Learning from Synthetic Data without Model Collapse in Iterative Instruction Tuning

# Learning from Synthetic Data without Model Collapse in Iterative Instruction Tuning
Source: [https://arxiv.org/html/2607.17043](https://arxiv.org/html/2607.17043)
Xiaonan Luo1,Yue Huang1,Kehan Guo1,Ping He2,Chuan Zou3,Ting Hua1,Xiangliang Zhang1

1University of Notre Dame,2Vanderbilt University,3University of Pennsylvania

###### Abstract

Model collapse is a central challenge in learning from synthetic data: as later\-generation large language models \(LLMs\) are trained on an increasing proportion of model\-generated data, performance can degrade due to narrowed coverage and accumulated bias\. Existing work mainly studies how to bound this degradation\. In iterative model evolution, however, the more meaningful objective is to ensure that each successive model improves over its predecessor, which requires diagnosing collapse at a granularity that is actionable for data curation\. We study this problem in synthetic data self\-improving for instruction tuning\. We show that collapse in this setting is not simply uniform performance degradation, but can appear as a polarization of competence, where synthetic training reinforces already strong skills while further degrading weak ones\. Motivated by this observation, we proposeKITE\(Knowledge\-boundaryInstructionTuning viaExploration\), a two\-stage framework that combines failure\-guided data generation with boundary\-aware uncertainty curation\. Experiments across several datasets and multiple open\-source LLMs show that KITE yields more stable improvement than strong synthetic\-data baselines\.

Learning from Synthetic Data without Model Collapse in Iterative Instruction Tuning

Xiaonan Luo1, Yue Huang1, Kehan Guo1, Ping He2, Chuan Zou3, Ting Hua1, Xiangliang Zhang1††thanks:Corresponding author\.1University of Notre Dame,2Vanderbilt University,3University of Pennsylvania

## 1Introduction

Synthetic data has become an increasingly important component in the post\-training of large language models \(LLMs\)\(Liuet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib8); Schick and Schütze,[2021](https://arxiv.org/html/2607.17043#bib.bib9); Riabiet al\.,[2021](https://arxiv.org/html/2607.17043#bib.bib10); Zhaoet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib11); Weiet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib15); Huanget al\.,[2025a](https://arxiv.org/html/2607.17043#bib.bib16),[b](https://arxiv.org/html/2607.17043#bib.bib20); Leeet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib19)\)\. However, the growing dependence on synthetic data introduces a fundamental challenge:*model collapse*\(Shumailovet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib1); Dohmatobet al\.,[2024b](https://arxiv.org/html/2607.17043#bib.bib2); Gerstgrasseret al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib3); Zhuet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib4); Fenget al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib5); Draysonet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib6); Aminet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib7); Kazdanet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib34); Dohmatobet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib35)\)\. As synthetic data becomes an increasingly large component of future training corpora, new generations of LLMs will be optimized on data that partially reflects the outputs of previous generations\. This can distort the effective training distribution, gradually reducing its diversity and fidelity to real\-world data\(Shumailovet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib1)\)\. Rather than preserving the full richness of human\-produced corpora, the model may place increasing emphasis on recurring, high\-probability patterns inherited from earlier synthetic generations\. Such drift can accumulate across iterative training, ultimately reducing coverage of long\-tail capabilities and harming performance on held\-out tasks\.

![Refer to caption](https://arxiv.org/html/2607.17043v1/x1.png)Figure 1:Selective skill\-mastery profiles of Llama\-3\-8B\-Instruct on GSM8K, comparing the base model with the model fine\-tuned on accumulated synthetic data\.Recent studies advance the understanding of model collapse under synthetic data\.Shumailovet al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib1)\)show that recursive training on model\-generated data can induce a degenerative feedback loop, whileDohmatobet al\.\([2024b](https://arxiv.org/html/2607.17043#bib.bib2)\)connect this phenomenon to changes in scaling laws driven by the loss of long\-tail support\.Gerstgrasseret al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib3)\)further show that retaining real data alongside accumulated synthetic data can keep test error bounded, andZhuet al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib4)\)propose token\-level editing as a practical way to synthesize text data while reducing collapse\. Despite these advances, important gaps remain\.First, existing work mainly asks how to control collapse so that degradation does not become unbounded\. In iterative model evolution, however, the more meaningful objective is to ensure that each successive model improves over the previous generation\.Second, existing analyses usually characterize collapse at an aggregate level, such as overall accuracy drop or long\-tail narrowing, which is too coarse to guide synthetic data construction\. For example, collapse may arise either because the data becomes too narrow and repetitive, which calls for data diversification, or because the model has specific weaknesses on a subset of skills, which calls for more targeted data construction\. This motivates the need to*diagnose collapse at a granularity that is actionable for data curation*\. We study model collapse in the self\-improving regime, where the*instructions*that shape the training distribution are drawn from the model’s own generations\. Following common practice, we use a strong external model only as a*verifier*that checks and normalizes answers, and we apply it identically to every method we compare\. This requires addressing two challenges at once:\(1\)diagnosing model collapse at a granularity that is actionable for data curation, and\(2\)designing synthetic data construction strategies that support sustained model improvement\.

To address the first question, we empirically examine how synthetic data changes the*skill profile*of evolving models\. Using GSM8K\(Cobbeet al\.,[2021](https://arxiv.org/html/2607.17043#bib.bib21)\)as an illustrative case, we apply the Deterministic Inputs Noisy And gate model \(DINA\)\(De La Torre,[2009](https://arxiv.org/html/2607.17043#bib.bib22)\)from the family of cognitive assessment models in education science to estimate*skill mastery*for both the base model and the fine\-tuned model trained with accumulated synthetic data\. Figure[1](https://arxiv.org/html/2607.17043#S1.F1)visualizes a subset of resulting skill profiles \(full results on other datasets and models, fine\-tuning settings, and data synthesis procedures are detailed in the Appendix[8\.2](https://arxiv.org/html/2607.17043#S8.SS2)\.\)\. We observe from Figure[1](https://arxiv.org/html/2607.17043#S1.F1), on the one hand, synthetic data improves several skills that were previously strong enough, includingDataandMoney & Finance\. On the other hand, some weak skills not fully mastered by the base model become even weaker in the fine\-tuned model, such asAlgebraandTime & Scheduling\. This result suggests that model collapse in instruction tuning is not simply a uniform degradation of overall performance, but a*polarization of competence*: synthetic data may further reinforce already strong capabilities while degrading weakly supported skills\.

Motivated by the above finding, we address the second question by proposingKnowledge\-boundaryInstructionTuning viaExploration \(KITE\), a two\-stage framework for synthetic data construction in iterative model evolution\. Overall, KITE is designed to provide useful learning signal so that newly added synthetic data improves rather than destabilizes the next\-generation model\. In the first stage, KITE constructs a large candidate bank by extracting weakness profiles from model failures guided by the DINA model, and then generating new data with rank\-based noise injection, which encourages exploration beyond dominant high\-probability regions\. In the second stage, it curates this bank using Kernel Boundary Uncertainty \(KBU\), a likelihood\-weighted semantic uncertainty score that identifies data near the model’s semantic knowledge boundary\. Experiments for instruction tuning on several reasoning datasets show that KITE enables stable and effective model evolution\.

In summary, we make the following key contributions:1\)we study model collapse in iterative synthetic data instruction tuning, and show that the failure is not uniform degradation but a*polarization of competence*;2\)we proposeKITE, whose two stages directly reverse the two arms of polarization via failure\-guided generation and boundary\-aware KBU curation\. and empirically show across several datasets and multiple open\-source LLMs that it yields more stable multi\-generation improvement than strong synthetic data baselines\.

## 2Related Work

### 2\.1Model Collapse

Model collapse studies the failure mode that arises when later\-generation models are trained on data increasingly contaminated by outputs from earlier generations\. Early work byShumailovet al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib1)\)shows that recursive training on model\-generated data can progressively distort the learned distribution and erase low\-probability modes, whileDohmatobet al\.\([2024b](https://arxiv.org/html/2607.17043#bib.bib2)\)analyze the same phenomenon through the lens of scaling laws and argue that the loss of tail support leads to a cutoff in continued improvement, and further show that collapse can persist as long as the synthetic fraction does not vanish\(Dohmatobet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib35)\)\.Gerstgrasseret al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib3)\)andKazdanet al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib34)\)show that collapse is not inevitable when synthetic data are accumulated together with the original real data, in which case the test error can remain bounded\.Zhuet al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib4)\)propose token\-level editing to construct semi\-synthetic text\.Fenget al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib5)\)argue that scaling with synthesized data requires verification, showing that verifier\-based selection can prevent collapse;Draysonet al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib6)\)show that machine\-generated text detection can be used to reduce collapse; andAminet al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib7)\)show that even weak but properly curated data can support continued model improvement\. However, most existing work treats the problem primarily from the perspective of degeneration control, where the goal is to keep degradation bounded\. In contrast, we study iterative model evolution for instruction tuning, where the key challenge is to diagnose collapse at a granularity that can guide data curation and support continued improvement from one generation to the next\.

### 2\.2LLMs for Synthetic Data

Large language models have become a powerful tool for synthetic data generation\(Liuet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib8)\), substantially extending earlier approaches based on conventional language models\(Schick and Schütze,[2021](https://arxiv.org/html/2607.17043#bib.bib9)\)\. Recent work has used LLMs to construct synthetic data for a wide range of purposes, including multilingual question answering\(Riabiet al\.,[2021](https://arxiv.org/html/2607.17043#bib.bib10)\), dialogue systems\(Zhaoet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib11)\), instruction tuning\(Xuet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib12); Zhanget al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib13); Zhonget al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib14)\), factuality improvement\(Weiet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib15)\), scientific reasoning\(Huanget al\.,[2025a](https://arxiv.org/html/2607.17043#bib.bib16)\), and dataset diversification\(Daiet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib17); Riazet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib18)\)\. More recent frameworks such as DataGen\(Huanget al\.,[2025b](https://arxiv.org/html/2607.17043#bib.bib20)\)and Janus\(Leeet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib19)\)further highlight the potential of LLMs to produce high\-quality synthetic corpora for model training, evaluation, and alignment\. However, these works study synthetic data primarily as a means to improve downstream performance, often in settings where data are generated by stronger external models and used in a single\-round training pipeline\. This is different from the model collapse phenomenon, where later\-generation models are trained on data generated from their own distribution\. We focus on iterative model evolution under self\-generated data\.

![Refer to caption](https://arxiv.org/html/2607.17043v1/x2.png)Figure 2:Top: iterative model evolution with KITE synthesis\. Bottom: the two stages of KITE, failure\-guided candidate bank construction and Kernel Boundary Uncertainty \(KBU\) data curation\.

## 3Methodology

### 3\.1Notation and Formulation

LetMtM\_\{t\}denote the instruction\-following model at evolution stepttwith output distributionpMt​\(y∣x\)p\_\{M\_\{t\}\}\(y\\mid x\), wherexxis an instruction andyyis an answer\. We start from an initial verified training setD0⋆=\{\(xi,yi⋆\)\}i=1n0D\_\{0\}^\{\\star\}=\\\{\(x\_\{i\},y\_\{i\}^\{\\star\}\)\\\}\_\{i=1\}^\{n\_\{0\}\}\(e\.g\., human data\) and mimic real\-world model evolution by*accumulating*training data over time: at steptt, our pipeline produces a new synthetic curated setDt=\{\(xi\)\}i=1BD\_\{t\}=\\\{\(x\_\{i\}\)\\\}\_\{i=1\}^\{B\}under a labeling budgetBB, which is then labeled toDt⋆=\{\(xi,yi⋆\)\}i=1BD\_\{t\}^\{\\star\}=\\\{\(x\_\{i\},y\_\{i\}^\{\\star\}\)\\\}\_\{i=1\}^\{B\}, and the next model is trained on the union of all verified data so far under the supervised fine\-tuning \(SFT\) objective\. We emphasize that eachMtM\_\{t\}is obtained by fine\-tuning an identical base model onDt⋆D\_\{t\}^\{\\star\}, rather than fromMt−1M\_\{t\-1\}\.

