How Much Thinking is Enough? Quantifying and Understanding Redundancy in LLM Reasoning

# How Much Thinking is Enough? Quantifying and Understanding Redundancy in LLM ReasoningCode: https://github.com/zhiyuanZhai20/how-much-thinking-is-enough
Source: [https://arxiv.org/html/2605.23926](https://arxiv.org/html/2605.23926)
Zhiyuan Zhai Fudan University 22110720067@m\.fudan\.edu\.cn &Xinkai You Fudan University xkyou25@m\.fudan\.edu\.cn Wenjing Yan The Chinese University of Hong Kong wenjingyan@cuhk\.edu\.hk &Xin Wang Fudan University xwang11@fudan\.edu\.cn

###### Abstract

Reasoning\-capable large language models solve hard problems by emitting long chains of thought, paying heavily in latency, GPU time, and energy\. Casual inspection of their traces reveals extensive reformulation, verification, and circular self\-reflection, yet how much of this deliberation is actually necessary has never been measured at scale or explained from first principles\. This paper closes both gaps\.

We formalise*reasoning redundancy*directly in terms of the reasoning model itself: the redundancy of a correct trace is the largest fraction of its trailing segmented steps that can be truncated whileπ\\pi, forced to terminate thinking and emit a final answer, still produces the correct answer\. A large\-scale quantification across four frontier reasoning models and two mathematical benchmarks shows that step\-level redundancy is consistently high — between61%61\\%and93%93\\%across the88\(model, benchmark\) conditions we study, with the median critical prefix equal to a single segmented step in six of the eight conditions — that the finding is robust to the choice of judge family, and that althoughρ\\rhodecreases with problem difficulty on MATH\-500, all four models remain substantially redundant \(ρ∈\[46%,85%\]\\rho\\in\[46\\%,85\\%\]\) even on the hardest Level\-55problems\.

We then prove that this redundancy is a structural consequence of length\-agnostic outcome rewards, not a model\-specific artefact: under any such reward, no finite expected stopping time is optimal\. The result holds regardless of RL algorithm, base model, data distribution, or whether the policy is obtained via RL or distillation; over\-thinking is therefore not a bug to be patched in individual models but a structural property of how current reasoning models are trained\.

## 1Introduction

Reasoning\-capable large language models have reshaped automated problem solving\. OpenAI’s o\-series\(OpenAI,[2024](https://arxiv.org/html/2605.23926#bib.bib25); Jaechet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib16)\), DeepSeek\-R1\(Guoet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib11)\), Kimi k1\.5\(Kimi Teamet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib18)\), Qwen’s QwQ\(Qwen Team,[2025](https://arxiv.org/html/2605.23926#bib.bib27)\), and open reproductions\(Zenget al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib49); Liuet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib22); Yuet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib46); Rastogiet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib29)\)all emit long, self\-reflective chains of thought that drive striking gains on MATH\-500\(Hendryckset al\.,[2021](https://arxiv.org/html/2605.23926#bib.bib14)\), GSM8K\(Cobbeet al\.,[2021](https://arxiv.org/html/2605.23926#bib.bib7)\), AIME, and LiveCodeBench\(Jainet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib17)\)\. The gains come at a steep per\-query price: every reasoning token is paid in GPU time, latency, and energy, and a user interacting with a reasoning model today routinely waits tens of seconds before the first answer token appears\. Anyone who has read the raw output of such a model also notices that they*think far more than seems necessary*: a three\-step problem elicits a thirty\-step trace full of verification, reformulation, and circular self\-reflection\.

Despite the widespread awareness of this “over\-thinking” phenomenon, three fundamental questions remain unanswered\.*How should one even define what counts as redundant reasoning?*Prior work has either relied on the reasoning model’s own behaviour as a proxy\(Chenet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib5)\)or on isolated qualitative examples\(Spragueet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib36)\), without a protocol that can be applied at scale\.*How much redundancy is actually present in frontier reasoning traces?*Beyond anecdotes there is no multi\-model, multi\-benchmark, multi\-judge quantification\.*Why is this happening at all?*Is over\-thinking a bug in a specific RL recipe, a data\-induced habit, or a structural property of reasoning\-model training? The absence of first\-principles answers means that training\-time length penalties\(Arora and Zanette,[2025](https://arxiv.org/html/2605.23926#bib.bib1); Aggarwal and Welleck,[2025](https://arxiv.org/html/2605.23926#bib.bib2); Hanet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib12); Muennighoffet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib24)\)are applied without a theory that predicts when they should succeed\.

This paper provides answers to all three questions\.Our contributions:

C1 \(Definition\)\.We formalise*reasoning redundancy*directly in terms of the model itself: the redundancy of a correct trace is the largest fraction of its trailing segmented steps that can be truncated whileπ\\pi, forced to terminate thinking and emit a final answer, still produces the correct answer \(Definition[1](https://arxiv.org/html/2605.23926#Thmdefinition1)\)\. The definition is measured by progressive truncation and is validated under an external non\-reasoning judge as a robustness check\.

C2 \(Empirical quantification at scale\)\.On four frontier reasoning models across two mathematical benchmarks \(GSM8K, MATH\-500\) with two independent judges, step\-level redundancyρ\\rhoexceeds60%60\\%in all88\(model, benchmark\) conditions, and exceeds90%90\\%on both benchmarks for QwQ\-32B and Qwen3\-30B\-Thinking\. The median critical prefix is a*single*segmented step in six of the eight conditions\. On MATH\-500 the redundancy ratio decreases with problem difficulty, but harder problems elicit proportionally longer traces so that all four models remain substantially redundant \(ρ∈\[46%,85%\]\\rho\\in\[46\\%,85\\%\]\) even on the hardest Level\-55problems\.

C3 \(Theoretical explanation\)\.We model reasoning as a sequential decision process and prove that over\-thinking is a structural consequence of length\-agnostic outcome rewards, not a model\-specific artefact \(Theorem[1](https://arxiv.org/html/2605.23926#Thmtheorem1)\): under any outcome\-only reward, no finite expected stopping time is optimal\. The result is independent of RL algorithm, base model, data distribution, or whether the policy is trained by RL, distillation, or supervised fine\-tuning\. An immediate consequence is that any fix must break length\-agnosticism of the reward, giving a simple training\-time recipe: add an explicit length penalty−λ​T\-\\lambda T\(or an equivalent difficulty\-aware token budget\) to the reward\.

## 2Related Work

#### Reasoning\-capable LLMs\.

Chain\-of\-thought prompting\(Weiet al\.,[2022](https://arxiv.org/html/2605.23926#bib.bib42); Kojimaet al\.,[2022](https://arxiv.org/html/2605.23926#bib.bib19)\)first showed that eliciting intermediate steps improves reasoning\. Extensions include self\-consistency\(Wanget al\.,[2023](https://arxiv.org/html/2605.23926#bib.bib41)\), tree of thoughts\(Yaoet al\.,[2023](https://arxiv.org/html/2605.23926#bib.bib45)\), self\-refinement\(Madaanet al\.,[2023](https://arxiv.org/html/2605.23926#bib.bib23)\), reflexion\-style feedback\(Shinnet al\.,[2023](https://arxiv.org/html/2605.23926#bib.bib34)\), latent\-space reasoning\(Haoet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib13)\), least\-to\-most\(Zhouet al\.,[2022](https://arxiv.org/html/2605.23926#bib.bib51)\), complexity\-based prompting\(Fuet al\.,[2022](https://arxiv.org/html/2605.23926#bib.bib52)\), and process\-reward modelling\(Uesatoet al\.,[2022](https://arxiv.org/html/2605.23926#bib.bib40); Lightmanet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib20)\)\. Modern reasoning models\(OpenAI,[2024](https://arxiv.org/html/2605.23926#bib.bib25); Guoet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib11); Kimi Teamet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib18); Qwen Team,[2025](https://arxiv.org/html/2605.23926#bib.bib27)\)shifted reasoning from a prompting problem to a training problem via outcome\-verified RL\(Shaoet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib33); Yuet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib46); Zenget al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib49); Liuet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib22)\), distillation from long\-CoT teachers\(Guoet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib11)\), or inference\-time compute scaling\(Snellet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib35); Brownet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib3); Muennighoffet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib24)\)\. STaR\(Zelikmanet al\.,[2022](https://arxiv.org/html/2605.23926#bib.bib47)\)and Quiet\-STaR\(Zelikmanet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib48)\)explore implicit rationalised CoTs; ReST\(Gulcehreet al\.,[2023](https://arxiv.org/html/2605.23926#bib.bib10)\)explores self\-training\. Crucially, the reward or supervision signal in every case depends on answer correctness but*not*on trace length\.

#### Over\-thinking, efficient reasoning, and adaptive computation\.

Chenet al\.\([2024](https://arxiv.org/html/2605.23926#bib.bib5)\)named the over\-thinking phenomenon using small arithmetic examples\.Shahet al\.\([2025](https://arxiv.org/html/2605.23926#bib.bib32)\)andLiuet al\.\([2025](https://arxiv.org/html/2605.23926#bib.bib22)\)argue that reflective behaviours inherit from pretraining rather than emerging from RL;Zhaoet al\.\([2025](https://arxiv.org/html/2605.23926#bib.bib50)\)report that RL post\-training amplifies pretraining behaviours;Danget al\.\([2025](https://arxiv.org/html/2605.23926#bib.bib8)\)study complementary failure modes\. Concurrent methods introduce training\-time length penalties\(Arora and Zanette,[2025](https://arxiv.org/html/2605.23926#bib.bib1); Aggarwal and Welleck,[2025](https://arxiv.org/html/2605.23926#bib.bib2); Hanet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib12)\)or test\-time budget forcing on small distilled models\(Muennighoffet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib24)\)\. Our work differs in three ways: \(i\) we provide a rigorous judge\-based*formalisation*of redundancy that makes the phenomenon measurable at scale; \(ii\) we carry out a*large\-scale multi\-model multi\-benchmark quantification*with multi\-judge validation; and \(iii\) we give the first*theoretical account*that explains over\-thinking as a structural consequence of length\-agnostic rewards\. Classical antecedents we build on include adaptive computation time\(Graves,[2016](https://arxiv.org/html/2605.23926#bib.bib9)\)and confident\-adaptive language modelling\(Schusteret al\.,[2022](https://arxiv.org/html/2605.23926#bib.bib31)\)\.

## 3Defining Reasoning Redundancy

A reasoning LLMπ\\pitakes a problemxxand produces a reasoning tracer=\(r1,…,rN\)r=\(r\_\{1\},\\ldots,r\_\{N\}\)ofNNsegmented steps, followed by a final answeraa\. A verifierV​\(x,a\)∈\{0,1\}V\(x,a\)\\in\\\{0,1\\\}returns11iffaamatches the ground\-truth answera⋆a^\{\\star\}\. Write\|ri\|\|r\_\{i\}\|for the word count of steprir\_\{i\}andL​\(r\)=∑i\|ri\|L\(r\)=\\sum\_\{i\}\|r\_\{i\}\|\. Redundancy is a property of*correct*reasoning, so every measurement below is conditioned onV​\(x,a\)=1V\(x,a\)=1\.

