BLADE: Boundary-Expanded and Layer-Adaptive Dynamic Exit for Efficient LLM Reasoning

# Boundary-Expanded and Layer-Adaptive Dynamic Exit for Efficient LLM Reasoning
Source: [https://arxiv.org/html/2607.28966](https://arxiv.org/html/2607.28966)
Keshu Fu1\\equalcontrib, Keqin Peng1\\equalcontrib, Jun Bai2\\equalcontrib, Shuhan Qin1, Chen Li1, Junzhu Liang3, Yefei Chen4 Jiaqi Li2, Yuanxin Ouyang1

###### Abstract

Large language models often improve task performance by generating long reasoning traces, but the resulting computation is frequently wasted on redundant verification and revision\. Existing probe\-based early\-exit approaches mainly inspect explicit self\-doubt expressions, leaving many earlier termination opportunities undetected\. Expanding inspection to ordinary reasoning boundaries improves coverage, but also exposes highly diverse intermediate states whose predictive information may reside in different hidden layers\. We present Boundary\-Expanded and Layer\-Adaptive Dynamic Exit for Efficient LLM Reasoning \(BLADE\), a lightweight framework that dynamically terminates reasoning by estimating whether the generated prefix is sufficient for correct answering\. BLADE constructs multi\-granular checkpoints from sentence, self\-doubt, and paragraph boundaries, and derives robust training labels through repeated answer completions\. It further learns a compact subset of informative probe layers instead of relying on fixed choices or expensive representations from all layers\. At inference time, calibrated predictions are combined with checkpoint\-specific confirmation rules to balance responsiveness and premature\-exit risk\. Experiments on five benchmarks and two Qwen3 reasoning models show that BLADE preserves near\-baseline accuracy while reducing generated tokens by 24\.8% on Qwen3\-8B and 15\.8% on Qwen3\-4B\. Ablation studies further confirm the benefits of diverse checkpoints and automatic layer selection, demonstrating an effective approach to more efficient LLM reasoning\.

## 1Introduction

Large Language Models \(LLMs\) achieve strong performance through extended reasoning traces\(Hanet al\.[2026](https://arxiv.org/html/2607.28966#bib.bib25)\)\. Early work showed that Chain\-of\-Thought \(CoT\) prompting improves arithmetic, commonsense, and symbolic reasoning\(Weiet al\.[2022](https://arxiv.org/html/2607.28966#bib.bib11)\)\. Recent studies show that additional inference\-time computation can improve reasoning performance\(Snellet al\.[2025](https://arxiv.org/html/2607.28966#bib.bib12); Wuet al\.[2025](https://arxiv.org/html/2607.28966#bib.bib24)\)\.

However, more computation is not always beneficial\. LLMs may overthink simple problems, repeatedly verify established conclusions, or continue generating after reaching a correct answer\(Chenet al\.[2024](https://arxiv.org/html/2607.28966#bib.bib21); Suiet al\.[2025](https://arxiv.org/html/2607.28966#bib.bib22); Penget al\.[2025](https://arxiv.org/html/2607.28966#bib.bib1)\)\. Such unnecessary reasoning increases inference cost and can degrade answer quality through redundant revision or deviation from a correct solution\(Akgület al\.[2025](https://arxiv.org/html/2607.28966#bib.bib4); Yanget al\.[2026](https://arxiv.org/html/2607.28966#bib.bib2)\)\.

![Refer to caption](https://arxiv.org/html/2607.28966v1/x1.png)Figure 1:A missed early\-exit opportunity: the correct answer appears before “Wait,” but self\-doubt\-only monitoring overlooks it and permits redundant verification\.A practical approach to mitigating overthinking is to enable reasoning early exit with a lightweight probe\(Zhanget al\.[2025a](https://arxiv.org/html/2607.28966#bib.bib3); Akgület al\.[2025](https://arxiv.org/html/2607.28966#bib.bib4)\)\. At selected checkpoints, the probe examines hidden states and predicts whether the current reasoning prefix is sufficient for a correct final answer\. This requires training only a small prediction module, without modifying or retraining the base model\. In this way, prior work uses self\-doubt markers, such as “wait,” “however,” and “let me reconsider,” as natural exit checkpoints\(Fuet al\.[2025](https://arxiv.org/html/2607.28966#bib.bib23); Yanget al\.[2026](https://arxiv.org/html/2607.28966#bib.bib2)\)\. These markers often signal a shift from problem solving to reflection or revision, indicating that a viable solution may already have been reached\. Probing internal confidence at such positions can therefore identify opportunities to terminate unnecessary reasoning\.

However, self\-doubt checkpoints miss many sufficient reasoning states\. As shown in Fig\.[1](https://arxiv.org/html/2607.28966#S1.F1), a regular sentence boundary may already complete a key calculation or establish the correct conclusion before any self\-doubt checkpoint appears\. We therefore expand the checkpoint set to include both self\-doubt markers and sentence boundaries, enabling earlier and broader detection of sufficient prefixes\. This broader coverage introduces heterogeneous reasoning states, ranging from incomplete derivations and intermediate conclusions to reflection and revision\. Their sufficiency signals may be most informative at different model depths, making fixed layer choices difficult to generalize\. Concatenating all layers avoids manual selection but creates a large, redundant probe input with substantial parameter and inference overhead\.

