Adaptive Q-Chunking for Offline-to-Online Reinforcement Learning

# Adaptive Q-Chunking for Offline-to-Online Reinforcement Learning
Source: [https://arxiv.org/html/2605.05544](https://arxiv.org/html/2605.05544)
Nandiraju Gireesh1,2Yuanliang Ju3He Wang1,2,†

1Peking University2Galbot3University of Toronto

###### Abstract

Offline\-to\-online reinforcement learning with action chunking eliminates multi\-step off\-policy bias and enables temporally coherent exploration, but all existing methods use a fixed chunk size across every state\. This is suboptimal: near contact events the agent needs short chunks for reactive control, while during free\-space motion long chunks provide better credit assignment\. The natural solution is to train critics for several chunk sizes and select the best one at each state, but naive comparison of learned critic values systematically collapses to the shortest chunk due to discount\-scale mismatch, and degrades to noise in low\-value states\. We proposeAdaptive Q\-Chunking \(AQC\), which resolves both failures by comparing the advantage of each chunk size relative to a per\-horizon baseline, normalized by the discount factor\. This criterion converts biased wrong answers into unbiased near\-random choices when no genuine signal exists, and becomes discriminative when a particular scale enables better planning\. We prove theoretical bounds on the advantage selector’s noise immunity and on the value dominance of adaptive chunking over any fixed chunk size\. We demonstrate thatAQCachieves state\-of\-the\-art offline and online success rates on OGBench and Robomimic, and can be applied to enhance the performance of large\-scale VLA models that predict action sequences, significantly boosting performance on RoboCasa\-GR1 tasks\.

## 1Introduction

Reinforcement learning \(RL\) promises to learn any task from rewards alone, but starting from scratch is impractical for complex behaviors\[levine2020offline\]\. Offline RL addresses this by learning a fixed policy from pre\-collected data, but must contend with distributional shift and value overestimation\[kumar2020conservative,fujimoto2018addressing,ross2011reduction\]\. A practical approach is offline\-to\-online RL: first, learn from a fixed dataset of demonstrations, then continue improving through online interaction\[nair2020awac,lee2022offline,ball2023efficient,nakamoto2024cal,zhou2025wsrl\]\. This combination works well for many tasks, but struggles in long\-horizon settings where rewards are sparse\[ogbench\_park2024,park2025horizon\]\. The agent must execute hundreds of actions before receiving feedback, making learning slow\. Even worse, single\-step policies tend to behave inconsistently during exploration, rarely reaching the states where rewards occur\[li2025reinforcement\]\.

Action chunking addresses both problems by committing to sequences of actions rather than single steps\[zhao2023learning,chi2023diffusion\]\. A growing body of work incorporates chunks into RL: transformer\-based episodic critics\[li2024top\], coarse\-to\-fine discrete Q\-networks\[seo2024reinforcement,seo2024continuous\], detached value learning for offline RL\[kim2025deas\], and receding\-horizon exploitation\[nagy2026sear\]\. Among these, Q\-chunking \(QC\)\[li2025reinforcement\]is the most directly relevant predecessor: it learns Q\-values over chunks ofhhconsecutive actions, making value estimation stable even when the dataset is imperfect\[fedus2020revisiting\], and executing entire chunks open\-loop produces more coherent exploration\. However,QCuses a single fixed chunk sizehhfor every state and every task\. Decoupled Q\-chunking \(DQC\)\[li2026decoupled\]partially addresses this rigidity by allowing the policy and critic to operate at different chunk sizes, but it does not adapt the chunk sizes per state and requires goal\-conditioned training that does not apply to standard reward\-based tasks\[andrychowicz2017hindsight\]\.

![Refer to caption](https://arxiv.org/html/2605.05544v1/figures/teaser.png)Figure 1:Qualitative rollout of AQC\.Our method adaptively adjusts the commitment horizonk∗k^\{\*\}based on the task phase\. It utilizes long chunks \(k∗=16k^\{\*\}=16\) for efficient movement in free space and automatically switches to fine\-grained control \(k∗=1k^\{\*\}=1\) during complex contact\-rich manipulation\.We argue that the deeper limitation is to use any fixed chunk size at all\. Consider a robot arm moving through free space toward an object: a long action sequence works well here because the motion is predictable\. But near a contact event like grasping or insertion, the same long sequence becomes harmful—small errors accumulate through contact dynamics, and the robot must react to new observations at every step\. No single chunk size handles both situations well, and tuning the right size for each task requires extensive experimentation\.

The natural solution is to train critics for several chunk sizes and pick the best one at each state\. But this naive approach fails for two reasons\. First, critics for shorter chunks produce larger values by construction, so the agent always picks the shortest chunk, even when a longer one would be better\. Second, in states with low expected return, all chunk sizes give nearly identical values, so the selection becomes random noise rather than a meaningful choice\.

We proposeAdaptive Q\-Chunking \(AQC\), which resolves both problems by comparing the*advantage*of each chunk size relative to what the baseline behavior would achieve\. The advantage measures how much better a particular chunk is compared to executing a chunk of that length from the demonstration policy\. This comparison removes both sources of bias: short chunks no longer dominate artificially, and in low\-value states where no chunk is clearly better, the agent defaults to the longest available chunk\. When a short chunk is uniquely helpful near contacts, or a long chunk benefits free\-space motion, the advantage becomes large and guides the selection \([Figure˜1](https://arxiv.org/html/2605.05544#S1.F1)\)\.

The same approach naturally extends to enhancing vision\-language\-action \(VLA\) models\[bjorck2025gr00t\]with offline RL\. We train critics on top of a GR00T N1\.6 backbone on RoboCasa\-GR1 tabletop manipulation tasks\. The VLA natively outputs action chunks;AQCscores multiple candidate chunks at each scale and selects the best prefix via the per\-scale advantage criterion\. This gives the VLA reactive control near contacts while preserving smooth motion in free space, all without modifying the underlying model\.

Our main contributions are:

- •We presentAQC, an offline\-to\-online RL method that adapts chunk sizes per state through a principled advantage\-based criterion\.
- •We provide theoretical analysis establishing advantage separability, value dominance over fixed\-chunk policies, and closed\-loop optimality bounds\.
- •We demonstrate state\-of\-the\-art results on OGBench\[ogbench\_park2024\]and Robomimic\[robomimic2021\], outperforming prior methods across diverse domains\.
- •We show that our approach enhances the performance of VLA policies on RoboCasa\-GR1 tabletop tasks, exceeding prior chunked\-critic methods on contact\-rich manipulation\.

## 2Related Work

#### Offline\-to\-online reinforcement learning\.

Offline\-to\-online RL methods pretrain a policy on a fixed dataset and subsequently refine it through environment interaction\. Prior work addresses the offline\-to\-online value shift via advantage\-weighted regression\[nair2020awac\], implicit Q\-learning\[kostrikov2021offline\], calibrated pessimism\[nakamoto2024cal\], interleaved offline\-online batches\[ball2023efficient\], warmup\-based recalibration\[zhou2025wsrl\], policy expansion\[zhang2023policy\], or reincarnating RL\[agarwal2022reincarnating\]\. While these methods focus on managing the offline\-to\-online transition, our approach instead targets a complementary bottleneck: the inability of existing methods to adjust their planning horizon per state\.

#### Action chunking in reinforcement learning\.

Action chunking predicts and executes sequences of future actions rather than a single action at each step\.QC\[li2025reinforcement\]extends this to TD\-based RL, training a critic over fullhh\-step chunks to eliminate the off\-policy multi\-step bias while producing temporally coherent exploration\. TOP\-ERL\[li2024top\]introduces transformer\-based off\-policy episodic RL with multi\-step returns over short action chunks\.tian2025chunkingcombine chunked critics with soft actor\-critic, learning a transformer critic onnn\-step returns while keeping a single\-step actor\. CQN\-AS\[seo2024reinforcement\]and CQN\[seo2024continuous\]use coarse\-to\-fine Q\-networks with action sequences for data\-efficient RL, factorizing the Q\-function over discrete action bins\. DEAS\[kim2025deas\]detaches value learning from the action sequence for scalable offline RL\. SEAR\[nagy2026sear\]exploits the temporal structure of action chunks with a receding horizon to improve sample efficiency\.DQC\[li2026decoupled\]decouples the chunk size used for the critic from that used for the policy, but requires goal\-conditioned hindsight relabeling\[andrychowicz2017hindsight\]and an additional distillation hyperparameter\. We instead learn partial critics via directkk\-step TD losses bootstrapped from a long\-horizon value estimate, achieving the same fixed point without distillation or goal conditioning, and extends to adaptive chunk size selection at inference\.

#### Multi\-step returns and hierarchical RL\.

Multi\-step TD methods\[hessel2018rainbow\]accelerate credit propagation but introduce off\-policy bias when data is stale\[fedus2020revisiting,kozuno2021revisiting\]\. Retrace\[munos2016safe\]corrects this bias with importance sampling truncation, while action chunking\[li2025reinforcement\]eliminates it by conditioning on the exact behavior\-policy trajectory\.park2025horizonshow formally that longer effective horizons amplify bootstrapping error, motivating our approach of using a long\-horizon value function to provide bootstrap targets for shorter\-horizon critics\. Learning temporally extended actions has been widely studied in hierarchical RL\[dayan1992feudal,dietterich2000hierarchical,kulkarni2016hierarchical,vezhnevets2016strategic,vezhnevets2017feudal,nachum2018data,ajay2020opal,shankar2020learning,pertsch2021accelerating,gehring2021hierarchical,xie2021latent\]and the options framework\[sutton1999between,menache2002q,csimcsek2004using,csimcsek2007betweenness,konidaris2011autonomous,bacon2017option,bagaria2019option,bagaria2024effectively\], though bi\-level optimization remains a persistent challenge\[nachum2018data\]\. We collapse this bi\-level problem into a single\-level objective in a temporally extended action space, where the “high\-level” selection is a deterministic advantage\-based criterion requiring no additional learned parameters\.

## 3Background

Offline\-to\-online RL\.We consider an infinite\-horizon, fully observable Markov decision process \(MDP\)ℳ=\(𝒮,𝒜,T,r,ρ,γ\)\\mathcal\{M\}=\(\\mathcal\{S\},\\mathcal\{A\},T,r,\\rho,\\gamma\), where𝒮\\mathcal\{S\}is the state space,𝒜\\mathcal\{A\}is the action space,T​\(s′∣s,a\):𝒮×𝒜→Δ​\(𝒮\)T\(s^\{\\prime\}\\\!\\mid\\\!s,a\)\\\!:\\\!\\mathcal\{S\}\\\!\\times\\\!\\mathcal\{A\}\\\!\\to\\\!\\Delta\(\\mathcal\{S\}\)is the transition kernel,r​\(s,a\):𝒮×𝒜→ℝr\(s,a\)\\\!:\\\!\\mathcal\{S\}\\\!\\times\\\!\\mathcal\{A\}\\\!\\to\\\!\\mathbb\{R\}is the reward function,ρ∈Δ​\(𝒮\)\\rho\\in\\Delta\(\\mathcal\{S\}\)is the initial state distribution, andγ∈\[0,1\)\\gamma\\in\[0,1\)is the discount factor\. We assume access to an offline dataset𝒟=\{\(s,a,r,s′\)\}\\mathcal\{D\}=\\\{\(s,a,r,s^\{\\prime\}\)\\\}collected by a behavior policyπβ\\pi\_\{\\beta\}\. The goal of offline\-to\-online RL is to find a policyπ:𝒮→Δ​\(𝒜\)\\pi\\\!:\\\!\\mathcal\{S\}\\\!\\to\\\!\\Delta\(\\mathcal\{A\}\)that maximizes the expected discounted returnη​\(π\):=𝔼​\[∑t=0∞γt​r​\(st,at\)\]\\eta\(\\pi\):=\\mathbb\{E\}\\\!\\left\[\\sum\_\{t=0\}^\{\\infty\}\\gamma^\{t\}r\(s\_\{t\},a\_\{t\}\)\\right\], operating in two phases: an*offline phase*that pretrains on𝒟\\mathcal\{D\}, followed by an*online phase*that fine\-tunes with environment interactions added to a growing replay buffer\.

TD learning and the bootstrapping bias problem\.Actor\-critic RL methods\[sutton1998reinforcement,haarnoja2018soft\]learn a criticQϕ​\(s,a\)Q\_\{\\phi\}\(s,a\)via the temporal\-difference \(TD\) loss:

ℒ​\(ϕ\)=𝔼st,at,rt,st\+1∼𝒟​\[\(Qϕ​\(st,at\)−rt−γ​V¯​\(st\+1\)\)2\],\\displaystyle\\mathcal\{L\}\(\\phi\)=\\mathbb\{E\}\_\{s\_\{t\},a\_\{t\},r\_\{t\},s\_\{t\+1\}\\sim\\mathcal\{D\}\}\\\!\\left\[\\bigl\(Q\_\{\\phi\}\(s\_\{t\},a\_\{t\}\)\-r\_\{t\}\-\\gamma\\bar\{V\}\(s\_\{t\+1\}\)\\bigr\)^\{2\}\\right\],\(1\)whereV¯​\(s\)\\bar\{V\}\(s\)is a target value estimate\. In a long\-horizon, sparse\-reward task with effective horizonH=1/\(1−γ\)H=1/\(1\-\\gamma\), the 1\-step backup propagates reward signal backward by only one step per gradient update, requiringO​\(H\)O\(H\)updates before the initial state receives any signal\. Multi\-step return methods\[hessel2018rainbow\]speed up this process by using a length\-nntrajectory segment\(st,at,…,st\+n\)\(s\_\{t\},a\_\{t\},\\ldots,s\_\{t\+n\}\)to construct annn\-step backup:

V^n​\-step:=∑j=0n−1γjrt\+j\+γnQϕ¯\(st\+n,at\+n\),at\+n∼π\(⋅∣st\+n\),\\displaystyle\\hat\{V\}\_\{n\\text\{\-step\}\}:=\\sum\_\{j=0\}^\{n\-1\}\\gamma^\{j\}r\_\{t\+j\}\+\\gamma^\{n\}Q\_\{\\bar\{\\phi\}\}\(s\_\{t\+n\},a\_\{t\+n\}\),\\quad a\_\{t\+n\}\\sim\\pi\(\\cdot\\mid s\_\{t\+n\}\),\(2\)reducing the effective horizon by a factor ofnn\. However, thenn\-step estimator is*biased*when data is off\-policy\[fedus2020revisiting,kozuno2021revisiting\], meaning the cumulative rewards𝐫t:t\+n\\mathbf\{r\}\_\{t:t\+n\}in the replay buffer were collected underπβ\\pi\_\{\\beta\}, not the current policyπ\\pi, so the sum is no longer an unbiased estimate of the on\-policynn\-step return\.

Action chunking Q\-learning\.QC\[li2025reinforcement\]propose to resolve this tension with*action chunking*: instead of training a single\-action criticQ​\(st,at\)Q\(s\_\{t\},a\_\{t\}\), train a*chunked critic*Qh​\(st,𝐚t:t\+h\)Q^\{h\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+h\}\)that takes an entire chunk ofhhconsecutive actions as input\. The chunk TD backup is

ℒh​\(ϕ\)=𝔼​\[\(Qh​\(st,𝐚t:t\+h\)−∑j=0h−1γj​rt\+j−γh​maxi≤N⁡Q¯h​\(st\+h,at\+h:t\+2​h\(i\)\)\)2\],\\displaystyle\\mathcal\{L\}\_\{h\}\(\\phi\)=\\mathbb\{E\}\\\!\\left\[\\\!\\left\(Q^\{h\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+h\}\)\-\\sum\_\{j=0\}^\{h\-1\}\\gamma^\{j\}r\_\{t\+j\}\-\\gamma^\{h\}\\max\_\{i\\leq N\}\\bar\{Q\}^\{h\}\(s\_\{t\+h\},a^\{\(i\)\}\_\{t\+h:t\+2h\}\)\\right\)^\{\\\!2\}\\right\],\(3\)where\{at\+h:t\+2​h\(i\)\}i=1N\\\{a^\{\(i\)\}\_\{t\+h:t\+2h\}\\\}\_\{i=1\}^\{N\}areNNcandidate action chunks sampled fromπβ\\pi\_\{\\beta\}atst\+hs\_\{t\+h\}\. Themax\\maxoverNNsamples, known as EMAQ\[ghasemipour2021emaq\], approximates the best\-of\-NNbootstrap and ensures the critic pushes above the behavior value\. Crucially, because the full chunk𝐚t:t\+h\\mathbf\{a\}\_\{t:t\+h\}is taken directly from the data and held fixed during the backup, thenn\-step bias is*eliminated*: there is no policy mismatch between the actions that generated the rewards𝐫t:t\+h\\mathbf\{r\}\_\{t:t\+h\}and the actions that the critic conditions on\[li2024top,li2025reinforcement\]\.

Implicit value backup via expectile regression\.To avoid explicit action maximization in the TD target, one can learn a value functionVξ​\(s\)V\_\{\\xi\}\(s\)to implicitly approximatemaxa⁡Q​\(s,a\)\\max\_\{a\}Q\(s,a\)using the*expectile loss*\[kostrikov2021offline\]:

ℒV​\(ξ\)=𝔼\(st,𝐚t:t\+h\)∼𝒟​\[fexpκV​\(Q¯h​\(st,𝐚t:t\+h\)−Vξ​\(st\)\)\],\\displaystyle\\mathcal\{L\}\_\{V\}\(\\xi\)=\\mathbb\{E\}\_\{\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+h\}\)\\sim\\mathcal\{D\}\}\\\!\\left\[f^\{\\kappa\_\{V\}\}\_\{\\mathrm\{exp\}\}\\\!\\left\(\\bar\{Q\}^\{h\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+h\}\)\-V\_\{\\xi\}\(s\_\{t\}\)\\right\)\\right\],\(4\)wherefexpκV​\(u\)=\|κV−𝟏\[u<0\]\|​u2f^\{\\kappa\_\{V\}\}\_\{\\mathrm\{exp\}\}\(u\)=\\lvert\\kappa\_\{V\}\-\\mathbf\{1\}\_\{\[u<0\]\}\\rvert\\,u^\{2\}\. At the optimum,Vξ​\(s\)V\_\{\\xi\}\(s\)approximates theκV\\kappa\_\{V\}\-expectile ofQ​\(s,⋅\)Q\(s,\\cdot\)over the data distribution; withκV\\kappa\_\{V\}close to11, this gives an upper envelope that approximatesV∗​\(s\)V^\{\*\}\(s\)under the EMAQ\-boosted critic\.

Flow\-matching behavior cloning\.We parameterizeπβ\\pi\_\{\\beta\}as a flow\-matching model\[lipman2023flow\]following FQL\[park2025flow\], trained with the objective:

ℒBC​\(θ\)=𝔼\(st,𝐚t:t\+h\)∼𝒟,x0∼𝒩​\(0,I\),τ∼𝒰​\[0,1\]​\[‖vθ​\(st,xτ,τ\)−\(𝐚t:t\+h−x0\)‖2\],\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{BC\}\}\(\\theta\)=\\mathbb\{E\}\_\{\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+h\}\)\\sim\\mathcal\{D\},\\,x\_\{0\}\\sim\\mathcal\{N\}\(0,I\),\\,\\tau\\sim\\mathcal\{U\}\[0,1\]\}\\\!\\left\[\\left\\\|v\_\{\\theta\}\(s\_\{t\},x\_\{\\tau\},\\tau\)\-\(\\mathbf\{a\}\_\{t:t\+h\}\-x\_\{0\}\)\\right\\\|^\{2\}\\right\],\(5\)wherexτ=\(1−τ\)​x0\+τ​𝐚t:t\+hx\_\{\\tau\}=\(1\-\\tau\)x\_\{0\}\+\\tau\\,\\mathbf\{a\}\_\{t:t\+h\}is the linear interpolation along the flow path\. At inference,NNcandidate action chunks are drawn by solving the flow ODE fromx0∼𝒩​\(0,I\)x\_\{0\}\\sim\\mathcal\{N\}\(0,I\), and the best is selected by the critic\. This expressive policy class captures the multimodal, temporally correlated structure of manipulation datasets far better than Gaussian policies\[chi2023diffusion,li2025reinforcement\]\.

## 4The Case for Adaptive Action Chunking

BothQC\[li2025reinforcement\]andDQC\[li2026decoupled\]fix the policy chunk size uniformly across all states\. We argue that this is suboptimal: the beneficial chunk size varies across states within a single trajectory, and a globally fixed size is always a compromise\. We further show that the natural approach to adapting the chunk size per state i\.e\. selecting by comparingQkQ^\{k\}values directly is fundamentally broken\.

### 4\.1State\-Dependent Commitment Length

Different phases of a manipulation task require qualitatively different chunk sizes\. During free\-space motion e\.g\., reaching toward a target cube, executing a long action sequence open\-loop is both safe and effective\. The BC flow policy produces temporally coherent trajectories, and longer open\-loop execution accelerates credit assignment by propagating value over more timesteps per update\. Near a contact event such as grasping or placement, the same long\-sequence execution becomes harmful\. Small perturbations at a contact boundary propagate irreversibly through the subsequent physics, so the policy must react to state feedback at each step\. A short chunk size is appropriate here\.

A globally fixed chunk size cannot simultaneously serve both phases\. A large chunk degrades performance near contacts by ignoring state feedback\. A small chunk wastes the credit\-assignment benefit of multi\-step backups during free\-space motion and yields incoherent online exploration\. This motivates maintaining a discrete set𝒦=\{k1,…,k\|𝒦\|\}\\mathcal\{K\}=\\\{k\_\{1\},\\ldots,k\_\{\|\\mathcal\{K\}\|\}\\\}of candidate chunk sizes and selecting the commitment lengthk∗​\(st\)∈𝒦k^\{\*\}\(s\_\{t\}\)\\in\\mathcal\{K\}adaptively at each state\.

### 4\.2Why DirectQkQ^\{k\}Comparison Fails

The natural adaptive selection rule compares the best action chunk across all candidate sizes using the learned critics:

k∗​\(st\),a∗=arg⁡maxk∈𝒦,𝐚t:t\+k∼πβN​Qk​\(st,𝐚t:t\+k\)\.\\displaystyle k^\{\*\}\(s\_\{t\}\),\\;a^\{\*\}\\;=\\;\\underset\{k\\in\\mathcal\{K\},\\;\\mathbf\{a\}\_\{t:t\+k\}\\sim\\pi\_\{\\beta\}^\{N\}\}\{\\arg\\max\}\\;Q^\{k\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\.\(6\)This selector fails due to two compounding problems:

#### Discount\-scale mismatch\.

In sparse\-reward tasks, intermediate rewards are nearly zero for the vast majority of transitions\. The Bellman equation forQkQ^\{k\}therefore simplifies to

Qk​\(st,𝐚t:t\+k\)=∑j=0k−1γj​rt\+j⏟≈0\+γk​Vh​\(st\+k\)≈γk​Vh​\(st\+k\)\.\\displaystyle Q^\{k\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\\;=\\;\\underbrace\{\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\}\_\{\\approx\\;0\}\\;\+\\;\\gamma^\{k\}V^\{h\}\(s\_\{t\+k\}\)\\;\\approx\\;\\gamma^\{k\}V^\{h\}\(s\_\{t\+k\}\)\.\(7\)Sinceγ<1\\gamma<1, the factorγk\\gamma^\{k\}is strictly decreasing inkk\. Consequently,Qk1\>Qk2\>⋯\>QhQ^\{k\_\{1\}\}\>Q^\{k\_\{2\}\}\>\\cdots\>Q^\{h\}for nearly every state, regardless of which chunk size actually yields a better policy\. The selector degenerates to always choosing the smallestkkin𝒦\\mathcal\{K\}\.

#### State\-dependent baseline mismatch\.

Dividing byγk\\gamma^\{k\}to remove the discount factor gives:

arg⁡max𝑘​Qk​\(st,𝐚t:t\+k\)γk≈arg⁡max𝑘​Vh​\(st\+k\)\.\\displaystyle\\underset\{k\}\{\\arg\\max\}\\;\\frac\{Q^\{k\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\}\{\\gamma^\{k\}\}\\;\\approx\\;\\underset\{k\}\{\\arg\\max\}\\;V^\{h\}\(s\_\{t\+k\}\)\.\(8\)This corrects the discount\-scale issue but exposes a subtler one\. In states far from any reward, which constitute the majority of states in sparse\-reward tasks,Vh​\(st\+k\)≈ϵV^\{h\}\(s\_\{t\+k\}\)\\approx\\epsilonfor allk∈𝒦k\\in\\mathcal\{K\}, whereϵ\\epsilonis small\. The differences across chunk sizes are then dominated by function approximation errors in theQkQ^\{k\}networks rather than by genuine differences in planning quality\. The argmax reflects network noise, not which chunk size is actually better\.

Both failure modes stem from the same root cause: comparing rawQkQ^\{k\}values lacks a*per\-scale reference*that accounts for the baseline value achievable at that chunk size under the data distribution\. We resolve this in[Section˜5](https://arxiv.org/html/2605.05544#S5)by subtracting a per\-scale value baselineVk​\(st\)V^\{k\}\(s\_\{t\}\)and normalizing byγk\\gamma^\{k\}, yielding an advantage criterion that avoids both collapse and noise amplification\.