Concretely, we first generate a large synthetic candidate bankℬt=\{xi\}i=1N\\mathcal\{B\}\_\{t\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}withN≫BN\\gg B, selectDt⊆ℬtD\_\{t\}\\subseteq\\mathcal\{B\}\_\{t\}with\|Dt\|=B\|D\_\{t\}\|=B, and apply a strong verified labeler to obtainDt⋆D\_\{t\}^\{\\star\}for training\.

We define*model collapse*in our setting as a*degradation of held\-out performance*\. Our objective is to design the data\-construction operator that generates and curates training data so that model evolution is*self\-improving*\. Iterative model evolution and the KITE framework are shown in Figure[2](https://arxiv.org/html/2607.17043#S2.F2)\. The rest of this section specifies our two\-stage pipeline for constructingℬt\\mathcal\{B\}\_\{t\}and selectingDtD\_\{t\}so that each newly added sliceDt⋆D\_\{t\}^\{\\star\}is both \(i\) targeted at current weaknesses and \(ii\) reliably improves the next model in the accumulated training regime\.

### 3\.2Stage 1: Failure\-Guided Candidate Bank Construction

At evolution steptt, our first stage constructs a large synthetic*candidate bank*ℬt\\mathcal\{B\}\_\{t\}of sizeN≫BN\\gg Bwhose role is to \(i\)expand coveragebeyond the model’s internal high\-probability modes and \(ii\) concentrate generation on*current weaknesses*so that subsequent curation can spend verifier budget on examples with real learning value\.

Concretely, we generateℬt\\mathcal\{B\}\_\{t\}by \(1\) extracting a weakness profile from training failures guided by the DINA model and \(2\) sampling new instructions with rank\-based noise injection\.

##### 1\) Failure\-guided weakness profiling\.

A core driver of synthetic collapse is*self\-reinforcement*: unconstrained generation tends to over\-sample skills the model already performs well on, so the synthetic distribution drifts toward easy regions and provides diminishing learning signal\. To counter this drift, we explicitly tie generation to*observed deficiencies*of the current model\.

Rather than using raw failed examples directly, we build a diagnosis pipeline based on the DINA model from cognitive science \(details in Appendix[8\.3](https://arxiv.org/html/2607.17043#S8.SS3)\)\. DINA is useful here because it does not treat each failure as an isolated instance, instead, it infers which latent skills are likely under\-mastered from the model’s response pattern over many examples\. This gives a more structured weakness estimate than simple example failure counting\.

Concretely, each training instance is associated with a binary skill requirement vector from a Q\-matrix, indicating which fine\-grained skills are needed to solve it\. In our implementation, the Q\-matrix is constructed by LLM\-assisted skill tagging followed by consistency checking, while the full procedure is deferred to Appendix[8\.3](https://arxiv.org/html/2607.17043#S8.SS3)\. Given the model’s correct and incorrect responses on the current training set, DINA estimates posterior mastery for each skill, and we convert low\-mastery skills into short natural\-language*weakness descriptors*\(e\.g\., missing reasoning type or domain knowledge\)\. We aggregate these descriptors into a weakness profileSt=\{sk\}k=1KS\_\{t\}=\\\{s\_\{k\}\\\}\_\{k=1\}^\{K\}, and use eachs∈Sts\\in S\_\{t\}to instantiate a question\-generation prompt templateπQ​\(s\)\\pi\_\{Q\}\(s\)that asks for new instructions specifically targeting that weakness\.

##### 2\) Rank\-based noise injection\.

Even under weakness conditioning, standard decoding often produces easy or templated instructions\. A natural baseline is to increase sampling temperature\. However, in our experiments, temperature tuning improves surface diversity but does*not*reliably yield informative training signal\. Related supporting experiment results are detailed in the Section[4\.5](https://arxiv.org/html/2607.17043#S4.SS5)\. We therefore use a rank\-based perturbation that explicitly flattens the*ordering*of token probabilities\.

Letℓt​\(c\)∈ℝ\|V\|\\ell\_\{t\}\(c\)\\in\\mathbb\{R\}^\{\|V\|\}be the logits ofMtM\_\{t\}at decoding contextccandqt​\(v∣c\)=softmax​\(ℓt​\(c\)\)vq\_\{t\}\(v\\mid c\)=\\mathrm\{softmax\}\(\\ell\_\{t\}\(c\)\)\_\{v\}\. Define the rank of tokenvvby its logit order,rt​\(v∣c\)=1\+\|\{u:ℓt,u​\(c\)\>ℓt,v​\(c\)\}\|r\_\{t\}\(v\\mid c\)=1\+\\big\|\\\{u:\\ell\_\{t,u\}\(c\)\>\\ell\_\{t,v\}\(c\)\\\}\\big\|\. We then sample from the modified distribution

qt\(α\)​\(v∣c\)\\displaystyle q^\{\(\\alpha\)\}\_\{t\}\(v\\mid c\)=softmax\(ℓt\(c\)\+αlogrt\(⋅∣c\)\)v\\displaystyle=\\operatorname\{softmax\}\\\!\\Bigl\(\\ell\_\{t\}\(c\)\+\\alpha\\log r\_\{t\}\(\\cdot\\mid c\)\\Bigr\)\_\{v\}\(1\)∝qt​\(v∣c\)​rt​\(v∣c\)α\.\\displaystyle\\propto q\_\{t\}\(v\\mid c\)\\,r\_\{t\}\(v\\mid c\)^\{\\alpha\}\.which smoothly upweights lower\-ranked tokens and encourages exploration without hard truncation\. Empirically, we find that rank\-noise decoding produces candidate instructions with more useful learning signal for subsequent training, which in turn leads to more stable self\-improvement\.

For eachs∈Sts\\in S\_\{t\}, we generate candidate instructions by decoding underqt\(αQ\)\(⋅∣πQ\(s\)\)q^\{\(\\alpha\_\{Q\}\)\}\_\{t\}\(\\cdot\\mid\\pi\_\{Q\}\(s\)\)and collectℬt=\{xi\}i=1N\\mathcal\{B\}\_\{t\}=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}by repeating across descriptors until reaching total bank sizeNN\.

### 3\.3Stage 2: Uncertainty\-Based Data Curation

Given the candidate bankℬt\\mathcal\{B\}\_\{t\}, Stage 2 selects a budgeted subsetDt⊆ℬtD\_\{t\}\\subseteq\\mathcal\{B\}\_\{t\}with\|Dt\|=B\|D\_\{t\}\|=Bfor labeling and training\. The key idea is to prioritize candidates that provide*useful learning signal*: not examples that are too easy forMtM\_\{t\}, and not examples that fall far outside the model’s knowledge boundary, where failure is primarily due to missing capabilities inherited from pre\-training and thus is less amenable to post\-training correction, but examples that lie near the model’s*semantic knowledge boundary*, where multiple plausible answers coexist\. Motivated by the growing use of uncertainty to characterize the model’s capacity, and in particular by Kernel Language Entropy \(KLE\) for hallucination detection\(Nikitinet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib28)\), we instantiate this intuition through a semantic uncertainty score over multiple sampled answers\.

##### Semantic uncertainty as curation signal\.

Token\-level uncertainty is often too local for instruction tuning, since it can be inflated by surface variation or paraphrasing without reflecting genuine uncertainty about the underlying answer\. Instead, for each candidate instructionx∈ℬtx\\in\\mathcal\{B\}\_\{t\}, we probeMtM\_\{t\}by samplingmmanswersy\(1\),…,y\(m\)∼Mt\(⋅∣x\)y^\{\(1\)\},\\dots,y^\{\(m\)\}\\sim M\_\{t\}\(\\cdot\\mid x\)and measure uncertainty in*semantic space*\. Lethi=ϕ​\(y\(i\)\)∈ℝdh\_\{i\}=\\phi\(y^\{\(i\)\}\)\\in\\mathbb\{R\}^\{d\}be the embedding of answery\(i\)y^\{\(i\)\}, and define a Radial Basis Function \(RBF\) kernel

Ki​j\\displaystyle K\_\{ij\}=exp⁡\(−∥hi−hj∥22​σ2\),\\displaystyle=\\exp\\\!\\left\(\-\\frac\{\\lVert h\_\{i\}\-h\_\{j\}\\rVert^\{2\}\}\{2\\sigma^\{2\}\}\\right\),\(2\)σ2\\displaystyle\\sigma^\{2\}=median⁡\{∥hi−hj∥2:i<j\}\.\\displaystyle=\\operatorname\{median\}\\left\\\{\\lVert h\_\{i\}\-h\_\{j\}\\rVert^\{2\}:i<j\\right\\\}\.This kernel captures whether sampled answers belong to the same semantic mode or to distinct modes\.

##### Boundary\-aware uncertainty via likelihood weighting\.

A purely semantic dispersion score is not sufficient by itself: under stochastic decoding, some sampled answers may be semantically distinct simply because they are low\-probability off\-manifold generations\. Treating all samples equally would therefore overestimate uncertainty and bias selection toward prompts that are noisy or far outside the model’s actionable knowledge boundary\. For post\-training data curation, we want disagreement among*plausible*answers, since that is more indicative of a prompt lying near the model’s semantic knowledge boundary and thus more likely to yield useful learning signal\.

To instantiate this idea, we weight each sampled answer by its likelihood underMtM\_\{t\}\. For a candidate instructionxx, lety\(1\),…,y\(m\)∼Mt\(⋅∣x\)y^\{\(1\)\},\\dots,y^\{\(m\)\}\\sim M\_\{t\}\(\\cdot\\mid x\)bemmsampled answers\. We compute the length\-normalized log\-likelihood of each sample:

ℓi\\displaystyle\\ell\_\{i\}=1\|y\(i\)\|∑klogpMt\(yk\(i\)\|x,y<k\(i\)\)\.\\displaystyle=\\frac\{1\}\{\\lvert y^\{\(i\)\}\\rvert\}\\sum\_\{k\}\\log p\_\{M\_\{t\}\}\\\!\\left\(y\_\{k\}^\{\(i\)\}\\,\\middle\|\\,x,y\_\{<k\}^\{\(i\)\}\\right\)\.\(3\)We then convert these scores into normalized weights:

w~i=exp⁡\(γ​ℓi\)∑jexp⁡\(γ​ℓj\)\.\\tilde\{w\}\_\{i\}=\\frac\{\\exp\(\\gamma\\ell\_\{i\}\)\}\{\\displaystyle\\sum\_\{j\}\\exp\(\\gamma\\ell\_\{j\}\)\}\.\(4\)This weighting suppresses low\-probability generations that would otherwise inflate uncertainty, while preserving disagreement among answers that the model itself assigns substantial probability mass\.

We then combine likelihood weighting with the semantic kernel defined above\. This kernelized formulation avoids making hard clustering decisions over sampled answers, and instead captures uncertainty through their continuous semantic similarity structure\. LetW=diag​\(w~1,…,w~m\)W=\\mathrm\{diag\}\(\\sqrt\{\\tilde\{w\}\_\{1\}\},\\dots,\\sqrt\{\\tilde\{w\}\_\{m\}\}\), and form the unit\-trace matrix

A=W​K​Wtr⁡\(W​K​W\)\.A=\\frac\{WKW\}\{\\operatorname\{tr\}\(WKW\)\}\.\(5\)We define the*Kernel Boundary Uncertainty*\(KBU\) score of instructionxxas the Rényi\-2 entropy ofAA:

UKBU​\(x\)=−log⁡tr⁡\(A2\)\.U\_\{\\mathrm\{KBU\}\}\(x\)=\-\\log\\operatorname\{tr\}\\\!\\left\(A^\{2\}\\right\)\.\(6\)We choose Rényi\-2 entropy because it provides a stable measure of the effective number of semantic modes represented in the weighted kernel, while remaining computationally lightweight\.