To address these issues, we proposeBLADE, aBoundary\-Expanded andLayer\-AdaptiveDynamicExit framework for efficient LLM reasoning\. BLADE formulates early exit as prefix\-sufficiency prediction and coordinates three decisions: where to inspect the reasoning trajectory, which hidden layers to probe, and when to terminate generation\. Because sparse self\-doubt checkpoints can miss earlier sufficient states,Multi\-Granular Reasoning Checkpoints \(MGRC\)augments them with sentence and paragraph boundaries, providing broader coverage at complementary reasoning granularities\.

The resulting checkpoints represent heterogeneous reasoning states, whose sufficiency signals may be distributed differently across model layers\. To avoid both brittle manual layer choices and costly all\-layer concatenation,Adaptive Probe\-Layer Selection \(APLS\)identifies a compact subset of informative layers for a lightweight sufficiency probe\. Finally, because dense sentence checkpoints are less semantically reliable than sparse self\-doubt checkpoints, BLADE adopts a calibrated, checkpoint\-aware stopping policy: confident self\-doubt predictions can trigger immediate exit, whereas sentence\-level predictions require consecutive confirmation to prevent premature termination\.

Experiments on five mathematical reasoning benchmarks with two Qwen3 backbones\(Yanget al\.[2025](https://arxiv.org/html/2607.28966#bib.bib19)\)show that BLADE achieves a favorable accuracy–efficiency trade\-off among the evaluated early\-exit methods\. Compared with full CoT generation, BLADE reduces generated tokens by24\.8%24\.8\\%on Qwen3\-8B and15\.8%15\.8\\%on Qwen3\-4B while largely preserving accuracy\. Further analyses validate both adaptive designs\. As shown in Supplementary Table 3, on MATH\-500\(Hendryckset al\.[2021](https://arxiv.org/html/2607.28966#bib.bib20); Lightmanet al\.[2024](https://arxiv.org/html/2607.28966#bib.bib29)\)with Qwen3\-8B, combining sentence and self\-doubt checkpoints improves accuracy from84\.89%84\.89\\%to85\.56%85\.56\\%while reducing generated tokens by17\.8%17\.8\\%compared with self\-doubt checkpoints alone\. At approximately86%86\\%accuracy, APLS uses17\.6%17\.6\\%fewer tokens than random four\-layer selection, demonstrating the importance of both broad checkpoint coverage and informative layer selection\.

Our main contributions are as follows:

- •We reveal a fundamental trade\-off in checkpoint design: self\-doubt\-only probing provides limited coverage, whereas broader checkpoint coverage introduces heterogeneous reasoning states\.
- •We propose BLADE, which combines multi\-granular reasoning checkpoints, adaptive probe\-layer selection, and checkpoint\-aware stopping to enable reliable and lightweight dynamic reasoning exit\.
- •Experiments on multiple benchmarks and backbones demonstrate favorable accuracy\-efficiency trade\-offs, reducing generated tokens by up to24\.8%24\.8\\%while largely preserving accuracy\. Further analyses validate the benefits of checkpoint diversity and automatic layer selection\.

![Refer to caption](https://arxiv.org/html/2607.28966v1/x2.png)Figure 2:Overview of BLADE\. MGRC constructs multi\-granular checkpoints and retains unanimous forced\-completion outcomes as prefix\-sufficiency labels\. APLS freezes the dense cross\-layer modeling, learns hard Top\-KKmasks across multiple runs, aggregates the selected subsets, and refits a compact probe\. At inference, the calibrated probe applies checkpoint\-type\-aware temporal stopping\.
## 2Related Work

#### Overthinking in Long\-CoT Reasoning\.

Long\-CoT reasoning can improve performance through additional test\-time computation, but may continue after sufficient information for a correct answer has already been obtained\. Prior work analyzes this overthinking behavior from outcome\- and process\-level perspectives\(Chenet al\.[2024](https://arxiv.org/html/2607.28966#bib.bib21); Suiet al\.[2025](https://arxiv.org/html/2607.28966#bib.bib22)\)\. In particular,Penget al\.\([2025](https://arxiv.org/html/2607.28966#bib.bib1)\)identify self\-doubt\-driven repeated verification as a major source of redundancy and mitigate it through prompting\. However, self\-doubt does not mark every sufficient reasoning state\. BLADE therefore treats it as one signal among multiple checkpoints and performs online prefix\-sufficiency prediction rather than globally shortening reasoning\.

