Crypto\.

Coverage:The ten training input\-output pairs are the observation\. Treat each demonstrated string transformation as a detail and assess whether the reasoning explains the character\-level pattern across all examples\.Directionality:The observed string input\-output pairs are the evidence, and the inferred string transformation is the explanation\. Assess whether the reasoning moves from the examples to the rule\.

### D\.5Reward Metric Selection

To select process\-level metrics for reward shaping, we screened each candidate metric along two axes: task applicability and reward\-hacking risk\. Applicability measures whether improving a metric is likely to support the reasoning behavior required by each training source\. Reward\-hacking risk measures how easily a policy could increase the metric through superficial changes in trace style, verbosity, or formatting rather than through genuine improvements in reasoning\.

Table[9](https://arxiv.org/html/2608.14791#A4.T9)summarizes this screening process\. Dark green cells indicate that a metric has a direct and consistent effect on the corresponding dataset, while hatched light green cells indicate a moderate or context\-dependent effect\. White cells indicate that the metric is negligible or not applicable for that dataset\.

This screening led us to select Evidence Coverage and Evidence–Explanation Directionality as our primary process rewards\. Evidence Coverage is broadly applicable because it encourages the model to explicitly account for the observations provided in the prompt across selection, verification, and generation settings\. Evidence–Explanation Directionality complements this by encouraging the reasoning trace to preserve the abductive flow from evidence toward explanation\. In contrast, although metrics such asBacktrackingandUncertainty Markerscan be useful for analysis, they carry substantially higher reward\-hacking risk: a policy could increase them by producing artificial self\-corrections or excessive hedging without improving the underlying reasoning process\.

MetricUniADILRCOPACause
LogicsCLIMATEAbduction
RulesList
Func\.CryptoReward\-hacking
riskBranchinessHighBacktrackingV\. highDifferential EliminationMod\.Prior InvocationHighEvidence CoverageLow–med\.Evidence–Explanation DirectionalityLow–med\.Uncertainty MarkersV\. high
Legend\.= effective candidate signal;= somewhat effective or context\-dependent signal;= not applicable or negligible\. Reward\-hacking risk estimates how easily a metric could be maximized by changing the style or verbosity of the trace rather than improving the underlying reasoning behavior\.

Table 9:Screening candidate process\-level metrics as possible reward terms\. Applicability is evaluated per training source, while reward\-hacking risk is assessed at the metric level\. This screening motivates our reward\-composition choice: Evidence Coverage and Evidence–Explanation Directionality are selected because they are broadly applicable across datasets and comparatively resistant to superficial optimization\.

## Appendix EProcess\-Level Metrics

### E\.1Evaluation Framework

All seven process\-level metrics are evaluated using an LLM\-as\-judge approach\. Each reasoning trace is passed toGemini 3 Flashtogether with a structured system prompt specific to the metric and a short dataset\-specific contextual framing\. The judge is required to return a fixed JSON schema; output parsing is fully deterministic and involves no free\-text post\-processing\. No few\-shot demonstrations are used in any judge prompt\. \(Note that this offline evaluation setup uses Gemini 3 Flash to independently verify trace quality, which is distinct from the training phase wheregpt\-oss\-120bwas used to compute dynamic RL rewards\)\.

Three families of metric are used\.Counting metrics\(Uncertainty Markers, Branchiness, Backtracking, Differential Elimination, Prior Invocation\) instruct the judge to return a list of concrete⟨excerpt,explanation⟩\\langle\\textit\{excerpt\},\\textit\{explanation\}\\ranglepairs, one per identified occurrence\. The reported value per trace is the total number of extracted instances, averaged over all traces in a dataset\.Coverage metrics\(Evidence Coverage\) ask the judge to enumerate every atomic observation detail and mark each one as explicitly addressed or not; the reported score per trace is the proportion of total details that are explicitly addressed, averaged over the dataset\.Score\-based metrics\(Evidence–Explanation Directionality\) ask the judge to assign one of three discrete values\(0\.0,0\.5,1\.0\)\(0\.0,\\ 0\.5,\\ 1\.0\); the reported value is the per\-trace score averaged over the dataset\.

### E\.2Metric Definitions

#### E\.2\.1Uncertainty Markers

Type:Counting\.

Core question:How many individual probabilistic or hedging words and phrases appear in the reasoning trace?

Uncertainty Markers extracts every distinct occurrence of a word or phrase that signals epistemic hedging or probabilistic qualification, rather than asking a binary question about their presence\. Each occurrence is logged as an independent example\.

Counted:

- •Probability/likelihood qualifiers:probably, likely, unlikely, possibly, in all likelihood, with high probability, most likely
- •Epistemic modal verbs:might, may, could, seems to, appears to, tends to
- •Hedging phrases:I believe, I think, I suspect, this suggests, this may indicate
- •Degree/approximation qualifiers:approximately, roughly, to some extent, somewhat, fairly
- •Epistemic uncertainty statements:we cannot be sure, it is unclear, the evidence is inconclusive, this remains to be confirmed
- •Frequency and scope limiters:often, typically, generally, sometimes, in some cases, less common

Not Counted:

- •Objective risk or statistical labels \(*e\.g\.*“high risk”\)
- •Phrases that establish a premise boundary \(*e\.g\.*“based on the information provided”\)
- •Evaluative or affective state descriptions \(*e\.g\.*“there is concern”\)

Reported statistic:Mean count of marker occurrences per trace, averaged over the dataset\.

#### E\.2\.2Branchiness

Type:Counting\.

Core question:How many times does the reasoning genuinely explore multiple distinct candidate explanations for the same observation?

Branchiness measures whether the model considers substantively different explanatory candidates rather than following a single linear chain\. The competing candidates must differ in underlying causal mechanism, agent, domain, or interpretation — not merely in phrasing or confidence level\.

Counted:

- •Exploring two diagnoses with separate evidential implications \(“If condition X we would expect F; if condition Y we would expect G”\)
- •Identifying different causal mechanisms for the same narrative event
- •Building and comparing competing hypotheses

Not counted:

- •Refinements or phrasings of the same explanation
- •Forward\-branching planning logic
- •Procedural trial\-and\-error
- •The final answer selection
- •A brief mention of an alternative immediately rejected
- •Simple step\-by\-step narration
- •Restating the supplied answer options

Reported statistic:Mean count of branching instances per trace, averaged over the dataset\.

#### E\.2\.3Backtracking

Type:Counting\.

Core question:How many times does the reasoning explicitly identify an error or flaw and change direction?

Backtracking \(self\-correction\) captures deliberate revision: the model recognises that something it said or computed is wrong and reverses course\. This is distinct from Branchiness, which measures exploration among valid alternatives\.

Counted:

- •Explicit admission of error \(“Wait, that’s wrong”, “I made a mistake”\)
- •Deliberate pausing and restarting \(“Hold on, let me re\-read the problem”\)
- •Mid\-reasoning strategy change \(“On second thought…”, “That approach doesn’t work, so…”\)
- •Realisation of a missed detail \(“I realise I forgot to account for…”\)

Not counted:

- •Comparing two valid paths \(Branchiness\)
- •A “however” contrast without admitting an error
- •The final answer selection

Reported statistic:Mean count of backtracking instances per trace, averaged over the dataset\.

#### E\.2\.4Differential Elimination

Type:Counting\.

Core question:How many explicit elimination or refutation moves against alternatives appear in the reasoning trace?

Differential Elimination extracts each distinct case where the model rules out an alternative hypothesis, answer option, or interpretation with an explicit evidence\-grounded reason\. The refutation must supply a*why*, not merely assert that an option is incorrect\.

Counted:

- •“We can rule out A because it contradicts symptom X”
- •Conditional falsification \(“If B were true we would see Y, but we do not”\)
- •Explicit contradiction between an alternative and the observed details

Not counted:

- •Empty dismissals without a context\-grounded reason
- •Pure positive support for the chosen hypothesis
- •Listing options without evaluating them
- •Vague preference statements

Reported statistic:Mean count of elimination instances per trace, averaged over the dataset\.

#### E\.2\.5Prior Invocation

Type:Counting\.

Core question:How many times does the reasoning explicitly invoke prior probability, typicality, or base\-rate knowledge?

Prior Invocation captures explicit references to background frequency or likelihood that the model brings to bear on the problem before or alongside the specific evidence in the trace\.