When the KBU score is small, the sampled answers concentrate on a single semantic mode\. This typically indicates thatxxis either too easy for the model, or the model is confidently wrong, but in either case it offers little curation signal\. Intermediate values indicate disagreement among multiple plausible answers, suggesting thatxxlies near the model’s semantic knowledge boundary and is likely to provide useful learning signal\. Excessively large values, however, may indicate that the prompt lies beyond the model’s actionable knowledge boundary\. We therefore select candidates whose uncertainty falls within a target percentile range\(umin,umax\)\(u\_\{\\min\},u\_\{\\max\}\), whereumin,umax∈\(0,1\)u\_\{\\min\},u\_\{\\max\}\\in\(0,1\)denote lower and upper quantile thresholds\.

ModelMethodDataset \(%\)AverageGSM8KMMLU\-ProMATHGPQAQwen\-3\-4B\-InstructInitial Model93\.63±\\pm0\.1869\.55±\\pm0\.2580\.80±\\pm0\.3240\.40±\\pm0\.6571\.10Human data94\.28±\\pm0\.2270\.40±\\pm0\.3181\.60±\\pm0\.4041\.41±\\pm0\.7171\.92Self\-Instruct93\.79±\\pm0\.2870\.12±\\pm0\.3581\.00±\\pm0\.4540\.90±\\pm0\.8171\.45Few\-shot syn\.93\.58±\\pm0\.3070\.01±\\pm0\.3880\.80±\\pm0\.4240\.40±\\pm0\.7671\.20CDS94\.03±\\pm0\.2570\.50±\\pm0\.2981\.20±\\pm0\.3841\.41±\\pm0\.6871\.79ToEdit94\.50±\\pm0\.2470\.40±\\pm0\.3381\.60±\\pm0\.4141\.41±\\pm0\.7471\.98Ours95\.04±\\pm0\.1971\.87±\\pm0\.2682\.60±\\pm0\.3541\.92±\\pm0\.6672\.86Qwen\-3\-1\.7BInitial Model85\.97±\\pm0\.2251\.10±\\pm0\.3172\.80±\\pm0\.3528\.28±\\pm0\.7559\.54Human data87\.50±\\pm0\.2852\.49±\\pm0\.3274\.00±\\pm0\.4228\.79±\\pm0\.7360\.70Self\-Instruct87\.01±\\pm0\.2951\.97±\\pm0\.2673\.40±\\pm0\.4727\.78±\\pm0\.8260\.04Few\-shot syn\.86\.73±\\pm0\.3152\.40±\\pm0\.3973\.00±\\pm0\.4427\.78±\\pm0\.7859\.98CDS87\.04±\\pm0\.2652\.88±\\pm0\.3074\.40±\\pm0\.4029\.29±\\pm0\.6960\.90ToEdit87\.50±\\pm0\.2552\.49±\\pm0\.3474\.00±\\pm0\.4328\.79±\\pm0\.7560\.70Ours88\.48±\\pm0\.2753\.30±\\pm0\.2975\.00±\\pm0\.3829\.29±\\pm0\.7161\.52Llama\-3\.2\-3B\-InstructInitial Model75\.28±\\pm0\.3038\.11±\\pm0\.3441\.60±\\pm0\.4530\.30±\\pm0\.8246\.32Human data76\.97±\\pm0\.2839\.03±\\pm0\.3342\.60±\\pm0\.4931\.31±\\pm0\.7447\.48Self\-Instruct76\.18±\\pm0\.3538\.20±\\pm0\.4141\.60±\\pm0\.5630\.80±\\pm0\.8346\.70Few\-shot syn\.75\.90±\\pm0\.3738\.65±\\pm0\.4341\.80±\\pm0\.5230\.80±\\pm0\.8046\.79CDS76\.97±\\pm0\.3039\.15±\\pm0\.3542\.60±\\pm0\.4631\.81±\\pm0\.7147\.63ToEdit77\.03±\\pm0\.2939\.03±\\pm0\.3442\.60±\\pm0\.4732\.32±\\pm0\.7347\.75Ours78\.30±\\pm0\.3239\.62±\\pm0\.3843\.00±\\pm0\.4332\.83±\\pm0\.6848\.44Llama\-3\-8B\-InstructInitial Model78\.32±\\pm0\.2538\.75±\\pm0\.2924\.40±\\pm0\.4232\.32±\\pm0\.7543\.45Human data79\.98±\\pm0\.2739\.38±\\pm0\.3226\.00±\\pm0\.4832\.83±\\pm0\.7244\.55Self\-Instruct79\.15±\\pm0\.3439\.11±\\pm0\.4025\.00±\\pm0\.5532\.32±\\pm0\.8043\.90Few\-shot syn\.77\.97±\\pm0\.3638\.75±\\pm0\.4225\.60±\\pm0\.5132\.32±\\pm0\.7843\.66CDS79\.91±\\pm0\.2939\.35±\\pm0\.3426\.20±\\pm0\.4532\.83±\\pm0\.6944\.57ToEdit80\.05±\\pm0\.2839\.38±\\pm0\.3326\.00±\\pm0\.4632\.83±\\pm0\.7144\.57Ours81\.12±\\pm0\.2339\.85±\\pm0\.2826\.60±\\pm0\.4233\.33±\\pm0\.6445\.23Gemma\-3\-4B\-InstructInitial Model84\.69±\\pm0\.2048\.30±\\pm0\.2866\.60±\\pm0\.3529\.29±\\pm0\.7757\.22Human data85\.98±\\pm0\.2149\.50±\\pm0\.3068\.00±\\pm0\.4130\.30±\\pm0\.7058\.45Self\-Instruct85\.20±\\pm0\.2748\.84±\\pm0\.3467\.40±\\pm0\.4628\.79±\\pm0\.8057\.56Few\-shot syn\.85\.57±\\pm0\.2948\.84±\\pm0\.3767\.40±\\pm0\.4328\.79±\\pm0\.7557\.65CDS85\.25±\\pm0\.2449\.50±\\pm0\.2868\.00±\\pm0\.3930\.81±\\pm0\.6758\.39ToEdit85\.98±\\pm0\.2349\.50±\\pm0\.3268\.00±\\pm0\.4230\.30±\\pm0\.7358\.44Ours86\.52±\\pm0\.2550\.01±\\pm0\.2969\.40±\\pm0\.3830\.81±\\pm0\.6959\.19

Table 1:Main results across models and synthetic data construction methods\.We report evaluation accuracy \(%\) on four reasoning benchmarks after five iterations of accumulated self\-improvement, averaged over 3 random seeds\.

## 4Experiments

### 4\.1Experimental Setting

##### Dataset\.

To evaluate our method, we use four widely adopted reasoning datasets covering complementary forms of difficulty\.GSM8K\(Cobbeet al\.,[2021](https://arxiv.org/html/2607.17043#bib.bib21)\)tests grade\-school arithmetic and multi\-step numerical reasoning;MMLU\-Pro\(Wanget al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib23)\)extends MMLU\(Hendryckset al\.,[2021a](https://arxiv.org/html/2607.17043#bib.bib24)\)across diverse academic subjects;MATH\(Hendryckset al\.,[2021b](https://arxiv.org/html/2607.17043#bib.bib25)\)contains competition\-level problems requiring symbolic reasoning and longer derivations; andGPQA\(Reinet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib26)\)consists of graduate\-level science questions difficult even for strong LLMs\. Full evaluation protocols are in Appendix[8\.2](https://arxiv.org/html/2607.17043#S8.SS2)\.

##### Baselines\.

We compare our method with baselines that adopt different synthetic data construction strategies for iterative self\-improvement\. For all synthetic\-data baselines, we follow the same accumulated training protocol: at each iteration, new synthetic data are generated, labeled, appended to the training set, and a new model is trained from the same base model on the accumulated labeled data\. We report results after five iterations\.Human data onlyserves as a non\-synthetic control, where training uses only the initial verified human dataset\.Self\-Instruct\(Wanget al\.,[2023](https://arxiv.org/html/2607.17043#bib.bib27)\)generates new instruction–response pairs by prompting the model to bootstrap synthetic tasks from a small seed set\.Few\-shot synthesisgenerates synthetic instruction–response pairs by few\-shot prompting from seed examples\.Cognitive Diagnosis\-based Synthesis \(CDS\)\(Zhaoet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib43)\)uses cognitive\-diagnosis\-derived weakness profiles to guide targeted data generation but omits the uncertainty\-based curation stage\.Token\-Level Editing \(ToEdit\)\(Zhuet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib4)\)produces semi\-synthetic data by diversifying model\-generated data through token\-level perturbation and editing\.

##### Others\.

Due to the space limitations,*Training*and*Evaluation*setting are detailed in the Appendix[8\.2](https://arxiv.org/html/2607.17043#S8.SS2)\.*Cost Analysis*and*Theoretical Analysis*are detailed in the Appendix[8\.4](https://arxiv.org/html/2607.17043#S8.SS4)and[8\.6](https://arxiv.org/html/2607.17043#S8.SS6)

### 4\.2Main Results

Table[1](https://arxiv.org/html/2607.17043#S3.T1)summarizes the performance of different synthetic data construction strategies across datasets and backbone LLMs after five iterations of model evolution\. Overall, KITE achieves the best average performance within every model group, demonstrating its effectiveness for synthetic data driven instruction tuning\. For example, on Qwen\-3\-4B\-Instruct, KITE reaches 95\.04% on GSM8K, 71\.87% on MMLU\-Pro, and 82\.60% on MATH, consistently outperforming the corresponding baseline settings\. We also note that the gains on MMLU\-Pro and GPQA are generally smaller than those on GSM8K and MATH, which is expected since these benchmarks are more challenging and often require broader knowledge and reasoning capacity that are harder to improve through instruction tuning alone\.

In contrast, other synthetic data baselines provide only limited gains\. Because we retain all verified real data across iterations \(the accumulate setting ofGerstgrasseret al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib3)\)\), catastrophic accuracy collapse is bounded by construction; the relevant failure mode here is therefore the inability of self\-generated data to push the model past the human\-data\-only ceiling\. This is exactly what the baselines exhibit\. Self\-Instruct and Few\-shot avoid severe degradation relative to the Initial Model, but they remain*below*the Human data setting\. CDS performs better than generic synthesis but below KITE, indicating that weakness\-guided generation alone is insufficient without uncertainty curation\. ToEdit can improve over simpler baselines, but remains consistently below KITE, showing that increasing diversity at the generation stage does not by itself guarantee the most useful learning signal\.

### 4\.3Ablation Studies

We modify KITE’s components to conduct an ablation study and validate each design choice\. Table[2](https://arxiv.org/html/2607.17043#S4.T2)reports results on Llama\-3\-8B\-Instruct\. The full method achieves the best average performance at 45\.2%, compared with 43\.2% without weakness profiling and 43\.9% when KBU is replaced with token entropy\.

To assess candidate\-bank construction, we remove weakness profiling and rank\-noise injection\. The performance drop inw/o weakness\.shows that targeting model deficiencies is important for generating useful synthetic data, while the drop inw/o rank\-noise\.suggests that sampling only from dominant model modes is insufficient to preserve coverage\. Replacing KBU with token entropy also consistently underperforms the full method, supporting the use of*semantic*rather than token\-level uncertainty\. Removinglikelihoodweighting further degrades performance, confirming that uncertainty should emphasize disagreement among*plausible*answers rather than low\-probability, off\-manifold generations\. We also vary band\-pass curation range\. Bothumin=0\.4,;umax=1\.0u\_\{\\min\}=0\.4,;u\_\{\\max\}=1\.0andumin=0\.0,;umax=0\.6u\_\{\\min\}=0\.0,;u\_\{\\max\}=0\.6underperform the full method, indicating that selecting only high\- or low\-uncertainty examples is suboptimal\. This supports our hypothesis that the most useful learning signal lies near an intermediate semantic knowledge boundary\.