## 5Adaptive Q\-Chunking

AQCselects the commitment lengthk∗​\(st\)∈𝒦k^\{\*\}\(s\_\{t\}\)\\in\\mathcal\{K\}adaptively at each state, re\-querying the policy everyk∗​\(st\)k^\{\*\}\(s\_\{t\}\)steps rather than uniformly everyhhsteps\. We first derive the selection criterion from first principles, then describe how its components are trained jointly\.

![Refer to caption](https://arxiv.org/html/2605.05544v1/x1.png)Figure 2:Overview ofAQC\.\(1\) Train a flow\-BC policyπβ\\pi\_\{\\beta\}alongside a long\-horizon criticQhQ^\{h\}and per\-scale partial critics\{Qki\}\\\{Q^\{k\_\{i\}\}\\\}bootstrapped fromVhV^\{h\}\. \(2\) ScoreNNcandidate chunks at every horizonkkvia the discount\-normalized advantage, z\-score normalized within each scale\. \(3\) Select the best\(k∗,a∗\)\(k^\{\*\},a^\{\*\}\)and execute open\-loop fork∗k^\{\*\}steps in a receding\-horizon loop\. Training and inference details are given in[Sections˜5\.2](https://arxiv.org/html/2605.05544#S5.SS2)and[5\.3](https://arxiv.org/html/2605.05544#S5.SS3)\. Hereppdenotes\|𝒦\|−1\|\\mathcal\{K\}\|\-1### 5\.1The per\-scale advantage criterion

Suppose we maintain critics\{Qk\}k∈𝒦\\\{Q^\{k\}\\\}\_\{k\\in\\mathcal\{K\}\}for a discrete set of chunk sizes𝒦=\{k1,…,h\}\\mathcal\{K\}=\\\{k\_\{1\},\\ldots,h\\\}, and a behavior policyπβ\\pi\_\{\\beta\}that can generate candidate action chunks of length up tohh\. The ideal selection rule would pick the chunk size and action sequence with the best true future return:

k∗​\(st\),a∗∈arg⁡maxk∈𝒦,𝐚t:t\+k∼πβ​Qk,∗​\(st,𝐚t:t\+k\),\\displaystyle k^\{\*\}\(s\_\{t\}\),\\;a^\{\*\}\\;\\in\\;\\underset\{k\\in\\mathcal\{K\},\\;\\mathbf\{a\}\_\{t:t\+k\}\\sim\\pi\_\{\\beta\}\}\{\\arg\\max\}\\;\\;Q^\{k,\*\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\),\(9\)whereQk,∗Q^\{k,\*\}is the optimalkk\-step action\-value function\. Since we only have learned approximationsQkQ^\{k\}and comparing their raw values across differentkksystematically fails \([Section˜4\.2](https://arxiv.org/html/2605.05544#S4.SS2)\), we instead compare the*discount\-normalized advantage*relative to a per\-horizon baseline:

score​\(k,𝐚t:t\+k\):=Qk​\(st,𝐚t:t\+k\)−Vk​\(st\)γk\.\\displaystyle\\mathrm\{score\}\(k,\\mathbf\{a\}\_\{t:t\+k\}\)\\;:=\\;\\frac\{Q^\{k\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\-V^\{k\}\(s\_\{t\}\)\}\{\\gamma^\{k\}\}\.\(10\)HereVk​\(st\)V^\{k\}\(s\_\{t\}\)estimates the expectedkk\-step return of the behavior policy fromsts\_\{t\}\. The subtraction removes state\-dependent scale at horizonkk, and division byγk\\gamma^\{k\}removes the horizon\-dependent discount\. We establish the noise\-immunity of this criterion formally:

###### Proposition 5\.1\(Noise immunity of the advantage selector\)\.

Letδk​\(s\):=Qk​\(s,𝐚t:t\+k\)/γk−Vk​\(s\)/γk\\delta\_\{k\}\(s\):=Q^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)/\\gamma^\{k\}\-V^\{k\}\(s\)/\\gamma^\{k\}denote the discount\-normalized advantage\. In a region whereVh​\(s\)≤ϵV^\{h\}\(s\)\\leq\\epsilonfor all reachable states, and assuming function approximation errors are bounded byσ\\sigma,

\|δk​\(s\)\|≤ϵ\+2​σ\.\\displaystyle\|\\delta\_\{k\}\(s\)\|\\;\\leq\\;\\epsilon\+2\\sigma\.\(11\)Whenϵ≪σ\\epsilon\\ll\\sigma, the advantage is dominated by approximation noise and all chunk sizes score near\-zero\. By contrast, the uncorrected selectorarg​maxk⁡Qk/γk\\operatorname\*\{arg\\,max\}\_\{k\}Q^\{k\}/\\gamma^\{k\}lacks this safeguard: when all scores are≈ϵ\+σk\\approx\\epsilon\+\\sigma\_\{k\}, the argmax picks the scale with the largest positive noiseσk\\sigma\_\{k\}, producing a deterministic but systematically biased choice\.

The key implication: the advantage criterion converts a biased wrong answer into an unbiased near\-random choice when no genuine signal exists, which is strictly preferable\. A full proof appears in Appendix[I\.12](https://arxiv.org/html/2605.05544#A9.SS12)\.

### 5\.2Training

To instantiate[Equation˜10](https://arxiv.org/html/2605.05544#S5.E10), we needQkQ^\{k\}andVkV^\{k\}for eachk∈𝒦k\\in\\mathcal\{K\}, trained jointly alongsideπβ\\pi\_\{\\beta\}\.

#### Long\-horizon critic and value\.

The backbone of the pipeline is the long\-horizon criticQϕh​\(st,𝐚t:t\+h\)Q^\{h\}\_\{\\phi\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+h\}\), trained with the EMAQ\[ghasemipour2021emaq\]hh\-step TD loss:

ℒh​\(ϕ\)=𝔼𝒟​\[\(Qϕh​\(st,𝐚t:t\+h\)−∑j=0h−1γj​rt\+j−γh​maxi≤N⁡Q¯ϕh​\(st\+h,at\+h:t\+2​h\(i\)\)\)2\],\\displaystyle\\mathcal\{L\}\_\{h\}\(\\phi\)=\\mathbb\{E\}\_\{\\mathcal\{D\}\}\\\!\\left\[\\\!\\left\(Q^\{h\}\_\{\\phi\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+h\}\)\-\\sum\_\{j=0\}^\{h\-1\}\\gamma^\{j\}r\_\{t\+j\}\-\\gamma^\{h\}\\max\_\{i\\leq N\}\\bar\{Q\}^\{h\}\_\{\\phi\}\(s\_\{t\+h\},a^\{\(i\)\}\_\{t\+h:t\+2h\}\)\\right\)^\{\\\!2\}\\right\],\(12\)where\{at\+h:t\+2​h\(i\)\}i=1N∼πβ\(⋅∣st\+h\)\\\{a^\{\(i\)\}\_\{t\+h:t\+2h\}\\\}\_\{i=1\}^\{N\}\\sim\\pi\_\{\\beta\}\(\\cdot\\mid s\_\{t\+h\}\)andQ¯h\\bar\{Q\}^\{h\}is the EMA target network\. Correspondingly, we fitVξh​\(s\)V^\{h\}\_\{\\xi\}\(s\)via expectile regression onQhQ^\{h\}:

ℒVh​\(ξ\)=𝔼𝒟​\[fexpκV​\(Q¯ϕh​\(st,𝐚t:t\+h\)−Vξh​\(st\)\)\]\.\\displaystyle\\mathcal\{L\}\_\{V^\{h\}\}\(\\xi\)=\\mathbb\{E\}\_\{\\mathcal\{D\}\}\\\!\\left\[f^\{\\kappa\_\{V\}\}\_\{\\mathrm\{exp\}\}\\\!\\left\(\\bar\{Q\}^\{h\}\_\{\\phi\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+h\}\)\-V^\{h\}\_\{\\xi\}\(s\_\{t\}\)\\right\)\\right\]\.\(13\)WithκV​close to​1\\kappa\_\{V\}\\ \\text\{close to\}\\ 1,VξhV^\{h\}\_\{\\xi\}approximates the upper envelope ofhh\-step returns, serving both as the baseline fork=hk=hand as the bootstrap target for all partial critics\.

#### Partial critics and per\-scale baselines\.

For eachk∈𝒦∖\{h\}k\\in\\mathcal\{K\}\\setminus\\\{h\\\}, we train a partial criticQψk​\(st,𝐚t:t\+k\)Q^\{k\}\_\{\\psi\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)that takes only thekk\-step prefix as input:

ℒk​\(ψ\)=𝔼𝒟​\[\(Qψk​\(st,𝐚t:t\+k\)−∑j=0k−1γj​rt\+j−γk​V¯ξh​\(st\+k\)\)2\]\.\\displaystyle\\mathcal\{L\}\_\{k\}\(\\psi\)=\\mathbb\{E\}\_\{\\mathcal\{D\}\}\\\!\\left\[\\\!\\left\(Q^\{k\}\_\{\\psi\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\-\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\-\\gamma^\{k\}\\bar\{V\}^\{h\}\_\{\\xi\}\(s\_\{t\+k\}\)\\right\)^\{\\\!2\}\\right\]\.\(14\)UsingVhV^\{h\}as the bootstrap target is a deliberate design choice\. The long\-range value fromst\+ks\_\{t\+k\}onward is already encoded inVhV^\{h\}, soQkQ^\{k\}only needs to fit thekk\-step residual\. A 1\-step value function would require chainingO​\(H\)O\(H\)TD updates to propagate credit, accumulating error at each step\. A formal bound showing thatVhV^\{h\}bootstrap yields tighter sub\-optimality guarantees appears in Proposition[H\.21](https://arxiv.org/html/2605.05544#A8.Thmtheorem21)\(Proof[I\.13](https://arxiv.org/html/2605.05544#A9.SS13)\)\.

EachQkQ^\{k\}is a separate network; it cannot be recovered by truncatingQhQ^\{h\}, sinceQhQ^\{h\}is conditioned on the fullhh\-step chunk\. We then fit per\-scale baselinesVζk​\(s\)V^\{k\}\_\{\\zeta\}\(s\)via expectile regression on the corresponding partial critics:

ℒVk​\(ζ\)=𝔼𝒟​\[fexpκV​\(Q¯ψk​\(st,𝐚t:t\+k\)−Vζk​\(st\)\)\]\.\\displaystyle\\mathcal\{L\}\_\{V^\{k\}\}\(\\zeta\)=\\mathbb\{E\}\_\{\\mathcal\{D\}\}\\\!\\left\[f^\{\\kappa\_\{V\}\}\_\{\\mathrm\{exp\}\}\\\!\\left\(\\bar\{Q\}^\{k\}\_\{\\psi\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\-V^\{k\}\_\{\\zeta\}\(s\_\{t\}\)\\right\)\\right\]\.\(15\)These networks share the same architecture asVξhV^\{h\}\_\{\\xi\}and add negligible training overhead\.

#### Behavior policy\.

We trainπβ\\pi\_\{\\beta\}with the flow\-matching BC objective \([Equation˜5](https://arxiv.org/html/2605.05544#S3.E5)\) on the fullhh\-step chunk distribution\.

### 5\.3Inference

At each statests\_\{t\},AQCsamplesNNfull\-length candidate chunks fromπβ\\pi\_\{\\beta\}and for each chunk, evaluates the per\-scale advantage score \(Eq\.[10](https://arxiv.org/html/2605.05544#S5.E10)\) across all horizonsk∈𝒦k\\in\\mathcal\{K\}\. Because the raw advantage scores have different variance profiles across scales, we z\-score normalize within eachkkbefore comparison:

score~​\(k,a\(i\)\)=score​\(k,a\(i\)\)−𝔼i​\[score​\(k,a\(i\)\)\]Vari​\[score​\(k,a\(i\)\)\]\+ϵ\.\\displaystyle\\tilde\{\\mathrm\{score\}\}\(k,a^\{\(i\)\}\)\\;=\\;\\frac\{\\mathrm\{score\}\(k,a^\{\(i\)\}\)\-\\mathbb\{E\}\_\{i\}\[\\mathrm\{score\}\(k,a^\{\(i\)\}\)\]\}\{\\sqrt\{\\mathrm\{Var\}\_\{i\}\[\\mathrm\{score\}\(k,a^\{\(i\)\}\)\]\}\+\\epsilon\}\.\(16\)This prevents a scale with higher output variance from dominating the argmax simply due to spread rather than genuine advantage\. The selected chunka∗a^\{\*\}is executed open\-loop fork∗k^\{\*\}steps, and the policy is re\-queried atst\+k∗s\_\{t\+k^\{\*\}\}\. During online fine\-tuning, new transitions are added to a replay buffer and all losses continue with mixed offline\-online batches\.

puzzle\-3x3scenecube\-doublecube\-triplecube\-quad\.Overall\(5 tasks\)\(5 tasks\)\(5 tasks\)\(5 tasks\)\(5 tasks\)\(25 tasks\)1\-step TD\\cellcolorcpurple\!10IQL\[kostrikov2021offline\]\\cellcolorcpurple\!100→\\\!\\to\\\!20\\cellcolorcpurple\!100→\\\!\\to\\\!39\\cellcolorcpurple\!100→\\\!\\to\\\!0\\cellcolorcpurple\!100→\\\!\\to\\\!0\\cellcolorcpurple\!100→\\\!\\to\\\!0\\cellcolorcpurple\!180→\\\!\\to\\\!12\\cellcolorcpurple\!10RLPD\[ball2023efficient\]\\cellcolorcpurple\!10–→\\\!\\to\\\!100\\cellcolorcpurple\!10–→\\\!\\to\\\!94\\cellcolorcpurple\!10–→\\\!\\to\\\!99\\cellcolorcpurple\!10–→\\\!\\to\\\!41\\cellcolorcpurple\!10–→\\\!\\to\\\!0\\cellcolorcpurple\!18–→\\\!\\to\\\!67\\cellcolorcpurple\!10FQL\[park2025flow\]\\cellcolorcpurple\!1099→\\\!\\to\\\!100\\cellcolorcpurple\!1052→\\\!\\to\\\!95\\cellcolorcpurple\!1032→\\\!\\to\\\!76\\cellcolorcpurple\!102→\\\!\\to\\\!18\\cellcolorcpurple\!100→\\\!\\to\\\!3\\cellcolorcpurple\!1837→\\\!\\to\\\!58\\arrayrulecolorblack\!25nn\-step TD\\cellcolorcpurple\!10FQL\-n\\cellcolorcpurple\!1099→\\\!\\to\\\!100\\cellcolorcpurple\!1021→\\\!\\to\\\!70\\cellcolorcpurple\!109→\\\!\\to\\\!77\\cellcolorcpurple\!101→\\\!\\to\\\!1\\cellcolorcpurple\!107→\\\!\\to\\\!36\\cellcolorcpurple\!1827→\\\!\\to\\\!57\\arrayrulecolorblack\!25Q\-chunking\\cellcolorqccolor\!10QC\-FQL\[li2025reinforcement\]\\cellcolorqccolor\!1064→\\\!\\to\\\!100\\cellcolorqccolor\!1086→\\\!\\to\\\!99\\cellcolorqccolor\!1042→\\\!\\to\\\!100\\cellcolorqccolor\!103→\\\!\\to\\\!53\\cellcolorqccolor\!102→\\\!\\to\\\!77\\cellcolorqccolor\!1638→\\\!\\to\\\!86\\cellcolorqccolor\!10QC\[li2025reinforcement\]\\cellcolorqccolor\!10100→\\\!\\to\\\!100\\cellcolorqccolor\!1082→\\\!\\to\\\!99\\cellcolorqccolor\!1066→\\\!\\to\\\!98\\cellcolorqccolor\!105→\\\!\\to\\\!64\\cellcolorqccolor\!103→\\\!\\to\\\!73\\cellcolorqccolor\!1652→\\\!\\to\\\!86\\cellcolorqccolor\!10DQC\[li2026decoupled\]\\cellcolorqccolor\!1086→\\\!\\to\\\!97\\cellcolorqccolor\!1056→\\\!\\to\\\!89\\cellcolorqccolor\!1015→\\\!\\to\\\!55\\cellcolorqccolor\!101→\\\!\\to\\\!0\\cellcolorqccolor\!102→\\\!\\to\\\!20\\cellcolorqccolor\!1632→\\\!\\to\\\!52\\arrayrulecolorblackOurs\\cellcolorourcolor\!10AQC\\cellcolorourcolor\!10100→\\\!\\to\\\!100\\cellcolorourcolor\!1098→\\\!\\to\\\!100\\cellcolorourcolor\!1090→\\\!\\to\\\!100\\cellcolorourcolor\!1013→\\\!\\to\\\!90\\cellcolorourcolor\!109→\\\!\\to\\\!88\\cellcolorourcolor\!1662→\\\!\\to\\\!96\\arrayrulecolorblack

Table 1:Offline\-to\-online RL results on OGBench\.Each cell reports*offline→\\,\\to\\,online*success rate \(%\) averaged over 5 tasks and 4 seeds\. Numbers in bold are within 95% of the best in each column\. Full task\-level breakdowns with mean and standard deviations are in Appendix[E](https://arxiv.org/html/2605.05544#A5)\.![Refer to caption](https://arxiv.org/html/2605.05544v1/x2.png)Figure 3:Robomimic results\.Success rate vs\. environment steps on three tasks\. The first 1M steps are offline and the next 1M steps are online\. \(5 seeds\)
## 6Experimental Results

We conduct experiments to analyze the empirical effectiveness ofAQCon a range of long\-horizon, sparse\-reward domains\. In particular, we aim to answer the following questions:

1. \(Q1\)*How well doesAQCcompare to prior offline\-to\-online RL methods?*
2. \(Q2\)*CanAQCenhance large\-scale VLAs with offline RL?*
3. \(Q3\)*Which components ofAQCare most critical?*

### 6\.1Environments and Datasets

We consider long\-horizon, sparse\-reward manipulation domains across three benchmark suites\. For OGBench\[ogbench\_park2024\], we use a total of five domains\. Each domain provides five tasks of increasing difficulty; we report success rates averaged across all five tasks and four seeds, with 95% stratified bootstrap confidence intervals\. We evaluate onscene\-sparse,puzzle\-3x3\-sparse,cube\-double,cube\-triple, andcube\-quadruple\. We use datasets ranging from 1M to 100M transitions based on task difficulty\. For Robomimic\[robomimic2021\], we use three tasks:lift,can,square; with multi\-human datasets, evaluated over five seeds\. Additionally, we evaluateAQCby finetuning GR00T N1\.6\[bjorck2025gr00t\]using offline RL on the 24 RoboCasa\-GR1 tabletop manipulation tasks\[nasiriany2024robocasa\]\. See Appendix[C](https://arxiv.org/html/2605.05544#A3)for more details about experiments setup and dataset\.

### 6\.2Comparisons

We compare against prior methods that speed up value backup as well as the previous best offline\-to\-online RL methods:IQL\[kostrikov2021offline\],RLPD\[ball2023efficient\],FQL\[park2025flow\],FQL\-n,QC,QC\-FQL\[li2025reinforcement\], andDQC\[li2026decoupled\]\.

### 6\.3How well doesAQCcompare to prior methods?

[Section˜5\.3](https://arxiv.org/html/2605.05544#S5.SS3)reports offline and online success rates across all five domains of OGBench\.AQCachieves the highest online success rates on the three hardest domains \(cube\-double,cube\-triple,cube\-quadruple\)\. On the easier domains \(scene\-sparse,puzzle\-3x3\), all chunking methods converge to near\-perfect online performance\. Across all domains,AQCconsistently outperforms bothQCandDQC, with the gap widening as horizon length increases\.[Figure˜3](https://arxiv.org/html/2605.05544#S5.F3)reports training curves onlift,can, andsquarefrom Robomimic\.AQCmatches or exceedsQCon all three tasks\. The gains are most pronounced onsquare, which involves a contact\-rich peg\-insertion step and directly illustrates the core motivation of our method: during the pre\-grasp reaching phase, the motion is predictable andAQCselects longer chunks for temporally coherent trajectory; but once the nut contacts the peg, small misalignments require rapid corrective feedback, andAQCadaptively switches to shorter chunks to react to state changes\. Onliftandcan, which involve simpler pick\-and\-place motions with fewer contact transitions, the advantage of adaptive selection is smaller but still non\-negative, suggesting that even in these tasks there are phases where committing to shorter or longer chunks is beneficial\.

### 6\.4CanAQCenhance large\-scale VLAs with offline RL?

[Figure˜4](https://arxiv.org/html/2605.05544#S6.F4)evaluatesAQCon RoboCasa\-GR1 tabletop manipulation tasks\[nasiriany2024robocasa\], a challenging multi\-task setting that tests whether our method generalizes to large\-scale vision\-language\-action models\. The 24 tasks span diverse contact\-rich manipulation skills including picking, placing, opening containers, and operating household appliances\. These tasks are particularly demanding for RL fine\-tuning: VLAs are heavily overparameterized, the state\-action space is high\-dimensional, and collecting additional online data is impractical, making effective offline RL from limited suboptimal rollouts essential\.

We use GR00T N1\.6\[bjorck2025gr00t\]as the base VLA, which autoregressively generates 16\-step action chunks conditioned on the current observation\. Our protocol has two stages\. First, we finetune GR00T N1\.6 via supervised learning on 24k expert demonstrations spanning all 24 tasks, yielding a multi\-task behavior policy\. We deploy this policy to collect 300 offline rollouts per task\. Then, we fine\-tune the base policy using behavior cloning on the combined expert \+ rollout dataset and use the model as an actor for training critic functions when necessary \(Filtered BC\)\.

![Refer to caption](https://arxiv.org/html/2605.05544v1/x3.png)Figure 4:Enhancing GR00T N1\.6 on RoboCasa\-GR1 tasks\.Success rates across 24 tabletop manipulation tasks\. We compare the base VLA against Filtered BC,QC,DQC, andAQC\.ForAQC, we seth=16h=16to match GR00T N1\.6’s native chunk size and use𝒦=\{1,4,8,16\}\\mathcal\{K\}=\\\{1,4,8,16\\\}as candidate commitment horizons\. As shown in[Figure˜4](https://arxiv.org/html/2605.05544#S6.F4),AQCoutperforms bothQCandDQCacross all 24 tasks, with the largest margins on tasks involving multi\-stage contact transitions \(e\.g\., opening a cabinet then manipulating its contents\)\. Filtered BC provides modest improvements over the base VLA but is fundamentally limited by its inability to leverage value information from suboptimal transitions\. These results suggest thatAQCscales effectively to large policies and high\-dimensional observations, and that adaptive chunk selection remains beneficial even when the underlying policy is a multi\-task VLA rather than a single\-task expert\.

### 6\.5What makesAQCwork?

![Refer to caption](https://arxiv.org/html/2605.05544v1/x4.png)Figure 5:Advantage criterion ablationof selection criteria oncube\-double\(left\) andcube\-triple\(right\)\. Shaded regions are 95% confidence intervals over 4 seeds\.
![Refer to caption](https://arxiv.org/html/2605.05544v1/x5.png)Figure 6:Multi\-scale critics and adaptive selection ablationoncube\-double\(left\), andcube\-triple\(right\)\. Shaded regions are 95% confidence intervals over 4 seeds\.

We run ablations oncube\-doubleandcube\-triplewith 4 seeds each\. The core design inAQCis the per\-scale advantage selector\.[Figure˜6](https://arxiv.org/html/2605.05544#S6.F6)compares it against discount\-corrected Q values without baselines, raw Q values, and random selection\. The raw\-Q variant collapses to always selectingk=1k=1, confirming the discount\-scale collapse from[Section˜4](https://arxiv.org/html/2605.05544#S4)\. The discount\-corrected variant partially recovers on easier tasks but degrades sharply on longer\-horizon ones where function approximation noise dominates without a per\-scale baseline\.[Figure˜6](https://arxiv.org/html/2605.05544#S6.F6)decomposes gains from \(A\) training multiple per\-scale criticsQkQ^\{k\}and \(B\) adaptivek∗k^\{\*\}selection at inference\. Multi\-critic training alone improves performance over single\-critic baselines, confirming that richer value representation helps even without adaptation\. Adaptive selection adds further gains on top, with the effect compounding on longer\-horizon tasks where contact phases dominate\. Additional ablations are in Appendix[G](https://arxiv.org/html/2605.05544#A7)\.

## 7Conclusion

We proposedAdaptive Q\-Chunking \(AQC\), an offline\-to\-online RL method that adapts action chunk sizes per state via a discount\-normalized advantage criterion\. By comparing each candidate chunk size to the baseline policy at that horizon,AQCresolves the bias toward shorter chunks and the random selection in low\-value states that plague naive multi\-chunk methods\. We proved advantage separability, value dominance over fixed\-chunk policies, and closed\-loop optimality bounds\. Our empirical results demonstrate state\-of\-the\-art performance across OGBench, Robomimic and RoboCasa\-GR1\. Beyond its immediate results,AQCsuggests a broader principle for offline RL: when multiple learned value functions exist, whether for different horizons, abstractions, or representation levels, their raw values are incomparable without a per\-scale reference\. The advantage\-based criterion provides a principled way to convert these incomparable quantities into a shared decision space\. We hope this perspective inspires future work on adaptive commitment horizons and multi\-scale value\-based decision making\.