Counted:

- •Population base rates
- •Domain\-knowledge priors \(“Most patients with these symptoms have condition X”\)
- •Comparative likelihoods \(“X is far more common than Y”\)
- •General tendencies or patterns \(“Usually, this symptom indicates…”\)
- •Explicit Bayesian\-style reasoning that weighs a prior against new evidence

Not counted:

- •Conclusions drawn only from the specific observations in the problem \(posterior inferences\)
- •World\-knowledge facts without frequency or likelihood content
- •The model’s own hedging expressions, which are captured by Uncertainty Markers
- •Restatements of information given in the prompt

Reported statistic:Mean count of prior\-invocation instances per trace, averaged over the dataset\.

#### E\.2\.6Evidence Coverage

Type:Coverage\.

Core question:What fraction of the specific observation details are explicitly accounted for by the chosen hypothesis?

Evidence Coverage enumerates every atomic observation detail present in the input and assesses whether the reasoning trace explicitly connects each one to the chosen hypothesis\. Atomicity is enforced throughout: compound statements are split into indivisible facts \(*e\.g\.*“headache and vomiting” becomes two items; “Kernig and Brudzinski signs present” becomes two items; every medication, vital sign, and lab value is its own item\)\. A detail is marked*addressed*only when the trace makes a clear logical link between that detail and the chosen hypothesis — restating the detail or merely acknowledging its existence is not sufficient\.

The coverage score per trace is

scov=\#​addressed details\#​total details∈\[0,1\]\.s\_\{\\mathrm\{cov\}\}\\;=\\;\\frac\{\\\#\\text\{addressed details\}\}\{\\\#\\text\{total details\}\}\\;\\in\[0,1\]\.
Reported statistic:Meanscovs\_\{\\mathrm\{cov\}\}over all traces in the dataset, expressed as a percentage in tables\.

#### E\.2\.7Evidence–Explanation Directionality

Type:Score\-based\.

Core question:How strongly does the reasoning respect the abductive direction from evidence to explanation?

This metric assesses whether the model treats the given observations as the fixed starting point and reasons forward to the best explanatory hypothesis\. The core failure mode the metric is designed to detect is the logical reversal in which a model assumes a hypothesis first and then deductively checks whether it predicts the evidence — a pattern corresponding to the prediction format “If hypothesisHHis true, we would expect observationOO; we observeOO, thereforeHH\.”

The judge assigns exactly one of three discrete scores:

- •1\.0— Clear evidence→\\toexplanation reasoning: the trace treats the observations as given, asks what best explains them, and builds its argument from the evidence upward to the conclusion\.
- •0\.5— Ambiguous or mixed directionality: the reasoning connects evidence and explanation but the logical flow is unclear or loosely structured, wavering between both directions\.
- •0\.0— Backward or deductive direction: the trace assumes a conclusion first and then verifies that the evidence confirms it\.

Reported statistic:Mean directionality score over all traces in the dataset \(range 0\.0–1\.0\)\.

### E\.3Dataset\-Specific Judge Notes

Every judge prompt injects a short dataset\-specific note providing contextual framing for the metric definition based on the format and content of each evaluation split\. Table[11](https://arxiv.org/html/2608.14791#A5.T11)gives the notes for all seven metrics \(the five counting metrics, Evidence Coverage, and Evidence–Explanation Directionality\)\.

### E\.4Judge Prompts

The subsections below give the verbatim system prompt and user\-prompt template sent to Gemini 3 Flash for each metric\. The placeholder\{dataset\_specific\_note\}is replaced at runtime with the corresponding cell from Table[11](https://arxiv.org/html/2608.14791#A5.T11)\. All evaluations are strictly zero\-shot\.

#### E\.4\.1Uncertainty Markers

System Prompt:

YouareanexpertlinguisticanalystevaluatingAI\-generatedreasoningtraces\.

\#\#WhatisanUncertaintyMarker?

Anuncertaintymarkerisa\*\*specificwordorphrase\*\*thatsignalsthemodelis

expressingadegreeofbelief,possibility,probability,ratherthanstating

somethingasanabsolute,universal,orestablishedfact\.Yourjobistolocate

everyindividualmarkerthatappearsinthereasoningtrace\.

\#\#Categoriesofuncertaintymarkerstoextract

\#\#\#1\.Probability/likelihoodqualifiers

Wordsorphrasesthatplacesomethingonaprobabilityscale\.

Examples:"probably","likely","unlikely","possibly","conceivably",

"inalllikelihood","thereisachance","withhighprobability",

"mostlikely","leastlikely","moreprobablethan"\.

\#\#\#2\.Epistemicmodalsandverbsofpotential

Verbsthatexpresspossibility,tentativejudgement,orpotentialratherthan

aguaranteedoutcome\.

Examples:"might","may","could","would","can"\(whenusedas’hasthe

potentialto’,e\.g\.,"canhelp"\),"seemsto","appearsto","tendsto"\.

\#\#\#3\.Hedgingphrases\(first\-personorimpersonal\)

Phrasesthatexplicitlyframeastatementasabelieforestimate\.

Examples:"Ibelieve","Ithink","Isuspect","I’mnotcertainbut",

"itispossiblethat","itseemsthat","itappearsthat",

"thissuggests","thismayindicate"\.

\#\#\#4\.Degree/approximationqualifiers

Phrasesthatsoftenaclaimbyexpressingpartialknowledgeorapproximation\.

Examples:"approximately","roughly","around","about","orso",

"tosomeextent","inpart","somewhat","fairly","relatively"\.

\#\#\#5\.Epistemicuncertaintystatements

Explicitacknowledgementsthatsomethingisunknownorunconfirmed\.

Examples:"wecannotbesure","itisuncertainwhether","itisunclear",

"theevidenceisinconclusive","thisisnotdefinitivelyestablished",

"thisremainstobeconfirmed"\.

\#\#\#6\.Frequencyandscopelimiters

Wordsthatsoftenauniversalassertionbylimitingitsfrequencyorscope,

leavingroomforexceptions\.

Examples:"often","typically","generally","frequently","lesscommon",

"insomecases","sometimes","usually"\.

\#\#WhatNOTtoExtract\(FalsePositives\)

Do\*\*NOT\*\*extractthefollowinglinguisticconstructs,astheydonotrepresent

epistemicuncertainty:

\-\*\*Objectiveriskorstatisticalmetrics:\*\*Mentionsof"risk"describean

objectivestateorclassification,notthespeaker’sdoubt\.

\(e\.g\.,DoNOTextract"highrisk","reducestherisk"\)\.

\-\*\*Evidentialattributions/Premiseboundaries:\*\*Phrasesthatciteasource

orestablishtheboundaryofthepremise\.

\(e\.g\.,DoNOTextract"Basedontheinformationprovided",

"Accordingtothetext"\)\.

\-\*\*Evaluativeoraffectivestates:\*\*Statingthatanemotionorclinical

attitudeexistsisafactualclaimaboutastateofaffairs\.

\(e\.g\.,DoNOTextract"Thereisconcern","Itisalarming"\)\.

\#\#Extractionrules

\-Extract\*\*eachindividualmarkeroccurrence\*\*asaseparateexample,evenif

thesamewordappearsmultipletimes\.Everyoccurrenceisitsownentry\.

\-The‘excerpt‘mustbea\*\*short,directquote\*\*fromthetext\-\-ideallythe

singlewordorshortphraseitself,plusjustenoughsurroundingcontext

\(<=15words\)tomakeitreadable\.

\-The‘explanation‘mustnamethemarkercategory\(fromthelistabove\)and

brieflystatewhatbelief,probability,orlimitationthemarkerexpresses

incontext\.

\-Ifthesamesentencecontainstwodistinctmarkers,extractthemastwo

separateentries\.

\-Do\*\*not\*\*paraphraseoralterthequotedtext\.

\#\#Dataset\-specificnote\(currentdatasetonly\)

\{dataset\_specific\_note\}

\#\#JSONoutputformat

ReturnONLYvalidJSONwiththisstructure:

\{

"overall\_analysis":"Briefanalysisofuncertaintymarkersdensityinthis

reasoningtrace",

"examples":\[

\{

"excerpt":"Quoteoftheuncertaintymarkerfromthereasoningtrace",

"explanation":"Categoryandmeaningofthisuncertaintymarker"

\}

\]

\}

User Prompt Template:

Dataset:\{dataset\}

Extracteveryindividualuncertaintymarkerfromthefollowingreasoningtrace\.

Returnoneentrypermarkeroccurrence\.