#### Dynamic Early Exit for Efficient LLM Reasoning\.

Existing methods differ in where they inspect reasoning and how they decide to stop\. DEER, LYNX, and DTSR monitor reflection\-related positions, such asWaitand other self\-doubt cues, using trial answers, hidden\-state probes, or prompted sufficiency judgments to trigger early exit\(Yanget al\.[2026](https://arxiv.org/html/2607.28966#bib.bib2); Akgület al\.[2025](https://arxiv.org/html/2607.28966#bib.bib4); Xianget al\.[2026](https://arxiv.org/html/2607.28966#bib.bib5)\)\. Reasoning Path Deviation Monitoring instead tracks high\-entropy token transitions to detect departures from productive reasoning paths\(Guanet al\.[2026](https://arxiv.org/html/2607.28966#bib.bib6)\), while AdaQR performs query\-level routing between lightweight and deeper reasoning modes\(Zhanget al\.[2025b](https://arxiv.org/html/2607.28966#bib.bib32)\)\. These approaches are often restricted to sparse reflection cues, local token statistics, or coarse query\-level decisions\. BLADE combines self\-doubt and sentence boundaries with checkpoint\-aware calibration and temporal confirmation, broadening checkpoint coverage while controlling premature exits\.

#### Prefix\-Sufficiency Probing\.

The hidden states of a reasoning model encode signals predictive of answer correctness, enabling self\-verification and early exit\(Zhanget al\.[2025a](https://arxiv.org/html/2607.28966#bib.bib3)\)\. However, probe effectiveness depends on the layers used: task\-relevant information is distributed unevenly across model depth and may span multiple layers\(Peterset al\.[2018](https://arxiv.org/html/2607.28966#bib.bib7); Tenneyet al\.[2019](https://arxiv.org/html/2607.28966#bib.bib8); de Vrieset al\.[2020](https://arxiv.org/html/2607.28966#bib.bib9)\)\. Using all layers is costly, while fixed subsets may omit informative representations\. BLADE addresses this trade\-off with APLS, which learns a compact, adaptive layer subset through dense cross\-layer modeling and uses only the selected representations at inference\.

## 3Methodology

We formulate reasoning early exit as prefix\-sufficiency prediction: determining whether an intermediate reasoning prefix already supports a reliable correct answer\. As shown in Fig\.[2](https://arxiv.org/html/2607.28966#S1.F2), BLADE consists of two training components and a checkpoint\-aware inference policy\. MGRC constructs diverse candidate states with low\-noise sufficiency labels, while APLS selects a compact set of hidden layers for efficient prediction\. At inference, the resulting probe applies checkpoint\-specific stopping rules to terminate unnecessary reasoning\.

### 3\.1Problem Formulation

Given a problemxix\_\{i\}, a reasoning model generates a token sequence

ri=\(ri,1,…,ri,Ti\)\.r\_\{i\}=\(r\_\{i,1\},\\ldots,r\_\{i,T\_\{i\}\}\)\.\(1\)At a candidate boundarytt, the current reasoning prefix is denoted byri,≤tr\_\{i,\\leq t\}, and its layer\-wise hidden states are

Hi,t=\(hi,t,0,…,hi,t,L−1\),hi,t,ℓ∈ℝd\.H\_\{i,t\}=\\left\(h\_\{i,t,0\},\\ldots,h\_\{i,t,L\-1\}\\right\),\\qquad h\_\{i,t,\\ell\}\\in\\mathbb\{R\}^\{d\}\.\(2\)
We define a binary prefix\-sufficiency variableYi,t∈\{0,1\}Y\_\{i,t\}\\in\\\{0,1\\\}, whereYi,t=1Y\_\{i,t\}=1indicates that the current prefix can reliably support a correct final answer\. A lightweight probe estimates

pθ​\(i,t\)=Pθ​\(Yi,t=1∣Hi,t\)=σ​\(fθ​\(Φ​\(Hi,t\)\)\),p\_\{\\theta\}\(i,t\)=P\_\{\\theta\}\(Y\_\{i,t\}=1\\mid H\_\{i,t\}\)=\\sigma\\\!\\left\(f\_\{\\theta\}\\\!\\left\(\\Phi\(H\_\{i,t\}\)\\right\)\\right\),\(3\)whereΦ\\Phiextracts representations from the compact layer subset selected by APLS\.

### 3\.2Multi\-Granular Checkpoint Supervision

#### Multi\-granular checkpoint construction\.

MGRC constructs training candidates from three complementary types of reasoning checkpoints:

𝒞itrain=𝒞isent∪𝒞idoubt∪𝒞ipara\.\\mathcal\{C\}^\{\\mathrm\{train\}\}\_\{i\}=\\mathcal\{C\}^\{\\mathrm\{sent\}\}\_\{i\}\\cup\\mathcal\{C\}^\{\\mathrm\{doubt\}\}\_\{i\}\\cup\\mathcal\{C\}^\{\\mathrm\{para\}\}\_\{i\}\.\(4\)Sentence checkpoints broadly cover completed calculations and intermediate conclusions, self\-doubt checkpoints capture reflection and verification states, and paragraph checkpoints provide additional coarse\-grained supervision\. Their combination extends the coverage of potentially sufficient states beyond sparse explicit self\-doubt checkpoints\.