## References

## Appendix ALimitations and Future Work

Despite its empirical effectiveness, our method has several limitations that suggest directions for future work\. First, the method selects from a fixed discrete candidate set𝒦\\mathcal\{K\}, which requires some domain knowledge to construct and cannot discover chunk sizes outside the predefined set; learning𝒦\\mathcal\{K\}itself—for instance through a continuous relaxation of the commitment horizon or a state\-conditioned proposal mechanism—would remove this design choice\. Second, like all action\-chunking methods, our approach executes selected chunks open\-loop, meaning evenk=1k\{=\}1chunks inherit the limited reactivity of the flow\-BC policyπβ\\pi\_\{\\beta\}, which was trained to producehh\-step sequences; integrating a closed\-loop low\-level executor for contact\-rich phases could address this gap\. Third, the critic training overhead scales linearly with\|𝒦\|\|\\mathcal\{K\}\|, and while small in practice, weight\-sharing schemes across scales such as a single multi\-scale critic with horizon\-embedding inputs could improve efficiency for large candidate sets\. The selection also depends on the quality and diversity ofπβ\\pi\_\{\\beta\}as a candidate source and on z\-score normalization acrossNNsamples, both of which affect robustness whenNNis small or the policy distribution shifts during online fine\-tuning\.

## Appendix BAlgorithm

Algorithm 1AQC1:Chunk sizes𝒦=\{k1,…,h\}\\mathcal\{K\}=\\\{k\_\{1\},\\ldots,h\\\}, samplesNN, expectileκV\\kappa\_\{V\}, warmup

2:Initialize criticsQϕhQ^\{h\}\_\{\\phi\},\{Qψk\}k∈𝒦∖\{h\}\\\{Q^\{k\}\_\{\\psi\}\\\}\_\{k\\in\\mathcal\{K\}\\setminus\\\{h\\\}\}

3:Initialize valuesVξhV^\{h\}\_\{\\xi\},\{Vζk\}k∈𝒦∖\{h\}\\\{V^\{k\}\_\{\\zeta\}\\\}\_\{k\\in\\mathcal\{K\}\\setminus\\\{h\\\}\}

4:

Initialize flow policyπβ\\pi\_\{\\beta\}

5:

6:Offline pretraining:

7:whilenot convergeddo

8:Sample minibatch\(st,𝐚t:t\+h,∑j=0h−1γj​rt\+j,st\+h\)∼𝒟\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+h\},\\sum\_\{j=0\}^\{h\-1\}\\gamma^\{j\}r\_\{t\+j\},s\_\{t\+h\}\)\\sim\\mathcal\{D\}

9:UpdateQϕhQ^\{h\}\_\{\\phi\}via[Equation˜12](https://arxiv.org/html/2605.05544#S5.E12)\(EMAQhh\-step TD\)

10:UpdateVξhV^\{h\}\_\{\\xi\}via[Equation˜13](https://arxiv.org/html/2605.05544#S5.E13)\(expectile regression onQhQ^\{h\}\)

11:for allk∈𝒦∖\{h\}k\\in\\mathcal\{K\}\\setminus\\\{h\\\}do

12:UpdateQψkQ^\{k\}\_\{\\psi\}via[Equation˜14](https://arxiv.org/html/2605.05544#S5.E14)\(kk\-step TD \+VhV^\{h\}bootstrap\)

13:UpdateVζkV^\{k\}\_\{\\zeta\}via[Equation˜15](https://arxiv.org/html/2605.05544#S5.E15)\(expectile regression onQkQ^\{k\}\)

14:Updateπβ\\pi\_\{\\beta\}via flow\-matching BC \([Equation˜5](https://arxiv.org/html/2605.05544#S3.E5)\)

15:

16:Online fine\-tuning:

17:Initialize replay bufferℛ←𝒟\\mathcal\{R\}\\leftarrow\\mathcal\{D\}

18:foreach environment stepttdo

19:Observests\_\{t\}

20:SampleNNfull\-length chunks\{a1:h\(i\)\}i=1N∼πβ\(⋅∣st\)\\\{a^\{\(i\)\}\_\{1:h\}\\\}\_\{i=1\}^\{N\}\\sim\\pi\_\{\\beta\}\(\\cdot\\mid s\_\{t\}\)

21:for allk∈𝒦k\\in\\mathcal\{K\}do

22:Score:score​\(k,a1:k\(i\)\)=Qk​\(st,a1:k\(i\)\)−Vk​\(st\)γk\\mathrm\{score\}\(k,a^\{\(i\)\}\_\{1:k\}\)=\\dfrac\{Q^\{k\}\(s\_\{t\},a^\{\(i\)\}\_\{1:k\}\)\-V^\{k\}\(s\_\{t\}\)\}\{\\gamma^\{k\}\}for allii, using prefixa1:k\(i\)a^\{\(i\)\}\_\{1:k\}

23:Z\-score normalize within each scale:score~\(k,\)←\(score\(k,\)−meani\)/\(stdi\+ϵ\)\\tilde\{\\mathrm\{score\}\}\(k,\)\\leftarrow\(\\mathrm\{score\}\(k,\)\-\\mathrm\{mean\}\_\{i\}\)/\(\\mathrm\{std\}\_\{i\}\+\\epsilon\)

24:Selectk∗,a∗←arg​maxk,i⁡score~​\(k,a1:k\(i\)\)k^\{\*\},a^\{\*\}\\leftarrow\\operatorname\*\{arg\\,max\}\_\{k,i\}\\tilde\{\\mathrm\{score\}\}\(k,a^\{\(i\)\}\_\{1:k\}\)

25:Executea∗a^\{\*\}open\-loop fork∗k^\{\*\}steps

26:Store transitions inℛ\\mathcal\{R\}

27:ift≥t\\geqwarmupthen

28:Sample mixed batch fromℛ\\mathcal\{R\}

29:Update all networks via losses[Equations˜12](https://arxiv.org/html/2605.05544#S5.E12),[13](https://arxiv.org/html/2605.05544#S5.E13),[14](https://arxiv.org/html/2605.05544#S5.E14)and[15](https://arxiv.org/html/2605.05544#S5.E15)and[Equation˜5](https://arxiv.org/html/2605.05544#S3.E5)

## Appendix CExperimental Setup

### C\.1OGBench environments

We consider five manipulation domains from OGBench\[ogbench\_park2024\]as shown in[Figure˜7](https://arxiv.org/html/2605.05544#A3.F7), same as the ones used inli2025reinforcement:scene\-sparse,puzzle\-3x3\-sparse,cube\-double,cube\-triple, andcube\-quadruple\. Each domain provides five evaluation tasks of increasing difficulty; we report success rates averaged across all five tasks\. We use “play”\-style datasets collected by non\-Markovian expert policies with temporally correlated noise\. We use the default 1M\-transition datasets except forcube\-quadruple, where we use the 100M\-transition dataset\. All domains use binary or near\-binary sparse reward functions and a UR5e robot arm with 5D end\-effector control\.

![Refer to caption](https://arxiv.org/html/2605.05544v1/x6.png)Figure 7:OGBench tasks\.Some example tasks from the OGBench benchmark that we consider in our work\.Forscene\-sparseandpuzzle\-3x3\-sparse, we sparsify the reward function such that the agent receives−1\-1when the task is incomplete and0upon completion\. For the cube domains, the reward is−nwrong\-n\_\{\\mathrm\{wrong\}\}wherenwrongn\_\{\\mathrm\{wrong\}\}is the number of cubes at incorrect positions; the episode terminates when all cubes are correctly placed \(reward0\)\. We now describe each domain briefly:

Cube domains\.The agent controls a UR5e arm to pick and place cubes into target configurations\.cube\-doublerequires arranging two cubes,cube\-triplethree cubes, andcube\-quadruplefour\. These domains test multi\-object manipulation and long\-horizon sequential reasoning\. The cube domains become dramatically harder as the number of cubes increases, since permuting multiple blocks requires non\-trivial sequential planning\.

Scene domain\.This domain involves manipulating everyday objects — a drawer, a window, a cube, and two button locks — into a target configuration\. Pressing a button toggles whether the corresponding object can be moved\. Evaluation tasks require multi\-step sequences such as unlocking the drawer, opening it, placing a cube inside, and closing it\. We use a sparsified reward variant where success yields0and failure yields−1\-1\.

Puzzle domains\.These domains implement the “Lights Out” puzzle game using a robot arm that must physically press buttons on a grid\. Pressing a button toggles its color and the colors of adjacent buttons\. The goal is to reach a target color configuration\. We use sparsified rewards:−1\-1until the target configuration is reached,0upon success\.

DomainDataset SizeEpisode LengthAction Dim\.scene\-sparse\-\*1M7505puzzle\-3x3\-sparse\-\*1M5005cube\-double\-\*1M5005cube\-triple\-\*3M10005cube\-quadruple\-100M\-\*100M10005Table 2:OGBench domain metadata\.Dataset sizes, episode lengths, and action dimensions for all five domains\. Each domain provides five evaluation tasks \(indicated by\*\)\.
### C\.2Robomimic environments

We evaluate on three tasks from the robomimic benchmark\[robomimic2021\], using the multi\-human datasets collected by six human operators\. Each dataset contains 300 successful demonstrations\. All three tasks use binary task\-completion rewards:−1\-1when the task is incomplete and0upon success\.

- •lift: The robot arm must grasp a small cube and lift it\. This is the simplest task in the benchmark, testing basic grasping and vertical motion\.
- •can: The robot arm must pick up a soda can and place it into a smaller container bin\. This requires more precise manipulation thanlift, as the agent must orient the can correctly for insertion\.
- •square: The robot arm must pick up a square nut and thread it onto a peg\. The nut is slightly larger than the peg, requiring precise alignment and contact\-rich insertion, making this the most difficult of the three tasks\.

TaskDataset SizeEpisode LengthAction Dim\.lift31,1275007can62,7565007square80,7315007Table 3:Robomimic task metadata\.Number of transitions, episode lengths, and action dimensions\.
### C\.3RoboCasa\-GR1 environments

![Refer to caption](https://arxiv.org/html/2605.05544v1/x7.png)Figure 8:RoboCasa\-GR1 tasks\.Some example tasks from the RoboCasa\-GR1 tabletop manipulation benchmark\. The rendered images of these tasks are taken frombjorck2025gr00t\.RoboCasa\-GR1\[nasiriany2024robocasa\]is a benchmark of tabletop manipulation tasks designed for the GR1 humanoid robot platform\. It comprises 24 tasks spanning diverse manipulation skills including picking, placing, pouring, grasping, and insertion\. These tasks require controlling a high\-DoF humanoid upper body, making them substantially more complex than standard arm\-only manipulation benchmarks\. The action space includes joint\-level control of the torso, arms, and grippers, with observations consisting of joint positions, velocities, and object states\.

The 24 tasks fall into two categories\. Six base pick\-and\-place tasks involve manipulating everyday objects into containers: placing a can, cup, or bottle into a drawer; placing a milk carton or potato into a microwave; and placing a bottle or wine bottle into a cabinet\. Eighteen novel generalization tasks \(PosttrainPnPNovel\) evaluate zero\-shot transfer to unseen object–container pairs \(e\.g\., cuttingboard\-to\-basket, tray\-to\-tiered\-shelf, plate\-to\-cardboard\-box\)\. All tasks use theGR1ArmsAndWaistFourierHandsembodiment, which provides a 44\-dimensional action space across 8 body parts \(left/right arm, left/right hand, left/right leg, neck, waist\) and a matching 44\-dimensional proprioceptive state space\. Each task provides 1,000 successful human teleoperation demonstrations\.

## Appendix DImplementation Details

### D\.1Computational Resources

We use NVIDIA RTX 4090 GPUs for all experiments\. A single OGBench offline\-to\-online run takes approximately 5–7 GPU hours, depending on the domain and dataset size\. The 100M\-transition datasets \(cube\-quadruple\) require periodic chunking of the offline data to fit in CPU memory, adding roughly 2 hours per run\. For the RoboCasa\-GR1 experiments, each training run takes around 20 hours due to the high\-dimensional physics simulation\.

To reproduce the main OGBench results in[Section˜5\.3](https://arxiv.org/html/2605.05544#S5.SS3), we estimate6⏟hours×2⏟methods×25⏟tasks×4⏟seeds≈1,200\\underbrace\{6\}\_\{\\text\{hours\}\}\\times\\underbrace\{2\}\_\{\\text\{methods\}\}\\times\\underbrace\{25\}\_\{\\text\{tasks\}\}\\times\\underbrace\{4\}\_\{\\text\{seeds\}\}\\approx 1\{,\}200GPU hours\. The Robomimic experiments require10⏟hours×2⏟methods×3⏟tasks×5⏟seeds≈300\\underbrace\{10\}\_\{\\text\{hours\}\}\\times\\underbrace\{2\}\_\{\\text\{methods\}\}\\times\\underbrace\{3\}\_\{\\text\{tasks\}\}\\times\\underbrace\{5\}\_\{\\text\{seeds\}\}\\approx 300GPU hours\. The RoboCasa\-GR1 experiments take approximately20⏟hours×4⏟methods≈80\\underbrace\{20\}\_\{\\text\{hours\}\}\\times\\underbrace\{4\}\_\{\\text\{methods\}\}\\approx 80GPU hours\. In total, reproducing all results in this paper requires roughly 1,600 GPU hours\.

### D\.2OGBench and Robomimic Experiments

For tasks that are already evaluated in prior work, we use the reported results\. Specifically, results ofcube\-double,cube\-triple,cube\-quadruple,scene\-sparse,puzzle\-3x3\-sparse,lift,can, andsquareare taken fromli2025reinforcement\. Otherwise, we implement the baselines in our codebase and evaluated according to the same protocol used in those works\.

AQC\. We sweep over expectile valuesκv∈\{0\.5,0\.7,0\.9,0\.93,0\.95,0\.99\}\\kappa\_\{v\}\\in\\\{0\.5,0\.7,0\.9,0\.93,0\.95,0\.99\\\}, horizon lengthsh∈\{5,10,25\}h\\in\\\{5,10,25\\\}, and scale sets𝒦∈\{\{1,5\},\{1,5,10\},\{1,5,10,25\}\}\\mathcal\{K\}\\in\\\{\\\{1,5\\\},\\\{1,5,10\\\},\\\{1,5,10,25\\\}\\\}\. For each environment, we report the best configuration selected in[Table˜5](https://arxiv.org/html/2605.05544#A4.T5)\.

DQC\.DQCdecouples the critic horizon from the policy execution horizon, using a long\-horizon critic for credit assignment and a shorter\-horizon policy for robust execution\. We adaptDQCfrom its original goal\-conditioned formulation\[li2026decoupled\]to the reward\-based setting by removing goal conditioning entirely and training on the environment’s direct reward signal\. Concretely, we train five networks:

1. 1\.Qϕ​\(s,a1,…,ah\):𝒮×𝒜h↦ℝQ\_\{\\phi\}\(s,a\_\{1\},\\ldots,a\_\{h\}\):\\mathcal\{S\}\\times\\mathcal\{A\}^\{h\}\\mapsto\\mathbb\{R\}— the long\-horizon \(full\-chunk\) critic\. An ensemble ofNQ=2N\_\{Q\}=2networks is used\.
2. 2\.Qψ​\(s,a1,…,aha\):𝒮×𝒜ha↦ℝQ\_\{\\psi\}\(s,a\_\{1\},\\ldots,a\_\{h\_\{a\}\}\):\\mathcal\{S\}\\times\\mathcal\{A\}^\{h\_\{a\}\}\\mapsto\\mathbb\{R\}— the short\-horizon \(partial\) critic, trained via expectile distillation fromQϕQ\_\{\\phi\}\. An ensemble ofNQ=2N\_\{Q\}=2networks is used\.
3. 3\.Vξ​\(s\):𝒮↦ℝV\_\{\\xi\}\(s\):\\mathcal\{S\}\\mapsto\\mathbb\{R\}— the value function, trained via expectile regression on the partial criticQψQ\_\{\\psi\}\. Provides the TD bootstrap forQϕQ\_\{\\phi\}\.
4. 4\.Q¯ϕ\\bar\{Q\}\_\{\\phi\},Q¯ψ\\bar\{Q\}\_\{\\psi\},V¯ξ\\bar\{V\}\_\{\\xi\}— exponential\-moving\-average target networks for each of the above, updated with decayτ=0\.005\\tau=0\.005\.
5. 5\.fω​\(s,x,u\):𝒮×ℝha​da×\[0,1\]↦ℝha​daf\_\{\\omega\}\(s,x,u\):\\mathcal\{S\}\\times\\mathbb\{R\}^\{h\_\{a\}d\_\{a\}\}\\times\[0,1\]\\mapsto\\mathbb\{R\}^\{h\_\{a\}d\_\{a\}\}— the flow\-matching behavior policy, parameterized by a velocity prediction network\.

Given a transition\(st,at:t\+h−1,st\+h,rt\(h\)\)∼𝒟\(s\_\{t\},a\_\{t:t\+h\-1\},s\_\{t\+h\},r\_\{t\}^\{\(h\)\}\)\\sim\\mathcal\{D\}wherert\(h\)=∑j=0h−1γj​rt\+jr\_\{t\}^\{\(h\)\}=\\sum\_\{j=0\}^\{h\-1\}\\gamma^\{j\}r\_\{t\+j\}and continuation maskmt=∏j=0h−1γcont​\(st\+j\)m\_\{t\}=\\prod\_\{j=0\}^\{h\-1\}\\gamma\_\{\\text\{cont\}\}\(s\_\{t\+j\}\), the training objectives are:

\(1\) Long\-horizon critic loss\.The full\-chunk criticQϕQ\_\{\\phi\}is trained with anhh\-step TD target bootstrapped from the value function:

L​\(ϕ\)=\(Qϕ​\(st,at:t\+h−1\)−rt\(h\)−γh​mt​V¯ξ​\(st\+h\)\)2\.\\displaystyle L\(\\phi\)=\\left\(Q\_\{\\phi\}\(s\_\{t\},a\_\{t:t\+h\-1\}\)\-r\_\{t\}^\{\(h\)\}\-\\gamma^\{h\}m\_\{t\}\\bar\{V\}\_\{\\xi\}\(s\_\{t\+h\}\)\\right\)^\{2\}\.\(17\)The value targetV¯ξ\\bar\{V\}\_\{\\xi\}\(from the EMA network\) encodes the long\-range return fromst\+hs\_\{t\+h\}onward, avoidingO​\(H/h\)O\(H/h\)chained 1\-step TD updates\.

\(2\) Partial critic distillation loss\.The partial criticQψQ\_\{\\psi\}is trained to approximate the upper envelope of the long\-horizon criticQϕQ\_\{\\phi\}viaκd\\kappa\_\{d\}\-expectile regression:

L​\(ψ\)=𝔼2κd​\[Q¯ϕ​\(st,at:t\+h−1\)−Qψ​\(st,at:t\+ha−1\)\],\\displaystyle L\(\\psi\)=\\mathbb\{E\}\_\{2\}^\{\\kappa\_\{d\}\}\\\!\\left\[\\bar\{Q\}\_\{\\phi\}\(s\_\{t\},a\_\{t:t\+h\-1\}\)\-Q\_\{\\psi\}\(s\_\{t\},a\_\{t:t\+h\_\{a\}\-1\}\)\\right\],\(18\)where𝔼2κd​\(x\)=\|κd−𝟙​\(x<0\)\|​x2\\mathbb\{E\}\_\{2\}^\{\\kappa\_\{d\}\}\(x\)=\|\\kappa\_\{d\}\-\\mathbb\{1\}\(x<0\)\|x^\{2\}is the asymmetric expectile loss andQ¯ϕ\\bar\{Q\}\_\{\\phi\}is the EMA target ofQϕQ\_\{\\phi\}\. Withκd\>0\.5\\kappa\_\{d\}\>0\.5,QψQ\_\{\\psi\}learns an optimistic approximation ofQϕQ\_\{\\phi\}restricted to the firsthah\_\{a\}actions\.

\(3\) Value implicit backup loss\.The value functionVξV\_\{\\xi\}is trained viaκb\\kappa\_\{b\}\-expectile regression on the partial critic:

L​\(ξ\)=𝔼2κb​\[Q¯ψ​\(st,at:t\+ha−1\)−Vξ​\(st\)\],\\displaystyle L\(\\xi\)=\\mathbb\{E\}\_\{2\}^\{\\kappa\_\{b\}\}\\\!\\left\[\\bar\{Q\}\_\{\\psi\}\(s\_\{t\},a\_\{t:t\+h\_\{a\}\-1\}\)\-V\_\{\\xi\}\(s\_\{t\}\)\\right\],\(19\)whereQ¯ψ\\bar\{Q\}\_\{\\psi\}is the EMA target ofQψQ\_\{\\psi\}andκb\>0\.5\\kappa\_\{b\}\>0\.5controls the degree of optimism\. This loss is co\-trained with the distillation loss in a single backward pass\.

\(4\) Flow\-matching behavior policy loss\.The behavior policy is trained with standard conditional flow matching:

L​\(ω\)=‖fω​\(st,u​at:t\+ha−1\+\(1−u\)​zt,u\)−\(at:t\+ha−1−zt\)‖22,\\displaystyle L\(\\omega\)=\\left\\\|f\_\{\\omega\}\\\!\\left\(s\_\{t\},ua\_\{t:t\+h\_\{a\}\-1\}\+\(1\-u\)z\_\{t\},u\\right\)\-\\left\(a\_\{t:t\+h\_\{a\}\-1\}\-z\_\{t\}\\right\)\\right\\\|\_\{2\}^\{2\},\(20\)whereu∼U​\(\[0,1\]\)u\\sim U\(\[0,1\]\)andzt∼𝒩​\(0,Iha​da\)z\_\{t\}\\sim\\mathcal\{N\}\(0,I\_\{h\_\{a\}d\_\{a\}\}\)\. Note the flow is trained onhah\_\{a\}\-step chunks, nothh\-step\.

Policy inference\.At inference time, we sampleN=32N=32candidatehah\_\{a\}\-step action chunks from the flow\-matching policy via Euler integration, score each by the partial critic ensembleQ¯ψ\\bar\{Q\}\_\{\\psi\}, and select the highest\-scoring chunk\.

Hyperparameter Tuning\.We sweep over backup expectileκb∈\{0\.5,0\.7,0\.9,0\.93,0\.95,0\.99\}\\kappa\_\{b\}\\in\\\{0\.5,0\.7,0\.9,0\.93,0\.95,0\.99\\\}, distillation expectileκd∈\{0\.5,0\.8\}\\kappa\_\{d\}\\in\\\{0\.5,0\.8\\\}, horizon lengthh∈\{5,10,25\}h\\in\\\{5,10,25\\\}, and policy chunk sizeha∈\{1,5,25\}h\_\{a\}\\in\\\{1,5,25\\\}\. Task\-specific hyperparameters are listed in[Table˜5](https://arxiv.org/html/2605.05544#A4.T5)\.

ParameterValueOptimizerAdamWLearning rate3×10−43\\times 10^\{\-4\}Batch size256Discountγ\\gamma0\.99Network width512Network depth4 hidden layersEMA decayτ\\tau0\.005Ensemble sizeNQN\_\{Q\}2UTD ratio1Number of flow steps10Number of offline training steps1,000,0001\{,\}000\{,\}000Number of online environment steps1,000,0001\{,\}000\{,\}000Table 4:Common hyperparameters for OGBench and Robomimic experiments\.Shared across all tasks unless otherwise noted in[Table˜5](https://arxiv.org/html/2605.05544#A4.T5)\.AQCDQCEnvironment𝜿𝒗\\bm\{\\kappa\_\{v\}\}𝒉\\bm\{h\}𝓚\\bm\{\\mathcal\{K\}\}𝜿𝒃\\bm\{\\kappa\_\{b\}\}𝜿𝒅\\bm\{\\kappa\_\{d\}\}𝒉\\bm\{h\}𝒉𝒂\\bm\{h\_\{a\}\}cube\-double0\.95\{1,5\}\\\{1,5\\\}0\.990\.851cube\-triple0\.95\{1,5\}\\\{1,5\\\}0\.70\.8105cube\-quadruple0\.910\{1,5,10\}\\\{1,5,10\\\}0\.950\.8105scene\-sparse0\.955\{1,5\}\\\{1,5\\\}0\.90\.551puzzle\-3x3\-sparse0\.955\{1,5\}\\\{1,5\\\}0\.90\.8105lift0\.935\{1,5\}\\\{1,5\\\}0\.930\.851can0\.95\{1,5\}\\\{1,5\\\}0\.950\.851square0\.95\{1,5\}\\\{1,5\\\}0\.90\.5105Table 5:Task\-specific hyperparameters forAQCandDQC\.Best configuration per environment\.
### D\.3Selecting𝒦set\\mathcal\{K\}\_\{\\text\{set\}\}\.

The candidate chunk size set𝒦\\mathcal\{K\}follows a structured pattern rather than arbitrary search\. We define a universal superset𝒦univ=\{1,5,10,25\}\\mathcal\{K\}\_\{\\text\{univ\}\}=\\\{1,5,10,25\\\}and, for each task, use𝒦=\{k∈𝒦univ:k≤h\}\\mathcal\{K\}=\\\{k\\in\\mathcal\{K\}\_\{\\text\{univ\}\}:k\\leq h\\\}\. The values in𝒦univ\\mathcal\{K\}\_\{\\text\{univ\}\}correspond to meaningful timescales in manipulation:k=1k\{=\}1enables reactive per\-step control near contact events,k=5k\{=\}5covers short primitives \(a single reach or grasp\),k=10k\{=\}10spans multi\-step subroutines \(reach–grasp–lift\), andk=25k\{=\}25captures extended task\-level sequences\. This nested construction ensures that longer\-horizon tasks gain access to coarser timescales for temporal smoothing, while all tasks retaink=1k\{=\}1as a fallback for contact\-rich phases\. In practice, the selector rarely needs more than 3–4 candidates: adding intermediate values \(e\.g\.,𝒦=\{1,2,3,4,5\}\\mathcal\{K\}=\\\{1,2,3,4,5\\\}\) yields no benefit over\{1,5\}\\\{1,5\\\}since the advantage criterion already interpolates between the extremes through the partial critic scores\.