AblationGSM8KMMLU\-ProMATHGPQAAvg\.w/o weakness\.76\.239\.125\.032\.343\.2w/o rank\-noise\.80\.439\.926\.233\.344\.9token entropy\.79\.939\.325\.031\.343\.9w/o likelihood\.80\.039\.326\.632\.344\.6\(0\.4, 1\.0\)79\.739\.526\.233\.344\.7\(0\.0, 0\.6\)79\.539\.726\.232\.844\.6Full method81\.139\.926\.633\.345\.2

Table 2:Ablation study on Llama\-3\-8B\-Instruct\.We evaluate the contribution of each component of our method\.![Refer to caption](https://arxiv.org/html/2607.17043v1/x3.png)Figure 3:Perplexity\-based analysis of synthetic data under recursive training\.\(a\) Perplexity distributions on wikitext\-2 with Llama\-3\-8B\-Instruct for human\-written data, synthetic data generated with temperature=0\.3, synthetic data generated with temperature=1\.0, and synthetic data generated with KITE\. \(b\) Perplexity on the wikitext\-2 test set across recursive model generations when Llama\-3\-8B\-Instruct trained on synthetic data\.
### 4\.4Long\-Horizon Evolution Dynamics

To test whether KITE keeps improving beyond the five iterations of Table[1](https://arxiv.org/html/2607.17043#S3.T1)or eventually stagnates or reverses, we extend Llama\-3\.2\-3B\-Instruct to nine generations under the identical pipeline \(Figure[4](https://arxiv.org/html/2607.17043#S4.F4); Generation 0 is the initial model and Generation 5 matches the “Ours” row\)\. Performance increases*monotonically*on every benchmark with no degradation\-and\-reversal signature of collapse, and the gains*saturate*gradually\. This is the expected behavior of a well\-behaved self\-improvement loop, continued but diminishing returns rather than runaway gains or collapse\.

![Refer to caption](https://arxiv.org/html/2607.17043v1/x4.png)Figure 4:Long\-horizon evolution dynamics of KITE on Llama\-3\.2\-3B\-Instruct\.
### 4\.5Generalization to Non\-Reasoning Tasks

To evaluate KITE beyond reasoning tasks, we followShumailovet al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib1)\)and convert wikitext\-2\(Merityet al\.,[2016](https://arxiv.org/html/2607.17043#bib.bib41)\)into an instruction\-tuning dataset for recursive self\-training\. Because this task has no explicit skill taxonomy, we apply KITE without weakness profiling\. As shown in Figure[3](https://arxiv.org/html/2607.17043#S4.F3)\(a\), standard decoding at both low and high temperatures produces synthetic data overly concentrated in the low\-perplexity region, whereas KITE yields a broader distribution closer to human\-written data\. Figure[3](https://arxiv.org/html/2607.17043#S4.F3)\(b\) further examines the consequence of this phenomenon under iterative training\. When synthetic data generated with high\-temperature are used for repeated self\-training, the perplexity on the wikitext\-2 test set increases monotonically from Model 1 to Model 5, which is aligned with insights discussed byShumailovet al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib1)\), while KITE keeps perplexity stable across iterations\.

![Refer to caption](https://arxiv.org/html/2607.17043v1/x5.png)Figure 5:OOD generalization on math reasoning benchmarks\.We evaluate Qwen\-3\-4B\-Instruct\-2507, trained with the above in\-domain math subsets, on the unseen OlymMATH and AIME benchmarks\.
### 4\.6OOD Generalization

Figure[5](https://arxiv.org/html/2607.17043#S4.F5)shows whether KITE trained on all math subsets of the four in\-domain datasets described above can generalize to out\-of\-distribution \(OOD\) mathematical reasoning benchmarks\. We evaluate Qwen\-3\-4B\-Instruct onOlymMATH\(Sunet al\.,[2025](https://arxiv.org/html/2607.17043#bib.bib39)\)andAIME\(Art of Problem Solving,[2026](https://arxiv.org/html/2607.17043#bib.bib38)\), harder competition\-style problems never seen during training or synthetic data construction\. KITE improves on both, suggesting that synthetic self\-improvement enhances the model’s underlying reasoning skills rather than merely fitting the benchmark distribution\.

## 5Conclusion

We study model collapse in synthetic data\-driven instruction tuning under iterative model evolution, where the objective is to ensure that each successive model improves over its predecessor\. We show that collapse can manifest as a polarization of competence, and propose KITE, a two\-stage framework that combines failure\-guided exploration with boundary\-aware uncertainty curation\. Experiments across multiple LLMs and benchmarks show that KITE yields more stable improvement than strong synthetic\-data baselines, with ablations confirming the importance of both components\.

## 6Limitations

Our study has several limitations that suggest directions for future work\. First, all methods use a strong external model \(gpt\-5\-mini\) to verify and normalize answers; although it is applied identically to KITE and every baseline and thus does not confound the relative comparison \(Appendix[8\.2](https://arxiv.org/html/2607.17043#S8.SS2)\), the answer channel still carries an element of distillation, so our contribution is best read as a study of self\-generated*instruction*distributions rather than pure closed\-loop self\-improvement\. Second, our DINA\-based weakness profiling relies on an LLM\-tagged, non\-expert\-validated Q\-matrix with fixed slip and guess priors, so the mastery scores are a structured heuristic for steering generation rather than calibrated measurements, and we leave a per\-skill post\-KITE re\-diagnosis to future work\. Finally, our evaluation focuses on reasoning\-style instruction tuning and non\-reasoning perplexity analysis with1\.71\.7–88B backbones; scaling KITE to larger models, RL\-style post\-training, and a broader range of tasks is an important next step\.

## 7Ethical Considerations

This work studies data\-construction methods for instruction tuning and does not introduce new human\-subject data collection\. All datasets used are publicly released research benchmarks used in accordance with their licenses and intended research use\. The synthetic instructions do not target individuals or sensitive attributes\. Our use of an LLM as labeler and as evaluation judge can propagate biases or errors present in that model; we mitigate this by preferring deterministic answer extraction and exact\-match whenever possible and restricting the judge to formatting/equivalence decisions \(Appendix[8\.2](https://arxiv.org/html/2607.17043#S8.SS2)\), but residual judge bias cannot be fully excluded\.

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## 8Appendix

### 8\.1Additional Related Works

##### Self\-improving LLMs\.

A growing body of work improves LLMs from their own generations\.Zelikmanet al\.\([2022](https://arxiv.org/html/2607.17043#bib.bib45)\)propose STaR, which bootstraps reasoning by iteratively generating rationales, retaining those that lead to correct answers, and fine\-tuning on them\.Chenet al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib44)\)propose self\-play fine\-tuning \(SPIN\), where a model improves by discriminating its own responses against human\-annotated data\.Gulcehreet al\.\([2023](https://arxiv.org/html/2607.17043#bib.bib47)\)introduce Reinforced Self\-Training \(ReST\), which grows a dataset by sampling from the policy and filtering with a reward model, andYuanet al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib46)\)propose Self\-Rewarding Language Models, where the model acts as its own judge to supply training rewards\. More recently,Wuet al\.\([2026](https://arxiv.org/html/2607.17043#bib.bib48)\)formulate alignment as a self\-generative adversarial game within a single LLM that doubles as a synthetic\-data engine, whileRoeet al\.\([2026](https://arxiv.org/html/2607.17043#bib.bib49)\)show that iteratively fine\-tuning a model on its own outputs is*mostly idempotent*, with behavioral traits stabilizing rather than amplifying, an observation consistent with the diminishing\-returns saturation we report over nine generations \(Figure[4](https://arxiv.org/html/2607.17043#S4.F4)\)\. These works establish that self\-generated supervision can drive improvement, but they primarily introduce new*training paradigms*\(rationale bootstrapping, self\-play, reward\-filtered RL, adversarial games\)\. In contrast, our work is not a new training objective but a synthetic\-data generation and curation framework for iterative model evolution: we focus on how to construct the self\-generated*instruction*distribution so that later\-generation models can keep improving without collapse, and KITE is orthogonal to and composable with these paradigms\.

##### Synthetic data and data curation\.

Our second stage is closely related to work on weakness profiling and data selection\.Zhaoet al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib43)\)propose CDS, which uses Cognitive Diagnosis Theory to characterize model weaknesses at the knowledge\-component level and synthesize targeted data, closely aligned with our DINA\-based diagnosis; we include CDS as a baseline in Table[1](https://arxiv.org/html/2607.17043#S3.T1)\.Sobhaniet al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib50)\)show that LLMs can categorize system inputs into fine\-grained, natural\-language performance categories, andZenget al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib51)\)build hierarchical capability trees \(EvalTree\) to profile model weaknesses more precisely; both provide structured alternatives for identifying where a model underperforms, complementary to our psychometric weakness estimate\. On the curation side,Xiaet al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib53)\)\(LESS\) select influential data via low\-rank gradient similarity to a target capability,Liet al\.\([2024](https://arxiv.org/html/2607.17043#bib.bib54)\)\(Superfiltering\) use a small proxy model’s instruction\-following\-difficulty scores to filter data efficiently, andYanget al\.\([2025](https://arxiv.org/html/2607.17043#bib.bib52)\)study dynamic data selection jointly with data augmentation to accelerate training\. Very recent work continues this thread:Fanet al\.\([2026](https://arxiv.org/html/2607.17043#bib.bib55)\)\(Optimsyn\) use gradient\-based influence estimation to optimize synthetic\-data generation rubrics via reinforcement learning, andAminet al\.\([2026](https://arxiv.org/html/2607.17043#bib.bib56)\)analyze the limitations of empirical risk minimization when training on mixed natural and synthetic data, showing that appropriate reweighting rather than plain ERM is needed to avoid collapse\. These methods largely select from, reweight, or optimize generation over a pool using influence, difficulty, or quality heuristics\. KITE differs in two ways: it*generates*candidates targeted at diagnosed weaknesses rather than only filtering or reweighting existing data, and its Kernel Boundary Uncertainty \(KBU\) selects examples near the model’s*semantic knowledge boundary*, giving an explicit anti\-collapse curation mechanism tailored to iterative synthetic\-data self\-improvement\.

### 8\.2Detailed Experiments Setting and Additional Results

#### 8\.2\.1Models

We evaluate our method on five open\-source instruction\-tuned LLMs spanning different model families and parameter scales:Qwen3\-4B\-Instruct\-2507,Qwen\-3\-1\.7B\(Qwen\-Team,[2025](https://arxiv.org/html/2607.17043#bib.bib31)\),Llama\-3\.2\-3B\-Instruct,Llama\-3\-8B\-Instruct\(AI@Meta,[2024](https://arxiv.org/html/2607.17043#bib.bib32)\), andGemma\-3\-4B\-It\(Gemma\-Team,[2025](https://arxiv.org/html/2607.17043#bib.bib33)\)\. Unless otherwise specified, each model serves as the common initialization point for all compared methods, and every evolved model is trained from the same initial checkpoint under the corresponding accumulated data setting\.

#### 8\.2\.2Datasets

We evaluate onGSM8K,MMLU\-Pro,MATH, andGPQA\. GSM8K is used in its standard form\. ForMMLU\-Pro, we include only the science\- and mathematics\-related subject areas, and exclude domains such as law, business, and philosophy\. ForMATH, following prior practice, we construct a 500\-example evaluation subset by uniform random sampling from the benchmark\. ForGPQA, we evaluate only on theGPQA\-Diamondsplit\. These choices allow us to focus evaluation on challenging reasoning tasks while keeping the benchmark suite computationally manageable and aligned with the goals of this work\. Example instances for each dataset are included in Table[3](https://arxiv.org/html/2607.17043#S8.T3)\.

#### 8\.2\.3Baselines

Our baseline set is chosen to isolate the axis we study,*how synthetic instructions are constructed and selected*, and therefore holds the training objective fixed \(accumulated SFT\) across all methods\. We deliberately do*not*treat self\-improvement*training paradigms*such as STaR\(Zelikmanet al\.,[2022](https://arxiv.org/html/2607.17043#bib.bib45)\)or SPIN\(Chenet al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib44)\)as competitors: they modify the learning objective \(rationale bootstrapping, self\-play discrimination\) rather than the instruction distribution, and KITE’s data\-construction operator is orthogonal to and composable with them\. Comparing KITE against a different training paradigm would confound the two axes; instead, we compare against the data\-construction strategies \(generic synthesis, weakness targeting, and semi\-synthetic editing\) that are directly substitutable for our operator under an identical training pipeline\. ToEdit and CDS in particular are strong, recent points of comparison for the diversification and targeting intuitions, respectively\.