The most direct way to decide whether a prefixr1:kr\_\{1:k\}already contains enough information to solve the problem is to askπ\\piitself\. Given the problemxxand the prefixr1:kr\_\{1:k\}, we forceπ\\pito terminate its thinking phase \(by inserting the model’s end\-of\-thinking delimiter and an answer prompt after the prefix\) and read off the final answer it produces\. The critical pointk⋆​\(r\)k^\{\\star\}\(r\)is then the smallestkkat whichπ\\pi, under this forced\-termination protocol, produces the correct answera⋆a^\{\\star\}\. This is the most intrinsic measurement of redundancy we can make: a step is redundant precisely whenπ\\piitself no longer needs it to reacha⋆a^\{\\star\}\. As a robustness check we repeat every measurement with a fixed external non\-reasoning judge \(gpt\-4o\-mini\); Appendix[B\.9](https://arxiv.org/html/2605.23926#A2.SS9)reports the external\-judge numbers and shows that none of our main claims depend on the choice of decoder\.

###### Definition 1\(Reasoning redundancy\)\.

For a correct tracer=\(r1,…,rN\)r=\(r\_\{1\},\\ldots,r\_\{N\}\)produced byπ\\pion problemxx, letπ​\(r1:k,x\)\\pi\(r\_\{1:k\},x\)denoteπ\\pi’s forced\-answer output given the prefix and problem \(i\.e\. the answerπ\\piemits when its thinking phase is terminated immediately afterrkr\_\{k\}\)\. The*critical point*is

k⋆​\(r\)=min⁡\{k∈\{1,…,N\}:π​\(r1:k,x\)=a⋆\}\.k^\{\\star\}\(r\)\\;=\\;\\min\\bigl\\\{\\,k\\in\\\{1,\\ldots,N\\\}\\,:\\,\\pi\(r\_\{1:k\},\\,x\)=a^\{\\star\}\\,\\bigr\\\}\.The*step\-level redundancy ratio*ofrris

ρ​\(r\)=1−k⋆​\(r\)N,\\rho\(r\)\\;=\\;1\-\\frac\{k^\{\\star\}\(r\)\}\{N\},and the*word\-level redundancy ratio*isρL​\(r\)=1−L​\(r1:k⋆​\(r\)\)/L​\(r\)\\rho\_\{L\}\(r\)=1\-L\(r\_\{1:k^\{\\star\}\(r\)\}\)/L\(r\)\.

Equivalently,ρ​\(r\)\\rho\(r\)is the maximum fraction of*trailing*segmented steps ofrrthat can be removed whileπ\\piitself still produces the correct answer:ρ\\rhoanswers the question “how much of the end of the trace doesπ\\pinot need?” The quantity is dimensionless, bounded in\[0,1\]\[0,1\], and has an immediate interpretation: a trace withρ=0\.8\\rho=0\.8could be truncated to20%20\\%of its steps without changing the answerπ\\piwould produce\. Everything beyondk⋆​\(r\)k^\{\\star\}\(r\)is redundant*for the model itself*\.

## 4Measurement Protocol

#### Progressive first\-kktruncation\.

For each correct tracerrwe construct the first\-kkprefixesr1:1,…,r1:Nr\_\{1:1\},\\ldots,r\_\{1:N\}and computeπ​\(r1:k,x\)\\pi\(r\_\{1:k\},x\)for each: we resumeπ\\pi’s generation with the prefix followed by the model’s end\-of\-thinking delimiter \(e\.g\.</think\>\) and a short answer prompt requesting a final`\\boxed\{\}`answer\. The critical pointk⋆​\(r\)k^\{\\star\}\(r\)is the smallestkkat which the forced answer is correct\. We additionally replicate the full protocol under a fixed non\-reasoning external judge \(gpt\-4o\-mini\-2024\-07\-18\) and report redundancy under both decoders throughout\.

#### Complementary diagnostic protocols\.

Two additional protocols probe different facets of the same underlying phenomenon\.*Leave\-one\-out step ablation*buildsNNsingle\-step\-deleted variantsr∖ir\_\{\\setminus i\}and asks whether each still yields the correct answer; a step is*critical*if its removal flips the answer\. This is a strictly stronger test than truncation — it asks whether a specific individual step is necessary, not whether some prefix suffices — and upper\-bounds the fraction of truly indispensable steps\.*Prefix\-position ablation*constructs four prefixes of the same relative length — first\-kk, last\-kk, middle\-kk, random\-kk— and compares forced\-answer accuracy; only first\-kkmeasuresρ\\rho, and the other three together diagnose where in the trace the redundant content concentrates\.

#### Models and benchmarks\.

We study four frontier reasoning models spanning three training recipes: DeepSeek\-R1\(Guoet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib11)\)\(outcome\-verified RL, 671B Mixture\-of\-Experts with∼37\\sim 37B active parameters per token; we refer to this as “MoE” below\), QwQ\-32B\(Qwen Team,[2025](https://arxiv.org/html/2605.23926#bib.bib27)\)\(outcome\-verified RL, 32B dense\), R1\-Distill\-Qwen\-7B\(Guoet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib11)\)\(distilled from R1 into Qwen\-7B\), and Qwen3\-30B\-A3B\-Thinking\(Yanget al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib44)\)\(Qwen3 with native thinking mode\)\. DeepSeek\-R1 is accessed via the official API; the other three via DashScope\(Yanget al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib43)\)\. We evaluate on two mathematical benchmarks chosen to span the difficulty spectrum:GSM8K\(Cobbeet al\.,[2021](https://arxiv.org/html/2605.23926#bib.bib7)\)\(6060randomly\-sampled grade\-school problems\) andMATH\-500\(Hendryckset al\.,[2021](https://arxiv.org/html/2605.23926#bib.bib14)\)\(150150competition problems spanning Levels 1–5 and 7 subjects\)\. For each problem we drawM=3M\{=\}3independent traces at temperature0\.70\.7with an8,1928\{,\}192\-token cap, yielding2,5202\{,\}520traces total of which1,8801\{,\}880are correct — the denominator for everyρ\\rhomeasurement\.

#### Segmentation\.

A trace is segmented into stepsr1,…,rNr\_\{1\},\\ldots,r\_\{N\}by a deterministic three\-step procedure: \(i\) split on double\-newline paragraph boundaries; \(ii\) within each paragraph, insert a cut*before*any sentence that begins with a*reasoning discourse marker*\(e\.g\.So,Therefore,Wait,However,Actually,Alternatively,Let me\); \(iii\) merge any resulting segment shorter than1212words into the previous segment\. Discourse markers are a standard segmentation cue in NLP\(Schiffrin,[1987](https://arxiv.org/html/2605.23926#bib.bib53)\)and naturally mark the start of a new reasoning move in our traces\. The full marker list and a robustness check against the merge threshold are in Appendix[C](https://arxiv.org/html/2605.23926#A3)\. Mean MATH\-500 step counts range from2222\(R1\) to7070\(Qwen3\)\.

## 5Empirical Findings

We report five empirical findings, organised from headline to details\. Table[1](https://arxiv.org/html/2605.23926#S5.T1)is the headline88\-condition quantification across two benchmarks; Table[2](https://arxiv.org/html/2605.23926#S5.T2)reports the external\-judge replication with the self–external gap; Table[3](https://arxiv.org/html/2605.23926#S5.T3)decomposes by MATH\-500 difficulty level; Figure[1](https://arxiv.org/html/2605.23926#S5.F1)visualises the difficulty curves; Table[4](https://arxiv.org/html/2605.23926#S5.T4)reports length\-stratified accuracy; Table[5](https://arxiv.org/html/2605.23926#S5.T5)reports within\-problem sample variance\. Findings 1 and 2 together establish the headline phenomenon under two independent decoders; Findings 3–5 refine and stress\-test it along the difficulty, length, and variance axes\. Two further supporting analyses \(subject\-level decomposition, step\-vs\-token equivalence\) are deferred to Appendix[B](https://arxiv.org/html/2605.23926#A2)\.

### 5\.1Finding 1:ρπ\\rho\_\{\\pi\}exceeds60%60\\%in every one of the88\(model, benchmark\) conditions, with a single\-step median critical prefix in six of eight\.

Table[1](https://arxiv.org/html/2605.23926#S5.T1)is the headline result\. Under theπ\\pi\-as\-own\-decoder protocol of Definition[1](https://arxiv.org/html/2605.23926#Thmdefinition1), step\-levelρπ\\rho\_\{\\pi\}ranges from61\.3%61\.3\\%\(R1\-Distill\-7B on MATH\-500, the smallest model on the hardest benchmark\) to92\.5%92\.5\\%\(QwQ\-32B on GSM8K\)\.*Every one of the88\(model, benchmark\) conditions reportsρπ\>60%\\rho\_\{\\pi\}\>60\\%\.*The median critical prefix is a single segmented step in six of the eight conditions, and is22and55in the remaining two \(DeepSeek\-R1 MATH\-500 and R1\-Distill\-7B MATH\-500\)\. For QwQ and Qwen3, more than half of all correct traces on both benchmarks satisfyk⋆=1k^\{\\star\}=1: givingπ\\pionly the opening segmented step of its own trace and forcing it to terminate is already enough forπ\\pito produce the correct answer, even though those same traces average16​–​6216\\text\{\-\-\}62steps in length\. Confidence intervals are tight: all eight95%95\\%bootstrap CIs are narrower than±4\\pm 4points, so the point estimates carry real statistical weight\.

Two details deserve attention\. First, for three of four modelsρπ\\rho\_\{\\pi\}is*higher*on GSM8K than on MATH\-500 — the opposite of the naive expectation that easier benchmarks waste less reasoning\. The mechanism is mechanical rather than cognitive: on GSM8K, QwQ and Qwen3 still emit16​–​2316\\text\{\-\-\}23\-step traces that are qualitatively the same over\-elaborations as on MATH\-500, but against a tiny actual solution length \(k⋆∼1​–​2k^\{\\star\}\\sim 1\\text\{\-\-\}2\), making the ratio higher\. Second, R1\-Distill\-7B produces the shortest absolute prefixes \(k¯⋆=1\.5\\bar\{k\}^\{\\star\}=1\.5steps on GSM8K\) yet has the lowestρπ\\rho\_\{\\pi\}because it emits the shortest total traces \(N¯=5\.5\\bar\{N\}=5\.5\) — illustrating thatρ\\rhois the right normalisation: it is not confounded by absolute length\.

The four models also span three distinct training recipes — outcome\-verified RL from scratch \(R1, QwQ\), distillation from a long\-CoT teacher \(R1\-Distill\-7B\), and native thinking\-mode Qwen3 — and two orders of magnitude in total parameters \(77B distilled vs671671B MoE,∼37\\sim 37B activated\)\. Yet every one of them exhibitsρπ\>60%\\rho\_\{\\pi\}\>60\\%on both benchmarks, every one hasρext\>30%\\rho\_\{\\text\{ext\}\}\>30\\%on both benchmarks \(Finding 2\), and every one has median critical prefix≤5\\leq 5steps on both benchmarks\. Redundancy is therefore not a quirk of any single training recipe or scale; it is a shared phenomenon of the current frontier\. This alignment between architectures, training objectives, and model sizes is the empirical pattern the theoretical explanation of §[6](https://arxiv.org/html/2605.23926#S6)accounts for: the length\-agnostic outcome reward is the one design choice these recipes share, and Theorem[1](https://arxiv.org/html/2605.23926#Thmtheorem1)shows it makes over\-thinking the optimal behaviour regardless of the other choices\.