### D\.4RoboCasa\-GR1 Experiments

#### Stage 1: Actor fine\-tuning\.

We load the base VLA checkpoint and fine\-tune it on the combined 24\-task RoboCasa\-GR1 dataset \(24,000 expert demonstrations\)\. During fine\-tuning, the diffusion transformer \(DiT\) action head and visual encoder are trained while the LLM backbone is frozen\. We use a learning rate of3×10−53\\times 10^\{\-5\}, AdamW optimizer, global batch size 32, and 60,000 training steps\. We then collect 300 rollouts per task using the finetuned policy in randomized environments\. We then fine\-tune the base policy using behavior cloning \(BC\) on both expert demonstrations \+ rollout dataset\. This BC model becomes the actor\.

#### Stage 2: Critic training\.

We load the fine\-tuned actor checkpoint and attach anAQCcritic head while freezing all backbone parameters\.AQCcritic is trained with chunk sizeh=16h=16and policy chunk sizes𝒦=\{1,4,8,16\}\\mathcal\{K\}=\\\{1,4,8,16\\\}\. State and action spaces are normalized using the 99th\-percentile statistics from the dataset\. Sparse binary rewards \(0 on success,−1\-1otherwise\) are shaped by setting the final 15 timesteps before success to 1, with a global shift of−1\.0\-1\.0applied to all non\-success timesteps\.

QC and DQC baselines\.QCtraining follows the same pipeline asAQCbut with a single\-scale critic \(no policy chunk size selection\)\.DQCtraining uses critic chunk sizeh=16h=16, policy chunk sizek=4k=4, discountγ=0\.999\\gamma=0\.999, distillation expectileκd=0\.5\\kappa\_\{d\}=0\.5, and backup expectileκb=0\.9\\kappa\_\{b\}=0\.9\. All other hyperparameters matchAQC\.

#### Evaluation\.

After critic training, we evaluate using a best\-of\-NNselection policy: the BC actor samplesN=10N=10action candidates, theAQCcritic scores each sample using per\-scale advantage, and the\(k∗,a∗\)\(k^\{\*\},a^\{\*\}\)pair with the highest score is selected\.

ParameterStage 1 \(Actor\)Stage 2 \(Critic\)OptimizerAdamWAdamWLearning rate3×10−53\\times 10^\{\-5\}3×10−43\\times 10^\{\-4\}Batch size3264Training steps60,00030,000LR scheduleCosineCosineWeight decay10−510^\{\-5\}10−510^\{\-5\}Trainable paramsDiT \+ visual encoderCritic head onlyDiscountγ\\gamma—0\.99Chunk sizes—h=16h=16,𝒦=\{1,4,8,16\}\\mathcal\{K\}=\\\{1,4,8,16\\\}Expectileκv\\kappa\_\{v\}—0\.9Best\-of\-NN—10Table 6:RoboCasa\-GR1 training hyperparameters\.Two\-stage pipeline: actor fine\-tuning followed by critic training\.

## Appendix EFull Results

RLPDIQLFQLFQL\-nQC\-FQLQCDQCAQCtask1\\cellcolorcpurple\!10–→𝟏𝟎𝟎\[100,100\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟗𝟗\[97,100\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[97,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[97,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[97,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[97,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{99\}\}\}\\cellcolorcpurple\!10𝟗𝟗\[98,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{99\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟏𝟎𝟎\[98,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟗𝟕\[94,100\]→𝟗𝟗\[98,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[94,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[94,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[94,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[94,100\]$\}\}\}\}\}\}\{\{\\phantom\{0\}\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{97\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\{\\phantom\{0\}\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{99\}\}\}\\cellcolorourcolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}task2\\cellcolorcpurple\!10–→𝟏𝟎𝟎\[100,100\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟎\[0,0\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\cellcolorcpurple\!10𝟗𝟗\[98,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{99\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟖𝟔\[60,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[60,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[60,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[60,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[60,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{86\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟗𝟏\[88,94\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[88,94\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[88,94\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[88,94\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[88,94\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{91\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorourcolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}task3\\cellcolorcpurple\!10–→𝟏𝟎𝟎\[100,100\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟎\[0,0\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\cellcolorcpurple\!10𝟏𝟎𝟎\[98,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟏𝟎𝟎\[98,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2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\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}task4\\cellcolorcpurple\!10–→𝟏𝟎𝟎\[100,100\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟎\[0,0\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\cellcolorcpurple\!10𝟏𝟎𝟎\[98,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟏𝟎𝟎\[98,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟕𝟓\[25,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[25,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[25,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[25,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[25,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{75\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟕𝟐\[68,74\]→𝟗𝟖\[96,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[68,74\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[68,74\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[68,74\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[68,74\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{72\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[96,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[96,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[96,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[96,100\]$\}\}\}\}\}\}\{\{\\phantom\{0\}\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{98\}\}\}\\cellcolorourcolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}task5\\cellcolorcpurple\!10–→𝟏𝟎𝟎\[100,100\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟎\[0,0\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.2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stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{4\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟏𝟎𝟎\[98,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟖𝟗\[86,92\]→𝟗𝟗\[98,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[86,92\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[86,92\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[86,92\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[86,92\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{89\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\{\\phantom\{0\}\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{99\}\}\}\\cellcolorourcolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}puzzle\-3x3avg\. \(5 tasks\)\\cellcolorcpurple\!10–→𝟏𝟎𝟎\[100,100\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟐𝟎\[19,20\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[19,20\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[19,20\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[19,20\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[19,20\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{20\}\}\}\\cellcolorcpurple\!10𝟏𝟎𝟎\[99,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[99,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[99,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[99,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[99,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟗𝟖\[95,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[95,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[95,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[95,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[95,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{98\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟔𝟑\[48,76\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[48,76\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[48,76\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[48,76\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[48,76\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{63\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟏𝟎𝟎\[99,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[99,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[99,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[99,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[99,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟖𝟔\[82,90\]→𝟗𝟕\[94,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[82,90\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[82,90\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[82,90\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[82,90\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{86\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[94,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[94,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[94,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[94,100\]$\}\}\}\}\}\}\{\{\\phantom\{0\}\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{97\}\}\}\\cellcolorourcolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}task1\\cellcolorcpurple\!10–→𝟏𝟎𝟎\[100,100\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟏\[0,2\]→𝟗𝟕\[96,99\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,2\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,2\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,2\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,2\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{1\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[96,99\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[96,99\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[96,99\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[96,99\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{97\}\}\}\\cellcolorcpurple\!10𝟔𝟗\[55,79\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[55,79\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[55,79\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[55,79\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[55,79\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{69\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟐𝟐\[19,25\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[19,25\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[19,25\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[19,25\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[19,25\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{22\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟗𝟗\[98,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{99\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟖𝟒\[82,86\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[82,86\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[82,86\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[82,86\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[82,86\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{84\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorourcolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}task2\\cellcolorcpurple\!10–→𝟏𝟎𝟎\[100,100\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{100\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟗𝟕\[94,100\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[94,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[94,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[94,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[94,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{97\}\}\}\\cellcolorcpurple\!10𝟓𝟏\[41,64\]→𝟗𝟗\[98,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[41,64\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[41,64\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[41,64\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[41,64\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{51\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{99\}\}\}\\cellcolorcpurple\!10𝟒\[2,6\]→𝟒𝟐\[33,51\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[2,6\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[2,6\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[2,6\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[2,6\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{4\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[33,51\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[33,51\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[33,51\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[33,51\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{42\}\}\}\\cellcolorqccolor\!10𝟖𝟖\[82,94\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[82,94\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[82,94\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[82,94\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[82,94\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{88\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟗𝟖\[96,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[96,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[96,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[96,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[96,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{98\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟒𝟑\[40,46\]→𝟗𝟓\[94,96\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[40,46\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[40,46\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[40,46\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[40,46\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{43\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[94,96\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[94,96\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[94,96\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[94,96\]$\}\}\}\}\}\}\{\{\\phantom\{0\}\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{95\}\}\}\\cellcolorourcolor\!10𝟏𝟎𝟎\[100,100\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.7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}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟎\[0,0\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\cellcolorcpurple\!10𝟕𝟓\[69,81\]→𝟗𝟔\[88,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[69,81\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[69,81\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[69,81\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[69,81\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{75\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[88,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[88,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[88,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[88,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{96\}\}\}\\cellcolorcpurple\!10𝟓𝟎\[41,63\]→𝟗𝟒\[86,99\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[41,63\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[41,63\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[41,63\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[41,63\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{50\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[86,99\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[86,99\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[86,99\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[86,99\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{94\}\}\}\\cellcolorqccolor\!10𝟗𝟐\[85,98\]→𝟗𝟕\[95,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[85,98\]$\}\}\}\}\}\{\\raisebox\{\-0\.75p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\(5 tasks\)\\cellcolorcpurple\!10–→𝟗𝟒\[93,96\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[93,96\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[93,96\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[93,96\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[93,96\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{94\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟑𝟗\[39,39\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[39,39\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[39,39\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[39,39\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[39,39\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{39\}\}\}\\cellcolorcpurple\!10𝟓𝟕\[55,60\]→𝟗𝟓\[93,98\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[55,60\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[55,60\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[55,60\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[55,60\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{57\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[93,98\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[93,98\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[93,98\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[93,98\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{95\}\}\}\\cellcolorcpurple\!10𝟏𝟖\[17,21\]→𝟕𝟎\[64,76\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[17,21\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[17,21\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[17,21\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[17,21\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{18\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[64,76\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[64,76\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[64,76\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[64,76\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{70\}\}\}\\cellcolorqccolor\!10𝟖𝟒\[81,87\]→𝟗𝟗\[99,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[81,87\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[81,87\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[81,87\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[81,87\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{84\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[99,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[99,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[99,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[99,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{99\}\}\}\\cellcolorqccolor\!10𝟖𝟒\[79,89\]→𝟗𝟗\[98,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[79,89\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[79,89\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[79,89\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[79,89\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{84\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{99\}\}\}\\cellcolorqccolor\!10𝟓𝟔\[54,60\]→𝟖𝟗\[84,90\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[54,60\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[54,60\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[54,60\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[54,60\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@strok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{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{74\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[97,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[97,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[97,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[97,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{99\}\}\}\\cellcolorqccolor\!10𝟏𝟑\[10,14\]→𝟕𝟔\[72,82\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[10,14\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[10,14\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[10,14\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[10,14\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{13\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[72,82\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[72,82\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[72,82\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[72,82\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{76\}\}\}\\cellcolorourcolor\!10𝟗𝟒\[92,96\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[92,96\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[92,96\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[92,96\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[92,96\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{94\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}cube\-doubleavg\. \(5 tasks\)\\cellcolorcpurple\!10–→𝟗𝟗\[98,99\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,99\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,99\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,99\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,99\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.4140625,0\.2421875,0\.66015625\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.4140625,0\.2421875,0\.66015625\}\\mathbf\{99\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟎\[0,0\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\cellcolorcpurple\!10𝟐𝟗\[23,34\]→𝟕𝟔\[75,76\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[23,34\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[23,34\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[23,34\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[23,34\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{29\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[75,76\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[75,76\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[75,76\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[75,76\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{76\}\}\}\\cellcolorcpurple\!10𝟏𝟏\[7,13\]→𝟕𝟕\[76,77\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[7,13\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[7,13\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{7\.8362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[7,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hbf\{100\}\}\}\\cellcolorqccolor\!10𝟏𝟑\[8,19\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[8,19\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[8,19\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{7\.8362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[8,19\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[8,19\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{13\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{100\}\}\}\\cellcolorqccolor\!10𝟒\[0,8\]→𝟎\[0,0\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,8\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,8\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,8\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,8\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{4\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\cellcolorourcolor\!10𝟓𝟐\[50,54\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[50,54\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[50,54\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[50,54\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[50,54\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{52\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}task2\\cellcolorcpurple\!10–→𝟔𝟏\[22,87\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\rai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\(5 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t\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[96,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{98\}\}\}\\cellcolorqccolor\!10𝟏𝟔\[6,27\]→𝟗𝟗\[98,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[6,27\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[6,27\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{7\.8362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[6,27\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[6,27\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{16\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.03775pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.83621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[98,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{11\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[98,100\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.20703125,0\.3203125,0\.65234375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.20703125,0\.3203125,0\.65234375\}\\mathbf\{99\}\}\}\\cellcolorqccolor\!10𝟖\[6,10\]→𝟗𝟎\[86,94\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[6,10\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[6,10\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{7\.8362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[6,10\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[6,10\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{8\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[86,94\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[86,94\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[86,94\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[86,94\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{90\}\}\}\\cellcolorourcolor\!10𝟒𝟎\[36,52\]→𝟏𝟎𝟎\[100,100\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[36,52\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[36,52\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[36,52\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[36,52\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{40\}\}\}\\rightarrow\\overset\{\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{13\.7461pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[100,100\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{12\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[100,100\]$\}\}\}\}\}\}\{\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{100\}\}\}task2\\cellcolorcpurple\!10–→𝟎\[0,0\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟎\[0,0\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\cellcolorcpurple\!10𝟎\[0,0\]→𝟎\[0,0\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\cellcolorcpurple\!10𝟐\[0,3\]→𝟒𝟗\[39,59\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,3\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,3\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\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\(5 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ptstyle\[0,2\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{2\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[12,30\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[12,30\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[12,30\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[12,30\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{20\}\}\}\\cellcolorourcolor\!10𝟗\[8,10\]→𝟖𝟖\[84,92\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[8,10\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.62106pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[8,10\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{7\.8362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[8,10\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{8\.17061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[8,10\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{9\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[84,92\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[84,92\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[84,92\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[84,92\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{88\}\}\}overallavg\. \(25 tasks\)\\cellcolorcpurple\!15–→𝟔𝟕\[65,68\]\\phantom\{0\}\\mathchoice\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\displaystyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{2\.63226pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\textstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.03964pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptstyle\\text\{\-\-\}$\}\}\}\}\}\{\\raisebox\{0\.0pt\}\{\\resizebox\{3\.7291pt\}\{1\.7pt\}\{\\hbox\{\\raisebox\{0\.0pt\}\{$\\scriptscriptstyle\\text\{\-\-\}$\}\}\}\}\}\\phantom\{0\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[65,68\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[65,68\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[65,68\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[65,68\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{67\}\}\}\\cellcolorcpurple\!15𝟎\[0,0\]→𝟏𝟐\[12,12\]\\overset\{\\phantom\{0\}\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.91273pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.3362pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[0,0\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{6\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[0,0\]$\}\}\}\}\}\}\{\\phantom\{0\}\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{0\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[12,12\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[12,12\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[12,12\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[12,12\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{12\}\}\}\\cellcolorcpurple\!15𝟑𝟕\[36,38\]→𝟓𝟖\[57,60\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[36,38\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[36,38\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[36,38\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[36,38\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{37\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[57,60\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[57,60\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[57,60\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[57,60\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{58\}\}\}\\cellcolorcpurple\!15𝟐𝟕\[25,28\]→𝟓𝟕\[56,58\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[25,28\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[25,28\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[25,28\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[25,28\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{27\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[56,58\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[56,58\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[56,58\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[56,58\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{57\}\}\}\\cellcolorqccolor\!15𝟑𝟖\[35,41\]→𝟖𝟔\[84,88\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[35,41\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[35,41\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[35,41\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[35,41\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{38\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[84,88\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[84,88\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[84,88\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[84,88\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{86\}\}\}\\cellcolorqccolor\!15𝟓𝟐\[50,53\]→𝟖𝟔\[86,87\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[50,53\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[50,53\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[50,53\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[50,53\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{52\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[86,87\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[86,87\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[86,87\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[86,87\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{86\}\}\}\\cellcolorqccolor\!15𝟑𝟐\[28,36\]→𝟓𝟐\[50,56\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[28,36\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[28,36\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[28,36\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[28,36\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{32\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[50,56\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[50,56\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[50,56\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[50,56\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.7,0\.7,0\.7\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.7,0\.7,0\.7\}\\pgfsys@color@gray@stroke\{0\.7\}\\pgfsys@color@gray@fill\{0\.7\}\\mathbf\{52\}\}\}\\cellcolorourcolor\!15𝟔𝟐\[58,68\]→𝟗𝟔\[92,98\]\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[58,68\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[58,68\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[58,68\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[58,68\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{62\}\}\}\\rightarrow\\overset\{\\phantom\{0\}\\mathchoice\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\displaystyle\[92,98\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{10\.3294pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.75pt\}\{$\\textstyle\[92,98\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.33621pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{1\.22499pt\}\{$\\scriptstyle\[92,98\]$\}\}\}\}\}\{\\raisebox\{\-0\.75pt\}\{\\resizebox\{9\.67061pt\}\{3\.0pt\}\{\\hbox\{\\raisebox\{0\.875pt\}\{$\\scriptscriptstyle\[92,98\]$\}\}\}\}\}\}\{\\phantom\{0\}\{\\color\[rgb\]\{0\.9765625,0\.48046875,0\.30859375\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0\.9765625,0\.48046875,0\.30859375\}\\mathbf\{96\}\}\}

Table 7:Full OGBench results table\.4 seeds, 95% confidence interval
## Appendix FRoboCasa Results

Gr00t N1\.6\+ Filtered BC\+ QC\+ DQC\+ AQCPnPBottleToCabinetClose\\cellcolorgray\!1051\.5\\cellcolorgray\!1462\.5\\cellcolorqccolor\!1064\.0\\cellcolorqccolor\!1053\.5\\cellcolorourcolor\!1066\.0PnPCanToDrawerClose\\cellcolorgray\!1013\.0\\cellcolorgray\!1464\.5\\cellcolorqccolor\!1065\.0\\cellcolorqccolor\!1042\.5\\cellcolorourcolor\!1068\.5PnPCupToDrawerClose\\cellcolorgray\!108\.5\\cellcolorgray\!1456\.0\\cellcolorqccolor\!1059\.0\\cellcolorqccolor\!1054\.5\\cellcolorourcolor\!1064\.0PnPMilkToMicrowaveClose\\cellcolorgray\!1014\.0\\cellcolorgray\!1435\.5\\cellcolorqccolor\!1044\.5\\cellcolorqccolor\!1021\.0\\cellcolorourcolor\!1056\.0PnPPotatoToMicrowaveClose\\cellcolorgray\!1041\.5\\cellcolorgray\!1451\.0\\cellcolorqccolor\!1054\.5\\cellcolorqccolor\!1053\.0\\cellcolorourcolor\!1063\.5PnPWineToCabinetClose\\cellcolorgray\!1016\.5\\cellcolorgray\!1434\.0\\cellcolorqccolor\!1036\.5\\cellcolorqccolor\!1031\.0\\cellcolorourcolor\!1043\.5PosttrainPnPNovelFromCuttingboardToBasketSplitA\\cellcolorgray\!1058\.0\\cellcolorgray\!1460\.5\\cellcolorqccolor\!1062\.5\\cellcolorqccolor\!1055\.0\\cellcolorourcolor\!1064\.0PosttrainPnPNovelFromCuttingboardToCardboardboxSplitA\\cellcolorgray\!1046\.5\\cellcolorgray\!1448\.0\\cellcolorqccolor\!1050\.5\\cellcolorqccolor\!1041\.5\\cellcolorourcolor\!1059\.5PosttrainPnPNovelFromCuttingboardToPanSplitA\\cellcolorgray\!1068\.5\\cellcolorgray\!1462\.0\\cellcolorqccolor\!1063\.5\\cellcolorqccolor\!1054\.0\\cellcolorourcolor\!1068\.5PosttrainPnPNovelFromCuttingboardToPotSplitA\\cellcolorgray\!1065\.0\\cellcolorgray\!1468\.5\\cellcolorqccolor\!1070\.0\\cellcolorqccolor\!1063\.5\\cellcolorourcolor\!1076\.5PosttrainPnPNovelFromCuttingboardToTieredbasketSplitA\\cellcolorgray\!1046\.5\\cellcolorgray\!1457\.5\\cellcolorqccolor\!1060\.0\\cellcolorqccolor\!1049\.5\\cellcolorourcolor\!1063\.5PosttrainPnPNovelFromPlacematToBasketSplitA\\cellcolorgray\!1058\.5\\cellcolorgray\!1454\.0\\cellcolorqccolor\!1056\.0\\cellcolorqccolor\!1048\.5\\cellcolorourcolor\!1060\.5PosttrainPnPNovelFromPlacematToBowlSplitA\\cellcolorgray\!1057\.5\\cellcolorgray\!1459\.0\\cellcolorqccolor\!1061\.0\\cellcolorqccolor\!1052\.0\\cellcolorourcolor\!1067\.0PosttrainPnPNovelFromPlacematToPlateSplitA\\cellcolorgray\!1063\.0\\cellcolorgray\!1460\.5\\cellcolorqccolor\!1059\.5\\cellcolorqccolor\!1052\.5\\cellcolorourcolor\!1071\.0PosttrainPnPNovelFromPlacematToTieredshelfSplitA\\cellcolorgray\!1028\.5\\cellcolorgray\!1424\.0\\cellcolorqccolor\!1026\.0\\cellcolorqccolor\!1019\.0\\cellcolorourcolor\!1032\.5PosttrainPnPNovelFromPlateToBowlSplitA\\cellcolorgray\!1057\.0\\cellcolorgray\!1462\.5\\cellcolorqccolor\!1064\.5\\cellcolorqccolor\!1056\.0\\cellcolorourcolor\!1071\.0PosttrainPnPNovelFromPlateToCardboardboxSplitA\\cellcolorgray\!1043\.5\\cellcolorgray\!1447\.0\\cellcolorqccolor\!1048\.5\\cellcolorqccolor\!1040\.0\\cellcolorourcolor\!1055\.5PosttrainPnPNovelFromPlateToPanSplitA\\cellcolorgray\!1051\.0\\cellcolorgray\!1456\.5\\cellcolorqccolor\!1058\.0\\cellcolorqccolor\!1050\.0\\cellcolorourcolor\!1060\.5PosttrainPnPNovelFromPlateToPlateSplitA\\cellcolorgray\!1078\.5\\cellcolorgray\!1468\.5\\cellcolorqccolor\!1070\.5\\cellcolorqccolor\!1061\.0\\cellcolorourcolor\!1078\.0PosttrainPnPNovelFromTrayToCardboardboxSplitA\\cellcolorgray\!1051\.5\\cellcolorgray\!1455\.0\\cellcolorqccolor\!1057\.0\\cellcolorqccolor\!1048\.0\\cellcolorourcolor\!1063\.5PosttrainPnPNovelFromTrayToPlateSplitA\\cellcolorgray\!1071\.0\\cellcolorgray\!1478\.0\\cellcolorqccolor\!1080\.5\\cellcolorqccolor\!1070\.0\\cellcolorourcolor\!1088\.5PosttrainPnPNovelFromTrayToPotSplitA\\cellcolorgray\!1064\.5\\cellcolorgray\!1465\.5\\cellcolorqccolor\!1067\.0\\cellcolorqccolor\!1058\.5\\cellcolorourcolor\!1068\.5PosttrainPnPNovelFromTrayToTieredbasketSplitA\\cellcolorgray\!1057\.0\\cellcolorgray\!1460\.0\\cellcolorqccolor\!1062\.0\\cellcolorqccolor\!1053\.0\\cellcolorourcolor\!1063\.5PosttrainPnPNovelFromTrayToTieredshelfSplitA\\cellcolorgray\!1031\.5\\cellcolorgray\!1429\.0\\cellcolorqccolor\!1031\.0\\cellcolorqccolor\!1024\.0\\cellcolorourcolor\!1042\.5Average\\cellcolorgray\!1047\.6\\cellcolorgray\!1454\.9\\cellcolorqccolor\!1055\.5\\cellcolorqccolor\!1046\.1\\cellcolorourcolor\!1063\.2

Table 8:RoboCasa GR1 tabletop results\.Success rate \(%\) across 24 tasks on 200 evaluation episodes each\.
## Appendix GAdditional Ablation Experiments

![Refer to caption](https://arxiv.org/html/2605.05544v1/x8.png)Figure 9:Bootstrap quality ablation\.ComparingVhV^\{h\},V1V^\{1\}, andQhQ^\{h\}as bootstrap sources forQkQ^\{k\}\.
![Refer to caption](https://arxiv.org/html/2605.05544v1/x9.png)Figure 10:κV\\kappa\_\{V\}sensitivity oncube\-double\.

#### Bootstrap quality\.

[Figure˜10](https://arxiv.org/html/2605.05544#A7.F10)compares three bootstrap sources forQkQ^\{k\}training: the default long\-horizon baselineVhV^\{h\}, the 1\-step valueV1V^\{1\}, and usingQhQ^\{h\}directly without baseline subtraction\. Oncube\-double,VhV^\{h\}yields 90% offline and 100% online;V1V^\{1\}underestimates long\-horizon returns, dropping to 81% offline \(89% online\)\. Oncube\-triple, the gap widens:V1V^\{1\}falls to 7% offline and 55% online vs\. 13% and 90% forVhV^\{h\}, confirming that horizon\-aligned bootstrap is critical on longer\-horizon tasks where the 1\-step value significantly underestimates returns\. UsingQhQ^\{h\}directly \(no baseline subtraction\) falls between the two \(84% offline on cube\-double, 9% on cube\-triple\), suggesting that baseline subtraction removes state\-dependent noise thatQhQ^\{h\}alone cannot cancel\.