<reasoning\_trace\>

\{text\}

</reasoning\_trace\>

#### E\.4\.2Branchiness

System Prompt:

YouareanexpertreasoninganalystevaluatingAI\-generatedreasoningtraces\.

\#\#WhatisBranchiness?

Branchinessmeasureswhetherthereasoning\*\*genuinelyexploresmultiple

distinctcandidateexplanations\*\*forthesameobservationbeforesettlingon

one,ratherthanfollowingasinglelinearpath\.

Thekeydistinctionisthis:

\-Countmultiplecandidateexplanationsonlywhentheyaresubstantively

differentexplanationsoftheobservation\.

\-DoNOTcountmultipleversions,refinements,orrestatementsofthesame

underlyingexplanation\.

\#\#WhatCOUNTSasabranchingmoment

Extractanexamplewhenyousee:

1\.Exploringtwoormoregenuinelydistinctcandidateexplanationsforthesame

observationbeforesettlingonone\.

2\.Identifyingdifferentcausalmechanisms,agents,domains,orscenario

interpretationsthatcouldexplaintheobservation\.

3\.Buildingandcomparingcompetinghypotheseswiththeirimplications/evidence

\("IfdiagnosisXwe’dexpectF\.\.\.IfdiagnosisYwe’dexpectG\.\.\."\)\.

\#\#WhatdoesNOTcount

\-Multiplephrasings,refinements,orconfidenceadjustmentsofthesame

explanation\.

\-Amainexplanationplusasmallmodifierordetailaddedtothatsame

explanation\.

\-Strictlyforward\-branchingpredictivelogicorconditionalplanning

\(e\.g\.,"IfIdoX,thenYhappens"\)\.

\-Tryingdifferentproceduralsolutionmethods\(thisisnotabductive

branching\)\.

\-Thefinalanswerselectionorconclusion\.

\-Abriefmentionofanalternativefollowedbyimmediaterejectionwithno

exploration\.

\-Simplestep\-by\-stepnarration\(First/Next/Then\)\.

\-Listingthegivenansweroptionswithoutexploringthem\.

\#\#Dataset\-specificnote\(currentdatasetonly\)

\{dataset\_specific\_note\}

\#\#Extractionrules

\-Extracteachdistinctbranchingmomentasaseparateexample\.

\-Use‘excerpt‘asashortdirectquotefromthereasoningtrace

\(preferably<=25words\)\.

\-Use‘explanation‘tostatewhythatquotereflectsmultiplegenuinely

distinctcandidateexplanationsratherthanvariantsofthesameexplanation

orlinearnarration\.

\-Ifthesamebranchisrepeatedwithnonewreasoningcontent,extractit

once\.

\-Donotcountsuperficialvariationunlessthecompetingexplanationsdiffer

inunderlyingmechanism,agent,domain,orinterpretation\.

\-Donotparaphrasequotedtext\.

\#\#JSONoutputformat

ReturnONLYvalidJSONwiththisstructure:

\{

"overall\_analysis":"Briefanalysisofbranchinessinthisreasoningtrace",

"examples":\[

\{

"excerpt":"Quoteofthebranchingmomentfromthereasoningtrace",

"explanation":"Whythisrepresentsbranching"

\}

\]

\}

User Prompt Template:

Dataset:\{dataset\}

AnalyzethefollowingreasoningtraceforBranchinessandextractconcrete

examples\.

<reasoning\_trace\>

\{text\}

</reasoning\_trace\>

#### E\.4\.3Backtracking

System Prompt:

YouareanexpertreasoninganalystevaluatingAI\-generatedreasoningtraces\.

\#\#WhatisBacktracking?

Backtracking\(alsocalledSelf\-Correction\)occurswhenthereasoningexplicitly

updatesorabandonsapreviouslystatedexplanatoryhypothesisinlightof

newlyprocessedevidenceoradeeperlogicalmismatch,andthenchanges

direction\.

\#\#WhatCOUNTSasabacktrackingmoment

Extractanexamplewhenyousee:

\-Explicitadmissionoferror:"Wait,that’swrong","Imadeamistake",

"Actually,Ineedtoreconsider\.\.\."

\-Deliberatepausingandrestarting:"Holdon,letmere\-readtheproblem",

"Let’sgobacktostep2"

\-Achangeofstrategymid\-reasoning:"Instead,let’stry\.\.\.",

"Onsecondthought\.\.\.","Thatapproachdoesn’twork,so\.\.\."

\-Realisationofamisseddetail:"IrealiseIforgottoaccountfor\.\.\.",

"Thisdoesn’tlookrightbecause\.\.\."

\#\#WhatdoesNOTcount

\-Comparingtwovalidpaths\(thatisBranchiness\)\.

\-Asimple"However"thatintroducesacontrastwithoutadmittinganerror\.

\-Thefinalanswerselection\.

\#\#Dataset\-specificnote\(currentdatasetonly\)

\{dataset\_specific\_note\}

\#\#Extractionrules

\-Extracteachexplicitself\-correction/backtrackingeventasaseparate

example\.

\-Use‘excerpt‘asashortdirectquotefromthereasoningtrace

\(preferably<=25words\)\.

\-Use‘explanation‘toclarifywhatwasrevisedandwhythisistrue

backtracking\.

\-Donotcountsimplecontrastwordsunlesstheyindicateanactual

correction\.

\-Donotparaphrasequotedtext\.

\#\#JSONoutputformat

ReturnONLYvalidJSONwiththisstructure:

\{

"overall\_analysis":"Briefanalysisofbacktracking/self\-correctionin

thisreasoningtrace",

"examples":\[

\{

"excerpt":"Quoteofthebacktrackingmomentfromthereasoningtrace",

"explanation":"Whythisrepresentsbacktracking/self\-correction"

\}

\]

\}

User Prompt Template:

Dataset:\{dataset\}

AnalyzethefollowingreasoningtraceforBacktracking/Self\-Correctionand

extractconcreteexamples\.

<reasoning\_trace\>

\{text\}

</reasoning\_trace\>

#### E\.4\.4Differential Elimination

System Prompt:

Youareanexpertevaluatorofabductivereasoningtraces\.

\#\#WhatisDifferentialElimination?

DifferentialEliminationmeasureshowmanydistinctalternativesareexplicitly

rejectedorruledoutduringreasoning\.Unlikeabinarypresence/absencecheck,

thismetricextractseacheliminationinstanceasitsownexample\.

\#\#WhatCOUNTSasadifferentialeliminationinstance

Extractanexamplewhenthetraceexplicitly:

1\.Rulesoutanalternativehypothesis/optionwithaspecificreason\.

2\.Showscontradictionbetweenanalternativeandobserveddetails\.

3\.Usesconditionalfalsification

\("IfXweretrue,wewouldseeY,butwedon’t\."\)\.

4\.Comparesalternativesandexplicitlymarksoneaslessplausibleor

incompatible\.

\#\#WhatdoesNOTcount

\-Emptydismissals\(e\.g\.,simplystating"OptionBisincorrect"or

"OptionCisfalse"\)withoutprovidingacontext\-grounded"why"basedon

thespecificevidence\.

\-Puresupportforthechosenhypothesiswithoutdiscussingalternatives\.

\-Listingoptionswithoutevaluatingoreliminatingthem\.

\-Vaguepreferencestatementswithoutaconcreteeliminationreason\.

\-Finalanswerstatementsthatdonotincludeexplicitrefutationcontent\.

\#\#Dataset\-specificnote\(currentdatasetonly\)

\{dataset\_specific\_note\}

\#\#Extractionrules

\-Extracteachdistinctelimination/refutationeventasaseparateexample\.

\-Use‘excerpt‘asashortdirectquotefromthereasoningtrace

\(preferably<=30words\)\.

\-Use‘explanation‘tostatewhatalternativewaseliminatedandwhy\.

\-Ifthesameeliminationisrepeatedwithoutnewrationale,includeitonce\.

\-Donotparaphrasequotedtext\.

\#\#JSONoutputformat

ReturnONLYvalidJSONwiththisstructure:

\{

"overall\_analysis":"Briefanalysisofeliminationbehaviorinthis

reasoningtrace",

"examples":\[

\{

"excerpt":"Quoteshowingexpliciteliminationofanalternative",

"explanation":"Whatwaseliminatedandwhythisisavalidelimination

instance"

\}

\]

\}

User Prompt Template:

Dataset:\{dataset\}

ExtracteveryexplicitDifferentialEliminationinstancefromthefollowing

reasoningtrace\.Returnoneentryperdistincteliminationevent\.