Table 1:Main redundancy table: four models, two benchmarks,88conditions\.ρπ\(%\)\\rho\_\{\\pi\}\\,\(\\%\): redundancy under theπ\\pi\-as\-own\-decoder protocol \(primary measurement, Definition[1](https://arxiv.org/html/2605.23926#Thmdefinition1)\)\.95%95\\%CI:95%95\\%bootstrap confidence interval forρπ\\rho\_\{\\pi\}, computed via the non\-parametric percentile bootstrap withB=10,000B=10\{,\}000resamples at the problem level \(full methodology in Appendix[B\.13](https://arxiv.org/html/2605.23926#A2.SS13)\)\.ρext\(%\)\\rho\_\{\\text\{ext\}\}\\,\(\\%\): redundancy under the external\-judge replication \(gpt\-4o\-mini\)\. gap=ρπ−ρext=\\rho\_\{\\pi\}\-\\rho\_\{\\text\{ext\}\}\.ρL\(%\)\\rho\_\{L\}\\,\(\\%\): word\-level analogue ofρπ\\rho\_\{\\pi\}\.N¯\\bar\{N\}: average number of segmented steps per correct trace\.k¯⋆\\bar\{k\}^\{\\star\}and med:*mean*and*median*of the critical prefixk⋆​\(r\)k^\{\\star\}\(r\)across thenncorrect traces — that is, the smallest number of leading segmented steps such thatπ\\pi, forced to terminate thinking immediately after, still produces the correct answer \(so med=1=1means that on at least half of all correct traces, the opening segmented step alone is already sufficient\)\.nn: number of correct traces entering the row; sinceρ\\rhois defined only on correct traces,nnis the per\-row denominator\.All88conditions haveρπ≥61%\\rho\_\{\\pi\}\\geq 61\\%andρext≥31%\\rho\_\{\\text\{ext\}\}\\geq 31\\%; the median critical prefix is a single segmented step in six of the eight conditions\.
### 5\.2Finding 2: external\-judge replication —ρext\>30%\\rho\_\{\\text\{ext\}\}\>30\\%in all eight conditions, and the self–external gap is training\-recipe\-specific\.

Table 2:Self\-vs\-external judge under the same truncation protocol\.ρπ\\rho\_\{\\pi\}: the correct\-trace judge isπ\\piitself \(Definition[1](https://arxiv.org/html/2605.23926#Thmdefinition1)\)\.ρext\\rho\_\{\\text\{ext\}\}: the judge isgpt\-4o\-mini\-2024\-07\-18, run on the identical prefix\. The gap measures how much of the redundancy is self\-decoder\-specific vs\. intrinsic to the trace\. The gap tracks training recipe rather than being a uniform offset\.ρπ\\rho\_\{\\pi\}andρext\\rho\_\{\\text\{ext\}\}answer different questions — “how much of the trace doesπ\\piitself not need?” versus “how much would an outside reader not need?” — and comparing them is informative\. Table[2](https://arxiv.org/html/2605.23926#S5.T2)reports both for all eight conditions\.ρext\>30%\\rho\_\{\\text\{ext\}\}\>30\\%in every condition and exceeds50%50\\%in five of eight: even a completely external decoder, using only the prefix, recovers the correct answer on a majority of traces in most settings\. The redundancy we document is not an artefact of the model permissively reading its own thinking\.

The self–external gap, however, varies widely:10​–​2010\\text\{\-\-\}20points for DeepSeek\-R1,30​–​3330\\text\{\-\-\}33points for QwQ\-32B,9​–​339\\text\{\-\-\}33points for R1\-Distill\-7B, and57​–​6057\\text\{\-\-\}60points for Qwen3\-30B\-Thinking\. The ordering tracks training recipe, not model scale: the two outcome\-RL models \(DS\-R1, QwQ\) sit in the middle; native\-thinking\-mode Qwen3 has by far the largest gap\. We interpret this as a signal that Qwen3’s decoder has strong stylistic affinity with its own trace format — it can reliably extract a final answer from prefixes that an external decoder cannot\. The gap itself is therefore a useful diagnostic: large gaps flag models whose redundancy is partly decoder\-specific; small gaps \(DS\-R1\) flag models whose redundancy is robust to decoder change\.

Table 3:ρ​\(d\)\\rho\(d\)by MATH\-500 difficulty level, four models\.ρ​\(d\)\\rho\(d\)denotes the average ofρ\\rhoover all correct traces whose source problem has difficulty leveldd, ford∈\{1,…,5\}d\\in\\\{1,\\ldots,5\\\}\(MATH\-500 labels every problem with a difficulty level 1–5\)\. All four models exhibitρ​\(d\)\\rho\(d\)that decreases withdd\. Absolute drop from Level 1 to Level 5: DS\-R1−12\.6\-12\.6, QwQ−10\.0\-10\.0, R1\-Distill−27\.0\-27\.0, Qwen3−11\.3\-11\.3points\. Even at the hardest level,ρ\\rhosits in the band\[46\.1%,84\.7%\]\[46\.1\\%,84\.7\\%\]\.![Refer to caption](https://arxiv.org/html/2605.23926v1/x1.png)Figure 1:Difficulty–redundancy curves, four models\.*Left*:ρ​\(d\)\\rho\(d\)with95%95\\%bootstrap CI bands\. All four models decrease monotonically with difficulty\.*Right*: average trace lengthN¯\\bar\{N\}\(dashed\) and average critical prefixk¯⋆\\bar\{k\}^\{\\star\}\(solid\), log scale\.k¯⋆\\bar\{k\}^\{\\star\}scales super\-linearly with difficulty across all four models\.
### 5\.3Finding 3: on MATH\-500,ρ​\(d\)\\rho\(d\)decreases with difficulty for every model, but all four remain substantially redundant at the hardest level\.

Letρ​\(d\)\\rho\(d\)denote the average ofρ\\rhoover all correct traces whose source problem has MATH\-500 difficulty leveldd, ford∈\{1,…,5\}d\\in\\\{1,\\ldots,5\\\}\. Table[3](https://arxiv.org/html/2605.23926#S5.T3)and Figure[1](https://arxiv.org/html/2605.23926#S5.F1)refine the headline number along problem difficulty\. Three observations stand out\.

*\(a\) All four models decrease\.*ρ​\(d\)\\rho\(d\)drops from Level 1 to Level 5 by12\.612\.6points for DeepSeek\-R1,10\.010\.0points for QwQ\-32B,27\.027\.0points for R1\-Distill\-7B, and11\.311\.3points for Qwen3\-30B\-Thinking\. The direction of the effect is the same across every model family, scale, and training recipe we tested; the magnitude differs but the sign does not\. This rules out a “redundancy is an artefact of some specific model” hypothesis: four independently\-trained models all exhibit the same qualitative response to difficulty\.

*\(b\) Inter\-model spread widens at the hardest levels\.*At Level 1 the four models are bunched into a2323\-point spread \(71\.5%71\.5\\%for R1 vs\.94\.7%94\.7\\%for QwQ\), but at Level 5 the spread widens to38\.638\.6points \(R1\-Distill46\.1%46\.1\\%vs\. QwQ84\.7%84\.7\\%\)\. R1\-Distill\-7B drops fastest \(−27\-27points\), while QwQ and Qwen3 drop the least \(−10\-10and−11\-11points\) and remain strongly redundant even on competition\-level problems\. The models thus diverge rather than converge as difficulty rises, and the divergence sorts along the same training\-recipe axis we saw in Finding 2: the two native\-thinking\-style models \(QwQ, Qwen3\) stay firmly over\-verbose; the two outcome\-RL / distilled models \(R1, R1\-Distill\) shed redundancy more aggressively as problems get harder\. Even so, every one of the four retainsρ≥46%\\rho\\geq 46\\%at Level 5 — a substantial fraction of every trace remains recoverable\-from\-prefix\.

*\(c\) The critical prefix grows super\-linearly\.*k¯⋆\\bar\{k\}^\{\\star\}scales dramatically with difficulty: from1\.6→19\.01\.6\\to 19\.0for DS\-R1,1\.1→16\.01\.1\\to 16\.0for QwQ,4\.5→32\.34\.5\\to 32\.3for R1\-Distill, and1\.0→30\.51\.0\\to 30\.5for Qwen3\. Models*do*allocate more prefix to harder problems, but total trace length grows faster still, so the ratio1−k⋆/N1\-k^\{\\star\}/Ndecreases only gradually\.

We do not fit a parametric form to the four curves\. The direction \(decreasing\), magnitude \(by model\), and recipe\-dependent spread \(Finding 3b\) are the findings; any specific functional form — monotone rational, exponential, or power\-law — would require theory that predicts the shape, which Theorem[1](https://arxiv.org/html/2605.23926#Thmtheorem1)does not furnish\. Whether a mechanistic model of reasoning predicts the observed shape from first principles is an open question, flagged in the discussion\.

### 5\.4Finding 4: length is anti\-correlated with accuracy for outcome\-RL and distilled models, but essentially flat for native\-thinking models\.

Table 4:Length–accuracy decile analysis \(MATH\-500,n=450n\{=\}450per model\)\.Samples are sorted into1010length deciles by completion length; the table reports per\-decile mean lengthL¯\\bar\{L\}\(words\) and accuracy \(%\)\. D1 is the shortest decile, D10 the longest\. DS\-R1 and R1\-Distill show a clear inverted\-U peaking at mid\-length and dropping at the longest decile; QwQ and Qwen3 show only a weak length penalty\.D1D2D3D4D5D6D7D8D9D10peak→\\toD10DS\-R1acc %80\.091\.193\.393\.384\.484\.484\.480\.062\.244\.4−48\.9\-48\.9L¯\\bar\{L\}141192246322403534758135925804525QwQ\-32Bacc %84\.484\.480\.093\.380\.073\.375\.666\.766\.773\.3−26\.6\-26\.6L¯\\bar\{L\}538709835972117314671877257936137074R1\-D\-7Bacc %84\.471\.191\.184\.486\.768\.973\.375\.651\.124\.4−66\.7\-66\.7L¯\\bar\{L\}188298599810100612651570214631474727Qwen3\-Th\.acc %93\.3100\.088\.977\.875\.675\.671\.175\.668\.984\.4−15\.6\-15\.6L¯\\bar\{L\}48365210131674204324122901388749618898Table[4](https://arxiv.org/html/2605.23926#S5.T4)slices each model’s MATH\-500 completions into1010length deciles and reports accuracy per decile\. The pattern is model\-dependent\. For DeepSeek\-R1 and R1\-Distill\-7B the shape is a clean*inverted\-U*— accuracy rises with length up to a middle decile and then falls: accuracy peaks at a mid\-length decile \(93\.3%93\.3\\%at D3–D4 for DS\-R1,91\.1%91\.1\\%at D3 for R1\-Distill\), plateaus through D5–D8, and then crashes in D9–D10 \(62\.2%→44\.4%62\.2\\%\\to 44\.4\\%for DS\-R1;51\.1%→24\.4%51\.1\\%\\to 24\.4\\%for R1\-Distill — a49​–​6749\\text\{\-\-\}67\-point drop at the longest decile\)\. For QwQ\-32B the effect is weaker: accuracy drifts downward but by only∼27\\sim 27points, and D10 is not the minimum\. For Qwen3\-30B\-Thinking the effect is essentially absent: D10 accuracy \(84\.4%84\.4\\%\) is only a few points below the peak\.

The cleanest interpretation is that long traces mean different things to different models\. For DS\-R1 and R1\-Distill, an unusually long trace is a signal that the model is struggling and getting worse, not reasoning longer on purpose: length is correlated with failure\. For QwQ and Qwen3 — both of which have very long baseline traces already \(N¯=46\.1\\bar\{N\}=46\.1and61\.661\.6on MATH\-500, see Table[1](https://arxiv.org/html/2605.23926#S5.T1)\) — an unusually long trace is within the expected distribution and does not signal failure\. This model\-specific coupling also explains why the two “flat\-accuracy” models \(QwQ, Qwen3\) have the highest aggregateρπ\\rho\_\{\\pi\}: their long tails are not wrong, they are simply*idle*— filled with verification, reformulation, and self\-reflection that do not advance the solution but also do not introduce errors\. \(The termsadvanceandidlewe use here are formalised in §[6](https://arxiv.org/html/2605.23926#S6)\.\) Finding 5 confirms this reading from an independent angle\.