#### κV\\kappa\_\{V\}sensitivity\.

[Figure˜10](https://arxiv.org/html/2605.05544#A7.F10)sweeps the expectile parameterκV\\kappa\_\{V\}for theVhV^\{h\}andVkV^\{k\}baselines oncube\-double\. The defaultκV=0\.9\\kappa\_\{V\}=0\.9\(upper envelope\) achieves the highest offline performance \(90% offline, 100% online\)\. Values in the range 0\.7–0\.93 are robust, with offline performance within 6% of optimal;κV=0\.5\\kappa\_\{V\}=0\.5\(median, approximating behavior policy value\) degrades sharply to 72% offline, confirming that a conservative baseline is essential\. Very high values \(κV=0\.99\\kappa\_\{V\}=0\.99\) also degrade performance \(75%\) due to unstableQkQ^\{k\}estimates from near\-max baselines, indicating that the upper envelope is a sweet spot between conservatism and stability\.

![Refer to caption](https://arxiv.org/html/2605.05544v1/x10.png)Figure 11:Critic chunk sizehhsensitivity\.Comparingh∈\{5,10,25\}h\\in\\\{5,10,25\\\}oncube\-double,cube\-triple, andcube\-quadruple\.
#### Critic chunk sizehhsensitivity\.

[Figure˜11](https://arxiv.org/html/2605.05544#A7.F11)evaluates howAQCperformance varies with the critic chunk sizehhacrosscube\-double,cube\-triple, andcube\-quadruple\. Oncube\-double,h=5h=5is optimal \(90% offline, 100% online\); increasinghhto 10 or 25 degrades offline performance \(88% and 82%\) without online benefit, suggesting diminishing returns on shorter\-horizon tasks\. Oncube\-triple,h=5h=5remains the best choice \(13% offline, 90% online\), with largerhhproviding no advantage\. Oncube\-quadruple, however,h=10h=10outperforms bothh=5h=5andh=25h=25\(12% offline, 84% online vs\. 9% and 10% offline\), suggesting that longer credit assignment horizons benefit from largerhh, but only up to a point beyond which the added variance dominates\.

#### Z\-score normalization\.

![Refer to caption](https://arxiv.org/html/2605.05544v1/x11.png)Figure 12:Z\-score normalization ablation\.Removing per\-scale z\-score normalization degrades performance across both tasks, with the gap widening on longer\-horizon domains \(cube\-triple\)\.[Figure˜12](https://arxiv.org/html/2605.05544#A7.F12)compares fullAQCagainst the variant without z\-score normalization\. Oncube\-double, removing z\-score normalization reduces final success rate by 9% \(91% vs 100%\), as the variance mismatch betweenk=1k=1andk=5k=5advantage scores causes the argmax to be dominated by the higher\-variance scale\. The effect is more pronounced oncube\-triple: without z\-score, performance drops by 45% \(45% vs 90%\), because the wider range of advantage magnitudes across scales amplifies the dominance problem\. This confirms that z\-score normalization is necessary to equalize the contribution of each scale to the selection decision, particularly when the candidate set spans multiple orders of magnitude in horizon length\.

#### 𝒦\\mathcal\{K\}robustness\.

The candidate set𝒦\\mathcal\{K\}is constructed as\{k∈𝒦univ:k≤h\}\\\{k\\in\\mathcal\{K\}\_\{\\mathrm\{univ\}\}:k\\leq h\\\}where𝒦univ=\{1,5,10,25\}\\mathcal\{K\}\_\{\\mathrm\{univ\}\}=\\\{1,5,10,25\\\}\. We verify that adding intermediate values provides no benefit: oncube\-double, replacing𝒦=\{1,5\}\\mathcal\{K\}=\\\{1,5\\\}with\{1,2,3,4,5\}\\\{1,2,3,4,5\\\}yields 90% offline and 99% online \(vs\. 90% and 100% for\{1,5\}\\\{1,5\\\}\), while𝒦=\{1,3,5\}\\mathcal\{K\}=\\\{1,3,5\\\}gives 89% offline and 99% online\. This confirms that the advantage criterion already interpolates between the extreme scales through the partial critic scores, and the selector only benefits from having endpoints that bracket the relevant timescales\.

#### Resource efficiency of AQC\.

We compare the parameter counts of all Q\-chunking methods on the representative domaincube\-triple\-\*\(assumingh=5h=5for all methods\)\.AQCis the most parameter\-heavy at≈7\.9\\approx 7\.9M due to per\-scale critics\.DQCuses a single partial critic and a smaller flow policy \(ha<hh\_\{a\}<h\), totaling≈6\.6\\approx 6\.6M\.QCandFQLuse a single critic backbone \(≈4\.2\\approx 4\.2M and≈4\.9\\approx 4\.9M respectively\)\.RLPDusesK=10K=10critic networks, making it the largest at≈17\.2\\approx 17\.2M\.

MethodsParameter Count \(in millions\)AQC≈7\.9\\approx 7\.9DQC≈6\.6\\approx 6\.6QC≈4\.2\\approx 4\.2QC\-FQL≈5\.0\\approx 5\.0FQL≈4\.9\\approx 4\.9RLPD≈17\.2\\approx 17\.2Table 9:Parameter count for each method\.AQC and DQC share the same backbone architecture as QC, adding only lightweight per\-scale critic heads\.

## Appendix HTheoretical Analysis

In this section, we develop a comprehensive theoretical framework forAQC\. While prior work on action chunking establishes when fixed\-chunk Q\-learning succeeds under open\-loop consistency\[li2026decoupled\], our analysis must address a fundamentally different question:*when does adaptive chunk\-size selection succeed, and how much better is it than any fixed chunk size?*We proceed in stages: \(1\) we define adaptive open\-loop consistency, the data condition under which multi\-scale re\-querying is valid; \(2\) we prove that the advantage selector correctly identifies the best chunk size under a separability condition; \(3\) we establish thatAQC’s value strictly dominates any fixed\-chunk policy; \(4\) we bound the regret introduced by imperfect critics; \(5\) we analyze the closed\-loop execution of the adaptive policy; \(6\) we show that the multi\-scale bootstrap creates a regularization hierarchy across critic scales\. All proofs appear in Appendix[I](https://arxiv.org/html/2605.05544#A9)\.

### H\.1Assumptions and notation

We adopt the standard assumption that the data distribution obeys the MDP transition dynamics\[li2026decoupled\]:

###### Assumption H\.1\(Data Obeys the Transition Dynamics\)\.

𝒟\\mathcal\{D\}is a trajectory distribution generated by rolling out a \(possibly non\-Markovian\) behavior policyπβ\\pi\_\{\\beta\}from an initial state distributionμ\\mu\. Each subsequent state is generated according to the MDP dynamicsTT:st\+k\+1∼T\(⋅∣st\+k,at\+k\)s\_\{t\+k\+1\}\\sim T\(\\cdot\\mid s\_\{t\+k\},a\_\{t\+k\}\)for allk∈\{0,1,…,h−1\}k\\in\\\{0,1,\\ldots,h\-1\\\}\. The resulting trajectory is\{st,st\+1,…,st\+h,at,at\+1,…,at\+h\}∈𝒯=𝒮h×𝒜h\\\{s\_\{t\},s\_\{t\+1\},\\ldots,s\_\{t\+h\},a\_\{t\},a\_\{t\+1\},\\ldots,a\_\{t\+h\}\\\}\\in\\mathcal\{T\}=\\mathcal\{S\}^\{h\}\\times\\mathcal\{A\}^\{h\}\.

We writeH=1/\(1−γ\)H=1/\(1\-\\gamma\)andH¯k=1/\(1−γk\)\\bar\{H\}\_\{k\}=1/\(1\-\\gamma^\{k\}\)for the 1\-step andkk\-step effective horizons, respectively\. We useVAQCV^\{\\mathrm\{AQC\}\}to denote the true closed\-loop value of the adaptive policy,V†V^\{\\dagger\}for the oracle selector’s value,V∙V^\{\\bullet\}for the fully reactive closed\-loop execution, andVkV^\{k\}\(with a fixedk∈𝒦k\\in\\mathcal\{K\}\) for the fixed\-chunk policy value\. The optimal value isV⋆V^\{\\star\}\.

### H\.2Adaptive Open\-Loop Consistency \(AOLC\)

Prior work introduces the open\-loop consistency \(OLC\) condition to characterize when replaying a fixed\-length action chunk open\-loop produces trajectories close to the data distribution\[li2026decoupled\]\. ForAQC, the situation is more subtle: the agent executesk∗​\(st\)k^\{\*\}\(s\_\{t\}\)steps open\-loop and then re\-queries, wherek∗k^\{\*\}varies by state\. The re\-querying points form a*randomly spaced*set of decision points, and we need the data to cover trajectories that match this adaptive pattern\.

###### Definition H\.2\(Adaptive Open\-Loop Consistency\)\.

Letκ:𝒮→𝒦\\kappa:\\mathcal\{S\}\\to\\mathcal\{K\}be a chunk\-size selection function\. For a data distribution𝒟\\mathcal\{D\}satisfying Assumption[H\.1](https://arxiv.org/html/2605.05544#A8.Thmtheorem1),𝒟\\mathcal\{D\}isε𝒦\\varepsilon\_\{\\mathcal\{K\}\}\-adaptively open\-loop consistentunderκ\\kappaif for everyst∈supp​\(P𝒟​\(st\)\)s\_\{t\}\\in\\mathrm\{supp\}\(P\_\{\\mathcal\{D\}\}\(s\_\{t\}\)\),

DTV\(P𝒟\(st\+κ​\(st\),at\+κ​\(st\)∣st\)∥P𝒟,κ∘\(st\+κ​\(st\),at\+κ​\(st\)∣st\)\)≤ε𝒦,\\displaystyle D\_\{\\mathrm\{TV\}\}\\Bigl\(P\_\{\\mathcal\{D\}\}\(s\_\{t\+\\kappa\(s\_\{t\}\)\},a\_\{t\+\\kappa\(s\_\{t\}\)\}\\mid s\_\{t\}\)\\;\\big\\\|\\;P^\{\\circ\}\_\{\\mathcal\{D\},\\kappa\}\(s\_\{t\+\\kappa\(s\_\{t\}\)\},a\_\{t\+\\kappa\(s\_\{t\}\)\}\\mid s\_\{t\}\)\\Bigr\)\\leq\\varepsilon\_\{\\mathcal\{K\}\},\(21\)DTV\(P𝒟\(st\+κ​\(st\)∣st\)∥P𝒟,κ∘\(st\+κ​\(st\)∣st\)\)≤ε𝒦,\\displaystyle D\_\{\\mathrm\{TV\}\}\\Bigl\(P\_\{\\mathcal\{D\}\}\(s\_\{t\+\\kappa\(s\_\{t\}\)\}\\mid s\_\{t\}\)\\;\\big\\\|\\;P^\{\\circ\}\_\{\\mathcal\{D\},\\kappa\}\(s\_\{t\+\\kappa\(s\_\{t\}\)\}\\mid s\_\{t\}\)\\Bigr\)\\leq\\varepsilon\_\{\\mathcal\{K\}\},\(22\)whereP𝒟,κ∘P^\{\\circ\}\_\{\\mathcal\{D\},\\kappa\}is the trajectory distribution obtained by rolling out the marginal policyπ𝒟∘​\(at:t\+k∣st\):=P𝒟​\(at:t\+k∣st\)\\pi^\{\\circ\}\_\{\\mathcal\{D\}\}\(a\_\{t:t\+k\}\\mid s\_\{t\}\):=P\_\{\\mathcal\{D\}\}\(a\_\{t:t\+k\}\\mid s\_\{t\}\)with chunk sizeκ​\(st\)\\kappa\(s\_\{t\}\)at each re\-query point\.

Intuitively,ε𝒦\\varepsilon\_\{\\mathcal\{K\}\}measures how well the data distribution is preserved when actions from𝒟\\mathcal\{D\}are replayed open\-loop using the*adaptive*chunking scheduleκ\\kappa\. Whenε𝒦=0\\varepsilon\_\{\\mathcal\{K\}\}=0, the data perfectly matches the open\-loop rollouts under the adaptive schedule\. The key difference from standard OLC is that the re\-querying points are state\-dependent, so the total variation distance bound must hold uniformly over the distribution of re\-querying times induced byκ\\kappa\.

###### Proposition H\.3\(AOLC implies per\-scale OLC\)\.

If𝒟\\mathcal\{D\}isε𝒦\\varepsilon\_\{\\mathcal\{K\}\}\-adaptively open\-loop consistent underκ​\(s\)≡k\\kappa\(s\)\\equiv k\(constant selection\), then𝒟\\mathcal\{D\}isε𝒦\\varepsilon\_\{\\mathcal\{K\}\}\-open\-loop consistent for chunk sizekkin the sense ofli2026decoupled\.

This shows that AOLC is a strict generalization: it reduces to standard OLC when the selection function is constant, and imposes a stronger uniform condition whenκ\\kappavaries across states\.

### H\.3Advantage Separability and Selector Soundness

The core ofAQCis the adaptive selectork∗​\(s\)∈arg​maxk∈𝒦⁡max𝐚t:t\+k⁡Ak​\(s,𝐚t:t\+k\)k^\{\*\}\(s\)\\in\\operatorname\*\{arg\\,max\}\_\{k\\in\\mathcal\{K\}\}\\max\_\{\\mathbf\{a\}\_\{t:t\+k\}\}A^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\), whereAk​\(s,𝐚t:t\+k\):=\(Qk​\(s,𝐚t:t\+k\)−Vk​\(s\)\)/γkA^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\):=\(Q^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-V^\{k\}\(s\)\)/\\gamma^\{k\}is the per\-scale advantage\. We now formalize when this selector is*sound*— i\.e\., when it identifies the truly best chunk size\.

###### Definition H\.4\(Advantage Separability\)\.

LetAk,∗​\(s,𝐚t:t\+k\):=\(Qk,∗​\(s,𝐚t:t\+k\)−Vk,∗​\(s\)\)/γkA^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\):=\(Q^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-V^\{k,\*\}\(s\)\)/\\gamma^\{k\}denote the optimal per\-scale advantage for chunk sizekk\. Define the oracle\-best chunk size at statessas

k†​\(s\)∈arg​maxk∈𝒦⁡max𝐚t:t\+k⁡Ak,∗​\(s,𝐚t:t\+k\)\.\\displaystyle k^\{\\dagger\}\(s\)\\in\\operatorname\*\{arg\\,max\}\_\{k\\in\\mathcal\{K\}\}\\;\\max\_\{\\mathbf\{a\}\_\{t:t\+k\}\}A^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\.\(23\)The data𝒟\\mathcal\{D\}exhibitsΔ\\Delta\-advantage separabilityatssif

max𝐚t:t\+k†⁡Ak†,∗​\(s,𝐚t:t\+k†\)−maxk≠k†⁡max𝐚t:t\+k⁡Ak,∗​\(s,𝐚t:t\+k\)≥Δ​\(s\)\.\\displaystyle\\max\_\{\\mathbf\{a\}\_\{t:t\+k^\{\\dagger\}\}\}A^\{k^\{\\dagger\},\*\}\(s,\\mathbf\{a\}\_\{t:t\+k^\{\\dagger\}\}\)\\;\-\\;\\max\_\{k\\neq k^\{\\dagger\}\}\\max\_\{\\mathbf\{a\}\_\{t:t\+k\}\}A^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\\;\\geq\\;\\Delta\(s\)\.\(24\)We say𝒟\\mathcal\{D\}is globallyΔ\\Delta\-advantage separable ifΔ​\(s\)≥Δ\>0\\Delta\(s\)\\geq\\Delta\>0for alls∈supp​\(P𝒟⋆​\(s\)\)s\\in\\mathrm\{supp\}\(P\_\{\\mathcal\{D\}^\{\\star\}\}\(s\)\)\.

Advantage separability measures the*intrinsic discriminability*of the best chunk size\. WhenΔ​\(s\)\\Delta\(s\)is large, the optimal scale is clearly distinguishable; whenΔ​\(s\)≈0\\Delta\(s\)\\approx 0, multiple scales are nearly equally good and no selector can do much better than random\. In practice,Δ​\(s\)\\Delta\(s\)is large near contact events \(where short chunks are uniquely valuable\) and in free\-space motion \(where long chunks clearly dominate\), and small in intermediate states\.

###### Theorem H\.5\(Selector Soundness\)\.

Letεk:=‖Qk−Qk,∗‖∞\\varepsilon\_\{k\}:=\\\|Q^\{k\}\-Q^\{k,\*\}\\\|\_\{\\infty\}andδk:=‖Vk−Vk,∗‖∞\\delta\_\{k\}:=\\\|V^\{k\}\-V^\{k,\*\}\\\|\_\{\\infty\}be the critic estimation errors, and letε¯:=maxk∈𝒦⁡\(εk\+δk\)\\bar\{\\varepsilon\}:=\\max\_\{k\\in\\mathcal\{K\}\}\(\\varepsilon\_\{k\}\+\\delta\_\{k\}\)\. If𝒟\\mathcal\{D\}isΔ\\Delta\-advantage separable andε¯<Δ​γkmin/2\\bar\{\\varepsilon\}<\\Delta\\gamma^\{k\_\{\\min\}\}/2, wherekmin:=mink∈𝒦⁡kk\_\{\\min\}:=\\min\_\{k\\in\\mathcal\{K\}\}k, then for alls∈supp​\(P𝒟⋆​\(s\)\)s\\in\\mathrm\{supp\}\(P\_\{\\mathcal\{D\}^\{\\star\}\}\(s\)\), the empirical selector

k^​\(s\)∈arg​maxk∈𝒦⁡max𝐚t:t\+k⁡Qk​\(s,𝐚t:t\+k\)−Vk​\(s\)γk\\displaystyle\\hat\{k\}\(s\)\\in\\operatorname\*\{arg\\,max\}\_\{k\\in\\mathcal\{K\}\}\\;\\max\_\{\\mathbf\{a\}\_\{t:t\+k\}\}\\frac\{Q^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-V^\{k\}\(s\)\}\{\\gamma^\{k\}\}\(25\)agrees with the oracle selector:k^​\(s\)=k†​\(s\)\\hat\{k\}\(s\)=k^\{\\dagger\}\(s\)\.

###### Proof sketch\.

Letfk​\(s\):=max𝐚t:t\+k⁡Ak,∗​\(s,𝐚t:t\+k\)f\_\{k\}\(s\):=\\max\_\{\\mathbf\{a\}\_\{t:t\+k\}\}A^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)andf^k​\(s\):=max𝐚t:t\+k⁡A^k​\(s,𝐚t:t\+k\)\\hat\{f\}\_\{k\}\(s\):=\\max\_\{\\mathbf\{a\}\_\{t:t\+k\}\}\\hat\{A\}^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\. Since\|Qk−Qk,∗\|≤εk\|Q^\{k\}\-Q^\{k,\*\}\|\\leq\\varepsilon\_\{k\}and\|Vk−Vk,∗\|≤δk\|V^\{k\}\-V^\{k,\*\}\|\\leq\\delta\_\{k\}, we have\|f^k​\(s\)−fk​\(s\)\|≤\(εk\+δk\)/γk≤ε¯/γkmin\|\\hat\{f\}\_\{k\}\(s\)\-f\_\{k\}\(s\)\|\\leq\(\\varepsilon\_\{k\}\+\\delta\_\{k\}\)/\\gamma^\{k\}\\leq\\bar\{\\varepsilon\}/\\gamma^\{k\_\{\\min\}\}\. Advantage separability givesfk†​\(s\)−maxk≠k†⁡fk​\(s\)≥Δ​\(s\)f\_\{k^\{\\dagger\}\}\(s\)\-\\max\_\{k\\neq k^\{\\dagger\}\}f\_\{k\}\(s\)\\geq\\Delta\(s\)\. By the triangle inequality, ifε¯/γkmin<Δ​\(s\)/2\\bar\{\\varepsilon\}/\\gamma^\{k\_\{\\min\}\}<\\Delta\(s\)/2, thenf^k†​\(s\)\>f^k​\(s\)\\hat\{f\}\_\{k^\{\\dagger\}\}\(s\)\>\\hat\{f\}\_\{k\}\(s\)for allk≠k†k\\neq k^\{\\dagger\}, sok^​\(s\)=k†​\(s\)\\hat\{k\}\(s\)=k^\{\\dagger\}\(s\)\. A full proof appears in Appendix[I\.2](https://arxiv.org/html/2605.05544#A9.SS2)\. ∎

This theorem establishes that the advantage selector is*consistent*: as critic accuracy improves, the selected chunk size converges to the oracle choice\. The bound is non\-vacuous precisely whenΔ​\(s\)\>0\\Delta\(s\)\>0, i\.e\., when the problem is intrinsically discriminable\.

###### Proposition H\.6\(Selector Regret\)\.

Under the conditions of[Theorem˜H\.5](https://arxiv.org/html/2605.05544#A8.Thmtheorem5), the per\-state regret of the empirical selector relative to the oracle is bounded by

\|A^k^​\(s\)​\(s\)−Ak†,∗​\(s\)\|≤ε¯\+2​ε¯γkmin​Δ​\(s\)⋅maxk∈𝒦⁡diam​\(A\)k,\\displaystyle\\left\|\\hat\{A\}^\{\\hat\{k\}\(s\)\}\(s\)\-A^\{k^\{\\dagger\},\*\}\(s\)\\right\|\\;\\leq\\;\\bar\{\\varepsilon\}\+\\frac\{2\\bar\{\\varepsilon\}\}\{\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\}\\cdot\\max\_\{k\\in\\mathcal\{K\}\}\\mathrm\{diam\}\(A\)^\{k\},\(26\)wherediam​\(A\)k:=sups,aAk,∗​\(s,a\)−infs,aAk,∗​\(s,a\)\\mathrm\{diam\}\(A\)^\{k\}:=\\sup\_\{s,a\}A^\{k,\*\}\(s,a\)\-\\inf\_\{s,a\}A^\{k,\*\}\(s,a\)is the advantage range for scalekk\.

### H\.4AQC Value Dominance

We now prove the central theoretical claim:AQC’s value strictly dominates any fixed\-chunk policy\. This requires analyzing how the selector’s decisions compound over time\.

###### Definition H\.7\(AQC Effective Policy\)\.

The effective policy ofAQCis the closed\-loop policyπAQC\\pi^\{\\mathrm\{AQC\}\}that, at each statests\_\{t\}, selectsk^​\(st\)\\hat\{k\}\(s\_\{t\}\)via the advantage criterion and executesa1:k^​\(st\)∗a^\{\*\}\_\{1:\\hat\{k\}\(s\_\{t\}\)\}open\-loop\. The value function ofπAQC\\pi^\{\\mathrm\{AQC\}\}is

VAQC\(s\):=𝔼πAQC\[∑τ=0∞γτr\(sτ,aτ\)\|s0=s\]\.\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\):=\\mathbb\{E\}\_\{\\pi^\{\\mathrm\{AQC\}\}\}\\\!\\left\[\\sum\_\{\\tau=0\}^\{\\infty\}\\gamma^\{\\tau\}r\(s\_\{\\tau\},a\_\{\\tau\}\)\\;\\middle\|\\;s\_\{0\}=s\\right\]\.\(27\)

###### Theorem H\.8\(AQC Dominates Fixed\-Chunk Policies\)\.

LetVkV^\{k\}denote the true closed\-loop value of the fixed\-kkpolicyπk​\(s\):=arg​max𝐚t:t\+k⁡Qk,∗​\(s,𝐚t:t\+k\)\\pi^\{k\}\(s\):=\\operatorname\*\{arg\\,max\}\_\{\\mathbf\{a\}\_\{t:t\+k\}\}Q^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)with open\-loop execution\. Assume𝒟\\mathcal\{D\}isΔ\\Delta\-advantage separable withΔ\>0\\Delta\>0andε¯<Δ​γkmin/2\\bar\{\\varepsilon\}<\\Delta\\gamma^\{k\_\{\\min\}\}/2, wherekmin:=mink∈𝒦⁡kk\_\{\\min\}:=\\min\_\{k\\in\\mathcal\{K\}\}k\. Then for alls∈supp​\(P𝒟⋆​\(s\)\)s\\in\\mathrm\{supp\}\(P\_\{\\mathcal\{D\}^\{\\star\}\}\(s\)\)and any fixedk∈𝒦k\\in\\mathcal\{K\},

VAQC​\(s\)−Vk​\(s\)≥γkmin​\(1−2​ε¯/\(γkmin​Δ\)\)1−γ​𝔼s′∼dAQC​\[A¯k†,∗​\(s′\)−A¯k,∗​\(s′\)\],\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{k\}\(s\)\\;\\geq\\;\\frac\{\\gamma^\{k\_\{\\min\}\}\(1\-2\\bar\{\\varepsilon\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\)\)\}\{1\-\\gamma\}\\;\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\\\!\\left\[\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\-\\bar\{A\}^\{k,\*\}\(s^\{\\prime\}\)\\right\],\(28\)wheredAQCd^\{\\mathrm\{AQC\}\}is the discounted state visitation distribution ofπAQC\\pi^\{\\mathrm\{AQC\}\}andA¯j,∗​\(s\):=max𝐚t:t\+j⁡Aj,∗​\(s,𝐚t:t\+j\)\\bar\{A\}^\{j,\*\}\(s\):=\\max\_\{\\mathbf\{a\}\_\{t:t\+j\}\}A^\{j,\*\}\(s,\\mathbf\{a\}\_\{t:t\+j\}\)is the best advantage at scalejj\.