#### Low\-noise prefix\-sufficiency supervision\.

For each candidate boundaryt∈𝒞itraint\\in\\mathcal\{C\}^\{\\mathrm\{train\}\}\_\{i\}, we force the model to terminate its current reasoning trajectory and generate an answerN=16N=16times\. Let

bi,t,k=𝕀​\[Check⁡\(a^i,t,k,ai⋆\)=1\],k=1,…,N,b\_\{i,t,k\}=\\mathbb\{I\}\\left\[\\operatorname\{Check\}\\left\(\\hat\{a\}\_\{i,t,k\},a\_\{i\}^\{\\star\}\\right\)=1\\right\],\\qquad k=1,\\ldots,N,\(5\)wherea^i,t,k\\hat\{a\}\_\{i,t,k\}is the answer produced by thekk\-th forced completion andai⋆a\_\{i\}^\{\\star\}is the ground\-truth answer\. We retain only unanimous outcomes:

yi,t=\{0,∑k=1Nbi,t,k=0,1,∑k=1Nbi,t,k=N,⊥,otherwise\.y\_\{i,t\}=\\begin\{cases\}0,&\\sum\_\{k=1\}^\{N\}b\_\{i,t,k\}=0,\\\\ 1,&\\sum\_\{k=1\}^\{N\}b\_\{i,t,k\}=N,\\\\ \\bot,&\\text\{otherwise\}\.\\end\{cases\}\(6\)Candidates labeled with⊥\\botare excluded from probe training, reducing the label noise caused by stochastic completions\.

#### Boundary\-adaptive dynamic exit\.

At inference, BLADE evaluates sentence and self\-doubt checkpoints, which differ in representation and frequency\. Applying a uniform stopping rule can therefore cause premature exits at dense sentence boundaries\. BLADE instead exits immediately after an accepted self\-doubt checkpoint, but requires two consecutive acceptances at sentence checkpoints\. Once triggered, the current prefix is retained and followed by a final\-answer completion; otherwise, reasoning continues to its natural end\.

### 3\.3Adaptive Probe\-Layer Selection

Concatenating all hidden layers captures rich cross\-layer information but incurs substantial redundancy and overhead\. As shown in Table[4](https://arxiv.org/html/2607.28966#S4.T4), the APLS compact probe reduces parameters by approximately 64%, peak memory by 85%, and training time per epoch by 90% relative to dense cross\-layer modeling\. APLS achieves this efficiency by selecting a compact, adaptive subset ofKKlayers through dense modeling, hard Top\-KKselection, and multi\-seed aggregation\.

#### Dense cross\-layer sufficiency modeling\.

Each layer representation is independently normalized and mapped through a shared projection:

ui,t,ℓ=GELU⁡\(Wp​LNℓ⁡\(hi,t,ℓ\)\+bp\)\.u\_\{i,t,\\ell\}=\\operatorname\{GELU\}\\left\(W\_\{p\}\\operatorname\{LN\}\_\{\\ell\}\(h\_\{i,t,\\ell\}\)\+b\_\{p\}\\right\)\.\(7\)The projected features from all layers are concatenated and passed to a dense prediction head:

zi,tT=fT​\(\[ui,t,0;…;ui,t,L−1\]\)\.z\_\{i,t\}^\{T\}=f\_\{T\}\\left\(\[u\_\{i,t,0\};\\ldots;u\_\{i,t,L\-1\}\]\\right\)\.\(8\)The dense model is trained on the unanimous sufficiency labels using class\-balanced binary cross\-entropy\. After training, the dense model is frozen, and its output logits are used to guide the optimization of the layer selector\.

#### Budget\-constrained hard Top\-KKselection\.

After freezing the dense model, each layer is assigned a learnable gate logitαℓ\\alpha\_\{\\ell\}, which is normalized into a soft gate score:

𝝅=softmax⁡\(𝜶\)\.\\boldsymbol\{\\pi\}=\\operatorname\{softmax\}\(\\boldsymbol\{\\alpha\}\)\.\(9\)Under the fixed layer budgetKK, the forward pass constructs a hard binary Top\-KKmask:

mℓH=𝕀​\[ℓ∈TopK⁡\(𝝅,K\)\]\.m\_\{\\ell\}^\{H\}=\\mathbb\{I\}\\left\[\\ell\\in\\operatorname\{TopK\}\\left\(\\boldsymbol\{\\pi\},K\\right\)\\right\]\.\(10\)Since the discrete Top\-KKoperation is not directly differentiable, we use a straight\-through estimator\(Bengioet al\.[2013](https://arxiv.org/html/2607.28966#bib.bib14)\):

mℓ=mℓH\+πℓ−stopgrad⁡\(πℓ\)\.m\_\{\\ell\}=m\_\{\\ell\}^\{H\}\+\\pi\_\{\\ell\}\-\\operatorname\{stopgrad\}\(\\pi\_\{\\ell\}\)\.\(11\)Therefore, exactlyKKlayers are active in the forward pass, while the backward pass propagates gradients to the gate logits through the continuous soft gate scores\.