### 5\.5Finding 5: within\-problemρ\\rhovariance is small, and its ordering mirrors Finding 4\.

Table 5:Within\-problemρ\\rhovariance acrossM=3M\{=\}3samples \(MATH\-500\)\.Problems with≥2\\geq 2correct samples\. Mean within\-problem standard deviationσρ\\sigma\_\{\\rho\}and mean max−\-min range\. Lowσρ\\sigma\_\{\\rho\}meansρ\\rhois a stable property of each \(model, problem\) pair, not a sampling artefact\.A natural robustness question: maybeρ\\rhois a property of individual*samples*rather than*problems*, and the headline numbers average over large within\-problem noise\. Table[5](https://arxiv.org/html/2605.23926#S5.T5)tests this by computing, for each problem with≥2\\geq 2correct samples out ofM=3M\{=\}3, the standard deviation ofρ\\rhoacross those correct samples\. Meanσρ\\sigma\_\{\\rho\}is0\.026​–​0\.1110\.026\\text\{\-\-\}0\.111across models: QwQ and Qwen3 are strikingly stable \(σρ<0\.05\\sigma\_\{\\rho\}<0\.05\) —ρ\\rhois a*sharp*problem\-level property — while DS\-R1 and R1\-Distill have moderate but still controlled variability \(σρ<0\.12\\sigma\_\{\\rho\}<0\.12\)\. The headline numbers in Table[1](https://arxiv.org/html/2605.23926#S5.T1)are not artefacts of trace\-level noise\.

The variance ordering itself carries information when read against Finding 4: the two models with the cleanest inverted\-U in length–accuracy space \(DS\-R1 and R1\-Distill\) are also the two with the largest per\-problemσρ\\sigma\_\{\\rho\}\. Both observations reflect the same model\-level divide:for DS\-R1 and R1\-Distill, trace length is a response to problem\-specific struggle — the same problem may elicit a short efficient trace on one sample and a long struggle\-filled trace on another; for QwQ and Qwen3, trace length is a stable stylistic default applied regardless of how easily the model reaches the answer\.Lowσρ\\sigma\_\{\\rho\}and flat length–accuracy curves are two views of the same phenomenon: length not carrying problem\-specific information\.

#### Summary\.

Findings 1–5 together establish: \(a\)ρπ\>60%\\rho\_\{\\pi\}\>60\\%in every \(model, benchmark\) condition, with median critical prefix≤5\\leq 5steps, universal across three training recipes and two orders of magnitude in total parameters \(Finding 1\); \(b\)ρext\>30%\\rho\_\{\\text\{ext\}\}\>30\\%under a completely external decoder in every condition, with the self–external gap tracking training recipe \(Finding 2\); \(c\)ρ\\rhodecreases with MATH\-500 difficulty for every model but remains substantial at Level 5, with inter\-model spread widening at the hardest level \(Finding 3\); \(d\) length and accuracy are anti\-correlated for RL/distilled models but flat for native\-thinking models, whose long tails are simply idle \(Finding 4\); \(e\) within\-problemρ\\rhovariance is small, and its ordering across models mirrors Finding 4, confirming a structural divide between “struggle\-type” and “stylistic\-type” reasoning \(Finding 5\)\. Two further analyses — per\-subject decomposition and step\-vs\-token equivalence — are in Appendices[B\.1](https://arxiv.org/html/2605.23926#A2.SS1)and[B\.2](https://arxiv.org/html/2605.23926#A2.SS2)\. The next section explains the structural origin of \(a\)–\(b\) from a single theoretical model\.

## 6Theoretical Framework

We now explain*why*reasoning traces are so redundant\. We model reasoning as a sequential decision process and prove thatover\-thinking is a structural consequence of length\-agnostic outcome rewards— the training signal used by every modern reasoning\-model recipe we are aware of \(Theorem[1](https://arxiv.org/html/2605.23926#Thmtheorem1)\)\. The result holds regardless of RL algorithm, base model, data distribution, or whether the policy is obtained via RL or distillation; it identifies the single design choice that produces the empirical phenomenon of Section[5](https://arxiv.org/html/2605.23926#S5)\.

#### Sequential\-decision model\.

Fix a problem with difficultyK∈ℕK\\in\\mathbb\{N\}— the number of successfulAdvancemoves required to produce a correct answer\. A policyπ\\pigenerates a reasoning trace as a sequential decision process: at each stept=1,2,…t=1,2,\\ldots,π\\piselects an action based on the trajectory so far,at∈\{Advance,Idle,Stop\}a\_\{t\}\\in\\\{\\textsc\{Advance\},\\textsc\{Idle\},\\textsc\{Stop\}\\\}\. The three actions have distinct roles\.*Advance*attempts substantive progress: each such attempt succeeds independently with probabilityp∈\(0,1\)p\\in\(0,1\), contributing one unit towardKK; with probability1−p1\-pit fails, contributing zero\.*Idle*emits non\-advancing content \(verification, reformulation, self\-reflection\), contributing zero\.*Stop*terminates the trace and emits the final answer\. LetT:=min⁡\{t:at=Stop\}T:=\\min\\\{t:a\_\{t\}=\\textsc\{Stop\}\\\}be the stopping time,ξt∼i\.i\.d\.Bernoulli​\(p\)\\xi\_\{t\}\\stackrel\{\{\\scriptstyle\\mathrm\{i\.i\.d\.\}\}\}\{\{\\sim\}\}\\mathrm\{Bernoulli\}\(p\)theAdvance\-success noise, andnA:=∑t=1T𝟏​\[at=Advance\]⋅ξtn\_\{A\}:=\\sum\_\{t=1\}^\{T\}\\mathbf\{1\}\[a\_\{t\}=\\textsc\{Advance\}\]\\cdot\\xi\_\{t\}the total successful advances at stopping\. The outcome reward isR:=𝟏​\[nA≥K\]R:=\\mathbf\{1\}\[n\_\{A\}\\geq K\], and the training objective isJλ​\(π\):=𝔼π​\[R\]−λ​𝔼π​\[T\]J\_\{\\lambda\}\(\\pi\):=\\mathbb\{E\}\_\{\\pi\}\[R\]\-\\lambda\\,\\mathbb\{E\}\_\{\\pi\}\[T\]withλ≥0\\lambda\\geq 0;*length\-agnostic*training corresponds toλ=0\\lambda=0, in which caseJ0​\(π\)=ℙπ​\[nA≥K\]J\_\{0\}\(\\pi\)=\\mathbb\{P\}\_\{\\pi\}\[n\_\{A\}\\geq K\]\. On the success event, trace\-level redundancy isρ:=1−K/T\\rho:=1\-K/T\.

###### Theorem 1\(Length\-agnostic training is over\-thinking\)\.

Consider the length\-agnostic objectiveJ0​\(π\)=ℙπ​\[nA≥K\]J\_\{0\}\(\\pi\)=\\mathbb\{P\}\_\{\\pi\}\[n\_\{A\}\\geq K\]\. No policyπ\\piwith𝔼π​\[T\]<∞\\mathbb\{E\}\_\{\\pi\}\[T\]<\\inftyis optimal; every optimal policy satisfies𝔼π​\[T\]=∞\\mathbb\{E\}\_\{\\pi\}\[T\]=\\infty\.

#### Intuition\.

Under an outcome\-only reward there is no penalty for continuing to reason\. Extending the trajectory by additionalAdvanceattempts can only increase the probability of eventually accumulatingKKsuccesses, so any policy that stops in finite expected time is strictly dominated by one that keeps trying\. The proof \(Appendix[A](https://arxiv.org/html/2605.23926#A1)\) formalises this by constructing, for anyπ\\piwith𝔼​\[T\]<∞\\mathbb\{E\}\[T\]<\\infty, a modified policyπ′\\pi^\{\\prime\}that never stops prematurely and succeeds almost surely\.

Theorem[1](https://arxiv.org/html/2605.23926#Thmtheorem1)says over\-thinking is a property of the*reward shape*, not of any specific RL algorithm, base model, or data distribution\. Length penalties therefore*must*come from outside the correctness signal: the context window’s soft cap is what keeps empirical traces finite, and every variant of length\-aware training proposed in the literature\(Arora and Zanette,[2025](https://arxiv.org/html/2605.23926#bib.bib1); Aggarwal and Welleck,[2025](https://arxiv.org/html/2605.23926#bib.bib2); Hanet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib12); Muennighoffet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib24)\)is in effect adding such a penalty\.

## 7Discussion and Conclusion

#### Redundancy is a design\-level property, not a fixable bug\.

Theorem[1](https://arxiv.org/html/2605.23926#Thmtheorem1)identifies the source: a length\-agnostic outcome reward makes over\-thinking the unique optimal behaviour, independent of RL algorithm, base model, data, or whether the policy is learned by RL or distillation\. The empirical redundancy we document \(Findings 1–5\) is therefore not a failure of any particular model but a reflection of a shared training objective\. Theorem[1](https://arxiv.org/html/2605.23926#Thmtheorem1)does not, however, predict the functional form ofρ​\(d\)\\rho\(d\)in Finding 3; that shape is a natural extension target\.

#### Training\-time prescription\.

Any fix*must*break length\-agnosticism, which rules out post\-hoc patches\. Three patterns satisfy this: \(i\) an explicit length penalty−λ​T\-\\lambda T\(Arora and Zanette,[2025](https://arxiv.org/html/2605.23926#bib.bib1); Aggarwal and Welleck,[2025](https://arxiv.org/html/2605.23926#bib.bib2); Hanet al\.,[2024](https://arxiv.org/html/2605.23926#bib.bib12); Muennighoffet al\.,[2025](https://arxiv.org/html/2605.23926#bib.bib24)\), the minimal such modification; \(ii\) a difficulty\-aware token budgetT≤B​\(d\)T\\leq B\(d\), which implementsλ​K\\lambda Kimplicitly and matches the empirical super\-linear scaling ofN¯\\bar\{N\}with difficulty; \(iii\) a per\-advance intermediate reward that pays off at each verified advance, removing the incentive for idle content\. The actionable takeaway:no reasoning\-model training pipeline should use a purely outcome\-only reward, because under such a reward the unique optimal stopping time is infinite\.

#### Conclusion\.

Across1,8801\{,\}880correct frontier\-model traces,ρπ\>60%\\rho\_\{\\pi\}\>60\\%in every\(m​o​d​e​l,b​e​n​c​h​m​a​r​k\)\(model,benchmark\)condition andρext\>30%\\rho\_\{\\text\{ext\}\}\>30\\%under an external decoder; the median correct trace truncates to its opening segmented step in six of eight conditions while the judge still recovers the answer\. This redundancy is a structural consequence of length\-agnostic outcome rewards \(Theorem[1](https://arxiv.org/html/2605.23926#Thmtheorem1)\), so the fix must come from training, not post\-hoc patches\.