#### 8\.2\.4Training setting

We use LLaMA\-Factory\(Zhenget al\.,[2024](https://arxiv.org/html/2607.17043#bib.bib40)\)as the training framework for all supervised fine\-tuning experiments\. Unless otherwise specified, we fine\-tune each model with LoRA under the SFT stage, usinglora\_rank=8\. For optimization, we use a learning rate of1×10−51\\times 10^\{\-5\}, a warmup ratio of 0\.1, and train for 1 epoch\. All experiments are conducted inbf16precision\. Other implementation details follow the default LLaMA\-Factory settings unless explicitly noted\.

The initial verified setD0⋆D\_\{0\}^\{\\star\}is the human\-authored training split of each dataset, held strictly disjoint from the held\-out evaluation instances reported in Table[1](https://arxiv.org/html/2607.17043#S3.T1); for benchmarks without a large native training split we seedD0⋆D\_\{0\}^\{\\star\}from the released development/auxiliary items and never draw seeds from the evaluation split\. The “Human data” control reuses onlyD0⋆D\_\{0\}^\{\\star\}across all iterations, whereas every synthetic method addsBBnewly verified instructions per iteration on top ofD0⋆D\_\{0\}^\{\\star\}, so all methods see the same real data and differ only in the synthetic additions\.

For KITE\-specific data construction, we use rank\-noise decoding with question\-generation strengthαQ=0\.75\\alpha\_\{Q\}=0\.75, candidate\-bank sizeN=2,000N=2,000, and labeling budgetB=500B=500per iteration\. In Stage 2, we samplem=10m=10answers per candidate, use likelihood\-weighting temperatureγ=0\.8\\gamma=0\.8, and select candidates by KBU quantile thresholds\(umin,umax\)=\(0\.2,0\.8\)\(u\_\{\\min\},u\_\{\\max\}\)=\(0\.2,0\.8\)\. For DINA\-based weakness profiling, we select weak skills using the mastery thresholdτ=0\.30\\tau=0\.30\. The semantic embedding functionϕ​\(⋅\)\\phi\(\\cdot\)used to construct the kernel is text\-embedding\-3\-small\(OpenAI,[2026](https://arxiv.org/html/2607.17043#bib.bib29)\)\. Unless otherwise noted, these hyperparameters are shared across all datasets and model backbones\. We use an external labeler to verify correct candidate answers before training\. Concretely, we use gpt\-5\-mini\(OpenAI,[2026](https://arxiv.org/html/2607.17043#bib.bib29)\)as the verifier\. The detailed prompt template is provided in the Appendix[8\.7](https://arxiv.org/html/2607.17043#S8.SS7)\.

#### 8\.2\.5Evaluation

We evaluate model performance using an LLM\-as\-judge protocol\(Zhenget al\.,[2023](https://arxiv.org/html/2607.17043#bib.bib36)\), where the judged model output is compared against the ground\-truth answer to determine correctness\. Concretely, we use gpt\-5\-mini\(OpenAI,[2026](https://arxiv.org/html/2607.17043#bib.bib29)\)as the judge model\. We first apply deterministic answer extraction and exact\-match comparison whenever possible, and use the judge only to handle semantically equivalent variants that differ in formatting or phrasing\. The detailed evaluation prompt template is provided in Appendix[8\.7](https://arxiv.org/html/2607.17043#S8.SS7)\.

Our setting also let us rule out the most natural confound, that KITE’s gains merely reflect distillation from the external strong verifier\. Self\-Instruct, Few\-shot synthesis, and ToEdit all consume the*identical*gpt\-5\-mini verification channel that KITE uses, yet several of them sit*below*the human\-data ceiling \(e\.g\., Self\-Instruct and Few\-shot on Qwen\-3\-1\.7B\)\. If verified labels were the source of improvement, every method drawing on the same verifier would inherit it equally; instead, only the method that changes*which instructions are synthesized and selected*moves above the ceiling\. The verifier is thus better read as a shared, held\-constant labeling oracle than as the driver of the effect, which is the whole point of holding the answer channel fixed\. We return to the residual distillation caveat in the Limitations\.

#### 8\.2\.6Skill Profile

We provide additional skill\-profile results\. For each dataset and model, we construct a diagnostic Q\-matrix over fine\-grained skills and fit the DINA model\(De La Torre,[2009](https://arxiv.org/html/2607.17043#bib.bib22)\)to the response patterns of the initial model and the corresponding fine\-tuned model after accumulated synthetic training\. The resulting posterior mastery scores allow us to compare how synthetic data changes skill\-level competence beyond aggregate accuracy\. Figures[10](https://arxiv.org/html/2607.17043#S8.F10),[11](https://arxiv.org/html/2607.17043#S8.F11),[12](https://arxiv.org/html/2607.17043#S8.F12), and Table[4](https://arxiv.org/html/2607.17043#S8.T4)show representative skill profiles for Llama\-3\-8B\-Instruct on different datasets\.

Across settings, we observe a consistent qualitative pattern\. Synthetic training does not simply shift all skills upward or downward uniformly\. Instead, it tends to further strengthen skills that are already relatively well supported, while weak or under\-supported skills often improve little or even deteriorate\. These results support our claim that collapse under synthetic data is better understood as a*polarization of competence*rather than uniform degradation\.

### 8\.3Detailed Diagnosis Pipeline

We diagnose the weakness profile of a target modelMtM\_\{t\}using the DINA model from the cognitive diagnosis literature\. DINA assumes that each evaluation example requires a subset of latent skills, and infers skill mastery from the model’s pattern of correct and incorrect responses\. We then convert low\-mastery skills into the weakness profile used in Stage 1 of KITE\.

Let𝒯=\{\(xj,yj⋆\)\}j=1J\\mathcal\{T\}=\\\{\(x\_\{j\},y\_\{j\}^\{\\star\}\)\\\}\_\{j=1\}^\{J\}be a diagnostic set, which is the training dataset in our implementation, and let𝒦=\{1,…,K\}\\mathcal\{K\}=\\\{1,\\dots,K\\\}denote the skill inventory\. Each examplejjis associated with a binary Q\-matrix rowqj=\(qj​1,…,qj​K\)∈\{0,1\}Kq\_\{j\}=\(q\_\{j1\},\\dots,q\_\{jK\}\)\\in\\\{0,1\\\}^\{K\}, whereqj​k=1q\_\{jk\}=1indicates that itemjjrequires skillkk\. In practice, the Q\-matrix is constructed from LLM\-assisted skill tagging\. For the target modelMtM\_\{t\}, we evaluate it on𝒯\\mathcal\{T\}and record a binary response vectorr\(t\)∈\{0,1\}Jr^\{\(t\)\}\\in\\\{0,1\\\}^\{J\}, whererj\(t\)=𝟏​\[Eval​\(Mt​\(xj\),yj⋆\)=1\]\.r\_\{j\}^\{\(t\)\}=\\mathbf\{1\}\[\\mathrm\{Eval\}\(M\_\{t\}\(x\_\{j\}\),y\_\{j\}^\{\\star\}\)=1\]\.

Letα\(t\)=\(α1\(t\),…,αK\(t\)\)∈\{0,1\}K\\alpha^\{\(t\)\}=\(\\alpha\_\{1\}^\{\(t\)\},\\dots,\\alpha\_\{K\}^\{\(t\)\}\)\\in\\\{0,1\\\}^\{K\}denote the latent skill mastery vector ofMtM\_\{t\}, whereαk\(t\)=1\\alpha\_\{k\}^\{\(t\)\}=1indicates mastery of skillkk\. Under DINA, an item can be solved ideally only if all required skills are mastered, with ideal responseηj\(t\)=∏k=1K\(αk\(t\)\)qj​k\.\\eta\_\{j\}^\{\(t\)\}=\\prod\_\{k=1\}^\{K\}\(\\alpha\_\{k\}^\{\(t\)\}\)^\{q\_\{jk\}\}\.Each itemjjis associated with a slip parametersjs\_\{j\}and a guess parametergjg\_\{j\}, and DINA models the response probability asPr⁡\(rj\(t\)=1∣α\(t\)\)=\(1−sj\)ηj\(t\)​gj1−ηj\(t\)\.\\Pr\(r\_\{j\}^\{\(t\)\}=1\\mid\\alpha^\{\(t\)\}\)=\(1\-s\_\{j\}\)^\{\\eta\_\{j\}^\{\(t\)\}\}g\_\{j\}^\{\\,1\-\\eta\_\{j\}^\{\(t\)\}\}\.Unlike the standard psychometric setting, where slip/guess and the mastery distribution are jointly estimated across a*population*of respondents, we diagnose a*single*model\. To keep the mastery estimate identifiable from one response vector, we do*not*free\-estimate item parameters; instead we fix weak, shared slip/guess priors \(sj=gj=0\.1s\_\{j\}=g\_\{j\}=0\.1unless noted\) and a factorized skill prior, and compute the posterior over the skill\-mastery vector under the DINA likelihood, which factorizes over items\. Because many items share each skill through the Q\-matrix, aggregating correct/incorrect responses across all items requiring skillkkyields a stable posterior for that skill even from a single respondent\. We define the posterior mastery of skillkkasmk\(t\)=Pr⁡\(αk\(t\)=1∣r\(t\)\),m\_\{k\}^\{\(t\)\}=\\Pr\(\\alpha\_\{k\}^\{\(t\)\}=1\\mid r^\{\(t\)\}\),and the corresponding weakness score aswk\(t\)=1−mk\(t\)\.w\_\{k\}^\{\(t\)\}=1\-m\_\{k\}^\{\(t\)\}\.This is a deliberate simplification of full DINA; as noted in the Limitations, we treat the resulting scores as a structured heuristic for steering generation rather than as calibrated latent\-trait measurements, and KITE’s downstream gains do not require the diagnosis to be exact\.

Finally, we form the weakness profile by selecting the most under\-mastered skills, e\.g\.,𝒲t=\{k:wk\(t\)≥τ\}\\mathcal\{W\}\_\{t\}=\\\{k:w\_\{k\}^\{\(t\)\}\\geq\\tau\\\}or equivalently the top\-LLskills ranked bywk\(t\)w\_\{k\}^\{\(t\)\}\. These skills are then converted into natural\-language weakness descriptorsSt=\{s1,…,sL\}S\_\{t\}=\\\{s\_\{1\},\\dots,s\_\{L\}\\\}, which are used to guide candidate generation in Stage 1\. In this way, DINA provides a structured mechanism for translating fine\-grained diagnosis results into targeted synthetic data construction\.

### 8\.4Cost Analysis

We briefly report the computational cost of KITE\. For each model, a full 5\-iteration run consists of three stages: synthetic candidate generation, KBU\-based uncertainty curation, and supervised fine\-tuning\. Among them, the dominant cost comes from repeated fine\-tuning, while the additional overhead of KITE mainly arises from Stage II, where we sample multiple answers per candidate to estimate uncertainty\. In our implementation, candidate generation and curation account for approximately 10% and 12% of the total wall\-clock time, respectively, while supervised fine\-tuning accounts for the remaining 78%\. Although KITE introduces extra synthesis and curation cost compared with simpler data synthesis baselines, the overhead remains affordable relative to training\.

It is worth being explicit about the cost/benefit trade\-off, since the per\-iteration gains are modest\. All methods in our comparison pay the same dominant costs, the same labeling budgetBBand the same fine\-tuning, so KITE’s*marginal*overhead is confined to the roughly22%22\\%of wall\-clock spent on generation and KBU curation; it does*not*increase the number of labeled examples the model trains on\. The relevant comparison is therefore not “extra data” but “the same labeling budget spent on better\-chosen instructions\.” Under a*fixed*labeling budget, that is the regime practitioners actually face, KITE converts an otherwise\-wasted budget \(naive synthesis that sits below the human\-data ceiling\) into consistent, if small, forward progress\. We view this as the practical takeaway: when a verifier/labeling budget is already being spent on self\-improvement, KITE is a near\-free\-lunch reallocation of that budget rather than an additional data cost\.