A temporary selection head evaluates the masked cross\-layer representation:

zi,tM=fM​\(\[m0​ui,t,0;…;mL−1​ui,t,L−1\]\)\.z\_\{i,t\}^\{M\}=f\_\{M\}\\left\(\[m\_\{0\}u\_\{i,t,0\};\\ldots;m\_\{L\-1\}u\_\{i,t,L\-1\}\]\\right\)\.\(12\)The gates and selection head are jointly optimized using class\-balanced classification supervision and knowledge distillation from the frozen dense model\. The selection head is used only during layer search and is discarded afterward\.

#### Stability\-aware multi\-seed aggregation\.

Due to representation redundancy across Transformer layers\(Dalviet al\.[2020](https://arxiv.org/html/2607.28966#bib.bib17)\), independent selection runs may identify different but functionally similar layer subsets\. Let𝒮\(r\)\\mathcal\{S\}^\{\(r\)\}denote the Top\-KKsubset obtained from runrr\. We compute the selection frequency of each layer:

fℓ=1R​∑r=1R𝕀​\[ℓ∈𝒮\(r\)\]\.f\_\{\\ell\}=\\frac\{1\}\{R\}\\sum\_\{r=1\}^\{R\}\\mathbb\{I\}\\left\[\\ell\\in\\mathcal\{S\}^\{\(r\)\}\\right\]\.\(13\)The final adaptive subset consists of theKKmost frequently selected layers:

𝒮⋆=TopKℓ⁡\(fℓ,K\),\\mathcal\{S\}^\{\\star\}=\\operatorname\{TopK\}\_\{\\ell\}\\left\(f\_\{\\ell\},K\\right\),\(14\)with mean gate rank used only to break frequency ties\. After fixing𝒮⋆\\mathcal\{S\}^\{\\star\}, we discard the dense model, gates, and temporary selection head, and refit a compact probe directly on the concatenated raw hidden states of the selected layers\.

MethodGSM8KMATH\-500AMC23AIME24AIME25AverageAcc\.\#Tok\.Acc\.\#Tok\.Acc\.\#Tok\.Acc\.\#Tok\.Acc\.\#Tok\.Acc\.\#Tok\.AESQwen3\-8BBase92\.3170589\.8463988\.7782261\.41195651\.91306476\.878370\.000Ours\-Mixed93\.773385\.6262087\.5561060\.5947652\.21105575\.258960\.213Ours\-Doubt93\.483087\.8380988\.4582761\.71009252\.51143176\.466840\.187LYNX\-K1693\.881184\.2250787\.7562661\.71044851\.51098276\.265200\.188LYNX\-K193\.089287\.3338987\.0496160\.81066552\.21124776\.266500\.163Qwen3\-4BBase85\.3141592\.0432591\.4720558\.31199952\.21314875\.876180\.000Ours\-Mixed92\.470086\.9290788\.9451257\.71138252\.51138675\.664140\.175Ours\-Doubt90\.793788\.7350090\.0512258\.01105652\.21229975\.467570\.112LYNX\-K1690\.597587\.6291889\.3511658\.01041552\.21190375\.668430\.109LYNX\-K190\.996992\.0386390\.0506858\.31138052\.21166275\.967810\.127Table 1:Main results\. Accuracy \(%\), average generated tokens, and AES\. Benchmark\-specific entries use the highest\-AES operating point in the pre\-specifiedδ\\deltagrid; Average follows the protocol in Sec\.[4\.1](https://arxiv.org/html/2607.28966#S4.SS1)\.

## 4Experiments

### 4\.1Setup

#### Benchmarks and models\.

We evaluate BLADE on five mathematical reasoning benchmarks—GSM8K\-test\(Cobbeet al\.[2021](https://arxiv.org/html/2607.28966#bib.bib18)\), MATH\-500\(Lightmanet al\.[2024](https://arxiv.org/html/2607.28966#bib.bib29)\), AMC 2023\(AI\-MO Team[2024](https://arxiv.org/html/2607.28966#bib.bib26)\), AIME 2024\(Zhang and Math\-AI Team[2024](https://arxiv.org/html/2607.28966#bib.bib27)\), and AIME 2025\(Zhang and Math\-AI Team[2025](https://arxiv.org/html/2607.28966#bib.bib28)\)—using Qwen3\-8B and Qwen3\-4B\(Yanget al\.[2025](https://arxiv.org/html/2607.28966#bib.bib19)\)\. The suite contains 1,919 questions, split into 192 calibration and 1,727 held\-out test examples\.

#### Probe training and inference\.

For each backbone, we train the dense teacher and compact probe on a 6,000\-question corpus comprising 2,000 examples each from GSM8K train\(Cobbeet al\.[2021](https://arxiv.org/html/2607.28966#bib.bib18)\), the numeric\-answer subset of MATH train\(Hendryckset al\.[2021](https://arxiv.org/html/2607.28966#bib.bib20)\), and DeepScaleR train\(Luoet al\.[2025](https://arxiv.org/html/2607.28966#bib.bib30)\)\. Data are split at the question level\. Both models are trained for 100 epochs with K16 strict\-clean supervision, while checkpoint and layer selection use only the internal validation split\. At inference, the resulting probe scores sentence and self\-doubt boundaries\. Architecture and labeling details appear in Sec\.[3\.3](https://arxiv.org/html/2607.28966#S3.SS3)\.