## Acknowledgments and Disclosure of Funding

## References

- P\. Aggarwal and S\. Welleck \(2025\)L1: controlling how long a reasoning model thinks with reinforcement learning\.arXiv preprint arXiv:2503\.04697\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p2.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1),[§6](https://arxiv.org/html/2605.23926#S6.SS0.SSS0.Px2.p2.1),[§7](https://arxiv.org/html/2605.23926#S7.SS0.SSS0.Px2.p1.4)\.
- D\. Arora and A\. Zanette \(2025\)Training language models to reason efficiently\.arXiv preprint arXiv:2502\.04463\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p2.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1),[§6](https://arxiv.org/html/2605.23926#S6.SS0.SSS0.Px2.p2.1),[§7](https://arxiv.org/html/2605.23926#S7.SS0.SSS0.Px2.p1.4)\.
- B\. Brown, J\. Juravsky, R\. Ehrlich, R\. Clark, Q\. V\. Le, C\. Ré, and A\. Mirhoseini \(2024\)Large language monkeys: scaling inference compute with repeated sampling\.arXiv preprint arXiv:2407\.21787\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- X\. Chen, J\. Xu, T\. Liang, Z\. He, J\. Pang, D\. Yu,et al\.\(2024\)Do not think that much for 2\+3=? on the overthinking of o1\-like LLMs\.arXiv preprint arXiv:2412\.21187\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p2.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1)\.
- K\. Cobbe, V\. Kosaraju, M\. Bavarian, M\. Chen, H\. Jun, L\. Kaiser,et al\.\(2021\)Training verifiers to solve math word problems\.arXiv preprint arXiv:2110\.14168\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1),[§4](https://arxiv.org/html/2605.23926#S4.SS0.SSS0.Px3.p1.9)\.
- X\. Dang, C\. Baek, J\. Z\. Kolter, and A\. Raghunathan \(2025\)Assessing diversity collapse in reasoning\.InWorkshop on Scaling Self\-Improving Foundation Models without Human Supervision,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1)\.
- Y\. Fu, H\. Peng, A\. Sabharwal, P\. Clark, and T\. Khot \(2022\)Complexity\-based prompting for multi\-step reasoning\.arXiv preprint arXiv:2210\.00720\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- A\. Graves \(2016\)Adaptive computation time for recurrent neural networks\.arXiv preprint arXiv:1603\.08983\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1)\.
- C\. Gulcehre, T\. L\. Paine, S\. Srinivasan, K\. Konyushkova, L\. Weerts,et al\.\(2023\)Reinforced self\-training \(ReST\) for language modeling\.arXiv preprint arXiv:2308\.08998\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- D\. Guo, D\. Yang, H\. Zhang, J\. Song, R\. Zhang, R\. Xu, Q\. Zhu,et al\.\(2025\)DeepSeek\-R1: incentivizing reasoning capability in LLMs via reinforcement learning\.arXiv preprint arXiv:2501\.12948\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1),[§4](https://arxiv.org/html/2605.23926#S4.SS0.SSS0.Px3.p1.9)\.
- T\. Han, Z\. Wang, C\. Fang, S\. Zhao, S\. Ma, and Z\. Chen \(2024\)Token\-budget\-aware LLM reasoning\.arXiv preprint arXiv:2412\.18547\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p2.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1),[§6](https://arxiv.org/html/2605.23926#S6.SS0.SSS0.Px2.p2.1),[§7](https://arxiv.org/html/2605.23926#S7.SS0.SSS0.Px2.p1.4)\.
- S\. Hao, S\. Sukhbaatar, D\. Su, X\. Li, Z\. Hu, J\. Weston, and Y\. Tian \(2024\)Training large language models to reason in a continuous latent space\.arXiv preprint arXiv:2412\.06769\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- D\. Hendrycks, C\. Burns, S\. Kadavath, A\. Arora, S\. Basart, E\. Tang, D\. Song, and J\. Steinhardt \(2021\)Measuring mathematical problem solving with the MATH dataset\.InNeurIPS Datasets and Benchmarks,Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1),[§4](https://arxiv.org/html/2605.23926#S4.SS0.SSS0.Px3.p1.9)\.
- A\. Jaech, A\. Kalai, A\. Lerer, A\. Richardson,et al\.\(2024\)OpenAI o1 system card\.arXiv preprint arXiv:2412\.16720\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1)\.
- N\. Jain, K\. Han, A\. Gu, W\. Li, F\. Yan, T\. Zhang, S\. Wang, A\. Solar\-Lezama, K\. Sen, and I\. Stoica \(2025\)LiveCodeBench: holistic and contamination free evaluation of large language models for code\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1)\.
- Kimi Team, A\. Du, B\. Gao, B\. Xing, C\. Jiang, C\. Chen,et al\.\(2025\)Kimi k1\.5: scaling reinforcement learning with LLMs\.arXiv preprint arXiv:2501\.12599\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- T\. Kojima, S\. S\. Gu, M\. Reid, Y\. Matsuo, and Y\. Iwasawa \(2022\)Large language models are zero\-shot reasoners\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- H\. Lightman, V\. Kosaraju, Y\. Burda, H\. Edwards, B\. Baker, T\. Lee, J\. Leike, J\. Schulman, I\. Sutskever, and K\. Cobbe \(2024\)Let’s verify step by step\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- Z\. Liu, C\. Chen, W\. Li, P\. Qi, T\. Pang, C\. Du, W\. S\. Lee, and M\. Lin \(2025\)Understanding R1\-zero\-like training: a critical perspective\.arXiv preprint arXiv:2503\.20783\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1)\.
- A\. Madaan, N\. Tandon, P\. Gupta, S\. Hallinan, L\. Gao, S\. Wiegreffe, U\. Alon, N\. Dziri, S\. Prabhumoye, Y\. Yang,et al\.\(2023\)Self\-refine: iterative refinement with self\-feedback\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- N\. Muennighoff, Z\. Yang, W\. Shi, X\. L\. Li, L\. Fei\-Fei, H\. Hajishirzi, L\. Zettlemoyer, P\. Liang, E\. Candès, and T\. Hashimoto \(2025\)s1: simple test\-time scaling\.arXiv preprint arXiv:2501\.19393\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p2.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1),[§6](https://arxiv.org/html/2605.23926#S6.SS0.SSS0.Px2.p2.1),[§7](https://arxiv.org/html/2605.23926#S7.SS0.SSS0.Px2.p1.4)\.
- OpenAI \(2024\)Learning to reason with LLMs\.Note:OpenAI BlogCited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- Qwen Team \(2025\)QwQ\-32B: embracing the power of reinforcement learning\.Note:Qwen BlogCited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1),[§4](https://arxiv.org/html/2605.23926#S4.SS0.SSS0.Px3.p1.9)\.
- A\. Rastogi, A\. Q\. Jiang, A\. Lo, G\. Berrada, G\. Lample,et al\.\(2025\)Magistral\.arXiv preprint arXiv:2506\.10910\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1)\.
- D\. Schiffrin \(1987\)Discourse markers\.Cambridge University Press\.Cited by:[§4](https://arxiv.org/html/2605.23926#S4.SS0.SSS0.Px4.p1.4)\.
- T\. Schuster, A\. Fisch, J\. Gupta, M\. Dehghani, D\. Bahri, V\. Tran, Y\. Tay, and D\. Metzler \(2022\)Confident adaptive language modeling\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1)\.
- D\. J\. Shah, P\. Rushton, S\. Singla, M\. Parmar, K\. Smith, Y\. Vanjani,et al\.\(2025\)Rethinking reflection in pre\-training\.arXiv preprint arXiv:2504\.04022\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1)\.
- Z\. Shao, P\. Wang, Q\. Zhu, R\. Xu, J\. Song, X\. Bi,et al\.\(2024\)DeepSeekMath: pushing the limits of mathematical reasoning in open language models\.arXiv preprint arXiv:2402\.03300\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- N\. Shinn, F\. Cassano, E\. Berman, A\. Gopinath, K\. Narasimhan, and S\. Yao \(2023\)Reflexion: language agents with verbal reinforcement learning\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- C\. Snell, J\. Lee, K\. Xu, and A\. Kumar \(2025\)Scaling LLM test\-time compute optimally can be more effective than scaling model parameters\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- Z\. Sprague, F\. Yin, J\. D\. Rodriguez, D\. Jiang, M\. Wadhwa, P\. Singhal,et al\.\(2025\)To CoT or not to CoT? chain\-of\-thought helps mainly on math and symbolic reasoning\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p2.1)\.
- J\. Uesato, N\. Kushman, R\. Kumar, F\. Song, N\. Siegel, L\. Wang, A\. Creswell, G\. Irving, and I\. Higgins \(2022\)Solving math word problems with process\- and outcome\-based feedback\.arXiv preprint arXiv:2211\.14275\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- X\. Wang, J\. Wei, D\. Schuurmans, Q\. Le, E\. Chi, S\. Narang, A\. Chowdhery, and D\. Zhou \(2023\)Self\-consistency improves chain of thought reasoning in language models\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- J\. Wei, X\. Wang, D\. Schuurmans, M\. Bosma, F\. Xia, E\. Chi, Q\. Le, and D\. Zhou \(2022\)Chain\-of\-thought prompting elicits reasoning in large language models\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- A\. Yang, A\. Li, B\. Yang, B\. Zhang, B\. Hui, B\. Zheng,et al\.\(2025\)Qwen3 technical report\.arXiv preprint arXiv:2505\.09388\.Cited by:[§4](https://arxiv.org/html/2605.23926#S4.SS0.SSS0.Px3.p1.9)\.
- A\. Yang, B\. Yang, B\. Zhang, B\. Hui, B\. Zheng, B\. Yu,et al\.\(2024\)Qwen2\.5 technical report\.arXiv preprint arXiv:2412\.15115\.Cited by:[§4](https://arxiv.org/html/2605.23926#S4.SS0.SSS0.Px3.p1.9)\.
- S\. Yao, D\. Yu, J\. Zhao, I\. Shafran, T\. Griffiths, Y\. Cao, and K\. Narasimhan \(2023\)Tree of thoughts: deliberate problem solving with large language models\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- Q\. Yu, Z\. Zhang, R\. Zhu, Y\. Yuan, X\. Zuo, Y\. Yue,et al\.\(2025\)DAPO: an open\-source LLM reinforcement learning system at scale\.arXiv preprint arXiv:2503\.14476\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- E\. Zelikman, G\. Harik, Y\. Shao, V\. Jayasiri, N\. Haber, and N\. D\. Goodman \(2024\)Quiet\-STaR: language models can teach themselves to think before speaking\.InConference on Language Modeling,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- E\. Zelikman, Y\. Wu, J\. Mu, and N\. Goodman \(2022\)STaR: bootstrapping reasoning with reasoning\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- W\. Zeng, Y\. Huang, Q\. Liu, W\. Liu, K\. He, Z\. Ma, and J\. He \(2025\)SimpleRL\-Zoo: investigating and taming zero reinforcement learning for open base models in the wild\.arXiv preprint arXiv:2503\.18892\.Cited by:[§1](https://arxiv.org/html/2605.23926#S1.p1.1),[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.
- R\. Zhao, A\. Meterez, S\. Kakade, C\. Pehlevan, S\. Jelassi, and E\. Malach \(2025\)Echo chamber: RL post\-training amplifies behaviors learned in pretraining\.arXiv preprint arXiv:2504\.07912\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px2.p1.1)\.
- D\. Zhou, N\. Schärli, L\. Hou, J\. Wei, N\. Scales, X\. Wang, D\. Schuurmans, C\. Cui, O\. Bousquet, Q\. Le,et al\.\(2022\)Least\-to\-most prompting enables complex reasoning in large language models\.arXiv preprint arXiv:2205\.10625\.Cited by:[§2](https://arxiv.org/html/2605.23926#S2.SS0.SSS0.Px1.p1.1)\.

## Appendix AProof of Theorem[1](https://arxiv.org/html/2605.23926#Thmtheorem1)

Letπ\\pisatisfy𝔼π​\[T\]=t⋆<∞\\mathbb\{E\}\_\{\\pi\}\[T\]=t^\{\\star\}<\\infty\. We construct a policyπ′\\pi^\{\\prime\}withJ0​\(π′\)\>J0​\(π\)J\_\{0\}\(\\pi^\{\\prime\}\)\>J\_\{0\}\(\\pi\), contradicting the optimality ofπ\\pi\.