###### Proof sketch\.

The proof adapts the performance difference lemma ofkakade2002approximatelyto the chunk\-selection setting via a meta\-MDP construction:

1\.Meta\-MDP: We define a meta\-MDP whose “actions” are pairs\(k,𝐚t:t\+k\)\(k,\\mathbf\{a\}\_\{t:t\+k\}\)and whose transitions correspond to executingkksteps open\-loop\. In this meta\-MDP, the performance difference lemma gives:

VAQC​\(s\)−Vk​\(s\)=11−γ​𝔼s′∼dAQC​\[γk∗​\(s′\)​A¯metak∗​\(s′\),∗​\(s′\)−γk​A¯metak,∗​\(s′\)\]\.\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{k\}\(s\)=\\frac\{1\}\{1\-\\gamma\}\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\\\!\\left\[\\gamma^\{k^\{\*\}\(s^\{\\prime\}\)\}\\bar\{A\}^\{k^\{\*\}\(s^\{\\prime\}\),\*\}\_\{\\mathrm\{meta\}\}\(s^\{\\prime\}\)\-\\gamma^\{k\}\\bar\{A\}^\{k,\*\}\_\{\\mathrm\{meta\}\}\(s^\{\\prime\}\)\\right\]\.\(29\)
2\.Selector correctness: Under advantage separability withε¯<Δ​γkmin/2\\bar\{\\varepsilon\}<\\Delta\\gamma^\{k\_\{\\min\}\}/2, the selector always choosesk∗​\(s′\)=k†​\(s′\)k^\{\*\}\(s^\{\\prime\}\)=k^\{\\dagger\}\(s^\{\\prime\}\), soA¯k∗​\(s′\),∗​\(s′\)≥A¯k,∗​\(s′\)\\bar\{A\}^\{k^\{\*\}\(s^\{\\prime\}\),\*\}\(s^\{\\prime\}\)\\geq\\bar\{A\}^\{k,\*\}\(s^\{\\prime\}\)for any fixedkk\.

3\.Discount normalization: Sinceγk∗​\(s′\)≥γkmin\\gamma^\{k^\{\*\}\(s^\{\\prime\}\)\}\\geq\\gamma^\{k\_\{\\min\}\}, we lower\-boundγk∗​\(s′\)​A¯k†,∗​\(s′\)−γk​A¯k,∗​\(s′\)≥γkmin​\(A¯k†,∗​\(s′\)−A¯k,∗​\(s′\)\)\\gamma^\{k^\{\*\}\(s^\{\\prime\}\)\}\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\-\\gamma^\{k\}\\bar\{A\}^\{k,\*\}\(s^\{\\prime\}\)\\geq\\gamma^\{k\_\{\\min\}\}\(\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\-\\bar\{A\}^\{k,\*\}\(s^\{\\prime\}\)\)\.

4\.Error accounting: The\(1−2​ε¯/\(γkmin​Δ\)\)\(1\-2\\bar\{\\varepsilon\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\)\)factor accounts for the probability of mis\-selection\.

A full proof appears in Appendix[I\.4](https://arxiv.org/html/2605.05544#A9.SS4)\. ∎

The bound has a clean interpretation: the dominance gap is proportional to \(a\) how often the selector is correct \(1−2​ε¯/\(γkmin​Δ\)1\-2\\bar\{\\varepsilon\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\)\), \(b\) the effective horizon1/\(1−γ\)1/\(1\-\\gamma\), and \(c\) the average advantage gap between the oracle and the fixed\-kkpolicy\. When there exists a set of states where scalekkis suboptimal, the bound is strictly positive\.

###### Corollary H\.9\(Strict Dominance\)\.

If there exists a set of states𝒮diff⊆supp​\(P𝒟⋆\)\\mathcal\{S\}\_\{\\mathrm\{diff\}\}\\subseteq\\mathrm\{supp\}\(P\_\{\\mathcal\{D\}^\{\\star\}\}\)withdAQC​\(𝒮diff\)\>0d^\{\\mathrm\{AQC\}\}\(\\mathcal\{S\}\_\{\\mathrm\{diff\}\}\)\>0such that for eachs∈𝒮diffs\\in\\mathcal\{S\}\_\{\\mathrm\{diff\}\}, there existk1,k2∈𝒦k\_\{1\},k\_\{2\}\\in\\mathcal\{K\}withA¯k1,∗​\(s\)≠A¯k2,∗​\(s\)\\bar\{A\}^\{k\_\{1\},\*\}\(s\)\\neq\\bar\{A\}^\{k\_\{2\},\*\}\(s\), thenVAQC​\(s\)\>Vk​\(s\)V^\{\\mathrm\{AQC\}\}\(s\)\>V^\{k\}\(s\)for allk∈𝒦k\\in\\mathcal\{K\}wheneverε¯<Δmin/2\\bar\{\\varepsilon\}<\\Delta\_\{\\min\}/2, whereΔmin:=mins∈𝒮diff⁡Δ​\(s\)\\Delta\_\{\\min\}:=\\min\_\{s\\in\\mathcal\{S\}\_\{\\mathrm\{diff\}\}\}\\Delta\(s\)\.

### H\.5Error Propagation Under Imperfect Critics

When the critics\{Qk,Vk\}\\\{Q^\{k\},V^\{k\}\\\}are learned from finite data, the selector’s decisions are based on noisy advantage estimates\. We now analyze how this noise propagates through the closed\-loop execution\.

###### Definition H\.10\(Critic Error Profile\)\.

For eachk∈𝒦k\\in\\mathcal\{K\}, define the critic errorsεk:=sups,𝐚t:t\+k\|Qk​\(s,𝐚t:t\+k\)−Qk,∗​\(s,𝐚t:t\+k\)\|\\varepsilon\_\{k\}:=\\sup\_\{s,\\mathbf\{a\}\_\{t:t\+k\}\}\|Q^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-Q^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\|andδk:=sups\|Vk​\(s\)−Vk,∗​\(s\)\|\\delta\_\{k\}:=\\sup\_\{s\}\|V^\{k\}\(s\)\-V^\{k,\*\}\(s\)\|\. The composite error isε¯𝒦:=maxk∈𝒦⁡\(εk\+δk\)\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}:=\\max\_\{k\\in\\mathcal\{K\}\}\(\\varepsilon\_\{k\}\+\\delta\_\{k\}\)\.

###### Theorem H\.11\(Adaptive Selector Regret\)\.

LetπAQC\\pi^\{\\mathrm\{AQC\}\}be the policy using the learned selectork^​\(s\)\\hat\{k\}\(s\)andπ†\\pi^\{\\dagger\}be the oracle policy usingk†​\(s\)k^\{\\dagger\}\(s\)\. Then for alls∈supp​\(P𝒟⋆​\(s\)\)s\\in\\mathrm\{supp\}\(P\_\{\\mathcal\{D\}^\{\\star\}\}\(s\)\),

\|VAQC​\(s\)−V†​\(s\)\|≤2​ε¯𝒦\(1−γ\)​γkmin​Δ​\(s\)⋅maxk∈𝒦⁡R¯kγk,\\displaystyle\\left\|V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{\\dagger\}\(s\)\\right\|\\;\\leq\\;\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\}\\cdot\\max\_\{k\\in\\mathcal\{K\}\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\},\(30\)whereR¯k:=sups,𝐚t:t\+k\|Qk,∗​\(s,𝐚t:t\+k\)−Vk,∗​\(s\)\|\\bar\{R\}\_\{k\}:=\\sup\_\{s,\\mathbf\{a\}\_\{t:t\+k\}\}\|Q^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-V^\{k,\*\}\(s\)\|is the maximum advantage magnitude for scalekk\.

###### Proof sketch\.

The proof combines the selector soundness bound \([Theorem˜H\.5](https://arxiv.org/html/2605.05544#A8.Thmtheorem5)\) with a value difference analysis:

1\. At each re\-query statess, the selector incurs mis\-selection probability at most2​ε¯𝒦/\(γkmin​Δ​\(s\)\)2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\)\.

2\. When a mis\-selection occurs, the value loss per mistake is bounded byR¯k/γk\\bar\{R\}\_\{k\}/\\gamma^\{k\}\.

3\. Over the trajectory, the expected number of mis\-selections in the discounted horizon is at most\(1/\(1−γ\)\)⋅2​ε¯𝒦/\(γkmin​Δ​\(s\)\)\(1/\(1\-\\gamma\)\)\\cdot 2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\)\.

4\. The total regret is the product of expected mis\-selection count and per\-mis\-selection loss\.

A full proof appears in Appendix[I\.5](https://arxiv.org/html/2605.05544#A9.SS5)\. ∎

###### Corollary H\.12\(Uniform Regret Bound\)\.

IfΔ​\(s\)≥Δ\>0\\Delta\(s\)\\geq\\Delta\>0for allss, then

‖VAQC−V†‖∞≤2​ε¯𝒦\(1−γ\)​γkmin​Δ⋅maxk∈𝒦⁡R¯kγk\.\\displaystyle\\\|V^\{\\mathrm\{AQC\}\}\-V^\{\\dagger\}\\\|\_\{\\infty\}\\;\\leq\\;\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\max\_\{k\\in\\mathcal\{K\}\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(31\)

This bound shows that the regret scales linearly with the critic error and inversely with the advantage gap\. The1/\(1−γ\)1/\(1\-\\gamma\)factor reflects the compounding of selector mistakes over the effective horizon\.

### H\.6Closed\-Loop Analysis for Adaptive Execution

AQCexecutesk^​\(st\)\\hat\{k\}\(s\_\{t\}\)steps open\-loop and then re\-queries\. This adaptive re\-querying pattern is fundamentally different from both DQC’s fixed\-chunk execution and standard 1\-step closed\-loop control\. We now analyze the optimality gap of the adaptive closed\-loop policy\.

###### Definition H\.13\(Adaptive Closed\-Loop Value\)\.

LetπCLAQC\\pi^\{\\mathrm\{AQC\}\}\_\{\\mathrm\{CL\}\}execute the learned policy in closed\-loop: at each step, selectk^​\(s\)\\hat\{k\}\(s\), samplea1:k^​\(s\)∗a^\{\*\}\_\{1:\\hat\{k\}\(s\)\}, executea1∗a^\{\*\}\_\{1\}, and re\-query atst\+1s\_\{t\+1\}\. The closed\-loop value is

V∙\(s\):=𝔼πCLAQC\[∑τ=0∞γτr\(sτ,aτ\)\|s0=s\]\.\\displaystyle V^\{\\bullet\}\(s\):=\\mathbb\{E\}\_\{\\pi^\{\\mathrm\{AQC\}\}\_\{\\mathrm\{CL\}\}\}\\left\[\\sum\_\{\\tau=0\}^\{\\infty\}\\gamma^\{\\tau\}r\(s\_\{\\tau\},a\_\{\\tau\}\)\\;\\middle\|\\;s\_\{0\}=s\\right\]\.\(32\)

###### Theorem H\.14\(Adaptive Closed\-Loop Optimality\)\.

Assume𝒟\\mathcal\{D\}isε𝒦\\varepsilon\_\{\\mathcal\{K\}\}\-adaptively open\-loop consistent under the oracle selectork†k^\{\\dagger\}, viewed as a fixed selection functionκ​\(s\):=k†​\(s\)\\kappa\(s\):=k^\{\\dagger\}\(s\)defined by the advantage separability condition\. Then for alls∈supp​\(P𝒟⋆​\(s\)\)s\\in\\mathrm\{supp\}\(P\_\{\\mathcal\{D\}^\{\\star\}\}\(s\)\),

V⋆​\(s\)−VCL†​\(s\)≤ε𝒦​γ\(1−γ\)2​\[21−\(1−2​ε𝒦\)​γkmin\+11−\(1−ε𝒦\)​γkmin\],\\displaystyle V^\{\\star\}\(s\)\-V^\{\\dagger\}\_\{\\mathrm\{CL\}\}\(s\)\\;\\leq\\;\\frac\{\\varepsilon\_\{\\mathcal\{K\}\}\\gamma\}\{\(1\-\\gamma\)^\{2\}\}\\left\[\\frac\{2\}\{1\-\(1\-2\\varepsilon\_\{\\mathcal\{K\}\}\)\\gamma^\{k\_\{\\min\}\}\}\+\\frac\{1\}\{1\-\(1\-\\varepsilon\_\{\\mathcal\{K\}\}\)\\gamma^\{k\_\{\\min\}\}\}\\right\],\(33\)wherekmin:=mink∈𝒦⁡kk\_\{\\min\}:=\\min\_\{k\\in\\mathcal\{K\}\}k\. Furthermore, when the learned selector is used,

V⋆​\(s\)−V∙​\(s\)≤\(V⋆​\(s\)−VCL†​\(s\)\)\+2​ε¯𝒦\(1−γ\)​γkmin​Δ⋅maxk⁡R¯kγk\.\\displaystyle V^\{\\star\}\(s\)\-V^\{\\bullet\}\(s\)\\;\\leq\\;\(V^\{\\star\}\(s\)\-V^\{\\dagger\}\_\{\\mathrm\{CL\}\}\(s\)\)\+\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(34\)

###### Proof sketch\.

The proof combines three ingredients:

1\.AOLC decomposition: At each re\-query point, the state distribution deviates from the data by at mostε𝒦\\varepsilon\_\{\\mathcal\{K\}\}in TV distance\. Using standard total\-variation\-to\-value bounds, the per\-re\-query value error is bounded byε𝒦\\varepsilon\_\{\\mathcal\{K\}\}times the value range\.

2\.Geometric compounding: The number of re\-queries in the discounted horizon is bounded by1/\(1−γkmin\)1/\(1\-\\gamma^\{k\_\{\\min\}\}\), since at minimumkmink\_\{\\min\}steps elapse between re\-queries\. This gives the effective horizon factor\.

3\.Selector error addition: The learned selector introduces additional regret bounded by[Theorem˜H\.11](https://arxiv.org/html/2605.05544#A8.Thmtheorem11)\.

The bound is structurally similar to DQC’s closed\-loop bound but withkmink\_\{\\min\}replacinghh, reflecting that the adaptive policy re\-queries more frequently \(at worst everykmink\_\{\\min\}steps\), which improves reactivity\. A full proof appears in Appendix[I\.6](https://arxiv.org/html/2605.05544#A9.SS6)\. ∎

The bound reveals a trade\-off: a larger𝒦\\mathcal\{K\}with smallkmink\_\{\\min\}enables better reactivity near contacts \(since the policy can re\-query sooner\), but increases the number of re\-query points where TV distance error can accumulate\.

### H\.7Multi\-Scale Bootstrap Regularization

AQC’s unique architectural choice is that all partial critics\{Qk\}k<h\\\{Q^\{k\}\\\}\_\{k<h\}bootstrap from the same long\-horizon value functionVhV^\{h\}\. We now prove that this creates a*value flow hierarchy*that implicitly regularizes the critic ensemble\.

###### Theorem H\.15\(Multi\-Scale Bootstrap Regularization\)\.

Letεh:=‖Vh−Vh,∗‖∞\\varepsilon\_\{h\}:=\\\|V^\{h\}\-V^\{h,\*\}\\\|\_\{\\infty\}and letεk\\varepsilon\_\{k\}be thekk\-step TD fitting error forQkQ^\{k\}\. Then for allk∈𝒦k\\in\\mathcal\{K\},

‖Qk−Qk,∗‖∞≤γk1−γk​εh⏟bootstrap error\+11−γk​εk⏟fitting error\.\\displaystyle\\\|Q^\{k\}\-Q^\{k,\*\}\\\|\_\{\\infty\}\\;\\leq\\;\\underbrace\{\\frac\{\\gamma^\{k\}\}\{1\-\\gamma^\{k\}\}\\varepsilon\_\{h\}\}\_\{\\text\{bootstrap error\}\}\\;\+\\;\\underbrace\{\\frac\{1\}\{1\-\\gamma^\{k\}\}\\varepsilon\_\{k\}\}\_\{\\text\{fitting error\}\}\.\(35\)Moreover, the total critic ensemble error is bounded by

∑k∈𝒦‖Qk−Qk,∗‖∞≤εh​∑k∈𝒦γk1−γk\+∑k∈𝒦εk1−γk\.\\displaystyle\\sum\_\{k\\in\\mathcal\{K\}\}\\\|Q^\{k\}\-Q^\{k,\*\}\\\|\_\{\\infty\}\\;\\leq\\;\\varepsilon\_\{h\}\\sum\_\{k\\in\\mathcal\{K\}\}\\frac\{\\gamma^\{k\}\}\{1\-\\gamma^\{k\}\}\\;\+\\;\\sum\_\{k\\in\\mathcal\{K\}\}\\frac\{\\varepsilon\_\{k\}\}\{1\-\\gamma^\{k\}\}\.\(36\)

###### Proof sketch\.

The Bellman operator𝒯hk​Q=Rt:t\+k\+γk​Vh\\mathcal\{T\}^\{k\}\_\{h\}Q=R\_\{t:t\+k\}\+\\gamma^\{k\}V^\{h\}is aγk\\gamma^\{k\}\-contraction in the sup\-norm\. LetQfpkQ^\{k\}\_\{\\mathrm\{fp\}\}be its fixed point\. Then‖Qfpk−Qk,∗‖∞≤γk​‖Vh−Vh,∗‖∞=γk​εh\\\|Q^\{k\}\_\{\\mathrm\{fp\}\}\-Q^\{k,\*\}\\\|\_\{\\infty\}\\leq\\gamma^\{k\}\\\|V^\{h\}\-V^\{h,\*\}\\\|\_\{\\infty\}=\\gamma^\{k\}\\varepsilon\_\{h\}\. Since the TD loss drivesQkQ^\{k\}towardQfpkQ^\{k\}\_\{\\mathrm\{fp\}\}with fitting errorεk\\varepsilon\_\{k\}, the contraction mapping theorem gives‖Qk−Qfpk‖∞≤εk/\(1−γk\)\\\|Q^\{k\}\-Q^\{k\}\_\{\\mathrm\{fp\}\}\\\|\_\{\\infty\}\\leq\\varepsilon\_\{k\}/\(1\-\\gamma^\{k\}\)\. Triangle inequality yields the per\-scale bound; summing overkkgives the ensemble bound\. A full proof appears in Appendix[I\.7](https://arxiv.org/html/2605.05544#A9.SS7)\. ∎

The key insight is that a singleεh\\varepsilon\_\{h\}term controls the bootstrap error for*all*partial critics simultaneously, creating a hierarchical regularization: improvingVhV^\{h\}automatically improves allQkQ^\{k\}critics\.

###### Corollary H\.16\(Bootstrap Advantage over 1\-Step\)\.

IfQkQ^\{k\}were bootstrapped fromV1V^\{1\}instead ofVhV^\{h\}, the bound becomes

‖Qk−Qk,∗‖∞≤H​γk1−γk​ε1\+11−γk​εk,\\displaystyle\\\|Q^\{k\}\-Q^\{k,\*\}\\\|\_\{\\infty\}\\;\\leq\\;\\frac\{H\\gamma^\{k\}\}\{1\-\\gamma^\{k\}\}\\varepsilon\_\{1\}\\;\+\\;\\frac\{1\}\{1\-\\gamma^\{k\}\}\\varepsilon\_\{k\},\(37\)whereH=1/\(1−γ\)H=1/\(1\-\\gamma\)andε1:=‖V1−V1,∗‖∞\\varepsilon\_\{1\}:=\\\|V^\{1\}\-V^\{1,\*\}\\\|\_\{\\infty\}\. The ratio of bootstrap terms isH​ε1/εhH\\varepsilon\_\{1\}/\\varepsilon\_\{h\}, which isO​\(H\)O\(H\)larger whenε1≈εh\\varepsilon\_\{1\}\\approx\\varepsilon\_\{h\}\.

This shows that theVhV^\{h\}bootstrap is strictly superior when the long\-horizon value function is comparably accurate to the 1\-step value — a condition that holds in practice sinceVhV^\{h\}is trained via expectile regression on the EMAQ\-boostedQhQ^\{h\}, which has already accumulated multi\-step credit\.

###### Theorem H\.17\(Value Flow Monotonicity\)\.

Under the training objective[Equation˜14](https://arxiv.org/html/2605.05544#S5.E14), the critic hierarchy exhibits*value flow monotonicity*: for anyk1<k2k\_\{1\}<k\_\{2\}in𝒦\\mathcal\{K\},

‖Qk1−Qk1,∗‖∞≤γk1γk2​‖Qk2−Qk2,∗‖∞\+max⁡\(0,εk1−γk1γk2​εk21−γk1\)\.\\displaystyle\\\|Q^\{k\_\{1\}\}\-Q^\{k\_\{1\},\*\}\\\|\_\{\\infty\}\\;\\leq\\;\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\\|Q^\{k\_\{2\}\}\-Q^\{k\_\{2\},\*\}\\\|\_\{\\infty\}\\;\+\\;\\max\\left\(0,\\frac\{\\varepsilon\_\{k\_\{1\}\}\-\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\varepsilon\_\{k\_\{2\}\}\}\{1\-\\gamma^\{k\_\{1\}\}\}\\right\)\.\(38\)When all critics are fit to equal precision \(εk≡ε\\varepsilon\_\{k\}\\equiv\\varepsilon\), shorter\-horizon critics are at least as accurate as longer\-horizon ones up to theγk1/γk2\\gamma^\{k\_\{1\}\}/\\gamma^\{k\_\{2\}\}scaling factor\.

###### Proof sketch\.

BothQk1Q^\{k\_\{1\}\}andQk2Q^\{k\_\{2\}\}bootstrap from the sameVhV^\{h\}, so they share the same bootstrap errorεh\\varepsilon\_\{h\}\. The ratio of total errors is

‖Qk1−Qk1,∗‖∞‖Qk2−Qk2,∗‖∞≤γk1/\(1−γk1\)γk2/\(1−γk2\)≈γk1γk2,\\displaystyle\\frac\{\\\|Q^\{k\_\{1\}\}\-Q^\{k\_\{1\},\*\}\\\|\_\{\\infty\}\}\{\\\|Q^\{k\_\{2\}\}\-Q^\{k\_\{2\},\*\}\\\|\_\{\\infty\}\}\\;\\leq\\;\\frac\{\\gamma^\{k\_\{1\}\}/\(1\-\\gamma^\{k\_\{1\}\}\)\}\{\\gamma^\{k\_\{2\}\}/\(1\-\\gamma^\{k\_\{2\}\}\)\}\\;\\approx\\;\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\},\(39\)since1/\(1−γk\)1/\(1\-\\gamma^\{k\}\)varies slowly inkkwhenγ\\gammais close to 1\. This means that the shorter\-horizon critics inherit a fractionγk1−k2\\gamma^\{k\_\{1\}\-k\_\{2\}\}of the longer\-horizon critics’ error, making them inherently more stable\. A full proof appears in Appendix[I\.9](https://arxiv.org/html/2605.05544#A9.SS9)\. ∎

This theorem reveals a structural property of AQC’s critic hierarchy: information flows “downward” fromVhV^\{h\}to all partial critics, with shorter horizons benefiting from stronger discount attenuation of the bootstrap error\.

### H\.8Comparison to Prior Methods

We now establish formal relationships betweenAQC,DQC, and fixed\-chunk baselines\.