<reasoning\_trace\>

\{text\}

</reasoning\_trace\>

#### E\.4\.5Prior Invocation

System Prompt:

YouareanexpertanalystevaluatingAI\-generatedreasoningtraces\.

\#\#WhatisaPrior?

Aprior\(orpriorprobability/baserate\)isapre\-existingprobability,

frequency,orbackgroundknowledgeabouthowcommonorlikelysomethingis

ingeneral,\*before\*consideringthespecificobservationsathand\.The

reasonerusesthispriortoadjusttheirfinaljudgment\.

Yourtaskistoidentifyeveryinstancewherethemodelexplicitlybrings

insuchpriorinformationtoinformitsreasoning\.

\#\#Categoriesofpriorstoextract

\#\#\#1\.Populationbaserates

Statisticalinformationabouthowcommonacondition,event,oroutcomeis

inarelevantpopulation\.

Examples:"Thisdiseaseaffects1in10,000people","Theprevalenceof

thisconditionisapproximately5%","Thisisararedisorder","Thisis

acommonoccurrenceinthisagegroup"\.

\#\#\#2\.Priorprobabilitiesfromdomainknowledge

Generalknowledgeaboutlikelihoodsthatthemodelbringstobearonthe

problem,notderivedfromthespecificobservations\.

Examples:"MostpatientswiththesesymptomshaveconditionX","Typically,

thistypeoffailureiscausedbyY","Ingeneral,ZismorelikelythanW"\.

\#\#\#3\.Comparativelikelihoods

Explicitcomparisonsofhowlikelydifferentpossibilitiesare,basedon

backgroundknowledgeratherthanthespecificevidence\.

Examples:"XisfarmorecommonthanY","Thisexplanationismoreprobable

apriori","Withoutspecificevidence,wewouldexpectZ"\.

\#\#\#4\.Referencetogeneraltendenciesorpatterns

Referencestowhat"usually"or"typically"happens,usedasapriorto

guidereasoning\.

Examples:"Usually,thissymptomindicates\.\.\.","Typically,patientswith

thisprofile\.\.\.","Inmostcaseslikethis\.\.\."\.

\#\#\#5\.ExplicitBayesian\-stylereasoning

Caseswherethemodelexplicitlyweighspriorprobabilityagainstnewevidence\.

Examples:"Eventhoughthetestispositive,giventhelowbaserate\.\.\.",

"Thepriorprobabilityislow,soweneedstrongevidence\.\.\.","Combining

thepriorwiththeseobservations\.\.\."\.

\#\#Extractionrules

\-Extract\*\*eachdistinctpriorconsideration\*\*asaseparateexample\.

\-The‘excerpt‘mustbea\*\*short,directquote\*\*fromthetextthatshows

themodelreferencingpriorinformation\(<=25wordsofcontext\)\.

\-The‘explanation‘mustidentifythetypeofprior\(fromthecategories

above\)andbrieflyexplainwhatpriorprobabilityorbaserateisbeing

referenced\.

\-Ifthemodelmentionsthesamepriormultipletimesindifferentparts

ofthereasoning,extracteachoccurrenceseparately\.

\-Do\*\*not\*\*paraphraseoralterthequotedtext\.

\#\#WhatdoesNOTcountasaprior

\-Conclusionsdrawn\*only\*fromthespecificobservationsintheproblem

\(theseareposteriorinferences,notpriors\)\.

\-Generalknowledgethatdoesn’tinvolveprobabilityorfrequency\(e\.g\.,

"Theheartpumpsblood"isafact,notaprior\)\.

\-Hypothesesgeneratedduringreasoningwithoutreferencetotheirgeneral

likelihood\.

\-Themodel’sownuncertaintyexpressions\(e\.g\.,"Ithink","probably"\)\-\-

thosearecapturedbytheuncertainty\_markersmetric\.

\-Restatementsofinformationgivenintheproblemprompt\.

\#\#Dataset\-specificnote\(currentdatasetonly\)

\{dataset\_specific\_note\}

\#\#JSONoutputformat

ReturnONLYvalidJSONwiththisstructure:

\{

"overall\_analysis":"Briefanalysisofpriorprobabilityusageinthis

reasoningtrace",

"examples":\[

\{

"excerpt":"Quoteofthepriorprobabilityconsiderationfromthe

reasoningtrace",

"explanation":"Typeofpriorandwhatprobability/frequencyisbeing

referenced"

\}

\]

\}

User Prompt Template:

Dataset:\{dataset\}

Extracteveryinstancewherethemodelconsidersapriorprobabilityor

baserateinthefollowingreasoningtrace\.Returnoneentryperprior

consideration\.

<reasoning\_trace\>

\{text\}

</reasoning\_trace\>

#### E\.4\.6Evidence Coverage

System Prompt:

Youareanexpertevaluatorofabductivereasoningtraces\.

\#\#YourTask

Givenareasoning\_traceinwhichamodelselectsonehypothesistoexplainan

observation,youmust:

1\.\*\*Extracteveryspecificdetail\*\*thatappearsintheobservation\(orthe

model’sdescriptionoftheobservation\)\-\-notjustthemainevent,but

alsoperipheralfacts,contextualclues,timingdetails,quantities,

locations,namedentities,andanyotherparticularsmentioned\.

2\.\*\*Foreachdetail\*\*,decidewhetherthereasoning\_trace\*explicitly\*

connectsthatdetailtothechosenhypothesis\.Adetailis"addressed"only

ifthereasoning\_tracemakesaclearlogicallinkbetweenthatdetailand

thehypothesis\-\-notmerelyrestatingitoracknowledgingitexists\.

3\.\*\*Provideevidence\*\*foreveryaddresseddetail:quotetheexactshort

passagefromthereasoning\_tracethatdemonstratestheconnection\.

4\.\*\*Writeabriefoverallanalysis\*\*summarisinghowfullythehypothesis

accountsforthecompleteobservation\.

\#\#Gradingcriteria

\-\*\*Addressed\(True\)\*\*:Thetracecontainsadirectexplanationof\*why\*or

\*how\*thechosenhypothesisaccountsforthisspecificdetail\.

\-\*\*Notaddressed\(False\)\*\*:Thedetailispresentintheobservationbutthe

traceeitherignoresit,onlyrestatesit,ortreatsitasirrelevantwithout

justification\.

\#\#Importantrules

\-Beexhaustive:donotskipminororbackgrounddetails\.

\-Donotrewardvaguegestures\.

\-Ahypothesisthatexplainsthemaineventbutignoressupportingdetails

shouldreceivealowcoveragescore\.

\-Baseeveryjudgementsolelyonwhatiswritteninthereasoning\_trace\-\-

donotinferorassumeanythingthatisnotstated\.

\#\#Dataset\-specificnote\(currentdatasetonly\)

\{dataset\_specific\_note\}

\#\#Extractionrules

\-\*\*EnforceAtomicity:\*\*Breakdowncompoundsentencesandlistsintoatomic

\(single,indivisible\)facts\.

\-Example:"headacheandvomiting"mustbesplitintotwoseparatedetails:

"headache"and"vomiting"\.

\-Example:"KernigandBrudzinskisignsarepresent"mustbesplitinto

"Kernigsignpresent"and"Brudzinskisignpresent"\.

\-Example:Separateeverysinglemedication,vitalsign,andlabvalueinto

itsownitem\.

\-Extracteachatomicobservationfactasoneitemin‘observation\_details‘\.

\-Use‘detail‘fortheobservationfacttext,‘addressed‘forexplicit

linkagestatus,and‘evidence‘forasupportingquote\.

\-Set‘addressed‘totrueonlywhenthereasoning\_traceexplicitlylinks

thedetailtothechosenhypothesis\.

\-If‘addressed‘isfalse,leave‘evidence‘asanemptystring\.

\-Beexhaustiveacrossmainandperipheraldetails\.

\#\#JSONoutputformat

ReturnONLYvalidJSONwiththisstructure:

\{

"overall\_analysis":"Briefanalysisofobservationcoverageinthis

reasoning\_trace",

"observation\_details":\[

\{

"detail":"Specificatomicdetailfromtheobservation",

"addressed":true,

"evidence":"Quotefromreasoning\_traceshowinghowthisdetailis

explained"

\}

\]

\}

User Prompt Template:

Dataset:\{dataset\}

Analysethefollowingobservationandreasoning\_traceandproducethestructured

observation\-coverageevaluation\.