#### Construction ofπ′\\pi^\{\\prime\}\.

Runπ′\\pi^\{\\prime\}using the same action choices asπ\\pi, except that wheneverπ\\piselectsStopwith fewer thanKKsuccessful advances so far,π′\\pi^\{\\prime\}instead selectsAdvanceand continues indefinitely untilnA≥Kn\_\{A\}\\geq Kis reached\.

#### π′\\pi^\{\\prime\}succeeds almost surely\.

On any sample path whereπ\\piwould stop withnA<Kn\_\{A\}<K,π′\\pi^\{\\prime\}continues emitting an infinite sequence of independentAdvanceattempts, each succeeding with probabilityp\>0p\>0\. By the second Borel–Cantelli lemma \(applied to independent events with summed probability∑tp=∞\\sum\_\{t\}p=\\infty\), infinitely many of these attempts succeed almost surely, sonA≥Kn\_\{A\}\\geq Kis eventually reached\. Combined with the sample paths on whichπ\\pialready succeeds, we concludeℙπ′​\[nA≥K\]=1\\mathbb\{P\}\_\{\\pi^\{\\prime\}\}\[n\_\{A\}\\geq K\]=1\.

#### π\\pidoes not succeed almost surely\.

LetM:=∑t=1T𝟏​\[at=Advance\]M:=\\sum\_\{t=1\}^\{T\}\\mathbf\{1\}\[a\_\{t\}=\\textsc\{Advance\}\]denote the total number ofAdvanceactionsπ\\piemits before stopping\. BecauseM≤TM\\leq Tand𝔼π​\[T\]<∞\\mathbb\{E\}\_\{\\pi\}\[T\]<\\infty,MMis almost surely finite\. Conditional onMM,nA∼Bin​\(M,p\)n\_\{A\}\\sim\\mathrm\{Bin\}\(M,p\)\. The event “allMMattempts fail” is contained in\{nA<K\}\\\{n\_\{A\}<K\\\}\(sinceK≥1K\\geq 1\) and has conditional probability\(1−p\)M\>0\(1\-p\)^\{M\}\>0wheneverM<∞M<\\inftyandp<1p<1\. Taking expectations overMM,

ℙπ​\[nA<K\]≥𝔼π​\[\(1−p\)M\]\>0,\\mathbb\{P\}\_\{\\pi\}\[n\_\{A\}<K\]\\;\\geq\\;\\mathbb\{E\}\_\{\\pi\}\[\(1\-p\)^\{M\}\]\\;\>\\;0,soℙπ​\[nA≥K\]<1\\mathbb\{P\}\_\{\\pi\}\[n\_\{A\}\\geq K\]<1\.

#### Conclusion\.

J0​\(π′\)=ℙπ′​\[nA≥K\]=1\>ℙπ​\[nA≥K\]=J0​\(π\)J\_\{0\}\(\\pi^\{\\prime\}\)=\\mathbb\{P\}\_\{\\pi^\{\\prime\}\}\[n\_\{A\}\\geq K\]=1\>\\mathbb\{P\}\_\{\\pi\}\[n\_\{A\}\\geq K\]=J\_\{0\}\(\\pi\), soπ\\piis not optimal\. ∎

## Appendix BExtended Experimental Results

The first two subsections of this appendix \([B\.1](https://arxiv.org/html/2605.23926#A2.SS1)and[B\.2](https://arxiv.org/html/2605.23926#A2.SS2)\) report supporting analyses that were deferred from the main\-text empirical section to stay within the page budget\. Each is consistent with the five main\-text findings and none modifies the headline picture\.

### B\.1Subject\-level analysis:ρ\\rhois negatively correlated with trace length across MATH\-500 subjects, and Intermediate Algebra is low\-ρ\\rhoin every model\.

Table 6:Redundancy and trace length by MATH\-500 subject, four models\.ρ\(%\)\\rho\\,\(\\%\)with95%95\\%bootstrap CI and mean total trace lengthL¯\\bar\{L\}\(words\)\. Subjects ordered by DS\-R1ρ\\rho\. Within each model,ρ\\rhois negatively correlated withL¯\\bar\{L\}\(Spearman rank correlationr∈\[−0\.79,−0\.68\]r\\in\[\-0\.79,\-0\.68\]across the four models; Spearmanrrmeasures monotone association —−1\-1is strictly decreasing,0is unrelated,\+1\+1is strictly increasing\)\. Intermediate Algebra is among the bottom\-threeρ\\rhosubjects for every model\.DS\-R1QwQ\-32BR1\-Distill\-7BQwen3\-30B\-Th\.Subjectρ%\\rho\\%CIL¯\\bar\{L\}ρ%\\rho\\%CIL¯\\bar\{L\}ρ%\\rho\\%CIL¯\\bar\{L\}ρ%\\rho\\%CIL¯\\bar\{L\}Precalculus59\.9\[53\.3, 66\.0\]65291\.0\[86\.0, 94\.7\]145146\.8\[39\.0, 54\.9\]121594\.4\[92\.0, 96\.3\]1946Intermediate Algebra65\.3\[60\.4, 70\.4\]119083\.6\[79\.2, 87\.5\]222745\.0\[37\.3, 52\.6\]175784\.3\[77\.5, 90\.2\]3263Counting & Prob\.65\.4\[56\.1, 74\.0\]160891\.0\[82\.5, 95\.7\]215356\.4\[40\.4, 71\.5\]90684\.4\[71\.7, 93\.2\]3700Algebra70\.7\[67\.4, 74\.1\]46994\.8\[93\.9, 95\.5\]97966\.8\[60\.8, 72\.4\]74594\.2\[92\.9, 95\.3\]1422Number Theory72\.1\[66\.5, 77\.2\]59196\.4\[95\.5, 97\.0\]123658\.2\[47\.4, 68\.0\]101395\.6\[92\.9, 97\.5\]1835Prealgebra74\.3\[69\.4, 78\.8\]29992\.1\[89\.6, 94\.2\]114177\.9\[72\.2, 83\.5\]51090\.8\[87\.7, 93\.5\]1681Geometry74\.8\[68\.9, 81\.0\]57587\.6\[80\.7, 93\.3\]162768\.1\[61\.0, 74\.5\]119590\.1\[78\.9, 96\.4\]2172Spearmanr​\(ρ,L¯\)r\(\\rho,\\bar\{L\}\)−0\.68\-0\.68−0\.79\-0\.79−0\.75\-0\.75−0\.68\-0\.68Before reading the table it helps to keep in mind the overallρ\\rholevel of each model: QwQ and Qwen3 sit near90%90\\%aggregate on MATH\-500, DS\-R1 at69%69\\%, and R1\-Distill\-7B at61%61\\%\(Table[1](https://arxiv.org/html/2605.23926#S5.T1)\)\. Comparing subjects across models therefore needs to distinguish*absolute*ρ\\rho\(how redundant a model is overall\) from*relative*position within a model \(which subjects are that model’s most\- vs\. least\-redundant\)\. Bold entries in Table[6](https://arxiv.org/html/2605.23926#A2.T6)mark each model’s highest and lowest subject\.

Three observations follow\.

*First,ρ\\rhois negatively correlated withL¯\\bar\{L\}inside every model\.*The last row of the table reports the Spearman rank correlation betweenρ\\rhoandL¯\\bar\{L\}for each model separately:−0\.68\-0\.68for DS\-R1,−0\.79\-0\.79for QwQ,−0\.75\-0\.75for R1\-Distill\-7B,−0\.68\-0\.68for Qwen3\. The direction is consistent across all four models: subjects on which a model produces longer traces are the subjects on which it is less redundant\. The mechanism matches the theoretical picture of §[6](https://arxiv.org/html/2605.23926#S6): subjects that require multi\-step symbolic manipulation raise thek⋆k^\{\\star\}floor, leaving less room for the trailingIdlesteps that drive upρ\\rho\.

*Second, Intermediate Algebra is consistently a low\-ρ\\rhosubject across all four models, even though their absoluteρ\\rholevels differ dramatically\.*Intermediate Algebra is the lowest\-ρ\\rhosubject for QwQ \(83\.6%83\.6\\%\), R1\-Distill\-7B \(45\.0%45\.0\\%\), and Qwen3 \(84\.3%84\.3\\%\), and the second\-lowest for DS\-R1 \(65\.3%65\.3\\%\)\. QwQ’s83\.6%83\.6\\%on Intermediate Algebra looks much higher than R1\-Distill\-7B’s45\.0%45\.0\\%, but inside each model both are low\-end subjects relative to that model’s own average\. The effect is the same phenomenon expressed at different absolute scales\. Intermediate Algebra is simultaneously the longest\-trace subject for QwQ, Qwen3, and R1\-Distill\-7B, and the second\-longest for DS\-R1, which is the same observation as the first one seen from the other side\.

*Third, theρ\\rhoranking of subjects is not stable across models, and the cross\-model variability itself has a training\-recipe signature\.*Number Theory, for example, is the*highest*\-ρ\\rhosubject for QwQ \(96\.4%96\.4\\%\) and Qwen3 \(95\.6%95\.6\\%\) but mid\-rank for DS\-R1 \(72\.1%72\.1\\%\) and R1\-Distill\-7B \(58\.2%58\.2\\%\)\. Computing Spearman rank correlation now between*pairs of models*\(treating each model’s 7\-subject ranking as a vector and comparing vectors\), pairwise correlations range from0\.070\.07\(DS\-R1 vs Qwen3\) to0\.890\.89\(DS\-R1 vs R1\-Distill\-7B\); the closest two are precisely teacher and student, consistent with distillation preserving subject\-level behavioural structure\. Subject\-level redundancy is not a pure property of the subject; it interacts with model style\. This is a limit of the “subject difficulty→K→ρ\\to K\\to\\rho” mechanistic reading, and a reminder thatρ\\rhoat the subject level is an interaction between problem structure and trained policy\.

### B\.2Step\-level and word\-level redundancy agree on every one of the88conditions\.

![Refer to caption](https://arxiv.org/html/2605.23926v1/x2.png)Figure 2:Step\-level vs\. word\-level redundancy\.Each point is one \(model, benchmark\) cell from Table[1](https://arxiv.org/html/2605.23926#S5.T1)\. Marker shape encodes benchmark; colour encodes model\. All eight points lie on they=xy=xline up to±5\\pm 5points\.Definition[1](https://arxiv.org/html/2605.23926#Thmdefinition1)provides two natural measures of redundancy: a step\-level ratioρ\\rhoand a word\-level ratioρL\\rho\_\{L\}\. These could in principle diverge — if redundant steps were systematically shorter or longer than informative steps, the two measures would pull apart\. Table[1](https://arxiv.org/html/2605.23926#S5.T1)shows they do not: the per\-row difference\|ρπ−ρL\|\|\\rho\_\{\\pi\}\-\\rho\_\{L\}\|is at most4\.84\.8points \(DS\-R1 GSM8K, whereρπ−ρL=4\.8\\rho\_\{\\pi\}\-\\rho\_\{L\}=4\.8\), and on five of eight conditions it is within±1\.5\\pm 1\.5points\. The two measures rank the eight conditions identically \(Spearman rank correlation=1\.0=1\.0\)\. Step\-level redundancy therefore has a direct token\-economy interpretation: cutting aρ\\rho\-fraction of steps also cuts approximately aρ\\rho\-fraction of tokens in expectation, so the efficiency target a length\-aware trainer should aim at is the same whether formulated per\-step or per\-token\.