###### Theorem H\.18\(AQC vs\. DQC\)\.

LetVDQCV^\{\\mathrm\{DQC\}\}be the closed\-loop value of DQC’s policy with fixed partial chunk sizehah\_\{a\}\. Assume there exists a set of states𝒮adapt\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}withdAQC​\(𝒮adapt\)≥ρ\>0d^\{\\mathrm\{AQC\}\}\(\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}\)\\geq\\rho\>0such that for eachs∈𝒮adapts\\in\\mathcal\{S\}\_\{\\mathrm\{adapt\}\},

maxk∈𝒦⁡A¯k,∗​\(s\)−A¯ha,∗​\(s\)≥δadapt​\(s\)\.\\displaystyle\\max\_\{k\\in\\mathcal\{K\}\}\\bar\{A\}^\{k,\*\}\(s\)\-\\bar\{A\}^\{h\_\{a\},\*\}\(s\)\\geq\\delta\_\{\\mathrm\{adapt\}\}\(s\)\.\(40\)Then

VAQC​\(s\)−VDQC​\(s\)≥γkmin​ρ1−γ​𝔼s′∼dAQC​\[δadapt​\(s′\)∣s′∈𝒮adapt\]−2​ε¯𝒦\(1−γ\)​γkmin​Δ⋅maxk⁡R¯kγk\.\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{\\mathrm\{DQC\}\}\(s\)\\;\\geq\\;\\frac\{\\gamma^\{k\_\{\\min\}\}\\rho\}\{1\-\\gamma\}\\;\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\[\\delta\_\{\\mathrm\{adapt\}\}\(s^\{\\prime\}\)\\mid s^\{\\prime\}\\in\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}\]\\;\-\\;\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(41\)

###### Proof sketch\.

The proof applies the dominance bound \([Theorem˜H\.8](https://arxiv.org/html/2605.05544#A8.Thmtheorem8)\) withk=hak=h\_\{a\}\. The first term captures the advantage of selecting the optimal scale over the fixedhah\_\{a\}scale at states where they differ\. The second term is the selector regret\. When the advantage gapδadapt\\delta\_\{\\mathrm\{adapt\}\}is large enough to overcome the selector regret, AQC strictly dominates DQC\. A full proof appears in Appendix[I\.10](https://arxiv.org/html/2605.05544#A9.SS10)\. ∎

###### Theorem H\.19\(AQC vs\.nn\-Step TD\)\.

LetVnV^\{n\}be the value of the policy learned viann\-step return backup on the same data𝒟\\mathcal\{D\}\. If𝒟\\mathcal\{D\}isδn\\delta\_\{n\}\-sub\-optimal \(in the sense ofli2026decoupled\) andΔ\\Delta\-advantage separable, then forδn\>3​ε𝒦​H​H¯kmin\\delta\_\{n\}\>3\\varepsilon\_\{\\mathcal\{K\}\}H\\bar\{H\}\_\{k\_\{\\min\}\},

VAQC​\(s\)−Vn​\(s\)≥δn​H¯n−3​ε𝒦​H​H¯kmin−2​ε¯𝒦\(1−γ\)​γkmin​Δ⋅maxk⁡R¯kγk,\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{n\}\(s\)\\;\\geq\\;\\delta\_\{n\}\\bar\{H\}\_\{n\}\-3\\varepsilon\_\{\\mathcal\{K\}\}H\\bar\{H\}\_\{k\_\{\\min\}\}\\;\-\\;\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\},\(42\)whereH¯k=1/\(1−γk\)\\bar\{H\}\_\{k\}=1/\(1\-\\gamma^\{k\}\)andH¯n=1/\(1−γn\)\\bar\{H\}\_\{n\}=1/\(1\-\\gamma^\{n\}\)\.

###### Proof sketch\.

By the dominance bound \([Theorem˜H\.8](https://arxiv.org/html/2605.05544#A8.Thmtheorem8)\) with any fixedk∈𝒦k\\in\\mathcal\{K\},VAQC​\(s\)≥Vk​\(s\)−selector\_regretV^\{\\mathrm\{AQC\}\}\(s\)\\geq V^\{k\}\(s\)\-\\text\{selector\\\_regret\}, where the regret term is bounded by[Theorem˜H\.11](https://arxiv.org/html/2605.05544#A8.Thmtheorem11)\. By DQC’s comparison theorem\[li2026decoupled\], the best fixed\-chunk policy satisfiesVk​\(s\)≥Vn​\(s\)\+δn​H¯n−3​ε𝒦​H​H¯kminV^\{k\}\(s\)\\geq V^\{n\}\(s\)\+\\delta\_\{n\}\\bar\{H\}\_\{n\}\-3\\varepsilon\_\{\\mathcal\{K\}\}H\\bar\{H\}\_\{k\_\{\\min\}\}, where the TV error term accounts for the adaptive re\-querying pattern withkmink\_\{\\min\}\. Chaining these inequalities yields the result\. A full proof appears in Appendix[I\.11](https://arxiv.org/html/2605.05544#A9.SS11)\. ∎

#### Structural properties\.

We highlight three structural properties of the AQC critic architecture\. The first restates the noise\-immunity result from[Proposition˜5\.1](https://arxiv.org/html/2605.05544#S5.Thmtheorem1)for completeness\. The remaining two characterize \(1\) the advantage ofVhV^\{h\}bootstrapping over a 1\-step baseline, and \(2\) the relationship betweenAQC’s partial critics and DQC’s distilled critic\.

###### Proposition H\.20\(Noise Immunity of the Advantage Selector; Restated from[Proposition˜5\.1](https://arxiv.org/html/2605.05544#S5.Thmtheorem1)\)\.

Letδk​\(s\):=Qk​\(s,𝐚t:t\+k\)/γk−Vk​\(s\)/γk\\delta\_\{k\}\(s\):=Q^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)/\\gamma^\{k\}\-V^\{k\}\(s\)/\\gamma^\{k\}\. In a region whereVh​\(s\)≤ϵV^\{h\}\(s\)\\leq\\epsilonfor all reachable states, and assuming function approximation errors for bothQkQ^\{k\}andVkV^\{k\}are bounded byσ\\sigma,

\|δk​\(s\)\|≤ϵ\+2​σ\.\\displaystyle\|\\delta\_\{k\}\(s\)\|\\;\\leq\\;\\epsilon\+2\\sigma\.\(43\)Whenϵ≪σ\\epsilon\\ll\\sigma\(far from rewards, the value signal is smaller than the noise floor\), the advantage is dominated by approximation noise and all chunk sizes score near\-zero\. By contrast, the uncorrected selectorarg​maxk⁡Qk/γk\\operatorname\*\{arg\\,max\}\_\{k\}Q^\{k\}/\\gamma^\{k\}lacks this safeguard: when all scores are≈ϵ\+σk\\approx\\epsilon\+\\sigma\_\{k\}, the argmax picks the scale with the largest positive noiseσk\\sigma\_\{k\}, producing a deterministic but systematically biased choice\.

#### Bootstrap target choice\.

The choice of bootstrap target in[Equation˜14](https://arxiv.org/html/2605.05544#S5.E14)is critical\. The following result quantifies theO​\(H\)O\(H\)improvement from usingVhV^\{h\}over a 1\-step value function\.

###### Proposition H\.21\(VhV^\{h\}Bootstrap Tightens the Sub\-optimality Bound; Restated\)\.

Letεh:=‖Vξh−V∗‖∞\\varepsilon\_\{h\}:=\\\|V^\{h\}\_\{\\xi\}\-V^\{\*\}\\\|\_\{\\infty\}andεk:=‖Qψk−Qfpk‖∞\\varepsilon\_\{k\}:=\\\|Q^\{k\}\_\{\\psi\}\-Q^\{k\}\_\{\\mathrm\{fp\}\}\\\|\_\{\\infty\}, whereQfpkQ^\{k\}\_\{\\mathrm\{fp\}\}is the fixed point of the Bellman operator𝒯hk\\mathcal\{T\}^\{k\}\_\{h\}usingV¯ξh\\bar\{V\}^\{h\}\_\{\\xi\}as bootstrap\. Then:

‖Qψk−Qk,∗‖∞≤γk1−γk​εh\+11−γk​εk\.\\displaystyle\\bigl\\\|Q^\{k\}\_\{\\psi\}\-Q^\{k,\*\}\\bigr\\\|\_\{\\infty\}\\;\\leq\\;\\frac\{\\gamma^\{k\}\}\{1\-\\gamma^\{k\}\}\\,\\varepsilon\_\{h\}\\;\+\\;\\frac\{1\}\{1\-\\gamma^\{k\}\}\\,\\varepsilon\_\{k\}\.\(44\)BootstrappingQkQ^\{k\}with a 1\-step value functionV1V^\{1\}satisfying‖V1−V∗‖∞≤ε1\\\|V^\{1\}\-V^\{\*\}\\\|\_\{\\infty\}\\leq\\varepsilon\_\{1\}yields:

‖Qψk−Qk,∗‖∞≤H​γk1−γk​ε1\+11−γk​εk,\\displaystyle\\bigl\\\|Q^\{k\}\_\{\\psi\}\-Q^\{k,\*\}\\bigr\\\|\_\{\\infty\}\\;\\leq\\;\\frac\{H\\,\\gamma^\{k\}\}\{1\-\\gamma^\{k\}\}\\,\\varepsilon\_\{1\}\\;\+\\;\\frac\{1\}\{1\-\\gamma^\{k\}\}\\,\\varepsilon\_\{k\},\(45\)whereH=1/\(1−γ\)H=1/\(1\-\\gamma\)is the effective horizon\.

#### Relationship to DQC’s critic\.

AQC’s partial critics recover DQC’s distilled critic as a special case, but without requiring goal\-conditioned hindsight relabeling or a distillation hyperparameter\.

###### Proposition H\.22\(QkQ^\{k\}ApproximatesDQC’s Distilled Critic; Restated\)\.

li2026decoupledconstruct a distilled partial criticQψPQ^\{P\}\_\{\\psi\}satisfying, at the fixed point of their distillation operator:

QψP​\(st,at:t\+ha\)≈∑j=0ha−1γj​rt\+j\+γha​V∗​\(st\+ha\)\.\\displaystyle Q^\{P\}\_\{\\psi\}\(s\_\{t\},a\_\{t:t\+h\_\{a\}\}\)\\;\\approx\\;\\sum\_\{j=0\}^\{h\_\{a\}\-1\}\\gamma^\{j\}r\_\{t\+j\}\\;\+\\;\\gamma^\{h\_\{a\}\}V^\{\*\}\(s\_\{t\+h\_\{a\}\}\)\.\(46\)OurQψkQ^\{k\}\_\{\\psi\}satisfies at its Bellman fixed point:

Qψk​\(st,at:t\+k\)=∑j=0k−1γj​rt\+j\+γk​V¯ξh​\(st\+k\),\\displaystyle Q^\{k\}\_\{\\psi\}\(s\_\{t\},a\_\{t:t\+k\}\)\\;=\\;\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\\;\+\\;\\gamma^\{k\}\\bar\{V\}^\{h\}\_\{\\xi\}\(s\_\{t\+k\}\),\(47\)which recovers DQC’s equation withha=kh\_\{a\}=kandV∗≈VξhV^\{\*\}\\approx V^\{h\}\_\{\\xi\}, up to error bounded byγk​εh\\gamma^\{k\}\\varepsilon\_\{h\}\.

### H\.9Summary of Theoretical Results

Table 10:Summary of theoretical results\. All bounds hold fors∈supp​\(P𝒟⋆​\(s\)\)s\\in\\mathrm\{supp\}\(P\_\{\\mathcal\{D\}^\{\\star\}\}\(s\)\)\.ResultConditionBoundSelector Soundness \([Theorem˜H\.5](https://arxiv.org/html/2605.05544#A8.Thmtheorem5)\)Δ\\Delta\-AS,ε¯<Δ​γkmin/2\\bar\{\\varepsilon\}<\\Delta\\gamma^\{k\_\{\\min\}\}/2k^​\(s\)=k†​\(s\)\\hat\{k\}\(s\)=k^\{\\dagger\}\(s\)Selector Regret \([Theorem˜H\.11](https://arxiv.org/html/2605.05544#A8.Thmtheorem11)\)Δ\\Delta\-AS2​ε¯\(1−γ\)​γkmin​Δ⋅R¯γkmin\\frac\{2\\bar\{\\varepsilon\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\frac\{\\bar\{R\}\}\{\\gamma^\{k\_\{\\min\}\}\}Value Dominance \([Theorem˜H\.8](https://arxiv.org/html/2605.05544#A8.Thmtheorem8)\)Δ\\Delta\-AS,ε¯<Δ​γkmin/2\\bar\{\\varepsilon\}<\\Delta\\gamma^\{k\_\{\\min\}\}/2VAQC​\(s\)≥Vk​\(s\)V^\{\\mathrm\{AQC\}\}\(s\)\\geq V^\{k\}\(s\)Closed\-Loop Optimality \([Theorem˜H\.14](https://arxiv.org/html/2605.05544#A8.Thmtheorem14)\)ε𝒦\\varepsilon\_\{\\mathcal\{K\}\}\-AOLCε𝒦​H2​H¯kmin\\varepsilon\_\{\\mathcal\{K\}\}H^\{2\}\\bar\{H\}\_\{k\_\{\\min\}\}Bootstrap Regularization \([Theorem˜H\.15](https://arxiv.org/html/2605.05544#A8.Thmtheorem15)\)VhV^\{h\}bootstrapγk1−γk​εh\+εk1−γk\\frac\{\\gamma^\{k\}\}\{1\-\\gamma^\{k\}\}\\varepsilon\_\{h\}\+\\frac\{\\varepsilon\_\{k\}\}\{1\-\\gamma^\{k\}\}Value Flow Monotonicity \([Theorem˜H\.17](https://arxiv.org/html/2605.05544#A8.Thmtheorem17)\)Equalεk\\varepsilon\_\{k\}‖Qk1‖≤γk1γk2​‖Qk2‖\\\|Q^\{k\_\{1\}\}\\\|\\leq\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\\|Q^\{k\_\{2\}\}\\\|AQC vs\. DQC \([Theorem˜H\.18](https://arxiv.org/html/2605.05544#A8.Thmtheorem18)\)δadapt\\delta\_\{\\mathrm\{adapt\}\}\-gapγkmin​ρ​δadapt1−γ−regret\\frac\{\\gamma^\{k\_\{\\min\}\}\\rho\\delta\_\{\\mathrm\{adapt\}\}\}\{1\-\\gamma\}\-\\text\{regret\}

## Appendix IProofs

We provide formal proofs for all results in Appendix[H](https://arxiv.org/html/2605.05544#A8)\.

### I\.1Proof of Proposition[H\.3](https://arxiv.org/html/2605.05544#A8.Thmtheorem3)

Whenκ​\(s\)≡k\\kappa\(s\)\\equiv k, the adaptive rollout distributionP𝒟,κ∘P^\{\\circ\}\_\{\\mathcal\{D\},\\kappa\}reduces to the standard open\-loop rollout distributionP𝒟∘P^\{\\circ\}\_\{\\mathcal\{D\}\}used in DQC’s definition of OLC\[li2026decoupled\]\. The TV distance constraints[Equations˜21](https://arxiv.org/html/2605.05544#A8.E21)and[22](https://arxiv.org/html/2605.05544#A8.E22)then become exactly the weak OLC constraints Equations 8 and 9 from DQC’s Definition 2\. Since DQC’s strong OLC Equation 10 imposes an additional per\-action\-sequence constraint, AOLC with constantκ\\kappaimplies weak OLC and is strictly weaker than strong OLC\.

### I\.2Proof of[Theorem˜H\.5](https://arxiv.org/html/2605.05544#A8.Thmtheorem5)

Letfk​\(s\):=max𝐚t:t\+k⁡Ak,∗​\(s,𝐚t:t\+k\)f\_\{k\}\(s\):=\\max\_\{\\mathbf\{a\}\_\{t:t\+k\}\}A^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)andf^k​\(s\):=max𝐚t:t\+k⁡A^k​\(s,𝐚t:t\+k\)\\hat\{f\}\_\{k\}\(s\):=\\max\_\{\\mathbf\{a\}\_\{t:t\+k\}\}\\hat\{A\}^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\), whereAk,∗A^\{k,\*\}andA^k\\hat\{A\}^\{k\}denote the true and estimated per\-scale advantages, respectively\.

For anyssand𝐚t:t\+k\\mathbf\{a\}\_\{t:t\+k\},

\|A^k​\(s,𝐚t:t\+k\)−Ak,∗​\(s,𝐚t:t\+k\)\|\\displaystyle\|\\hat\{A\}^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-A^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\|=\|Qk​\(s,𝐚t:t\+k\)−Vk​\(s\)γk−Qk,∗​\(s,𝐚t:t\+k\)−Vk,∗​\(s\)γk\|\\displaystyle=\\left\|\\frac\{Q^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-V^\{k\}\(s\)\}\{\\gamma^\{k\}\}\-\\frac\{Q^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-V^\{k,\*\}\(s\)\}\{\\gamma^\{k\}\}\\right\|\(48\)≤\|Qk​\(s,𝐚t:t\+k\)−Qk,∗​\(s,𝐚t:t\+k\)\|\+\|Vk​\(s\)−Vk,∗​\(s\)\|γk\\displaystyle\\leq\\frac\{\|Q^\{k\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-Q^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\|\+\|V^\{k\}\(s\)\-V^\{k,\*\}\(s\)\|\}\{\\gamma^\{k\}\}\(49\)≤εk\+δkγk≤ε¯γkmin\.\\displaystyle\\leq\\frac\{\\varepsilon\_\{k\}\+\\delta\_\{k\}\}\{\\gamma^\{k\}\}\\;\\leq\\;\\frac\{\\bar\{\\varepsilon\}\}\{\\gamma^\{k\_\{\\min\}\}\}\.\(50\)
Taking the maximum over𝐚t:t\+k\\mathbf\{a\}\_\{t:t\+k\}, we get\|f^k​\(s\)−fk​\(s\)\|≤ε¯/γkmin\|\\hat\{f\}\_\{k\}\(s\)\-f\_\{k\}\(s\)\|\\leq\\bar\{\\varepsilon\}/\\gamma^\{k\_\{\\min\}\}\.

ByΔ\\Delta\-advantage separability,fk†​\(s\)−maxk≠k†⁡fk​\(s\)≥Δ​\(s\)f\_\{k^\{\\dagger\}\}\(s\)\-\\max\_\{k\\neq k^\{\\dagger\}\}f\_\{k\}\(s\)\\geq\\Delta\(s\)\. For the empirical selector to mis\-select, we needf^k′​\(s\)\>f^k†​\(s\)\\hat\{f\}\_\{k^\{\\prime\}\}\(s\)\>\\hat\{f\}\_\{k^\{\\dagger\}\}\(s\)for somek′≠k†k^\{\\prime\}\\neq k^\{\\dagger\}\. By the triangle inequality:

f^k′​\(s\)−f^k†​\(s\)\\displaystyle\\hat\{f\}\_\{k^\{\\prime\}\}\(s\)\-\\hat\{f\}\_\{k^\{\\dagger\}\}\(s\)≤fk′​\(s\)−fk†​\(s\)\+\|f^k′​\(s\)−fk′​\(s\)\|\+\|f^k†​\(s\)−fk†​\(s\)\|\\displaystyle\\leq f\_\{k^\{\\prime\}\}\(s\)\-f\_\{k^\{\\dagger\}\}\(s\)\+\|\\hat\{f\}\_\{k^\{\\prime\}\}\(s\)\-f\_\{k^\{\\prime\}\}\(s\)\|\+\|\\hat\{f\}\_\{k^\{\\dagger\}\}\(s\)\-f\_\{k^\{\\dagger\}\}\(s\)\|\(51\)≤−Δ​\(s\)\+2​ε¯/γkmin\.\\displaystyle\\leq\-\\Delta\(s\)\+2\\bar\{\\varepsilon\}/\\gamma^\{k\_\{\\min\}\}\.\(52\)
Ifε¯<Δ​γkmin/2\\bar\{\\varepsilon\}<\\Delta\\gamma^\{k\_\{\\min\}\}/2, thenf^k′​\(s\)−f^k†​\(s\)<0\\hat\{f\}\_\{k^\{\\prime\}\}\(s\)\-\\hat\{f\}\_\{k^\{\\dagger\}\}\(s\)<0for allk′≠k†k^\{\\prime\}\\neq k^\{\\dagger\}, sok^​\(s\)=k†​\(s\)\\hat\{k\}\(s\)=k^\{\\dagger\}\(s\)\.

For the probabilistic bound whenε¯≥Δ​γkmin/2\\bar\{\\varepsilon\}\\geq\\Delta\\gamma^\{k\_\{\\min\}\}/2, a mis\-selection requires the estimation error to bridge at least half the advantage gap\. By Markov’s inequality applied to the estimation error:

ℙ​\(k^​\(s\)≠k†​\(s\)\)\\displaystyle\\mathbb\{P\}\(\\hat\{k\}\(s\)\\neq k^\{\\dagger\}\(s\)\)≤ℙ​\(\|A^k^​\(s\)​\(s\)−Ak†,∗​\(s\)\|≥Δ​\(s\)/2\)\\displaystyle\\leq\\mathbb\{P\}\\left\(\|\\hat\{A\}^\{\\hat\{k\}\(s\)\}\(s\)\-A^\{k^\{\\dagger\},\*\}\(s\)\|\\geq\\Delta\(s\)/2\\right\)\(53\)≤2​ε¯γkmin​Δ​\(s\)\.\\displaystyle\\leq\\frac\{2\\bar\{\\varepsilon\}\}\{\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\}\.\(54\)

### I\.3Proof of Proposition[H\.6](https://arxiv.org/html/2605.05544#A8.Thmtheorem6)

Decompose the regret into two cases: correct selection and mis\-selection\.

Whenk^​\(s\)=k†​\(s\)\\hat\{k\}\(s\)=k^\{\\dagger\}\(s\), the regret is bounded by the critic error:

\|A^k†​\(s\)−Ak†,∗​\(s\)\|≤ε¯/γkmin\.\\displaystyle\|\\hat\{A\}^\{k^\{\\dagger\}\}\(s\)\-A^\{k^\{\\dagger\},\*\}\(s\)\|\\leq\\bar\{\\varepsilon\}/\\gamma^\{k\_\{\\min\}\}\.\(55\)
Whenk^​\(s\)≠k†​\(s\)\\hat\{k\}\(s\)\\neq k^\{\\dagger\}\(s\)\(occurring with probability at most2​ε¯/\(γkmin​Δ​\(s\)\)2\\bar\{\\varepsilon\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\)by[Theorem˜H\.5](https://arxiv.org/html/2605.05544#A8.Thmtheorem5)\), the regret is bounded by the maximum advantage range:

\|A^k^​\(s\)​\(s\)−Ak†,∗​\(s\)\|≤ε¯/γkmin\+maxk⁡diam​\(A\)k\.\\displaystyle\|\\hat\{A\}^\{\\hat\{k\}\(s\)\}\(s\)\-A^\{k^\{\\dagger\},\*\}\(s\)\|\\leq\\bar\{\\varepsilon\}/\\gamma^\{k\_\{\\min\}\}\+\\max\_\{k\}\\mathrm\{diam\}\(A\)^\{k\}\.\(56\)
Combining:

𝔼​\[\|A^k^​\(s\)​\(s\)−Ak†,∗​\(s\)\|\]\\displaystyle\\mathbb\{E\}\\left\[\|\\hat\{A\}^\{\\hat\{k\}\(s\)\}\(s\)\-A^\{k^\{\\dagger\},\*\}\(s\)\|\\right\]≤\(1−perr\)⋅ε¯/γkmin\+perr⋅\(ε¯/γkmin\+maxk⁡diam​\(A\)k\)\\displaystyle\\leq\(1\-p\_\{\\mathrm\{err\}\}\)\\cdot\\bar\{\\varepsilon\}/\\gamma^\{k\_\{\\min\}\}\+p\_\{\\mathrm\{err\}\}\\cdot\(\\bar\{\\varepsilon\}/\\gamma^\{k\_\{\\min\}\}\+\\max\_\{k\}\\mathrm\{diam\}\(A\)^\{k\}\)\(57\)=ε¯/γkmin\+2​ε¯γkmin​Δ​\(s\)⋅maxk⁡diam​\(A\)k\.\\displaystyle=\\bar\{\\varepsilon\}/\\gamma^\{k\_\{\\min\}\}\+\\frac\{2\\bar\{\\varepsilon\}\}\{\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\}\\cdot\\max\_\{k\}\\mathrm\{diam\}\(A\)^\{k\}\.\(58\)

### I\.4Proof of[Theorem˜H\.8](https://arxiv.org/html/2605.05544#A8.Thmtheorem8)

We construct a meta\-MDPℳ~=\(𝒮,𝒜~,T~,r~,γ\)\\tilde\{\\mathcal\{M\}\}=\(\\mathcal\{S\},\\tilde\{\\mathcal\{A\}\},\\tilde\{T\},\\tilde\{r\},\\gamma\)where: \-𝒜~=⋃k∈𝒦\{\(k,𝐚t:t\+k\):𝐚t:t\+k∈𝒜k\}\\tilde\{\\mathcal\{A\}\}=\\bigcup\_\{k\\in\\mathcal\{K\}\}\\\{\(k,\\mathbf\{a\}\_\{t:t\+k\}\):\\mathbf\{a\}\_\{t:t\+k\}\\in\\mathcal\{A\}^\{k\}\\\}is the set of meta\-actions\. \-T~​\(s′∣s,\(k,𝐚t:t\+k\)\)\\tilde\{T\}\(s^\{\\prime\}\\mid s,\(k,\\mathbf\{a\}\_\{t:t\+k\}\)\)is the distribution ofst\+ks\_\{t\+k\}after executing𝐚t:t\+k\\mathbf\{a\}\_\{t:t\+k\}open\-loop forkksteps fromst=ss\_\{t\}=s\. \-r~​\(s,\(k,𝐚t:t\+k\)\)=∑j=0k−1γj​r​\(st\+j,at\+j\)\\tilde\{r\}\(s,\(k,\\mathbf\{a\}\_\{t:t\+k\}\)\)=\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\(s\_\{t\+j\},a\_\{t\+j\}\)is the cumulative discounted reward\.

In this meta\-MDP, bothπAQC\\pi^\{\\mathrm\{AQC\}\}andπk\\pi^\{k\}are valid policies\. The performance difference lemma\[kakade2002approximately\]gives:

VAQC​\(s\)−Vk​\(s\)=11−γ​𝔼s′∼dAQC​\[Qmetak,∗​\(s′,\(k∗​\(s′\),a∗\)\)−Qmetak,∗​\(s′,\(k,ak\)\)\],\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{k\}\(s\)=\\frac\{1\}\{1\-\\gamma\}\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\\\!\\left\[Q^\{k,\*\}\_\{\\mathrm\{meta\}\}\(s^\{\\prime\},\(k^\{\*\}\(s^\{\\prime\}\),a^\{\*\}\)\)\-Q^\{k,\*\}\_\{\\mathrm\{meta\}\}\(s^\{\\prime\},\(k,a^\{k\}\)\)\\right\],\(59\)whereQmetak,∗Q^\{k,\*\}\_\{\\mathrm\{meta\}\}is the optimal meta\-action value function\.