<observation\>

\{full\_input\}

</observation\>

<reasoning\_trace\>

\{text\}

</reasoning\_trace\>

#### E\.4\.7Evidence–Explanation Directionality \(Score\-Based\)

System Prompt:

Youareanexpertevaluatorassessingthelogicaldirectionalityofreasoning

chainsgeneratedbyAImodels\.

\#\#MetricDefinition

\*\*Evidence\-ExplanationDirectionality:\*\*Themodeldemonstrates

awarenessthatabductionrunsstrictlyfromevidencetoexplanation\(not

explanationtoevidence,asinprediction/deduction\)\.

Inproperabductivereasoning,themodelMUSTstartwiththeknownobservations

\(theevidence\)andlogicallyreasonforwardtodeterminethebesthypothesis

\(theexplanation\)\.Reversingthislogicalconnection\-\-assuminganexplanation

istrueandthendeductivelycheckingifitpredictsoralignswiththegiven

observations\-\-isacorelogicalflawforthisevaluation\.

\#\#ScoringRubric

\#\#\#Score1\.0:ProperAbductiveDirection\(Evidence\-\>Explanation\)

Thereasoningchainexplicitlytreatstheevidence/observationsasthe

foundationalstartingpoint\.

\-Themodellooksattheprovidedfactsandconstructivelyasks"whatbest

explainsthis?"or"wheredoesthisevidencelead?"

\-Itbuildsitsargumentupwardfromtheevidencetoreachtheexplanatory

conclusion\.

\-Itneverpresupposesthecorrectnessofanexplanationbeforeanalyzing

theevidence\.

\-Languagecues:"Giventheseobservations\.\.\.","BecauseweseeX,itsuggests

Y\.\.\.","Theevidencepointstoward\.\.\."

\#\#\#Score0\.5:AmbiguousorMixedDirectionality

Thereasoningconnectsevidenceandexplanation,butthelogicalflowis

fundamentallyunclearorlooselystructured\.

\-Themodelmightlistfactsandtheexplanationside\-by\-sidewithoutlinking

themdirectionally\.

\-Itmightwaverbackwardsandforwards,showingsignsofevidence\-\>explanation

logicmixedtightlywithexplanation\-\>evidencelogic\.

\-Theflowlackstherobustforwardmathematicalmomentumoftrueabduction

butisn’texplicitlydeductiveeither\.

\#\#\#Score0\.0:Backward/DeductiveDirection\(Explanation\-\>Evidence\)

Thereasoningchaincompletelyreversestheabductiveflow,workingbackward

fromanassumedconclusiontotheevidence\.

\-Themodeltreatsacandidatehypothesisasagiven,explicitlydeducingwhat

observations\*would\*follow,andthenchecksiftheevidencematches\.

\-Itoperatesusinga"prediction"format:"If\[Hypothesis\]istrue,wewould

expecttosee\[Observation\]\.Wesee\[Observation\],thereforeitistrue\."

\-Anyreasoningthatmerelyjustifiesapre\-selectedanswerviapost\-hoc

confirmationbiasfallshere\.

\#\#Dataset\-SpecificGuidelines

\{dataset\_specific\_note\}

\#\#OutputFormat

ReturnexactlyandONLYawell\-formattedJSONobjectcontainingtheexact

fieldsbelow:

\{

"reasoning\_analysis":"Aconcise\(1\-2sentence\)explanationofthelogical

directionalityobservedinthechain\.",

"directionality\_score":1\.0

\}

Note:‘directionality\_score‘muststrictlybe0\.0,0\.5,or1\.0\.Donotoutput

markdowncodeblocks\.

User Prompt Template:

Dataset:\{dataset\}

EvaluatethereasoningchainbelowforEvidence\-ExplanationDirectionality

andassignascoreof0\.0,0\.5,or1\.0\.

\*\*Observations/Evidence:\*\*

<observations\>

\{full\_input\}

</observations\>

\*\*Model’sReasoningChain:\*\*

<reasoning\_chain\>

\{text\}

</reasoning\_chain\>

Assignadirectionalityscore\(0\.0/0\.5/1\.0\)andbrieflyexplainyour

assessmentinthe‘reasoning\_analysis‘field\.

### E\.5Detailed Per\-Dataset Results

Table[10](https://arxiv.org/html/2608.14791#A5.T10)reports per\-dataset process\-level scores for DeepSeek\-R1\-Distill\-Qwen\-7B\. For counting metrics, values are mean occurrence counts per trace; for Evidence Coverage and Directionality, values are mean scores\. Values in bold are the maximum for that metric\-dataset cell across the three conditions\.

DatasetsART

B\-COPA

GoEmo\.

MedQA

DefNLI

MuSR\-M

MuSR\-O

MuSR\-T

NeuLR

StratQA

ConditionMetricCor\-GRPOBacktracking0\.50\.90\.20\.81\.50\.81\.60\.61\.50\.9Branchiness1\.20\.40\.91\.51\.61\.21\.31\.51\.40\.6Coverage \(%\)43\.041\.023\.034\.046\.049\.039\.044\.035\.037\.0Diff\. Elim\.1\.01\.00\.60\.71\.00\.91\.10\.71\.51\.2Directionality0\.20\.10\.30\.10\.10\.20\.10\.20\.20\.1Prior0\.40\.70\.10\.30\.50\.90\.70\.31\.00\.4Unc\. Markers0\.60\.81\.01\.01\.11\.30\.40\.81\.11\.1CEDAR\-GRPOBacktracking0\.60\.90\.20\.81\.70\.91\.90\.62\.31\.0Branchiness1\.30\.51\.02\.82\.51\.51\.81\.61\.60\.7Coverage \(%\)44\.047\.061\.048\.071\.052\.070\.048\.055\.033\.0Diff\. Elim\.1\.81\.10\.61\.31\.11\.51\.31\.21\.71\.3Directionality0\.710\.560\.490\.700\.440\.390\.700\.710\.580\.74Prior0\.70\.40\.40\.90\.51\.00\.90\.81\.10\.5Unc\. Markers1\.61\.01\.51\.11\.51\.32\.11\.41\.11\.1BaseBacktracking0\.60\.40\.50\.80\.70\.90\.80\.71\.00\.5Branchiness1\.20\.91\.11\.41\.31\.51\.41\.31\.60\.5Coverage \(%\)38\.022\.040\.025\.047\.034\.045\.036\.023\.021\.0Diff\. Elim\.0\.70\.50\.60\.90\.81\.00\.90\.81\.10\.6Directionality0\.10\.30\.20\.20\.20\.20\.10\.10\.20\.5Prior0\.50\.30\.40\.70\.60\.80\.70\.60\.90\.4Unc\. Markers0\.80\.50\.71\.00\.91\.11\.00\.91\.20\.6Table 10:Process\-level metric scores for DeepSeek\-R1\-Distill\-Qwen\-7B\. Values are bolded if they are the maximum for that specific metric and dataset across all three conditions\.DatasetUnc\. MarkersBranch\.Back\.Diff\. Elim\.PriorObs\. Cov\.Direct\.ARTFocus on uncertainty while reasoning about which hypothesis better explains observations\.Do not count simple option selection; count only internal exploration\.Look for true reconsideration of which hypothesis explains observations better\.Count explicit eliminations of the non\-chosen hypothesis or alternatives\.Do not count priors that merely restate hypotheses; count inferential base\-rate reasoning\.Treat ‘Observation 1’ and ‘Observation 2’ as complete; extract all details\.Check reasoning starts from observations and seeks hypothesis, not back\-fitting\.B\-COPAFocus on uncertainty in causal reasoning and option comparison\.Do not count simple choice selection; count exploration within cause\-effect reasoning\.Look for reconsideration of causal interpretation, not restating options\.Count explicit elimination of non\-selected option or causal alternatives\.Priors about typical everyday cause\-effect relationships\.Treat ‘Cause:’ as observation; extract relevant details\.‘Cause’ is evidence; model must evaluate which option is most plausible effect\.DefNLIUncertainty indicating defeasible or non\-certain inference relations\.Exploration of stronger vs\. weaker inferences and defeasible conclusions\.Reconsideration when inference appears defeated or less robust\.Explicit rejection of candidate inference relations shown inconsistent\.Priors about typical premise\-hypothesis relations and defaults\.Treat ‘Premise’, ‘Hypothesis’, ‘Update’ together as complete observation\.Reasoning evaluates logical impact of Update from given text\.GoEmo\.Uncertainty in label selection; avoid counting task\-domain terms\.Count only if model explores multiple emotion labels\.Count only genuine revisions in emotion\-label selection\.Explicit elimination of alternative labels with trace\-grounded justification\.Priors about emotions typically associated with contexts\.Treat ‘Text’ as complete observation; extract salient spans\.Reasoning extracts cues before selecting emotion label\.Table 11:Dataset\-specific notes for all process\-level metrics injected into the judge prompt\. Each cell gives the exact text appended to the shared system prompt for that dataset\.DatasetUnc\. MarkersBranch\.Back\.Diff\. Elim\.PriorObs\. Cov\.Direct\.MuSRUncertainty in narrative interpretation and conclusion drawing\.Exploration of different narrative interpretations or solutions\.Revised interpretations of narrative details or changed conclusions\.Explicit rejection of narrative interpretations or scenario explanations\.Priors about typical narrative behavior, motives, patterns\.Treat ‘Context’ and ‘Problem’ as complete; extract narrative details \(actors, timing, locations\)\.Reasoning builds explanations from details, not assumes conclusion and back\-fits\.MedQAExtract markers only from model’s own reasoning, not probabilistic symptoms\.Count only genuine differential exploration \(diagnoses, treatment paths\)\.Corrections in diagnostic/treatment reasoning or missed details\.Each answer choice explicitly ruled out with clinical rationale\.Priors only when clearly introduced as inferential reasoning, not restated data\.Treat ‘Problem’ as observation; extract all clinical details\.Reasoning starts from clinical evidence and moves toward diagnosis/treatment\.NeuLRUncertainty while weighing competing abductive explanations\.Do not count superficial option comparison; genuine exploration\.True reconsideration of which explanation best fits observations\.Explicit elimination moves using concrete mismatch evidence\.Priors justifying why one hypothesis more plausible\.Treat ‘Logical Rules and Known Facts’ and ‘Target Conclusion’ as complete\.Reasoning uses Rules/Facts to find Missing Fact, not work backwards\.StratQAUncertainty in multi\-step reasoning and evidence\-to\-conclusion transitions\.Exploration of multiple reasoning paths or inference chains\.Revisions in reasoning chains, flawed inferences, changed conclusions\.Explicit rejection of competing YES/NO reasoning lines\.World\-knowledge priors and typicality assumptions\.Treat ‘Question’ and ‘Evidence’ as complete; extract relevant facts\.Reasoning builds upon facts toward YES/NO answer, not cherry\-picks post\-hoc\.Table 11:Dataset\-specific notes for all process\-level metrics injected into the judge prompt\. Each cell gives the exact text appended to the shared system prompt for that dataset\. \(continued\)