#### Calibration, reporting, and metrics\.

For each frozen method, conformal thresholds are estimated on the calibration split atδ∈𝒟=\{0\.002,0\.003,0\.005,0\.01\}\\delta\\in\\mathcal\{D\}=\\\{0\.002,0\.003,0\.005,0\.01\\\}, whereδ\\deltacontrols the stringency of the stopping threshold: smaller values yield more conservative early\-exit decisions\. The resulting thresholds are applied unchanged to the held\-out test split\. We report answer accuracy, average generated response tokens, and Accuracy–Efficiency Score \(AES\)\(Luoet al\.[2026](https://arxiv.org/html/2607.28966#bib.bib10)\); prompt tokens are excluded because they are shared by all methods\. LetppandLLdenote the accuracy and average generated\-token count of an evaluated method, respectively, and letpbp\_\{b\}andLbL\_\{b\}denote those of the corresponding Full\-CoT baseline\. AES is defined as

AES=Lb−LLb\+\{3​p−pbpb,p≥pb,−5​pb−ppb,p<pb\.\\mathrm\{AES\}=\\frac\{L\_\{b\}\-L\}\{L\_\{b\}\}\+\\begin\{cases\}3\\frac\{p\-p\_\{b\}\}\{p\_\{b\}\},&p\\geq p\_\{b\},\\\\\[2\.0pt\] \-5\\frac\{p\_\{b\}\-p\}\{p\_\{b\}\},&p<p\_\{b\}\.\\end\{cases\}\(15\)The first term measures relative token savings, while the second rewards accuracy improvements and penalizes accuracy degradation\. Higher AES therefore indicates a better accuracy–efficiency trade\-off\.

For each benchmark, tables report the operating point with the highest AES over𝒟\\mathcal\{D\}\. The*Average*columns first macro\-average metrics across the five benchmarks at each operating point and then average them over𝒟\\mathcal\{D\}; they are therefore not averages of the displayed benchmark\-wise best points\.

![Refer to caption](https://arxiv.org/html/2607.28966v1/x3.png)Figure 3:Average AES on the five\-benchmark Qwen3\-8B evaluation suite\. Results are equal\-weight macro\-averages across the five benchmarks\.
#### Baselines\.

We compare BLADE with Full\-CoT and LYNX\-style baselines\(Akgület al\.[2025](https://arxiv.org/html/2607.28966#bib.bib4)\)\.LYNX\-K1uses self\-doubt checkpoints, the fixed LYNX layer subset, and forced\-exit labels derived from a single completion\.LYNX\-K16uses the same checkpoints and fixed layers, but replaces the single\-completion labels with K16 strict\-clean supervision\. For layer\-selection ablations, we further compare against final\-layer, validation\-selected single\-layer, fixed four\-layer, random four\-layer, adjacent\-middle, evenly spaced, and all\-layer probes\.

### 4\.2Main Results

#### Accuracy\-preserving token savings\.

Table[1](https://arxiv.org/html/2607.28966#S3.T1)summarizes performance at the best operating point for each benchmark\. On Qwen3\-8B, BLADE achieves the highest average AES \(0\.2130\.213\), exceeding LYNX\-K16 \(0\.1880\.188\) and LYNX\-K1 \(0\.1630\.163\)\. It reduces the average number of generated tokens from7,8377\{,\}837to5,8965\{,\}896, a24\.8%24\.8\\%reduction, while retaining75\.2%75\.2\\%accuracy compared with76\.8%76\.8\\%for Full\-CoT\. These results show that BLADE provides a stronger overall balance between answer accuracy and reasoning cost than the LYNX baselines\.

#### Performance across calibration levels\.

Fig\.[3](https://arxiv.org/html/2607.28966#S4.F3)reports the combined five\-benchmark results at everyδ∈𝒟\\delta\\in\\mathcal\{D\}\. BLADE achieves the strongest AES throughout the evaluated grid, showing that its token\-saving advantage is not tied to a single calibration setting\. In particular, its advantage remains under stricter stopping thresholds, where maintaining accuracy is most important\.

#### Generality across models\.

The same pattern holds for Qwen3\-4B: BLADE achieves an average AES of0\.1750\.175, exceeding LYNX\-K16 \(0\.1090\.109\) and LYNX\-K1 \(0\.1270\.127\)\. It reduces average generated tokens from7,6187\{,\}618to6,4146\{,\}414\(15\.8%15\.8\\%\) while maintaining nearly unchanged average accuracy \(75\.6%75\.6\\%versus75\.8%75\.8\\%for Full\-CoT\)\. Together with the Qwen3\-8B results, this finding shows that BLADE remains effective across the two evaluated Qwen3 backbones rather than relying on the reasoning behavior of a single model\.