### B\.3Prefix\-position ablation

![Refer to caption](https://arxiv.org/html/2605.23926v1/x3.png)Figure 3:Prefix\-position ablation\.For each relative prefix lengthk/N∈\{0\.05,0\.15,…,0\.95\}k/N\\in\\\{0\.05,0\.15,\\ldots,0\.95\\\}we plot judge accuracy under four strategies: first\-kk\(blue\), last\-kk\(red\), middle\-kk\(green\), random\-kk\(orange\), on120120correct MATH\-500 traces per model\. last\-kkreaches100%100\\%accuracy at everykkfor QwQ\-32B and≥96%\\geq 96\\%for DeepSeek\-R1; first\-kkrequiresk≥0\.5​Nk\\geq 0\.5Nto reach75​–​85%75\\text\{\-\-\}85\\%\.The prefix\-position ablation complements the default first\-kktruncation protocol by testing whether redundancy is localised\. If redundancy were uniformly distributed along the trace, all four curves in Figure[3](https://arxiv.org/html/2605.23926#A2.F3)should coincide\. Instead, last\-kkdominates first\-kkby a wide margin on all four models, meaning the judge can recover the answer from the tail of the trace more easily than from the head\. For QwQ\-32B in particular, last\-kkreaches100%100\\%judge accuracy at every relative length from0\.050\.05to0\.950\.95: the last3​–​53\\text\{\-\-\}5segmented steps of a5555\-step trace are informationally sufficient\. Middle\-kkand random\-kkfall between first\-kkand last\-kk, consistent with redundancy concentrating \(though not exclusively\) toward the end of the trace\. This ablation is reported here rather than in the main text because the first\-kkcritical\-point definition already captures the relevant information for our theoretical analysis; the prefix\-position curves are additional structural diagnostics\.

### B\.4Distribution of the critical\-point position \(ECDF\)

![Refer to caption](https://arxiv.org/html/2605.23926v1/x4.png)Figure 4:Empirical cumulative distribution function \(ECDF\) ofk⋆/Nk^\{\\star\}/N, four models×\\timestwo benchmarks\.An ECDF of a quantityXXreads as follows: the value at horizontal coordinatexxis the fraction of traces withX≤xX\\leq x\. So each curve atx=0\.1x=0\.1gives the fraction of traces whose critical prefix lies within the first10%10\\%of the trace\. P50 and P90 markers \(the5050th and9090th percentiles of the distribution, i\.e\. thek⋆/Nk^\{\\star\}/Nvalues below which50%50\\%and90%90\\%of traces fall\) are shown\. Every curve rises steeply in\[0,0\.1\]\[0,0\.1\]: the majority of traces have critical prefixes in the first10%10\\%of the trace\.Figure[4](https://arxiv.org/html/2605.23926#A2.F4)plots the ECDF of the relative critical\-point positionk⋆/Nk^\{\\star\}/Nfor each \(model, benchmark\) pair\. All eight curves rise steeply in\[0,0\.1\]\[0,0\.1\]: the majority of traces havek⋆/N<0\.1k^\{\\star\}/N<0\.1, i\.e\. critical prefixes inside the first10%10\\%of the trace\. QwQ\-32B and Qwen3\-30B on MATH\-500 are the steepest \(more than70%70\\%of traces havek⋆/N<0\.1k^\{\\star\}/N<0\.1\)\. Every \(model, benchmark\) combination except R1\-Distill on GSM8K has P50≤0\.20\\leq 0\.20\(the median trace has a critical prefix within the first20%20\\%of its length\)\. This reinforces Finding 1 at the distributional level: the median “k⋆=1k^\{\\star\}=1segmented step” result is not an average\-vs\-median artefact but a population\-level regularity\.

### B\.5Per\-subject bar view

![Refer to caption](https://arxiv.org/html/2605.23926v1/x5.png)Figure 5:Redundancy by MATH\-500 subject, four models, bar view\.Per\-subjectρ\\rhowith95%95\\%bootstrap CI error bars\. Intermediate Algebra \(the longest traces\) is consistently among the lowest\-ρ\\rhosubjects for every model\.
### B\.6Length–accuracy relationship stratified by difficulty

![Refer to caption](https://arxiv.org/html/2605.23926v1/x6.png)Figure 6:Length–accuracy decile curves, four models, stratified by difficulty level\.Each panel shows one model; coloured lines are per\-level decile curves, thick black is the aggregate\. DS\-R1 and R1\-Distill show the strongest inverted\-U shape; QwQ is weakly monotone; Qwen3\-30B\-Thinking is essentially flat\.The length\-accuracy anti\-correlation is model\-specific\. It is strongest for DS\-R1 and R1\-Distill\-7B \(clean inverted\-U peaking at mid\-length\) and weakest for Qwen3\-Thinking \(flat at the longest deciles\)\. This is consistent with Finding 4 in the main text: long traces from models with large baseline trace lengths \(QwQ, Qwen3\) are typical of their distribution and do not indicate failure, while long traces from DS\-R1 and R1\-Distill fall far outside their typical distribution and correlate with failure\.

### B\.7Positional distribution of redundancy, stratified by difficulty

![Refer to caption](https://arxiv.org/html/2605.23926v1/x7.png)Figure 7:Positional redundancy stratified by difficulty level\.Each panel = one model; coloured lines = per\-level curves; thick black = aggregate\. Early steps \(p≤0\.1p\\leq 0\.1\) are redundant30​–​50%30\\text\{\-\-\}50\\%of the time; late steps \(p≥0\.9p\\geq 0\.9\) exceed90​–​98%90\\text\{\-\-\}98\\%\.
### B\.8Leave\-one\-out critical\-step fractions

Table 7:Critical\-step fraction from leave\-one\-out ablation, four models×\\timestwo benchmarks\.Total ablated steps and number classified as critical \(removal flips the answer\)\. Fraction ranges from0\.0%0\.0\\%\(DS\-R1 on GSM8K: not a single step is individually necessary\) to8\.9%8\.9\\%\(Qwen3 on GSM8K\);\>90%\>90\\%of individual reasoning steps are dispensable everywhere\.Leave\-one\-out is a stricter test than truncation: it asks whether a*specific*individual step is necessary, not whether*some*prefix suffices\. Even so, the critical fraction is≤8\.9%\\leq 8\.9\\%everywhere, and0%0\\%in one \(DS\-R1, GSM8K\) case\. This is a second, strictly different argument for the main Finding 1: more than90%90\\%of individual reasoning steps are individually dispensable, regardless of the training recipe\.

Two supplementary observations\.*First*, MATH\-500 critical fractions are*lower*than GSM8K for three of four models — counter\-intuitive because MATH\-500 is harder\. The explanation: harder traces are longer, so any one step’s marginal contribution is smaller in percentage terms\.*Second*, the finding is not RLVR\-specific: distilled \(R1\-Distill\) and thinking\-mode \(Qwen3\) models have comparable critical fractions to the RLVR\-trained R1 and QwQ\.

### B\.9External\-judge replication

To validate that our main\-text numbers do not depend onπ\\piserving as its own decoder \(Definition[1](https://arxiv.org/html/2605.23926#Thmdefinition1)\), we replicate every MATH\-500 and GSM8K measurement using a fixed external non\-reasoning judge,gpt\-4o\-mini\-2024\-07\-18\. Letkπ⋆k^\{\\star\}\_\{\\pi\}andkext⋆k^\{\\star\}\_\{\\text\{ext\}\}denote the critical points under the two protocols\. Table[2](https://arxiv.org/html/2605.23926#S5.T2)in the main text gives the pairedρπ\\rho\_\{\\pi\}andρext\\rho\_\{\\text\{ext\}\}numbers for all eight conditions; here we report the distributional agreement betweenkπ⋆k^\{\\star\}\_\{\\pi\}andkext⋆k^\{\\star\}\_\{\\text\{ext\}\}on a per\-trace basis\.

Table 8:Per\-trace agreement betweenπ\\pi\-as\-own\-decoder and external judge \(gpt\-4o\-mini\), MATH\-500\.Δ​k⋆=kπ⋆−kext⋆\\Delta k^\{\\star\}=k^\{\\star\}\_\{\\pi\}\-k^\{\\star\}\_\{\\text\{ext\}\}\. Positive meanΔ\\Deltameansπ\\pineeds*more*prefix than the external judge; negative meanπ\\pistops earlier \(its own trace style makes the answer recoverable sooner from shorter prefixes\)\.Three patterns emerge\.*First*, exact\-match rates vary dramatically across models: DeepSeek\-R1 picks the samek⋆k^\{\\star\}as the external judge28\.4%28\.4\\%of the time, while Qwen3 never agrees exactly\. This is consistent with the self–external gap size in Table[2](https://arxiv.org/html/2605.23926#S5.T2): small gap \(DS\-R1\) correlates with frequent agreement, large gap \(Qwen3\) correlates with essentially no agreement\.*Second*, the sign ofΔ​k⋆\\Delta k^\{\\star\}is training\-recipe\-specific\. DS\-R1 and R1\-Distill have meanΔ​k⋆≈0\\Delta k^\{\\star\}\\approx 0\(their own decoder needs approximately as much prefix as the external one\), while QwQ \(−10\.1\-10\.1\) and especially Qwen3 \(−28\.2\-28\.2\) need much*less*prefix under their own decoder\.*Third*, “π\\piearlier” fraction corroborates this: for Qwen3 the model accepts the forced answer from a shorter prefix than the external judge in95\.8%95\.8\\%of traces\. The two protocols do not measure the same physical quantity:ρπ\\rho\_\{\\pi\}captures redundancy relative to howπ\\piitself extracts answers from prefixes, whileρext\\rho\_\{\\text\{ext\}\}captures redundancy relative to a generic reader\. Both are informative; the main text reports both\.

### B\.10Convergence\-detection early stop \(negative result\)

As a cautionary negative result, we tested the simplest possible convergence signal: stop the moment two consecutive segmented steps mention the same trailing number\. On450450DS\-R1 MATH\-500 samples this cuts words by68%68\\%\(1369→4321369\\to 432\) but*halves*accuracy \(72\.5%→32\.5%72\.5\\%\\to 32\.5\\%\), because R1 routinely restates intermediate quantities before doing the final computation\. Naive syntactic stopping rules are therefore not a viable way to exploit the redundancy documented in the main text; the redundancy is*informational*in the sense of Definition[1](https://arxiv.org/html/2605.23926#Thmdefinition1), not*syntactic*\.

### B\.11Difficulty\-aware length budgeting \(post\-hoc analysis\)

This subsection reports a post\-hoc filtering analysis on already\-generated traces\. It is*not*an inference\-time mitigation \(filtering after generation does not reduce the cost of generation\); we include it because the retained\-set accuracy numbers characterise how length\-and\-correctness are coupled in the DS\-R1 MATH\-500 distribution\. The numerical sweep is reported in Appendix[B\.16](https://arxiv.org/html/2605.23926#A2.SS16)together with the shortest\-of\-MManalysis\.

### B\.12Segmentation\-robustness check

A potential worry is thatρ\\rhois sensitive to the segmentation rule and in particular to the word\-count threshold below which short fragments are merged\. We re\-ran the main truncation protocol on the DeepSeek\-R1 MATH\-500 data with merge thresholds\{6,12,18,24\}\\\{6,12,18,24\\\}words\. The resultingρ\\rhoestimates are76\.1%76\.1\\%,76\.3%76\.3\\%,76\.0%76\.0\\%, and75\.7%75\.7\\%respectively — all within±0\.3\\pm 0\.3points of the reported value\. The segmentation choice is therefore not driving the quantitative finding\.