## Appendix FSFT Data Construction and Training Details

To compare CEDAR\-GRPO against supervised fine\-tuning, we construct an SFT dataset from exactly the same training sources used in the GRPO setting\. The original examples contain task inputs and gold final answers, but they do not consistently provide a complete rationale in the output format used during RL training\. We therefore generate synthetic reference rationales and insert them into the model response as the content of the<think\>block, while keeping the original gold label, sentence set, missing fact, or program as the<answer\>target\.

Synthetic rationales are generated with Gemini 3\.0 Flash Preview through the OpenRouter chat\-completions API\. For all datasets, we use temperature0\.20\.2, top\-p=0\.9p=0\.9, up to six retries per example, eight concurrent API workers, and a 90\-second request timeout\. The maximum completion budget is 350 tokens for UniADILR, Balanced COPA, CLIMATE\-FEVER, and AbductionRules; 450 tokens for CauseLogics; 900 tokens for Crypto; and 1000 tokens for List Function\. The generator is given the problem input together with the gold answer and is instructed to produce a concise rationale that justifies the gold output\. The resulting SFT target has the same response structure as the GRPO policy outputs:

<think\>

\{synthetic\_rationale\}

</think\>

<answer\>

\{gold\_answer\}

</answer\>

We apply strict validation before accepting a generated rationale\. Each completion must exactly match the required<think\>–<answer\>format, and the answer block must match the known gold answer\. For classification and selection datasets, this means exact matching against the normalized label or sentence\-number set\. For AbductionRules, the answer must exactly match the gold missing fact\. For the two code\-generation datasets, the generated answer is also compiled and executed against the held\-out test cases for that example; the rationale is accepted only if the generatedtransformfunction passes all tests\. The accepted dataset stores only the validated rationale as an added field, leaving the original gold answer or gold function as the final supervised answer\. Failed or malformed generations are retried; unresolved failures are left without a rationale and can be resumed later\.

The final mixed SFT dataset uses the same train/validation split as the GRPO data: 1,920 training examples and 480 validation examples\. UniADILR, Balanced COPA, CauseLogics, CLIMATE\-FEVER, and AbductionRules each contribute 320 training and 80 validation examples; Crypto and List Function each contribute 160 training and 40 validation examples\. Each sample is additionally tagged with its dataset name before the mixed split is shuffled with seed 42\.

### F\.1SFT optimization and model selection

We train the Qwen3\-4B and DeepSeek\-R1\-Distill\-Qwen\-7B SFT baselines with TRL’sSFTTraineron the full mixed split described above\. Each example uses the same dataset\-specific system and user prompts as in GRPO, followed by the validated<think\>–<answer\>target\. The cross\-entropy loss is applied only to assistant tokens; system and user tokens are masked\.

SettingSFT configurationData1,920 training and 480 validation examples; sequence packing disabledQuantization / LoRA4\-bit NF4; rankr=64r=64;α=64\\alpha=64; Unsloth gradient checkpointingLoRA targetsq\_proj,k\_proj,v\_proj,o\_proj,gate\_proj,up\_proj,down\_projSequence length4,096 tokensOptimizerAdamW \(Torch\), learning rate1×10−51\\times 10^\{\-5\},\(β1,β2\)=\(0\.9,0\.99\)\(\\beta\_\{1\},\\beta\_\{2\}\)=\(0\.9,0\.99\), weight decay0\.10\.1Schedule / clippingCosine decay, 2 warmup steps, maximum gradient norm0\.10\.1Batch / budgetSingle\-GPU train/eval batch size 4; gradient accumulation 1 \(effective train batch size 4\); five epoch \(2400 optimizer steps\)CheckpointingAdapter checkpoints every 128 optimizer stepsValidationAll 480 examples every 128 steps and at epoch end; one sampled completion per example \(T=0\.7T=0\.7, top\-p=0\.95p=0\.95, maximum 2,048 new tokens\)SelectionSaved checkpoint with the highest mean validation correctness under the dataset\-specific exact\-, set\-, or execution\-based verifierTable 12:Optimization and model\-selection setup for the SFT ablation\. Held\-out evaluation tasks are not used for checkpoint selection\.
### F\.2Synthetic\-rationale prompt templates

The prompts below are the dataset\-specific templates used to generate the SFT rationales\. In each case, the generator receives the original task input and the gold answer, and must return only a formatted SFT target\.

#### F\.2\.1Prompt Template: UniADILR

System Prompt:

Youaregeneratingsupervisedfine\-tuningdataforanabductiveevidence\-selectionmodel\.

Youwillreceive:

1\.Acontextwithnumberedsentences

2\.Ahypothesis

3\.Thegoldsupportingsentencenumbers

Writeaconciserationaleexplainingwhythegoldsentencesprovidethenecessaryevidenceforthehypothesis\.

Therationaleshould:

\-Startfromthehypothesis\.

\-Identifywhatneedstobeexplainedorsupported\.

\-Explainhoweachgoldsentencecontributestotheexplanation\.

\-Explainwhytheselectedsentencesworktogethertosupportthehypothesis\.

\-Ignoreunrelatedcontextsentences\.

\-Avoidexternalknowledgeandunsupportedspeculation\.

Outputexactly:

<think\>

\[Aconciseabductiveevidence\-selectionrationale,usually3\-5sentences\.\]

</think\>

<answer\>

\[Thegoldsentencenumbersonly,comma\-separated\.\]

</answer\>

Donotchangethegoldsentencenumbersintheanswersection\.

Donotincludetheword"sent"intheanswersection\.

Donotincludeanytext,punctuation,orexplanationotherthancomma\-separatednumbersintheanswersection\.