MethodGSM8KMATH\-500AMC23AIME24AIME25AverageAcc\.\#Tok\.Acc\.\#Tok\.Acc\.\#Tok\.Acc\.\#Tok\.Acc\.\#Tok\.Acc\.\#Tok\.AESQwen3\-8BBase92\.3170589\.8463988\.7782261\.41195651\.91306476\.878370\.000Ours: selected K493\.773385\.6262087\.5561060\.5947652\.21105575\.258960\.213Final layer92\.568284\.0260683\.6531758\.61009148\.81153171\.65643\-0\.006LYNX\-K493\.373586\.7334487\.7537860\.8984053\.11114175\.160170\.199Best single92\.971284\.9288687\.3546858\.61026951\.91171373\.357590\.100Worst single92\.899187\.6385485\.7657358\.31139748\.11242571\.46460\-0\.162Adjacent\-middle K493\.072985\.1277688\.9611961\.11110650\.31059875\.061660\.159Evenly\-spaced K492\.876484\.0247285\.9524361\.4988152\.51137073\.757550\.151All layers93\.882281\.8222086\.6517260\.81075151\.21204773\.760940\.103Random K4 \(10\-seed avg\.\)92\.672086\.1318187\.5569160\.61021151\.91179974\.659450\.166Qwen3\-4BBase85\.3141592\.0432591\.4720558\.31199952\.21314875\.876180\.000Ours: selected K492\.470086\.9290788\.9451257\.71138252\.51138675\.664140\.175Final layer92\.490789\.1363490\.7605158\.01177751\.21278875\.068750\.054LYNX\-K492\.273688\.9314190\.7575957\.71135552\.81160575\.564200\.169Best single92\.288688\.9370090\.5527258\.61193151\.91268675\.469070\.075Worst single89\.677988\.9382390\.5605057\.41176651\.21277474\.067350\.024Adjacent\-middle K491\.768689\.3297189\.3519857\.71077852\.21205375\.061690\.176Evenly\-spaced K492\.473889\.6351389\.1490457\.41125152\.51170176\.065910\.176All layers92\.876890\.2338190\.7597857\.11135251\.91160375\.464760\.147Random K4 \(10\-seed avg\.\)92\.775588\.7333289\.0487857\.61157351\.41173675\.766090\.151Table 2:Layer\-selection ablation\. Accuracy \(%\), average generated tokens, and AES\. All results follow the reporting protocol in Sec\.[4\.1](https://arxiv.org/html/2607.28966#S4.SS1); Random K4 is averaged over 10 layer seeds before operating\-point selection\.

### 4\.3Layer\-Selection Ablation

Table[2](https://arxiv.org/html/2607.28966#S4.T2)compares BLADE with single\-layer, all\-layer, predefined, and random strategies to assess the value of cross\-layer modeling and automatic compact layer selection\.

#### Comparison with single\-layer probes\.

BLADE substantially outperforms the validation\-selected best single\-layer probe, achieving average AES values of0\.2130\.213versus0\.1000\.100on Qwen3\-8B and0\.1750\.175versus0\.0750\.075on Qwen3\-4B\. Moreover, even the best single\-layer probe underperforms most of the evaluated four\-layer strategies on both backbones\. The gap between the best and worst single\-layer probes \(0\.1000\.100versus−0\.162\-0\.162on Qwen3\-8B and0\.0750\.075versus0\.0240\.024on Qwen3\-4B\) further shows that performance is sensitive to the chosen layer\. Together, these results indicate that a single layer provides an incomplete representation for prefix\-sufficiency prediction, whereas combining complementary information from multiple depths is beneficial\.

![Refer to caption](https://arxiv.org/html/2607.28966v1/x4.png)Figure 4:Runtime\-policy ablations on MATH\-500 with Qwen3\-8B\. All variants use the same frozen probe\. The left panel shows candidate streams, and the right panel shows stopping policies\.
#### Comparison with all\-layer representations\.

Using all layers is not necessarily optimal\. BLADE achieves average AES values of0\.2130\.213and0\.1750\.175on Qwen3\-8B and Qwen3\-4B, respectively, compared with0\.1030\.103and0\.1470\.147for the all\-layer probe\. Thus, simply concatenating representations from every layer does not improve prefix\-sufficiency prediction and may introduce redundant or less task\-relevant features that are harder to model effectively\. As shown in Sec\.[4\.6](https://arxiv.org/html/2607.28966#S4.SS6), retaining dense cross\-layer modeling at deployment incurs substantially greater parameter and hidden\-state access costs\. BLADE instead uses APLS to compress dense cross\-layer information into a compact subset that retains strong early\-exit performance\.

#### Comparison with Fixed and Random Layer Selection\.

BLADE avoids reliance on manually specified layer indices\. On Qwen3\-8B, it achieves the highest average AES among the evaluated strategies \(0\.2130\.213\), outperforming the fixed LYNX\-K4 subset \(0\.1990\.199\) and random four\-layer selection \(0\.1660\.166\)\. On Qwen3\-4B, BLADE reaches0\.1750\.175, exceeding LYNX\-K4 \(0\.1690\.169\) and random selection \(0\.1510\.151\), and remaining comparable to adjacent\-middle and evenly spaced subsets \(0\.1760\.176\)\. Although multiple four\-layer combinations can yield similar representations, BLADE consistently identifies a strong compact subset without manual tuning or random search, supporting automatic layer selection in APLS\.