### 8\.5Statistical Reliability

We briefly elaborate more on the statistical reliability of our method\. The reported cells are means over three seeds with standard deviations shown in Table[1](https://arxiv.org/html/2607.17043#S3.T1)\. On GSM8K, the benchmark with the most headroom, KITE’s improvement over the strongest baseline is separated by more than the sum of the two per\-cell standard deviations for every backbone \(e\.g\., Qwen\-3\-4B:95\.04±0\.1995\.04\\pm 0\.19vs\.94\.50±0\.2494\.50\\pm 0\.24; Llama\-3\-8B:81\.12±0\.2381\.12\\pm 0\.23vs\.80\.05±0\.2880\.05\\pm 0\.28\), so these gains are unlikely to be seed noise\. On MATH the improvement is consistently positive and clears this bar on some backbones \(e\.g\., Qwen\-3\-4B and Gemma\-3\-4B\) but is within seed noise on others, and on the knowledge\-heavy benchmarks we are more cautious still: on GPQA several methods, including KITE and CDS, fall within one standard deviation of one another, and we do not claim a significant improvement there\. We therefore view GSM8K as the setting where the effect is most robust, MATH as consistently positive, and MMLU\-Pro/GPQA as directional, and we flag this explicitly rather than averaging it away\.

Beyond per\-cell tests, the more compelling evidence is*consistency across independent backbones*\. KITE attains the highest average of any method in all five model families\. Treating each backbone as an independent trial and the strongest competing baseline as the null, a one\-sided sign test givesp=\(1/2\)5≈0\.031p=\(1/2\)^\{5\}\\approx 0\.031, so the probability that KITE tops every family by chance is below5%5\\%\. At the finer granularity of individual \(model, benchmark\) cells, KITE is best or tied\-best in all 20 cells and the*unique*best in 18; the only two exceptions to uniqueness are GPQA cells \(Qwen\-3\-1\.7B and Gemma\-3\-4B\) where CDS ties it, and we already decline to claim significance on GPQA\. This win\-rate pattern, rather than any single large margin, is what we mean by*reliable*improvement: the effect is small per cell but does not depend on a lucky backbone or benchmark\. We deliberately avoid pooling all cells into one average and reporting a single inflated significance number, since the cells are not exchangeable\.

### 8\.6Theoretical Analysis

We provide an analysis showing why noise injection can prevent model collapse at the distributional level\. Our analysis is conducted in a linear\-Gaussian proxy, following the setting and notations fromDohmatobet al\.\([2024a](https://arxiv.org/html/2607.17043#bib.bib42)\), while we formulate the collapse to the synthetic data distribution becoming increasingly concentrated in a low\-variance subspace\.

Letx∈ℝdx\\in\\mathbb\{R\}^\{d\}denote a feature representation of a training example\. Assume the target \(human\) data followx∼𝒩​\(0,Σ⋆\)x\\sim\\mathcal\{N\}\(0,\\Sigma\_\{\\star\}\), and labels are generated by

y\\displaystyle y=w⋆⊤​x\+ε,\\displaystyle=w\_\{\\star\}^\{\\top\}x\+\\varepsilon,ε\\displaystyle\\varepsilon∼𝒩​\(0,σ2\)\.\\displaystyle\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}\)\.Suppose a model is trained by ordinary least squares onTTi\.i\.d\. samples from a training distribution with covarianceΣtrain\\Sigma\_\{\\mathrm\{train\}\}, producing an estimatorw^\\hat\{w\}\. We evaluate generalization under the target distributionΣ⋆\\Sigma\_\{\\star\}\.

Theorem 1 \(Excess risk under covariance mismatch\)\.Assume the training data satisfy

xi\\displaystyle x\_\{i\}∼i\.i\.d\.​𝒩​\(0,Σtrain\),\\displaystyle\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\mathcal\{N\}\(0,\\Sigma\_\{\\mathrm\{train\}\}\),yi\\displaystyle y\_\{i\}=w⋆⊤​xi\+εi,\\displaystyle=w\_\{\\star\}^\{\\top\}x\_\{i\}\+\\varepsilon\_\{i\},εi\\displaystyle\\varepsilon\_\{i\}∼i\.i\.d\.​𝒩​\(0,σ2\),\\displaystyle\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\mathcal\{N\}\(0,\\sigma^\{2\}\),withT\>d\+1T\>d\+1\. Letw^\\hat\{w\}be the OLS estimator trained onTTsamples, and let the test point satisfyx∼𝒩​\(0,Σ⋆\)x\\sim\\mathcal\{N\}\(0,\\Sigma\_\{\\star\}\)\. Then the expected excess risk is

ℛ​\(Σtrain\)\\displaystyle\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{train\}\}\):=𝔼​\[\(x⊤​\(w^−w⋆\)\)2\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\bigl\(x^\{\\top\}\(\\hat\{w\}\-w\_\{\\star\}\)\\bigr\)^\{2\}\\right\]=σ2T−d−1​tr⁡\(Σ⋆​Σtrain−1\)\.\\displaystyle=\\frac\{\\sigma^\{2\}\}\{T\-d\-1\}\\,\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}\\right\)\.
*Proof\.*LetX∈ℝT×dX\\in\\mathbb\{R\}^\{T\\times d\}be the design matrix whoseii\-th row isxi⊤x\_\{i\}^\{\\top\}, and let

y\\displaystyle y=X​w⋆\+ε,\\displaystyle=Xw\_\{\\star\}\+\\varepsilon,ε\\displaystyle\\varepsilon=\(ε1,…,εT\)⊤\.\\displaystyle=\(\\varepsilon\_\{1\},\\ldots,\\varepsilon\_\{T\}\)^\{\\top\}\.The OLS estimator is

w^=\(X⊤​X\)−1​X⊤​y\.\\hat\{w\}=\(X^\{\\top\}X\)^\{\-1\}X^\{\\top\}y\.Substitutingy=X​w⋆\+εy=Xw\_\{\\star\}\+\\varepsilon, we obtain

w^\\displaystyle\\hat\{w\}=\(X⊤​X\)−1​X⊤​\(X​w⋆\+ε\)\\displaystyle=\(X^\{\\top\}X\)^\{\-1\}X^\{\\top\}\(Xw\_\{\\star\}\+\\varepsilon\)=w⋆\+\(X⊤​X\)−1​X⊤​ε,\\displaystyle=w\_\{\\star\}\+\(X^\{\\top\}X\)^\{\-1\}X^\{\\top\}\\varepsilon,and therefore

w^−w⋆=\(X⊤​X\)−1​X⊤​ε\.\\hat\{w\}\-w\_\{\\star\}=\(X^\{\\top\}X\)^\{\-1\}X^\{\\top\}\\varepsilon\.
We first compute the covariance of the estimation error conditional onXX\. Since𝔼​\[ε∣X\]=0\\mathbb\{E\}\[\\varepsilon\\mid X\]=0andCov⁡\(ε∣X\)=σ2​IT\\operatorname\{Cov\}\(\\varepsilon\\mid X\)=\\sigma^\{2\}I\_\{T\}, it follows that

Cov⁡\(w^−w⋆∣X\)\\displaystyle\\operatorname\{Cov\}\(\\hat\{w\}\-w\_\{\\star\}\\mid X\)=\(X⊤​X\)−1​X⊤​Cov⁡\(ε∣X\)​X​\(X⊤​X\)−1\\displaystyle\\quad=\(X^\{\\top\}X\)^\{\-1\}X^\{\\top\}\\operatorname\{Cov\}\(\\varepsilon\\mid X\)X\(X^\{\\top\}X\)^\{\-1\}=σ2​\(X⊤​X\)−1\.\\displaystyle\\quad=\\sigma^\{2\}\(X^\{\\top\}X\)^\{\-1\}\.Hence, conditional onXX, the error vectorw^−w⋆\\hat\{w\}\-w\_\{\\star\}has mean zero and covarianceσ2​\(X⊤​X\)−1\\sigma^\{2\}\(X^\{\\top\}X\)^\{\-1\}\.

Now consider a fresh test pointx∼𝒩​\(0,Σ⋆\)x\\sim\\mathcal\{N\}\(0,\\Sigma\_\{\\star\}\), independent of the training data\. Conditional onXX, we have

𝔼\[\(x⊤\(w^−w⋆\)\)2\|X\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\bigl\(x^\{\\top\}\(\\hat\{w\}\-w\_\{\\star\}\)\\bigr\)^\{2\}\\,\\middle\|\\,X\\right\]=𝔼\[\(w^−w⋆\)⊤xx⊤\(w^−w⋆\)\|X\]\.\\displaystyle\\quad=\\mathbb\{E\}\\\!\\left\[\(\\hat\{w\}\-w\_\{\\star\}\)^\{\\top\}xx^\{\\top\}\(\\hat\{w\}\-w\_\{\\star\}\)\\,\\middle\|\\,X\\right\]\.Taking expectation over the test point first and using𝔼​\[x​x⊤\]=Σ⋆\\mathbb\{E\}\[xx^\{\\top\}\]=\\Sigma\_\{\\star\}, we get

𝔼\[\(x⊤\(w^−w⋆\)\)2\|X\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\bigl\(x^\{\\top\}\(\\hat\{w\}\-w\_\{\\star\}\)\\bigr\)^\{2\}\\,\\middle\|\\,X\\right\]=𝔼\[\(w^−w⋆\)⊤Σ⋆\(w^−w⋆\)\|X\]\.\\displaystyle\\quad=\\mathbb\{E\}\\\!\\left\[\(\\hat\{w\}\-w\_\{\\star\}\)^\{\\top\}\\Sigma\_\{\\star\}\(\\hat\{w\}\-w\_\{\\star\}\)\\,\\middle\|\\,X\\right\]\.Using the identity𝔼​\[z⊤​A​z\]=tr⁡\(A​Cov⁡\(z\)\)\\mathbb\{E\}\[z^\{\\top\}Az\]=\\operatorname\{tr\}\(A\\,\\operatorname\{Cov\}\(z\)\)for a zero\-mean random vectorzz, this becomes

𝔼\[\(x⊤\(w^−w⋆\)\)2\|X\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\bigl\(x^\{\\top\}\(\\hat\{w\}\-w\_\{\\star\}\)\\bigr\)^\{2\}\\,\\middle\|\\,X\\right\]=tr⁡\(Σ⋆​Cov⁡\(w^−w⋆∣X\)\)\\displaystyle\\quad=\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\\operatorname\{Cov\}\(\\hat\{w\}\-w\_\{\\star\}\\mid X\)\\right\)=σ2​tr⁡\(Σ⋆​\(X⊤​X\)−1\)\.\\displaystyle\\quad=\\sigma^\{2\}\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\(X^\{\\top\}X\)^\{\-1\}\\right\)\.
It remains to take expectation over the random design matrixXX\. Since the rows ofXXare i\.i\.d\. Gaussian with covarianceΣtrain\\Sigma\_\{\\mathrm\{train\}\}, the sample covariance matrixX⊤​XX^\{\\top\}Xfollows a Wishart distribution with scale matrixΣtrain\\Sigma\_\{\\mathrm\{train\}\}andTTdegrees of freedom:

X⊤​X∼𝒲d​\(Σtrain,T\)\.X^\{\\top\}X\\sim\\mathcal\{W\}\_\{d\}\(\\Sigma\_\{\\mathrm\{train\}\},T\)\.A standard property of the inverse Wishart expectation states that, whenT\>d\+1T\>d\+1,

𝔼​\[\(X⊤​X\)−1\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\(X^\{\\top\}X\)^\{\-1\}\\right\]=1T−d−1​Σtrain−1\.\\displaystyle=\\frac\{1\}\{T\-d\-1\}\\,\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}\.Therefore,

ℛ​\(Σtrain\)\\displaystyle\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{train\}\}\)=𝔼\[𝔼\[\(x⊤\(w^−w⋆\)\)2\|X\]\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\mathbb\{E\}\\\!\\left\[\\bigl\(x^\{\\top\}\(\\hat\{w\}\-w\_\{\\star\}\)\\bigr\)^\{2\}\\,\\middle\|\\,X\\right\]\\right\]=σ2​𝔼​\[tr⁡\(Σ⋆​\(X⊤​X\)−1\)\]\.\\displaystyle=\\sigma^\{2\}\\,\\mathbb\{E\}\\\!\\left\[\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\(X^\{\\top\}X\)^\{\-1\}\\right\)\\right\]\.By linearity of trace and expectation,

ℛ​\(Σtrain\)\\displaystyle\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{train\}\}\)=σ2​tr⁡\(Σ⋆​𝔼​\[\(X⊤​X\)−1\]\)\\displaystyle=\\sigma^\{2\}\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\\mathbb\{E\}\\\!\\left\[\(X^\{\\top\}X\)^\{\-1\}\\right\]\\right\)=σ2T−d−1​tr⁡\(Σ⋆​Σtrain−1\)\.\\displaystyle=\\frac\{\\sigma^\{2\}\}\{T\-d\-1\}\\,\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}\\right\)\.Thus,

ℛ​\(Σtrain\)=σ2T−d−1​tr⁡\(Σ⋆​Σtrain−1\),\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{train\}\}\)=\\frac\{\\sigma^\{2\}\}\{T\-d\-1\}\\,\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}\\right\),as claimed\.