User Prompt:

Context:

\{context\_str\}

Hypothesis:

\{hypothesis\}

GoldSupportingSentences:

\{gold\_answer\}

GeneratetheSFTtarget\.

#### F\.2\.2Prompt Template: Balanced COPA

System Prompt:

Youaregeneratingsupervisedfine\-tuningdataforanabductivecausalreasoningmodel\.

Youwillreceive:

1\.Apremisedescribingasituationorevent

2\.Twopossiblecauses

3\.Thegoldanswer

Writeaconciserationaleexplainingwhythegoldansweristhemoreplausiblecauseofthepremise\.

Therationaleshould:

\-Startfromthepremise\.

\-Evaluatebothchoicesaspossiblecauses\.

\-Explainwhythegoldchoicemoredirectlyorplausiblyleadstothepremise\.

\-Explainwhytheotherchoiceisweaker,lessdirect,orlessplausible\.

\-Usecommonsenseandtypicalreal\-worldcausalrelationships\.

\-Avoidunrelatedspeculation\.

Outputexactly:

<think\>

\[Aconciseabductivecausalrationale,usually3\-5sentences\.\]

</think\>

<answer\>

\[Thegoldansweronly:either1or2\.\]

</answer\>

Donotchangethegoldanswerintheanswersection\.

Donotincludeanytext,punctuation,orexplanationintheanswersection\.

User Prompt:

Premise:\{premise\}

Choice1:\{choice1\}

Choice2:\{choice2\}

GoldAnswer:

\{gold\_answer\}

GeneratetheSFTtarget\.

#### F\.2\.3Prompt Template: CauseLogics

System Prompt:

Youaregeneratingsupervisedfine\-tuningdataforanabductivelogicalreasoningmodel\.

Youwillreceive:

1\.Asetofpremises

2\.Asetofrules

3\.Anobservedphenomenon

4\.Apossiblecause

5\.Thegoldanswer

Writeaconciserationaleexplainingwhyaddingthepossiblecausedoesordoesnotmakethephenomenonlogicallyinferable\.

Therationaleshould:

\-Startfromthephenomenon\.

\-Assumethepossiblecauseisaddedtothepremises\.

\-Identifytherelevantruleorrulechain\.

\-Statewhichrequiredfactsarealreadypresentorcanbederived\.

\-Explainwhetherforwardreasoningcaninferthephenomenon\.

\-Ifthephenomenoncannotbeinferred,brieflystatewheretheproofpathfails\.

\-Useonlythegivenpremises,rules,andpossiblecause\.

\-Avoidunrelatedfacts,unrelatedrules,andexternalknowledge\.

Outputexactly:

<think\>

\[Aconciseabductivelogicalrationale,usually3\-6sentences\.\]

</think\>

<answer\>

\[Thegoldansweronly:TRUEorFALSE\.\]

</answer\>

Donotchangethegoldanswerintheanswersection\.

Donotincludeanytext,punctuation,orexplanationintheanswersection\.

User Prompt:

Premises:

\{premises\_text\}

Rules:

\{rules\_text\}

Phenomenon:

\{phenomenon\}

PossibleCause:

\{possible\_cause\}

GoldAnswer:

\{gold\_answer\}

GeneratetheSFTtarget\.

#### F\.2\.4Prompt Template: CLIMATE\-FEVER

System Prompt:

Youaregeneratingsupervisedfine\-tuningdataforanevidence\-groundedfact\-checkingmodel\.

Youwillreceive:

1\.Aclaim

2\.Alistofevidences

3\.Thegoldanswer

Writeaconciserationaleexplainingwhytheevidencesupports,refutes,isinsufficienttoevaluate,ordisputestheclaim\.

Therationaleshould:

\-Startfromtheclaim\.

\-Identifythemostrelevantevidence\.

\-Explainhowtheevidencerelatestotheclaim\.

\-ForSUPPORTS,explainwhytheevidencemakestheclaimmorelikelytrue\.

\-ForREFUTES,explainwhytheevidencecontradictstheclaim\.

\-ForNOTENOUGHINFO,explainwhatkeyinformationismissing\.

\-ForDISPUTED,explainthattheprovidedevidenceismixed,conflicting,ordoesnotleadtoasingleclearverdict\.

\-Useonlytheprovidedevidence,notexternalknowledge\.

\-Avoidunrelatedevidenceandunsupportedspeculation\.

Outputexactly:

<think\>

\[Aconciseevidence\-groundedrationale,usually3\-5sentences\.\]

</think\>

<answer\>

\[Thegoldansweronly:SUPPORTS,REFUTES,NOTENOUGHINFO,orDISPUTED\.\]

</answer\>

Donotchangethegoldanswerintheanswersection\.

Donotincludeanytext,punctuation,orexplanationintheanswersection\.

User Prompt:

Claim:

\{claim\}

Evidence:

\{evidence\_text\}

GoldAnswer:

\{gold\_answer\}

GeneratetheSFTtarget\.

#### F\.2\.5Prompt Template: AbductionRules

System Prompt:

Youaregeneratingsupervisedfine\-tuningdataforanabductivereasoningmodel\.

Youwillreceive:

1\.Acontextwithfactsandrules

2\.Aquery

3\.Thegoldmissingfact

Writeaconciserationaleexplainingwhythegoldmissingfactmakesthequerydecidablefromthecontext\.

Therationaleshould:

\-Startfromthequery\.

\-Identifytherelevantruleorrulechain\.

\-Statewhichrequiredfactsarealreadypresent\.

\-Statewhatthegoldmissingfactadds\.

\-Explainwhetherthequerybecomesprovableordisprovable\.

\-Ignoreunrelatedfactsandrules\.

Outputexactly:

<think\>

\[Aconciseabductiverationale,usually3\-5sentences\.\]

</think\>

<answer\>

\[Thegoldmissingfactonly,asonesentenceendingwithaperiod\.\]

</answer\>

Donotchangethegoldmissingfactintheanswersection\.

Donotincludemorethanonefactintheanswersection\.

User Prompt:

Context:

\{context\}

Query:

\{query\}

GoldMissingFact:

\{gold\_answer\}

GeneratetheSFTtarget\.

#### F\.2\.6Prompt Template: Crypto

System Prompt:

Youaregeneratingsupervisedfine\-tuningdataforarule\-induction\-to\-codemodel\.

Youwillreceive:

1\.Severaltrainingexampleswithinputandoutputstrings

2\.AgoldPythonfunctionthatimplementstheintendedhiddentransformation

Writeaconciserationaleexplainingwhythegoldtransformationfitsthetrainingexamples,thenprovideaPythonimplementationofthesamerule\.

Therationaleshould:

\-Startfromtheinput\-outputexamples\.

\-Identifythehiddencharacter\-levelstringtransformation\.

\-Explainhowtheruleaccountsfortheoutputsacrossexamples\.

\-MentionwhethertheruleisaCaesarshift,Atbashmapping,oranotherexactcharactermappingwhenrelevant\.

\-Brieflyruleoutasimplerwrongpatternifhelpful\.

\-Emphasizethattheruleshouldgeneralizebeyondtheshownexamples\.

\-Avoidunrelatedspeculation\.

Outputexactly:

<think\>

\[Aconciserule\-inductionrationale,usually3\-5sentences\.\]

</think\>

<answer\>

\[Pythoncodeonly\.\]

</answer\>

Coderequirements:

\-Defineexactlyonefunctionnamedtransform\.

\-Thefunctiontakesoneargument:s\.

\-Thefunctionmustreturnastring\.

\-Usethegoldfunctiononlytoinfertheintendedrule\.

\-Preservethebehaviorimpliedbytheexamplesforlowercaseletters,uppercaseletters,andanynon\-lettercharacters\.

\-Donothardcodethetrainingexamples\.

\-Donotuseimports,printing,input\(\),orrandomness\.

\-Donotincludemarkdowncodeblocks\.

\-DonotincludeanytextoutsidethePythonfunctionintheanswersection\.

User Prompt:

Trainingexamples:

\{train\_prompt\}

\{split\_text\}

GoldFunction:

\{gold\_function\}

GeneratetheSFTtarget\.

For Crypto, at most the first ten normal training examples are included in the rationale\-generation prompt\. When available, the split metadata is included as a transformation\-type hint\. The generated code is accepted only if it passes the example’s held\-out test cases\.

#### F\.2\.7Prompt Template: List Function

System Prompt:

Youaregeneratingsupervisedfine\-tuningdataforarule\-induction\-to\-codemodel\.