![Refer to caption](https://arxiv.org/html/2607.28966v1/x5.png)

\(a\) Qwen3\-8B

![Refer to caption](https://arxiv.org/html/2607.28966v1/x6.png)

\(b\) Qwen3\-4B

Figure 5:Layer\-selection frequencies over 10 APLS runs on \(a\) Qwen3\-8B and \(b\) Qwen3\-4B\. Low cross\-run agreement indicates non\-unique layer subsets\.

### 4\.4Runtime\-Policy Ablations

We isolate the candidate stream and temporal stopping policy\. Fig\.[4](https://arxiv.org/html/2607.28966#S4.F4)shows their effects on the MATH\-500 accuracy–token frontier\.

#### Candidate stream\.

Fig\.[4](https://arxiv.org/html/2607.28966#S4.F4)\(a\) compares doubt\-only and mixed candidate streams on MATH\-500 with Qwen3\-8B\. By adding ordinary sentence boundaries, the mixed stream introduces more exit opportunities and improves the high\-accuracy region of the frontier\. This result suggests that self\-doubt checkpoints are informative but too sparse to cover all prefixes that already support a reliable answer, motivating broader checkpoint coverage for safe early exit\.

#### Temporal stopping policy\.

Fig\.[4](https://arxiv.org/html/2607.28966#S4.F4)\(b\) compares temporal stopping policies on MATH\-500 with Qwen3\-8B\. The asymmetric policy exits immediately after a positive self\-doubt prediction but requires two consecutive positive sentence predictions\. It yields a stronger high\-accuracy trade\-off than uniform immediate stopping: temporal confirmation reduces premature exits at heterogeneous sentence boundaries while retaining the broader coverage of the mixed candidate stream\.

Overall, the strongest runtime frontier is obtained by combining mixed candidate coverage and asymmetric temporal stopping\. Thus, candidate expansion alone is insufficient: its benefit depends on how the additional candidates are calibrated and converted into stopping decisions\.

### 4\.5Layer\-Selection Stability Analysis

Fig\.[5](https://arxiv.org/html/2607.28966#S4.F5), together with Table[3](https://arxiv.org/html/2607.28966#S4.T3), shows that independent APLS runs have low layer overlap and near\-zero rank correlations, while their AUROC variance remains consistently small across both model scales\. Panels \(a\) and \(b\) report the layer\-selection frequencies for Qwen3\-8B and Qwen3\-4B, respectively\. The frequency\-aggregated subsets are\[15,19,31,35\]\[15,19,31,35\]for Qwen3\-8B and\[19,21,22,27\]\[19,21,22,27\]for Qwen3\-4B, further indicating that the preferred absolute layer indices are model dependent\.

One plausible explanation is that prefix\-sufficiency information is redundantly distributed across model depth\. Because Transformer residual representations are progressively updated and partially preserve information from earlier layers, several different layer combinations may provide comparable evidence for the same sufficiency decision\. The selection objective may therefore admit multiple functionally similar solutions rather than a single uniquely optimal subset\. Accordingly, the selector should be interpreted as identifying an effective compact representation, rather than recovering a unique set of mechanistically critical layers\.

### 4\.6Efficiency and Resource Analysis

Finally, we examine whether the compact layer subset identified by APLS translates into practical resource savings\. We compare the deployed APLS compact probe with the dense cross\-layer model used during layer selection\. Both architectures are evaluated under the same controlled training protocol on Qwen3\-8B\.

Table 3:Layer\-selection stability across 10 fixed\-split runs\. Pairwise Jaccard and rank correlations are computed over 45 run pairs, and AUROC over individual runs\. Low subset agreement but small AUROC variance indicates stable performance despite non\-unique layer choices\.Table 4:Efficiency comparison between the dense cross\-layer model and the compact APLS probe\.As shown in Table[4](https://arxiv.org/html/2607.28966#S4.T4), the APLS compact probe substantially reduces the parameter count and peak allocated memory compared with the dense cross\-layer model\. It also reduces the per\-epoch training time from approximately 40 seconds to under 4 seconds\. These results show that APLS converts compact probe\-layer selection into concrete computational savings\.

## 5Conclusion

We propose BLADE, a boundary\-expanded, layer\-adaptive framework for dynamic early exit in efficient LLM reasoning\. Experiments on five benchmarks and two Qwen3 backbones show that BLADE reduces generated tokens while largely preserving accuracy\. Ablations show that combining sentence and self\-doubt boundaries uncovers exit opportunities missed by self\-doubt\-only monitoring, while automatic layer selection yields compact representations\. These findings highlight the complementary roles of checkpoint coverage and adaptive layer selection in efficient reasoning\.

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