Theorem 1 shows that generalization is controlled by the mismatch termtr⁡\(Σ⋆​Σtrain−1\)\\operatorname\{tr\}\(\\Sigma\_\{\\star\}\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}\)\. If the training covariance loses variance along some direction that still has nontrivial mass underΣ⋆\\Sigma\_\{\\star\}, then the corresponding eigenvalue ofΣtrain−1\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}becomes large, and the excess risk can grow sharply\. In this sense, model collapse can be viewed as*covariance collapse*: synthetic data become too concentrated, and the resulting model generalizes poorly to the broader target distribution\.

Now consider synthetic data with covarianceΣsyn\\Sigma\_\{\\mathrm\{syn\}\}\. We inject additive noise in feature space,

xtrain\\displaystyle x\_\{\\mathrm\{train\}\}=xsyn\+η,\\displaystyle=x\_\{\\mathrm\{syn\}\}\+\\eta,η\\displaystyle\\eta∼𝒩​\(0,Σnoise\),\\displaystyle\\sim\\mathcal\{N\}\(0,\\Sigma\_\{\\mathrm\{noise\}\}\),independently ofxsynx\_\{\\mathrm\{syn\}\}\. Since the two components are independent and centered, the covariance of the resulting training distribution is

Σtrain\\displaystyle\\Sigma\_\{\\mathrm\{train\}\}=Cov⁡\(xtrain\)\\displaystyle=\\operatorname\{Cov\}\(x\_\{\\mathrm\{train\}\}\)=Cov⁡\(xsyn\)\+Cov⁡\(η\)\\displaystyle=\\operatorname\{Cov\}\(x\_\{\\mathrm\{syn\}\}\)\+\\operatorname\{Cov\}\(\\eta\)=Σsyn\+Σnoise\.\\displaystyle=\\Sigma\_\{\\mathrm\{syn\}\}\+\\Sigma\_\{\\mathrm\{noise\}\}\.
Corollary 1 \(Unbounded collapse without noise injection\)\.Suppose recursive synthetic training induces a sequence of synthetic covariances\{Σsyn\(n\)\}n≥1\\\{\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\\\}\_\{n\\geq 1\}, and training at generationnnusesΣtrain\(n\)=Σsyn\(n\)\\Sigma\_\{\\mathrm\{train\}\}^\{\(n\)\}=\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}with no injected noise\. Assume thatλmin​\(Σsyn\(n\)\)→0\\lambda\_\{\\min\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)\\to 0, and letvnv\_\{n\}be a unit eigenvector associated withλmin​\(Σsyn\(n\)\)\\lambda\_\{\\min\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)\. If there existsc⋆\>0c\_\{\\star\}\>0such that

vn⊤​Σ⋆​vn≥c⋆,∀n,v\_\{n\}^\{\\top\}\\Sigma\_\{\\star\}v\_\{n\}\\geq c\_\{\\star\},\\qquad\\forall n,then

ℛ​\(Σsyn\(n\)\)⟶∞as​n→∞\.\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)\\longrightarrow\\infty\\qquad\\text\{as \}n\\to\\infty\.
*Proof\.*By Theorem 1,

ℛ​\(Σsyn\(n\)\)\\displaystyle\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)=σ2T−d−1​tr⁡\(Σ⋆​\(Σsyn\(n\)\)−1\)\.\\displaystyle=\\frac\{\\sigma^\{2\}\}\{T\-d\-1\}\\,\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)^\{\-1\}\\right\)\.Let the eigendecomposition ofΣsyn\(n\)\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}be

Σsyn\(n\)\\displaystyle\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}=Un​Λn​Un⊤,\\displaystyle=U\_\{n\}\\Lambda\_\{n\}U\_\{n\}^\{\\top\},Λn\\displaystyle\\Lambda\_\{n\}=diag⁡\(λ1\(n\),…,λd\(n\)\),\\displaystyle=\\operatorname\{diag\}\\\!\\left\(\\lambda\_\{1\}^\{\(n\)\},\\ldots,\\lambda\_\{d\}^\{\(n\)\}\\right\),withλmin​\(Σsyn\(n\)\)=λd\(n\)\\lambda\_\{\\min\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)=\\lambda\_\{d\}^\{\(n\)\}and corresponding unit eigenvectorvn=ud\(n\)v\_\{n\}=u\_\{d\}^\{\(n\)\}\. Then

\(Σsyn\(n\)\)−1=Un​Λn−1​Un⊤,\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)^\{\-1\}=U\_\{n\}\\Lambda\_\{n\}^\{\-1\}U\_\{n\}^\{\\top\},and therefore

tr⁡\(Σ⋆​\(Σsyn\(n\)\)−1\)\\displaystyle\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)^\{\-1\}\\right\)=∑i=1d\(ui\(n\)\)⊤​Σ⋆​ui\(n\)λi\(n\)\.\\displaystyle\\quad=\\sum\_\{i=1\}^\{d\}\\frac\{\(u\_\{i\}^\{\(n\)\}\)^\{\\top\}\\Sigma\_\{\\star\}u\_\{i\}^\{\(n\)\}\}\{\\lambda\_\{i\}^\{\(n\)\}\}\.Since every term is nonnegative,

tr⁡\(Σ⋆​\(Σsyn\(n\)\)−1\)\\displaystyle\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)^\{\-1\}\\right\)≥vn⊤​Σ⋆​vnλmin​\(Σsyn\(n\)\)\\displaystyle\\quad\\geq\\frac\{v\_\{n\}^\{\\top\}\\Sigma\_\{\\star\}v\_\{n\}\}\{\\lambda\_\{\\min\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)\}≥c⋆λmin​\(Σsyn\(n\)\)\.\\displaystyle\\quad\\geq\\frac\{c\_\{\\star\}\}\{\\lambda\_\{\\min\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)\}\.Becauseλmin​\(Σsyn\(n\)\)→0\\lambda\_\{\\min\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)\\to 0, the right\-hand side diverges to\+∞\+\\infty, and hence

ℛ​\(Σsyn\(n\)\)\\displaystyle\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)≥σ2T−d−1​c⋆λmin​\(Σsyn\(n\)\)\\displaystyle\\geq\\frac\{\\sigma^\{2\}\}\{T\-d\-1\}\\,\\frac\{c\_\{\\star\}\}\{\\lambda\_\{\\min\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)\}⟶∞\.\\displaystyle\\longrightarrow\\infty\.This proves the claim\. A particularly simple case is when the collapsed direction shrinks geometrically, i\.e\.,λmin​\(Σsyn\(n\)\)=ρn​λ0\\lambda\_\{\\min\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)=\\rho^\{n\}\\lambda\_\{0\}for someρ∈\(0,1\)\\rho\\in\(0,1\)\. Then Corollary 1 yields

ℛ​\(Σsyn\(n\)\)\\displaystyle\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{syn\}\}^\{\(n\)\}\)≥σ2T−d−1​c⋆λ0​ρ−n,\\displaystyle\\geq\\frac\{\\sigma^\{2\}\}\{T\-d\-1\}\\,\\frac\{c\_\{\\star\}\}\{\\lambda\_\{0\}\}\\,\\rho^\{\-n\},so the excess risk grows at least exponentially with the number of generations\.

Proposition 1 \(Noise injection prevents covariance collapse\)\.If the injected noise is chosen such that

Σtrain⪰λ¯​I,λ¯\>0,\\Sigma\_\{\\mathrm\{train\}\}\\succeq\\underline\{\\lambda\}I,\\qquad\\underline\{\\lambda\}\>0,then

ℛ​\(Σtrain\)\\displaystyle\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{train\}\}\)≤σ2T−d−1​tr⁡\(Σ⋆\)λ¯\.\\displaystyle\\leq\\frac\{\\sigma^\{2\}\}\{T\-d\-1\}\\,\\frac\{\\operatorname\{tr\}\(\\Sigma\_\{\\star\}\)\}\{\\underline\{\\lambda\}\}\.
*Proof\.*From Theorem 1,

ℛ​\(Σtrain\)\\displaystyle\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{train\}\}\)=σ2T−d−1​tr⁡\(Σ⋆​Σtrain−1\)\.\\displaystyle=\\frac\{\\sigma^\{2\}\}\{T\-d\-1\}\\,\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}\\right\)\.It therefore suffices to upper\-bound the mismatch termtr⁡\(Σ⋆​Σtrain−1\)\\operatorname\{tr\}\(\\Sigma\_\{\\star\}\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}\)\.

The assumptionΣtrain⪰λ¯​I\\Sigma\_\{\\mathrm\{train\}\}\\succeq\\underline\{\\lambda\}Imeans that every eigenvalue ofΣtrain\\Sigma\_\{\\mathrm\{train\}\}is at leastλ¯\\underline\{\\lambda\}\. Equivalently, if

Σtrain\\displaystyle\\Sigma\_\{\\mathrm\{train\}\}=U​Λ​U⊤,\\displaystyle=U\\Lambda U^\{\\top\},Λ\\displaystyle\\Lambda=diag⁡\(λ1,…,λd\),\\displaystyle=\\operatorname\{diag\}\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\}\),thenλi≥λ¯\\lambda\_\{i\}\\geq\\underline\{\\lambda\}for allii\. Hence

Σtrain−1\\displaystyle\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}=U​Λ−1​U⊤,\\displaystyle=U\\Lambda^\{\-1\}U^\{\\top\},Λ−1\\displaystyle\\Lambda^\{\-1\}=diag⁡\(λ1−1,…,λd−1\),\\displaystyle=\\operatorname\{diag\}\\\!\\left\(\\lambda\_\{1\}^\{\-1\},\\ldots,\\lambda\_\{d\}^\{\-1\}\\right\),and sinceλi≥λ¯\\lambda\_\{i\}\\geq\\underline\{\\lambda\}, we have

λi−1≤λ¯−1,∀i\.\\lambda\_\{i\}^\{\-1\}\\leq\\underline\{\\lambda\}^\{\-1\},\\qquad\\forall i\.Therefore,

Σtrain−1⪯λ¯−1​I\.\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}\\preceq\\underline\{\\lambda\}^\{\-1\}I\.
Now use the fact thatΣ⋆⪰0\\Sigma\_\{\\star\}\\succeq 0\. For positive semidefinite matrices, the Loewner order is preserved under multiplication inside the trace: ifA⪯BA\\preceq BandC⪰0C\\succeq 0, then

tr⁡\(C​A\)≤tr⁡\(C​B\)\.\\operatorname\{tr\}\(CA\)\\leq\\operatorname\{tr\}\(CB\)\.Applying this withA=Σtrain−1A=\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\},B=λ¯−1​IB=\\underline\{\\lambda\}^\{\-1\}I, andC=Σ⋆C=\\Sigma\_\{\\star\}, we obtain

tr⁡\(Σ⋆​Σtrain−1\)\\displaystyle\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\\Sigma\_\{\\mathrm\{train\}\}^\{\-1\}\\right\)≤tr⁡\(Σ⋆​λ¯−1​I\)\\displaystyle\\leq\\operatorname\{tr\}\\\!\\left\(\\Sigma\_\{\\star\}\\underline\{\\lambda\}^\{\-1\}I\\right\)=λ¯−1​tr⁡\(Σ⋆\)\.\\displaystyle=\\underline\{\\lambda\}^\{\-1\}\\operatorname\{tr\}\(\\Sigma\_\{\\star\}\)\.Substituting this into Theorem 1 gives

ℛ​\(Σtrain\)\\displaystyle\\mathcal\{R\}\(\\Sigma\_\{\\mathrm\{train\}\}\)≤σ2T−d−1​tr⁡\(Σ⋆\)λ¯,\\displaystyle\\leq\\frac\{\\sigma^\{2\}\}\{T\-d\-1\}\\,\\frac\{\\operatorname\{tr\}\(\\Sigma\_\{\\star\}\)\}\{\\underline\{\\lambda\}\},which proves the claim\.