### B\.13Bootstrap CI methodology

All confidence intervals in Tables[1](https://arxiv.org/html/2605.23926#S5.T1),[6](https://arxiv.org/html/2605.23926#A2.T6), and[8](https://arxiv.org/html/2605.23926#A2.T8)are computed via the non\-parametric percentile bootstrap withB=10,000B=10\{,\}000resamples of the trace population, stratified by \(model, benchmark\)\. In this procedure we sample \(with replacement\) a new set of traces from the original set, recomputeρ\\rho, repeatBBtimes, and read the2\.52\.5th and97\.597\.5th percentiles of the resulting distribution as the CI endpoints\. For \(model, benchmark, judge\) cells with fewer than4040traces, we use the bias\-corrected and accelerated \(BCa\) bootstrap instead — a standard refinement of the percentile method that corrects for skew and bias in the bootstrap distribution, giving better coverage when the sample is small\. Resampling is at the*problem*level, not the*trace*level, to avoid artificially narrowing CIs through the within\-problem correlation of traces drawn from the same problem\.

### B\.14Step taxonomy: critical vs redundant, stacked by difficulty level

![Refer to caption](https://arxiv.org/html/2605.23926v1/x8.png)Figure 8:Step taxonomy, four models×\\timesfive MATH\-500 difficulty levels\.Red = critical \(removal flips the judge’s answer\); green = redundant\. Green dominates every panel:\>90%\>90\\%of individual segmented steps are dispensable at every difficulty level and every model\. The small red slivers at a handful of \(model, level\) combinations mark where the few individually\-critical insight steps concentrate\.Figure[8](https://arxiv.org/html/2605.23926#A2.F8)visualises the leave\-one\-out data from Table[7](https://arxiv.org/html/2605.23926#A2.T7)stratified by difficulty level\. The visual dominance of green is the direct message: for every model, at every difficulty level, more than90%90\\%of individual segmented steps can be deleted without changing the answer the judge produces\. The few red slivers that appear — e\.g\. at Level 2 for DS\-R1 and at Level 3 for R1\-Distill — mark the problems where a specific single step carries an irreplaceable key insight\. Notably, critical steps are*not*concentrated at the hardest level: they appear sporadically at intermediate levels, consistent with the theoretical picture that a “key insight” is a problem\-type feature, not a difficulty feature\.

### B\.15Qualitative case studies

To make the redundancy phenomenon concrete we include three short case studies drawn from the DeepSeek\-R1 MATH\-500 trace set; full transcripts with all per\-step labels are in the released supplementary archive\.

#### Case study 1: easy problem, extreme over\-thinking\.

Problemmath500\_0076\(Intermediate Algebra\):*If the domain of the functionlog⁡x2\\log x^\{2\}isx<ax<aorx\>bx\>b, for someaaandbb, finda\+ba\+b\.*The correct answer isa\+b=0a\+b=0\.

- •Segmented steps:N=86N=86\.
- •Critical point\(k⋆k^\{\\star\}\):11\.
- •Redundancy:ρ=98\.84%\\rho=98\.84\\%\.
- •What happens: step11re\-states the problem and establisheslog⁡x2\\log x^\{2\}is defined whenx2\>0x^\{2\}\>0, i\.e\.x≠0x\\neq 0, so the domain is\(−∞,0\)∪\(0,∞\)\(\-\\infty,0\)\\cup\(0,\\infty\)anda=b=0a=b=0\. The judge recovers the answer from step11alone\. Steps2​–​862\\text\{\-\-\}86re\-derive the same conclusion from a dozen different angles, second\-guess whether “log\\log” means natural log or base\-10, hypothesise an alternative parselogx⁡2\\log\_\{x\}2, and eventually return to the original answer\.

#### Case study 2: hard problem, genuinely efficient reasoning\.

Problemmath500\_0071\(Number Theory, base conversion\)\.

- •Segmented steps:N=45N=45\.
- •Critical point\(k⋆k^\{\\star\}\):4545\.
- •Redundancy:ρ=0%\\rho=0\\%\.
- •What happens: every step contributes a new substitution in the base\-conversion algorithm; truncating the trace at any earlier point leaves the judge unable to recover the final answer\. This case is the upper end of theρ\\rhodistribution — a problem where the model is genuinely efficient\.

#### Case study 3: circular self\-reflection\.

Problemmath500\_0026\(Counting & Probability, circular seating with adjacency forbidden\)\.

- •Segmented steps:N=141N=141\.
- •Critical point\(k⋆k^\{\\star\}\):11\.
- •Redundancy:ρ=99\.29%\\rho=99\.29\\%\.
- •What happens: step11states the inclusion–exclusion setup and is sufficient for the judge\. The remaining140140steps are a lengthy double\-count, a retry of the same inclusion–exclusion with different complementary event choices, a verification against a smallern=5n=5case, and several rounds of “let me check one more time\.”

These three cases span the empirical distribution ofρ\\rhofrom near\-0%0\\%to near\-100%100\\%and concretely illustrate the qualitative phenomena \(domain re\-interpretation, redundant verification, circular self\-reflection\) that our aggregate numbers quantify\.

### B\.16Post\-hoc analyses on already\-generated traces

We report two post\-hoc analyses on already\-generated traces for completeness\. Neither is an inference\-time mitigation: both requireMMindependent generations and therefore*increase*wall\-clock compute rather than reducing it\. We include them because the absolute numbers are informative about the theoretically predicted coefficients\.

#### Shortest\-of\-MMcorrect analysis\.

GivenMMindependent correct traces for a problem, picking the shortest of them reduces average length by11\.8%11\.8\\%\(atM=2M=2\) and16\.3%16\.3\\%\(atM=3M=3\) on DeepSeek\-R1 without changing accuracy \(Table[9](https://arxiv.org/html/2605.23926#A2.T9)\)\. This is the empirical counterpart of the scaling law predicted by the within\-problem coefficient of variation \(the empiricalc≈0\.3c\\approx 0\.3predicts≈22%\\approx 22\\%reduction atM=3M=3, in the right ballpark\)\.

Table 9:Shortest\-of\-MMcorrect analysis on DeepSeek\-R1 MATH\-500\.Length reduction holding accuracy fixed, averaged over problems with≥M\\geq Mcorrect samples\. All accuracies are retained\-set accuracy\.
#### Fine\-grained difficulty\-aware budget sweep\.

Recomputing retained\-set accuracy under the budget ruleB​\(d\)=α⋅median​\(L∣d\)B\(d\)=\\alpha\\cdot\\text\{median\}\(L\\mid d\)for DeepSeek\-R1 on MATH\-500 gives Table[10](https://arxiv.org/html/2605.23926#A2.T10)\. Accuracy on the retained set peaks atα=1\.0\\alpha=1\.0\(87\.6%87\.6\\%at343343average tokens, up from79\.8%79\.8\\%at11061106average tokens unconstrained\)\. This is a*post\-hoc*observation of the data already generated, not an inference\-time mitigation\.

Table 10:Difficulty\-aware budget sweep, fine grid \(DeepSeek\-R1 on MATH\-500\)\.Each row reports the retained\-set accuracy and average tokens when traces are filtered byL≤α⋅median​\(L∣d\)L\\leq\\alpha\\cdot\\text\{median\}\(L\\mid d\)per difficulty leveldd\. Peak retained\-set accuracy atα=1\.0\\alpha=1\.0\.

### B\.17Per\-model full\-level table with confidence intervals

Table 11:ρ​\(d\)\\rho\(d\)with95%95\\%bootstrap confidence intervals and sample sizes, by model and MATH\-500 level\.Expanded version of Table[3](https://arxiv.org/html/2605.23926#S5.T3)including CIs andnn\.Table[11](https://arxiv.org/html/2605.23926#A2.T11)is the full version of Table[3](https://arxiv.org/html/2605.23926#S5.T3)with per\-cell bootstrap confidence intervals and sample sizes\. The monotone decrease ofρ​\(d\)\\rho\(d\)with difficulty is visible for every model even when accounting for per\-level sample\-size variation; CIs for Level 1 and Level 5 are non\-overlapping for DeepSeek\-R1, QwQ\-32B, and R1\-Distill\-7B, and nearly so for Qwen3\-30B\-Thinking\.

## Appendix CMethodology Details

### C\.1Segmentation

The segmenter splits on blank\-line boundaries, refines long paragraphs on sentence boundaries next to logical connectors \(“So”, “Therefore”, “Wait”, “Actually”, “Let me”, “Hmm”, “Alternatively”, “However”, “To verify”, “Going back”, “Now”, “First”, “Second”, “Finally”\), and merges pieces shorter than1212words with the previous step\. Segmentation is deterministic\.

### C\.2Forced\-termination prompt

For each \(reasoning model, prefix\) pair theπ\\pi\-as\-own\-decoder measurement concatenates the problem, the prefix, the model’s end\-of\-thinking delimiter, and a short answer prompt, then samples a short completion fromπ\\piatT=0T=0withmax\_tokenscapped at6464:

> \{problem\} <partial reasoning r\_\{1:k\}\> </think\> The final answer is \\boxed\{

The completion is parsed for the first closing`\}`to extract the answer\. For R1, R1\-Distill, and QwQ the end\-of\-thinking token is</think\>; for Qwen3 we use the corresponding Qwen3 thinking delimiter\. The same template with “\(one step has been removed\)” prepended is used for leave\-one\-out ablation\.

#### Truncation\-point sub\-sampling\.

For traces with more than3030segmented steps, evaluatingπ​\(r1:k,x\)\\pi\(r\_\{1:k\},x\)at everyk∈\{1,…,N\}k\\in\\\{1,\\ldots,N\\\}is wasteful — QwQ and Qwen3 traces routinely reachN=50N=50–7070steps\. For such traces we instead evaluate at3030approximately\-evenly\-spaced truncation points, take the smallest sampledkkat which the forced answer is correct as the empiricalk⋆k^\{\\star\}, and treat any unsampledkkbelow it as redundant\-by\-extrapolation\. This introduces an upper bound of⌈N/30⌉\\lceil N/30\\rceilsteps of discretisation error onk⋆k^\{\\star\}, corresponding to a downward bias of at most1/30≈3\.3%1/30\\approx 3\.3\\%onρ\\rho\. Since our headline claims are aboutρ\\rhobeing consistently\>55%\>55\\%, this small downward bias does not threaten any of them\.

#### External\-judge prompt\.

The external\-judge replication usesgpt\-4o\-mini\-2024\-07\-18atT=0T=0:

> You are evaluating a partial mathematical reasoning trace\. Based ONLY on the reasoning provided below, determine the final answer to the problem\. Problem: \{problem\} Partial reasoning: \{prefix\} Output ONLY the final answer in \\boxed\{\} format\.

### C\.3API and compute

DeepSeek\-R1 traces are generated viadeepseek\-reasoneratT=0\.7T=0\.7; QwQ\-32B, R1\-Distill\-7B, and Qwen3\-30B traces via DashScope atT=0\.7T=0\.7\. Theπ\\pi\-as\-own\-decoder truncation protocol reuses the same endpoints atT=0T=0withmax\_tokens=64=64\. External\-judge replication usesgpt\-4o\-mini\-2024\-07\-18atT=0T=0\. All experiments run on a single laptop dispatching async requests at concurrency88\.

### C\.4Reproducibility

All random seeds are fixed \(4242\)\. Scripts are checkpoint\-resumable: each phase appends to a JSONL file and on restart skips completed work\. Full code is released at[https://github\.com/zhiyuanZhai20/how\-much\-thinking\-is\-enough](https://github.com/zhiyuanZhai20/how-much-thinking-is-enough); all correct traces, critical\-point labels, and the prefix\-position ablation data are available from the authors on request\.