Youwillreceive:

1\.Severaltrainingexampleswithinputandoutputlists

2\.AgoldPythonfunctionthatimplementstheintendedhiddentransformation

Writeaconciserationaleexplainingwhythegoldtransformationfitsthetrainingexamples,thenprovideaPythonimplementationofthesamerule\.

Therationaleshould:

\-Startfromtheinput\-outputexamples\.

\-Identifythehiddenlisttransformation\.

\-Explainhowtheruleaccountsfortheoutputs\.

\-Brieflyruleoutasimplerwrongpatternifhelpful\.

\-Emphasizethattheruleshouldgeneralizebeyondtheshownexamples\.

\-Avoidunrelatedspeculation\.

Outputexactly:

<think\>

\[Aconciserule\-inductionrationale,usually3\-5sentences\.\]

</think\>

<answer\>

\[Pythoncodeonly\.\]

</answer\>

Coderequirements:

\-Defineexactlyonefunctionnamedtransform\.

\-Thefunctiontakesoneargument:lst\.

\-Thefunctionmustreturnalistofintegers\.

\-Usethegoldfunctiononlytoinfertheintendedrule;adaptthefunctionnameandargumentnametotransform\(lst\)\.

\-Donothardcodethetrainingexamples\.

\-Donotuseimports,printing,input\(\),orrandomness\.

\-Donotincludemarkdowncodeblocks\.

\-DonotincludeanytextoutsidethePythonfunctionintheanswersection\.

User Prompt:

Trainingexamples:

\{train\_prompt\}

GoldFunction:

\{gold\_function\}

GeneratetheSFTtarget\.

For List Function, the generated code must define a singletransform\(lst\)function and pass all held\-out test cases for that example before the accompanying rationale is accepted\.

## Appendix GAdditional Dataset Notes for the General\-Reasoning Ablation

For the general\-reasoning ablation, we construct a balanced auxiliary training mixture from six widely used reasoning benchmarks: GSM8K, FOLIO, CommonsenseQA, VitaminC, MMLU, and BIG\-Bench\([3](https://arxiv.org/html/2608.14791#bib.bib38);[11](https://arxiv.org/html/2608.14791#bib.bib40);[36](https://arxiv.org/html/2608.14791#bib.bib39);[30](https://arxiv.org/html/2608.14791#bib.bib41);[15](https://arxiv.org/html/2608.14791#bib.bib42);[34](https://arxiv.org/html/2608.14791#bib.bib43)\)\. The goal of this ablation is not to optimize for any single benchmark family, but to test whether improvements attributed to abductive training persist when the auxiliary signal is replaced by a broader mixture of mathematical, logical, commonsense, fact\-verification, and academic question\-answering tasks\. To keep the comparison controlled, each source contributes the same number of instances\. The resulting corpus contains 2,400 examples in total, split into 1,920 training examples and 480 validation examples\.

### G\.1Mixture composition

DatasetReasoning profileSamplesRole in the ablation mixtureGSM8K\([3](https://arxiv.org/html/2608.14791#bib.bib38)\)Arithmetic,multi\-step reasoning400Introduces numerical and procedural reasoning through grade\-school math word problems\. Final supervision is based on the extracted numerical answer\.FOLIO\([11](https://arxiv.org/html/2608.14791#bib.bib40)\)Formal logicalinference400Provides premise–conclusion judgments grounded in first\-order logical structure, adding a strongly symbolic component to the mixture\.CommonsenseQA\([36](https://arxiv.org/html/2608.14791#bib.bib39)\)Commonsensemultiple choice400Supplies broad everyday reasoning questions in a fixed five\-choice format, complementing the more formal logical tasks\.VitaminC\([30](https://arxiv.org/html/2608.14791#bib.bib41)\)Fact verificationunder contrastive evidence400Adds evidence\-sensitive classification in which small textual changes can reverse the correct label, making it useful for testing fine\-grained judgment\.MMLU\([15](https://arxiv.org/html/2608.14791#bib.bib42)\)Academicmultiple choice400Contributes subject\-diverse question answering from a reasoning\-oriented subset of academic domains\.BIG\-Bench\([34](https://arxiv.org/html/2608.14791#bib.bib43)\)Heterogeneousmultiple\-choice reasoning400Adds task diversity through four reasoning\-focused multiple\-choice tasks spanning deduction, state tracking, date reasoning, and tabular reasoning\.Table 13:Composition of the general\-reasoning ablation mixture\. Each source contributes 400 instances, producing a balanced 2,400\-example corpus with an 80/20 train/validation split\.
### G\.2Dataset\-specific notes

##### GSM8K\.

GSM8K is a benchmark of grade\-school mathematics problems intended to test multi\-step quantitative reasoning\([3](https://arxiv.org/html/2608.14791#bib.bib38)\)\. We sample 400 examples from the public training data and partition them into 320 training and 80 validation instances\. Although the benchmark includes worked solutions, the supervision target in this ablation is the final numeric answer, so this component primarily evaluates whether general reasoning fine\-tuning improves reliable answer derivation rather than free\-form explanation quality\.

##### FOLIO\.

FOLIO is a natural\-language reasoning benchmark with first\-order\-logic structure, where a model must determine whether a conclusion follows from a set of premises\([11](https://arxiv.org/html/2608.14791#bib.bib40)\)\. We use 320 training and 80 validation examples drawn from the released train and validation portions\. Since the original benchmark includes an uncertainty class, this component is especially useful for probing whether the model can preserve calibrated three\-way logical judgments under mixed\-task training\.

##### CommonsenseQA\.

CommonsenseQA is a five\-way multiple\-choice benchmark designed to require everyday background knowledge rather than shallow textual matching\([36](https://arxiv.org/html/2608.14791#bib.bib39)\)\. We include 320 training and 80 validation questions sampled from the released train and validation data\. Within the ablation mixture, CommonsenseQA serves as the main source of broad commonsense supervision, counterbalancing the more formal or domain\-specific benchmarks\.

##### VitaminC\.

VitaminC is a fact\-verification benchmark built around contrastive evidence edits, such that small revisions in wording, numbers, or negation may flip the correct label\([30](https://arxiv.org/html/2608.14791#bib.bib41)\)\. We sample 320 training and 80 validation instances while approximately preserving the original class proportions\. In the resulting subset, the training split contains 160SUPPORTS, 114REFUTES, and 46NOT ENOUGH INFOexamples, while the validation split contains 40, 29, and 11 examples respectively\. This component is included to test whether the model can maintain sensitivity to fine\-grained evidential distinctions under broader reasoning supervision\.

##### MMLU\.

MMLU is a large multitask benchmark spanning many academic subjects and levels of expertise\([15](https://arxiv.org/html/2608.14791#bib.bib42)\)\. For the ablation, we use a focused eight\-subject subset: abstract algebra, formal logic, logical fallacies, college computer science, high\-school statistics, high\-school physics, econometrics, and high\-school world history\. We sample 50 questions per subject and split each subject into 40 training and 10 validation examples, yielding 320 training and 80 validation instances overall\. This design preserves topical breadth while biasing the mixture toward subjects with a clearer reasoning component\.

##### BIG\-Bench\.

BIG\-Bench is a collaborative benchmark suite intended to probe a wide range of language\-model capabilities\([34](https://arxiv.org/html/2608.14791#bib.bib43)\)\. We restrict attention to four reasoning\-oriented multiple\-choice tasks:logical\_deduction,tracking\_shuffled\_objects,date\_understanding, andpenguins\_in\_a\_table\. Each task contributes 100 examples, split into 80 training and 20 validation instances\. This portion of the mixture broadens task diversity beyond standard question answering by incorporating explicit deduction, state tracking, temporal reasoning, and lightweight table reasoning\.

### G\.3Interpretation

This ablation mixture is intentionally heterogeneous\. GSM8K emphasizes quantitative reasoning; FOLIO emphasizes formal logical validity; CommonsenseQA emphasizes everyday inference; VitaminC emphasizes evidence\-conditioned judgment; and MMLU and BIG\-Bench introduce broader subject and task diversity\. The equal allocation of 400 examples per source prevents the ablation from being dominated by any single benchmark family and makes the comparison against abductive training easier to interpret\.

At the same time, this corpus should not be read as a benchmark in its own right\. It is a controlled training mixture designed for ablation analysis\. The purpose of the condition is to test whether gains arise specifically from abductive supervision, or whether a comparably sized but more general reasoning curriculum yields similar improvements\.