Proposition 1 formalizes the key intuition: noise injection prevents collapse by restoring variance in directions that synthetic data under\-represent\. As long as the training covariance is kept uniformly away from singularity, the excess risk remains bounded\.

Finally, we emphasize that our practical method does not inject Gaussian noise directly in feature space\. Instead, rank\-based logit perturbation serves as a discrete approximation in LLM modeling to the same principle\. It reduces over\-concentration on dominant modes and expands support toward under\-represented regions of the data distribution\. The analysis above, therefore, should be understood as a theoretical justification for the role of noise injection in preventing collapse, rather than as an exact model of our implementation\.

### 8\.7Prompt Templates

LLM Skill Tagging PromptYou are an expert educational assessment assistant tasked with annotating the latent skills required to solve a question\.Your goal is to identify which skills from a predefined skill inventory are necessary for correctly answering the given question\. The output will be used to construct a Q\-matrix for cognitive diagnosis\.Instructions:1\. Carefully read the question\. 2\. Select all skills from the provided skill inventory that are necessary to solve the question\. 3\. Only use skills from the provided inventory; do not invent new skill names\. 4\. Choose the minimal sufficient set of skills\. Do not include skills that are only loosely related\. 5\. Return the output strictly as a Python\-style list of strings\.Skill Inventory:\{skill\_inventory\}Question:\{question\}Output Format Example:\[ "Arithmetic", "Ratio and Proportion", "Time and Scheduling" \]Now identify the required skills for the question\.Figure 6:LLM skill tagging prompt used to construct the Q\-matrix for the DINA\-based diagnosis pipeline\. Given a predefined skill inventory and a question, the model selects the minimal set of required skills\.Few\-shot Synthesis PromptYou are an advanced AI assistant tasked with generating synthetic instructions for instruction tuning\.Your goal is to produce new examples that follow the style and difficulty of the provided demonstrations\.Instructions:1\. Carefully study the few\-shot examples below\. 2\. Infer the task style, answer format, and expected level of reasoning\. 3\. Generate exactly \{n\} new instructions that are similar in style but not duplicates of the demonstrations\. 4\. Each generated instruction should be self\-contained and clear\. 5\. Return the output strictly as a Python\-style list of dictionaries, where each dictionary has one key:"instruction"\.Few\-shot Examples:\{few\_shot\_examples\}Output Format Example:\[ \{"instruction": "Question 1"\}, \{"instruction": "Question 2"\} \]Now generate \{n\} new instructions\.Figure 7:Few\-shot synthesis prompt used in theFew\-shot synthesisbaseline and in Figure[1](https://arxiv.org/html/2607.17043#S1.F1)\. The model is prompted with a small set of seed examples and asked to generate new synthetic data in the same style\.Answer Judge PromptYou are an expert evaluator for question answering tasks\.Your goal is to determine whether the model answer is semantically correct with respect to the reference answer\. Focus on factual correctness and final meaning rather than surface wording\.Instructions:1\. Read the question, the reference answer, and the model answer carefully\. 2\. Judge whether the model answer is correct according to the reference answer\. 3\. Minor paraphrases, equivalent expressions, and formatting differences should be treated as correct\. 4\. If the model answer is incomplete, contradictory, or contains the wrong final conclusion, it should be treated as incorrect\. 5\. Return your judgment in the following JSON format only: \{"verdict": "correct" or "incorrect", "reason": "short explanation"\}Question:\{question\}Reference Answer:\{reference\_answer\}Model Answer:\{model\_answer\}Now output the judgment in JSON only\.Figure 8:Answer comparison prompt used in the LLM\-as\-judge evaluation pipeline\. The judge compares the model answer against the reference answer and returns a correctness verdict with a short explanation\.Labeling PromptYou are an expert data labeler for instruction\-tuning examples\.Your goal is to produce a verified reference answer for the given instruction\. Use the candidate answer as a helpful draft, but do not trust it blindly\. The final output should be a high\-quality, correct answer suitable for supervised fine\-tuning\.Instructions:1\. Read the instruction carefully\. 2\. Check whether the candidate answer is correct, complete, and consistent with the instruction\. 3\. If the candidate answer is correct, rewrite it into a clean canonical reference answer\. 4\. If the candidate answer is partially correct or incorrect, provide the correct answer instead\. 5\. Keep the final answer concise, faithful, and directly responsive to the instruction\. 6\. Return the output strictly as a JSON object with exactly the following keys:"is\_correct","verified\_answer", and"reason"\. 7\."is\_correct"must be eithertrueorfalse\. 8\."verified\_answer"must contain the final answer that should be used for training\. 9\."reason"must be a short explanation of your decision\.Instruction:\{instruction\}Candidate Answer:\{candidate\_answer\}Output Format Example:\{"is\_correct": true, "verified\_answer": "42", "reason": "The candidate answer is correct and has been normalized\."\}Now return the verified label in JSON only\.Figure 9:Labeling prompt used for the external verifier in the training\-time labeling stage\. Given a selected synthetic instruction and a candidate answer, the labeler returns a correctness decision together with a verified reference answer used to construct the labeled datasetDt⋆D\_\{t\}^\{\\star\}\.![Refer to caption](https://arxiv.org/html/2607.17043v1/x6.png)Figure 10:Skill mastery profile on MMLU\-Pro\-Physics estimated by DINA for the initial and fine\-tuned models\. Eachsis\_\{i\}denotes a dataset\-specific skill defined in Table[4](https://arxiv.org/html/2607.17043#S8.T4)\.![Refer to caption](https://arxiv.org/html/2607.17043v1/x7.png)Figure 11:Skill mastery profile on MATH estimated by DINA for the initial and fine\-tuned models\. Eachsis\_\{i\}denotes a dataset\-specific skill defined in Table[4](https://arxiv.org/html/2607.17043#S8.T4)\.![Refer to caption](https://arxiv.org/html/2607.17043v1/x8.png)Figure 12:Skill mastery profile on GPQA estimated by DINA for the initial and fine\-tuned models\. Eachsis\_\{i\}denotes a dataset\-specific skill defined in Table[4](https://arxiv.org/html/2607.17043#S8.T4)\.DatasetExample InputReference AnswerGSM8KJohn drives for 3 hours at a speed of 60 mph and then turns around because he realizes he forgot something very important at home\. He tries to get home in 4 hours but spends the first 2 hours in standstill traffic\. He spends the next half\-hour driving at a speed of 30mph, before being able to drive the remaining time of the 4 hours going at 80 mph\. How far is he from home at the end of those 4 hours?When he turned around he was 3\*60=180 miles from home He was only able to drive 4\-2=2 hours in the first four … Final Answer: 45MMLU\-ProDetermine the root\-mean\-square \(rms\) values of displacement, velocity, and acceleration for a damped forced harmonic oscillator operating at steady state\.xrms=xmax2,vrms=ω​xmax2​…x\_\{\\mathrm\{rms\}\}=\\frac\{x\_\{\\max\}\}\{\\sqrt\{2\}\},v\_\{\\mathrm\{rms\}\}=\\frac\{\\omega x\_\{\\max\}\}\{\\sqrt\{2\}\}\.\.\.MATHIf the seven\-digit number854​n​526854n526is divisible by1111, what isnn?n=5n=\\boxed\{5\}\.GPQAFind KE of product particles in,P​i​\(\+\)=m​u​\(\+\)\+n​uPi\(\+\)=mu\(\+\)\+nuhereP​i​\(\+\)Pi\(\+\)is stationary\. Rest mass ofP​i​\(\+\)Pi\(\+\)andm​u​\(\+\)mu\(\+\)is 139\.6 MeV and 105\.7 MeV respectively\.4\.12 MeV, 29\.8 MeVTable 3:Example instances from the evaluation datasets\.MATHMMLU\-Pro\-PhysicsSymbolSkillSymbolSkills1s\_\{1\}Integration & Summations1s\_\{1\}Quantum States & Wavefunctionss2s\_\{2\}Differentiation & Optimizations2s\_\{2\}Nuclear Reactions & Radioactivitys3s\_\{3\}Matrices, Vectors & Transformss3s\_\{3\}Special & General Relativitys4s\_\{4\}Complex Numbers & Polar Forms4s\_\{4\}Atomic Structure & Spectras5s\_\{5\}Logic, Proof & Problem\-Solvings5s\_\{5\}Maxwell’s Eqs & EM Wavess6s\_\{6\}Circle Theorems & Arc Lengths6s\_\{6\}Magnetic Fields & Inductions7s\_\{7\}Trigonometric Funcs & Identitiess7s\_\{7\}Geometric & Physical Opticss8s\_\{8\}Combinatorics & Countings8s\_\{8\}Rotational Dynamics & Torques9s\_\{9\}Triangle Properties & Congruences9s\_\{9\}Electric Fields & Coulomb’s Laws10s\_\{10\}Coordinate Geom\. & Conicss10s\_\{10\}Wave Properties & Interferences11s\_\{11\}Functions, Limits & Continuitys11s\_\{11\}Gravitation & Orbital Mech\.s12s\_\{12\}Modular Arith\. & Number Theorys12s\_\{12\}Fluid Statics & Dynamicss13s\_\{13\}Area, Volume & Measurements13s\_\{13\}Thermodynamic Laws & Cycless14s\_\{14\}Probability & Expected Values14s\_\{14\}Heat Transfer & Entropys15s\_\{15\}Prime Factorization & Divisibilitys15s\_\{15\}Circuits & Ohm’s Laws16s\_\{16\}Quadratic & Higher\-Deg\. Eqss16s\_\{16\}Conservation of Momentums17s\_\{17\}Sequences, Series & Summationss17s\_\{17\}Conservation of Energys18s\_\{18\}Polynomial Factoring & Rootss18s\_\{18\}Newton’s Laws & Forcess19s\_\{19\}Exponents, Logs & Radicalss19s\_\{19\}Kinematics & Projectile Motions20s\_\{20\}Linear Equations & Inequalitiess20s\_\{20\}Dimensional Analysis & Units
GPQASymbolSkills1s\_\{1\}Immunology & Microbiologys2s\_\{2\}Computational & Quantum Chem\.s3s\_\{3\}Nuclear & Particle Physicss4s\_\{4\}Astrophysics & Cosmologys5s\_\{5\}Condensed Matter & Materialss6s\_\{6\}Bioinformatics & Exp\. Designs7s\_\{7\}Biochemistry & Macromoleculess8s\_\{8\}Coordination & Inorganic Chem\.s9s\_\{9\}Quantum Mechanics & Spins10s\_\{10\}Molecular & Cell Biologys11s\_\{11\}Spectroscopy & Analytical Methodss12s\_\{12\}Organic Reaction Mechanismss13s\_\{13\}Genetics & Gene Expressions14s\_\{14\}Molecular Structure & Bondings15s\_\{15\}Chemical Kinetics & Equilibriums16s\_\{16\}Electromagnetism & Opticss17s\_\{17\}Classical Mechanics & Relativitys18s\_\{18\}Thermodynamics & Stat\. Mech\.

Table 4:Skill index mapping for the skill\-profile figures\.Each symbolsis\_\{i\}corresponds to a dataset\-specific fine\-grained skill\.