Using the relationship between meta\-action values and per\-scale advantages:

Qmetak,∗​\(s,\(k,𝐚t:t\+k\)\)\\displaystyle Q^\{k,\*\}\_\{\\mathrm\{meta\}\}\(s,\(k,\\mathbf\{a\}\_\{t:t\+k\}\)\)=∑j=0k−1γj​rt\+j\+γk​V∗​\(st\+k\)\\displaystyle=\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\+\\gamma^\{k\}V^\{\*\}\(s\_\{t\+k\}\)\(60\)=Vk,∗​\(s\)\+γk​Ak,∗​\(s,𝐚t:t\+k\),\\displaystyle=V^\{k,\*\}\(s\)\+\\gamma^\{k\}A^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\),\(61\)so the advantage in the meta\-MDP is:

Qmetak,∗​\(s,\(k,𝐚t:t\+k\)\)−maxj,𝐚t:t\+j⁡Qmetaj,∗​\(s,\(j,𝐚t:t\+j\)\)=γk​\(Ak,∗​\(s,𝐚t:t\+k\)−A¯k†,∗​\(s\)\)\.\\displaystyle Q^\{k,\*\}\_\{\\mathrm\{meta\}\}\(s,\(k,\\mathbf\{a\}\_\{t:t\+k\}\)\)\-\\max\_\{j,\\mathbf\{a\}\_\{t:t\+j\}\}Q^\{j,\*\}\_\{\\mathrm\{meta\}\}\(s,\(j,\\mathbf\{a\}\_\{t:t\+j\}\)\)=\\gamma^\{k\}\\left\(A^\{k,\*\}\(s,\\mathbf\{a\}\_\{t:t\+k\}\)\-\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s\)\\right\)\.\(62\)
Under advantage separability withε¯<Δ/2\\bar\{\\varepsilon\}<\\Delta/2, the selector choosesk∗​\(s′\)=k†​\(s′\)k^\{\*\}\(s^\{\\prime\}\)=k^\{\\dagger\}\(s^\{\\prime\}\), so:

𝔼s′∼dAQC​\[Qmetak,∗​\(s′,\(k∗​\(s′\),a∗\)\)−Qmetak,∗​\(s′,\(k,ak\)\)\]\\displaystyle\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\\\!\\left\[Q^\{k,\*\}\_\{\\mathrm\{meta\}\}\(s^\{\\prime\},\(k^\{\*\}\(s^\{\\prime\}\),a^\{\*\}\)\)\-Q^\{k,\*\}\_\{\\mathrm\{meta\}\}\(s^\{\\prime\},\(k,a^\{k\}\)\)\\right\]\(63\)=𝔼s′∼dAQC​\[γk∗​\(s′\)​A¯k†,∗​\(s′\)−γk​A¯k,∗​\(s′\)\]\.\\displaystyle\\quad=\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\\\!\\left\[\\gamma^\{k^\{\*\}\(s^\{\\prime\}\)\}\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\-\\gamma^\{k\}\\bar\{A\}^\{k,\*\}\(s^\{\\prime\}\)\\right\]\.\(64\)
Sinceγk∗​\(s′\)≥γkmin\\gamma^\{k^\{\*\}\(s^\{\\prime\}\)\}\\geq\\gamma^\{k\_\{\\min\}\}for alls′s^\{\\prime\}andγk≤γkmin\\gamma^\{k\}\\leq\\gamma^\{k\_\{\\min\}\}, and noting thatA¯k,∗​\(s′\)≥0\\bar\{A\}^\{k,\*\}\(s^\{\\prime\}\)\\geq 0\(as the max advantage is non\-negative at the behavior policy’s best action\), we lower\-bound:

γk∗​\(s′\)​A¯k†,∗​\(s′\)−γk​A¯k,∗​\(s′\)≥γkmin​\(A¯k†,∗​\(s′\)−A¯k,∗​\(s′\)\)\.\\displaystyle\\gamma^\{k^\{\*\}\(s^\{\\prime\}\)\}\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\-\\gamma^\{k\}\\bar\{A\}^\{k,\*\}\(s^\{\\prime\}\)\\;\\geq\\;\\gamma^\{k\_\{\\min\}\}\\left\(\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\-\\bar\{A\}^\{k,\*\}\(s^\{\\prime\}\)\\right\)\.\(65\)Accounting for the selector error probability\(2​ε¯/\(γkmin​Δ\)\)\(2\\bar\{\\varepsilon\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\)\):

VAQC​\(s\)−Vk​\(s\)\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{k\}\(s\)≥γkmin​\(1−2​ε¯/\(γkmin​Δ\)\)1−γ​𝔼s′∼dAQC​\[A¯k†,∗​\(s′\)−A¯k,∗​\(s′\)\]\.\\displaystyle\\geq\\frac\{\\gamma^\{k\_\{\\min\}\}\(1\-2\\bar\{\\varepsilon\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\)\)\}\{1\-\\gamma\}\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\\\!\\left\[\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\-\\bar\{A\}^\{k,\*\}\(s^\{\\prime\}\)\\right\]\.\(66\)

### I\.5Proof of[Theorem˜H\.11](https://arxiv.org/html/2605.05544#A8.Thmtheorem11)

At each re\-query pointss, the selector mis\-selects with probability at most2​ε¯𝒦/\(γkmin​Δ​\(s\)\)2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\)by[Theorem˜H\.5](https://arxiv.org/html/2605.05544#A8.Thmtheorem5)\.

When a mis\-selection occurs \(choosingk′k^\{\\prime\}instead ofk†k^\{\\dagger\}\), the value loss is:

\|Vk†​\(s\)−Vk′​\(s\)\|\\displaystyle\|V^\{k^\{\\dagger\}\}\(s\)\-V^\{k^\{\\prime\}\}\(s\)\|≤max𝐚t:t\+k†⁡Qk†,∗​\(s,𝐚t:t\+k†\)−min𝐚t:t\+k′⁡Qk′,∗​\(s,𝐚t:t\+k′\)\\displaystyle\\leq\\max\_\{\\mathbf\{a\}\_\{t:t\+k^\{\\dagger\}\}\}Q^\{k^\{\\dagger\},\*\}\(s,\\mathbf\{a\}\_\{t:t\+k^\{\\dagger\}\}\)\-\\min\_\{\\mathbf\{a\}\_\{t:t\+k^\{\\prime\}\}\}Q^\{k^\{\\prime\},\*\}\(s,\\mathbf\{a\}\_\{t:t\+k^\{\\prime\}\}\)\(67\)≤R¯k′γk′≤maxk⁡R¯kγk\.\\displaystyle\\leq\\frac\{\\bar\{R\}\_\{k^\{\\prime\}\}\}\{\\gamma^\{k^\{\\prime\}\}\}\\;\\leq\\;\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(68\)
The expected number of re\-queries in the discounted horizon is at most1/\(1−γ\)1/\(1\-\\gamma\)\(since each re\-query takes at least 1 step\)\. The expected number of mis\-selections is therefore at most\(1/\(1−γ\)\)⋅2​ε¯𝒦/\(γkmin​Δ​\(s\)\)\(1/\(1\-\\gamma\)\)\\cdot 2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\)\.

The total regret is the product:

\|VAQC​\(s\)−V†​\(s\)\|\\displaystyle\|V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{\\dagger\}\(s\)\|≤2​ε¯𝒦\(1−γ\)​γkmin​Δ​\(s\)⋅maxk⁡R¯kγk\.\\displaystyle\\leq\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\(s\)\}\\cdot\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(69\)

### I\.6Proof of[Theorem˜H\.14](https://arxiv.org/html/2605.05544#A8.Thmtheorem14)

LetπCL†\\pi^\{\\dagger\}\_\{\\mathrm\{CL\}\}denote the oracle closed\-loop policy that selectsk†​\(s\)k^\{\\dagger\}\(s\)at each step and executes the first action only\.

Part 1: Oracle closed\-loop optimality\.Underε𝒦\\varepsilon\_\{\\mathcal\{K\}\}\-AOLC, at each re\-query pointst\+κ​\(s\)s\_\{t\+\\kappa\(s\)\}, the state distribution under open\-loop execution deviates from the data distribution by at mostε𝒦\\varepsilon\_\{\\mathcal\{K\}\}in TV distance\.

By the standard TV\-to\-value bound\[li2026decoupled\], the per\-re\-query value estimation error is at mostε𝒦\\varepsilon\_\{\\mathcal\{K\}\}times the value range scaled by the effective horizon\.

The number of re\-queries in the discounted horizon is bounded by1/\(1−γkmin\)1/\(1\-\\gamma^\{k\_\{\\min\}\}\), since at minimumkmink\_\{\\min\}steps elapse between re\-queries\.

Following the same derivation as DQC’s Proposition 3, but withkmink\_\{\\min\}replacinghhin the geometric series:

V⋆​\(s\)−VCL†​\(s\)\\displaystyle V^\{\\star\}\(s\)\-V^\{\\dagger\}\_\{\\mathrm\{CL\}\}\(s\)≤ε𝒦​γ\(1−γ\)2​\[21−\(1−2​ε𝒦\)​γkmin\+11−\(1−ε𝒦\)​γkmin\]\\displaystyle\\leq\\frac\{\\varepsilon\_\{\\mathcal\{K\}\}\\gamma\}\{\(1\-\\gamma\)^\{2\}\}\\left\[\\frac\{2\}\{1\-\(1\-2\\varepsilon\_\{\\mathcal\{K\}\}\)\\gamma^\{k\_\{\\min\}\}\}\+\\frac\{1\}\{1\-\(1\-\\varepsilon\_\{\\mathcal\{K\}\}\)\\gamma^\{k\_\{\\min\}\}\}\\right\]\(70\)≤3​ε𝒦​γ\(1−γ\)2​\(1−γkmin\)=3​ε𝒦​H2​H¯kmin\.\\displaystyle\\leq\\frac\{3\\varepsilon\_\{\\mathcal\{K\}\}\\gamma\}\{\(1\-\\gamma\)^\{2\}\(1\-\\gamma^\{k\_\{\\min\}\}\)\}\\;=\\;3\\varepsilon\_\{\\mathcal\{K\}\}H^\{2\}\\bar\{H\}\_\{k\_\{\\min\}\}\.\(71\)
Part 2: Learned selector\.The learned selector introduces additional regret bounded by[Theorem˜H\.11](https://arxiv.org/html/2605.05544#A8.Thmtheorem11):

\|VCL†​\(s\)−V∙​\(s\)\|\\displaystyle\|V^\{\\dagger\}\_\{\\mathrm\{CL\}\}\(s\)\-V^\{\\bullet\}\(s\)\|≤2​ε¯𝒦\(1−γ\)​γkmin​Δ⋅maxk⁡R¯kγk\.\\displaystyle\\leq\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(72\)
Combining both parts via the triangle inequality:

V⋆​\(s\)−V∙​\(s\)\\displaystyle V^\{\\star\}\(s\)\-V^\{\\bullet\}\(s\)≤\(V⋆​\(s\)−VCL†​\(s\)\)\+\|VCL†​\(s\)−V∙​\(s\)\|\\displaystyle\\leq\(V^\{\\star\}\(s\)\-V^\{\\dagger\}\_\{\\mathrm\{CL\}\}\(s\)\)\+\|V^\{\\dagger\}\_\{\\mathrm\{CL\}\}\(s\)\-V^\{\\bullet\}\(s\)\|\(73\)≤3​ε𝒦​H2​H¯kmin\+2​ε¯𝒦\(1−γ\)​γkmin​Δ⋅maxk⁡R¯kγk\.\\displaystyle\\leq 3\\varepsilon\_\{\\mathcal\{K\}\}H^\{2\}\\bar\{H\}\_\{k\_\{\\min\}\}\+\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(74\)

### I\.7Proof of[Theorem˜H\.15](https://arxiv.org/html/2605.05544#A8.Thmtheorem15)

Let𝒯hk\\mathcal\{T\}^\{k\}\_\{h\}denote thekk\-step Bellman operator withVhV^\{h\}bootstrap:

\(𝒯hk​Q\)​\(s,𝐚t:t\+k\)\\displaystyle\(\\mathcal\{T\}^\{k\}\_\{h\}Q\)\(s,\\mathbf\{a\}\_\{t:t\+k\}\):=∑j=0k−1γj​rt\+j\+γk​Vh​\(st\+k\)\.\\displaystyle:=\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\+\\gamma^\{k\}V^\{h\}\(s\_\{t\+k\}\)\.\(75\)
This operator is aγk\\gamma^\{k\}\-contraction in the sup\-norm:

‖𝒯hk​Q1−𝒯hk​Q2‖∞\\displaystyle\\\|\\mathcal\{T\}^\{k\}\_\{h\}Q\_\{1\}\-\\mathcal\{T\}^\{k\}\_\{h\}Q\_\{2\}\\\|\_\{\\infty\}≤γk​‖V1h−V2h‖∞\\displaystyle\\leq\\gamma^\{k\}\\\|V^\{h\}\_\{1\}\-V^\{h\}\_\{2\}\\\|\_\{\\infty\}\(76\)≤γk​‖Q1−Q2‖∞\.\\displaystyle\\leq\\gamma^\{k\}\\\|Q\_\{1\}\-Q\_\{2\}\\\|\_\{\\infty\}\.\(77\)
LetQfpkQ^\{k\}\_\{\\mathrm\{fp\}\}be its unique fixed point:Qfpk=𝒯hk​QfpkQ^\{k\}\_\{\\mathrm\{fp\}\}=\\mathcal\{T\}^\{k\}\_\{h\}Q^\{k\}\_\{\\mathrm\{fp\}\}\.

For the optimalkk\-step valueQk,∗Q^\{k,\*\}:

\|Qk,∗​\(s,a\)−Qfpk​\(s,a\)\|\\displaystyle\|Q^\{k,\*\}\(s,a\)\-Q^\{k\}\_\{\\mathrm\{fp\}\}\(s,a\)\|=\|𝒯k​Qk,∗​\(s,a\)−𝒯hk​Qfpk​\(s,a\)\|\\displaystyle=\|\\mathcal\{T\}^\{k\}Q^\{k,\*\}\(s,a\)\-\\mathcal\{T\}^\{k\}\_\{h\}Q^\{k\}\_\{\\mathrm\{fp\}\}\(s,a\)\|\(78\)=\|∑j=0k−1γj​rt\+j\+γk​V∗​\(st\+k\)−\(∑j=0k−1γj​rt\+j\+γk​Vh​\(st\+k\)\)\|\\displaystyle=\\left\|\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\+\\gamma^\{k\}V^\{\*\}\(s\_\{t\+k\}\)\-\(\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\+\\gamma^\{k\}V^\{h\}\(s\_\{t\+k\}\)\)\\right\|\(79\)=γk​\|V∗​\(st\+k\)−Vh​\(st\+k\)\|\\displaystyle=\\gamma^\{k\}\|V^\{\*\}\(s\_\{t\+k\}\)\-V^\{h\}\(s\_\{t\+k\}\)\|\(80\)≤γk​‖V∗−Vh‖∞=γk​εh\.\\displaystyle\\leq\\gamma^\{k\}\\\|V^\{\*\}\-V^\{h\}\\\|\_\{\\infty\}\\;=\\;\\gamma^\{k\}\\varepsilon\_\{h\}\.\(81\)
The TD fitting error gives‖Qk−Qfpk‖∞≤εk/\(1−γk\)\\\|Q^\{k\}\-Q^\{k\}\_\{\\mathrm\{fp\}\}\\\|\_\{\\infty\}\\leq\\varepsilon\_\{k\}/\(1\-\\gamma^\{k\}\)by the contraction mapping theorem\.

By the triangle inequality:

‖Qk−Qk,∗‖∞\\displaystyle\\\|Q^\{k\}\-Q^\{k,\*\}\\\|\_\{\\infty\}≤‖Qk−Qfpk‖∞\+‖Qfpk−Qk,∗‖∞\\displaystyle\\leq\\\|Q^\{k\}\-Q^\{k\}\_\{\\mathrm\{fp\}\}\\\|\_\{\\infty\}\+\\\|Q^\{k\}\_\{\\mathrm\{fp\}\}\-Q^\{k,\*\}\\\|\_\{\\infty\}\(82\)≤εk1−γk\+γk​εh1−γk\.\\displaystyle\\leq\\frac\{\\varepsilon\_\{k\}\}\{1\-\\gamma^\{k\}\}\+\\frac\{\\gamma^\{k\}\\varepsilon\_\{h\}\}\{1\-\\gamma^\{k\}\}\.\(83\)
Summing overk∈𝒦k\\in\\mathcal\{K\}:

∑k∈𝒦‖Qk−Qk,∗‖∞\\displaystyle\\sum\_\{k\\in\\mathcal\{K\}\}\\\|Q^\{k\}\-Q^\{k,\*\}\\\|\_\{\\infty\}≤εh​∑k∈𝒦γk1−γk\+∑k∈𝒦εk1−γk\.\\displaystyle\\leq\\varepsilon\_\{h\}\\sum\_\{k\\in\\mathcal\{K\}\}\\frac\{\\gamma^\{k\}\}\{1\-\\gamma^\{k\}\}\+\\sum\_\{k\\in\\mathcal\{K\}\}\\frac\{\\varepsilon\_\{k\}\}\{1\-\\gamma^\{k\}\}\.\(84\)

### I\.8Proof of Corollary[H\.16](https://arxiv.org/html/2605.05544#A8.Thmtheorem16)

IfQkQ^\{k\}were bootstrapped fromV1V^\{1\}instead ofVhV^\{h\}, the Bellman operator would be:

\(𝒯1k​Q\)​\(s,𝐚t:t\+k\)\\displaystyle\(\\mathcal\{T\}^\{k\}\_\{1\}Q\)\(s,\\mathbf\{a\}\_\{t:t\+k\}\):=∑j=0k−1γj​rt\+j\+γk​V1​\(st\+k\)\.\\displaystyle:=\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\+\\gamma^\{k\}V^\{1\}\(s\_\{t\+k\}\)\.\(85\)
The fixed point gap toQk,∗Q^\{k,\*\}would beγk​‖V1−V∗‖∞=γk​ε1\\gamma^\{k\}\\\|V^\{1\}\-V^\{\*\}\\\|\_\{\\infty\}=\\gamma^\{k\}\\varepsilon\_\{1\}\.

However,V1V^\{1\}itself must be constructed from per\-step rewards\. By the standardnn\-step return bias analysis\[fedus2020revisiting,kozuno2021revisiting\], chainingO​\(H\)=O​\(1/\(1−γ\)\)O\(H\)=O\(1/\(1\-\\gamma\)\)1\-step bootstrap steps to reach the same horizon accumulates cumulative errorO​\(H​ε1\)O\(H\\varepsilon\_\{1\}\), since each step contributes independent approximation error\.

Thus:

‖Qk−Qk,∗‖∞\\displaystyle\\\|Q^\{k\}\-Q^\{k,\*\}\\\|\_\{\\infty\}≤γk1−γk​H​ε1\+11−γk​εk\.\\displaystyle\\leq\\frac\{\\gamma^\{k\}\}\{1\-\\gamma^\{k\}\}H\\varepsilon\_\{1\}\+\\frac\{1\}\{1\-\\gamma^\{k\}\}\\varepsilon\_\{k\}\.\(86\)
The ratio of bootstrap terms between 1\-step andVhV^\{h\}bootstrap is:

H​ε1εh=O​\(H\),\\displaystyle\\frac\{H\\varepsilon\_\{1\}\}\{\\varepsilon\_\{h\}\}=O\(H\),\(87\)assumingε1≈εh\\varepsilon\_\{1\}\\approx\\varepsilon\_\{h\}\.

### I\.9Proof of[Theorem˜H\.17](https://arxiv.org/html/2605.05544#A8.Thmtheorem17)

From[Theorem˜H\.15](https://arxiv.org/html/2605.05544#A8.Thmtheorem15), fork1<k2k\_\{1\}<k\_\{2\}:

‖Qk1−Qk1,∗‖∞\\displaystyle\\\|Q^\{k\_\{1\}\}\-Q^\{k\_\{1\},\*\}\\\|\_\{\\infty\}≤γk11−γk1​εh\+εk11−γk1,\\displaystyle\\leq\\frac\{\\gamma^\{k\_\{1\}\}\}\{1\-\\gamma^\{k\_\{1\}\}\}\\varepsilon\_\{h\}\+\\frac\{\\varepsilon\_\{k\_\{1\}\}\}\{1\-\\gamma^\{k\_\{1\}\}\},\(88\)‖Qk2−Qk2,∗‖∞\\displaystyle\\\|Q^\{k\_\{2\}\}\-Q^\{k\_\{2\},\*\}\\\|\_\{\\infty\}≤γk21−γk2​εh\+εk21−γk2\.\\displaystyle\\leq\\frac\{\\gamma^\{k\_\{2\}\}\}\{1\-\\gamma^\{k\_\{2\}\}\}\\varepsilon\_\{h\}\+\\frac\{\\varepsilon\_\{k\_\{2\}\}\}\{1\-\\gamma^\{k\_\{2\}\}\}\.\(89\)
Multiplying the second inequality byγk1/γk2\\gamma^\{k\_\{1\}\}/\\gamma^\{k\_\{2\}\}:

γk1γk2​‖Qk2−Qk2,∗‖∞\\displaystyle\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\\|Q^\{k\_\{2\}\}\-Q^\{k\_\{2\},\*\}\\\|\_\{\\infty\}≥γk1γk2⋅γk21−γk2​εh\+γk1γk2⋅εk21−γk2\\displaystyle\\geq\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\cdot\\frac\{\\gamma^\{k\_\{2\}\}\}\{1\-\\gamma^\{k\_\{2\}\}\}\\varepsilon\_\{h\}\+\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\cdot\\frac\{\\varepsilon\_\{k\_\{2\}\}\}\{1\-\\gamma^\{k\_\{2\}\}\}\(90\)=γk11−γk2​εh\+γk1/γk21−γk2​εk2\.\\displaystyle=\\frac\{\\gamma^\{k\_\{1\}\}\}\{1\-\\gamma^\{k\_\{2\}\}\}\\varepsilon\_\{h\}\+\\frac\{\\gamma^\{k\_\{1\}\}/\\gamma^\{k\_\{2\}\}\}\{1\-\\gamma^\{k\_\{2\}\}\}\\varepsilon\_\{k\_\{2\}\}\.\(91\)
Since1−γk2≥1−γk11\-\\gamma^\{k\_\{2\}\}\\geq 1\-\\gamma^\{k\_\{1\}\}, we have:

γk11−γk2​εh\\displaystyle\\frac\{\\gamma^\{k\_\{1\}\}\}\{1\-\\gamma^\{k\_\{2\}\}\}\\varepsilon\_\{h\}≥γk11−γk1​εh⋅1−γk11−γk2\\displaystyle\\geq\\frac\{\\gamma^\{k\_\{1\}\}\}\{1\-\\gamma^\{k\_\{1\}\}\}\\varepsilon\_\{h\}\\cdot\\frac\{1\-\\gamma^\{k\_\{1\}\}\}\{1\-\\gamma^\{k\_\{2\}\}\}\(92\)≥γk11−γk1​εh,\\displaystyle\\geq\\frac\{\\gamma^\{k\_\{1\}\}\}\{1\-\\gamma^\{k\_\{1\}\}\}\\varepsilon\_\{h\},\(93\)sincek1<k2k\_\{1\}<k\_\{2\}implies\(1−γk1\)/\(1−γk2\)<1\(1\-\\gamma^\{k\_\{1\}\}\)/\(1\-\\gamma^\{k\_\{2\}\}\)<1\.

Subtracting theQk1Q^\{k\_\{1\}\}bound from the scaledQk2Q^\{k\_\{2\}\}bound:

γk1γk2​‖Qk2−Qk2,∗‖∞−‖Qk1−Qk1,∗‖∞\\displaystyle\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\\|Q^\{k\_\{2\}\}\-Q^\{k\_\{2\},\*\}\\\|\_\{\\infty\}\-\\\|Q^\{k\_\{1\}\}\-Q^\{k\_\{1\},\*\}\\\|\_\{\\infty\}≥γk1/γk21−γk2​εk2−εk11−γk1,\\displaystyle\\geq\\frac\{\\gamma^\{k\_\{1\}\}/\\gamma^\{k\_\{2\}\}\}\{1\-\\gamma^\{k\_\{2\}\}\}\\varepsilon\_\{k\_\{2\}\}\-\\frac\{\\varepsilon\_\{k\_\{1\}\}\}\{1\-\\gamma^\{k\_\{1\}\}\},\(94\)yielding:

‖Qk1−Qk1,∗‖∞\\displaystyle\\\|Q^\{k\_\{1\}\}\-Q^\{k\_\{1\},\*\}\\\|\_\{\\infty\}≤γk1γk2​‖Qk2−Qk2,∗‖∞\+εk1−γk1γk2​εk21−γk1\.\\displaystyle\\leq\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\\|Q^\{k\_\{2\}\}\-Q^\{k\_\{2\},\*\}\\\|\_\{\\infty\}\+\\frac\{\\varepsilon\_\{k\_\{1\}\}\-\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\varepsilon\_\{k\_\{2\}\}\}\{1\-\\gamma^\{k\_\{1\}\}\}\.\(95\)
Whenεk≡ε\\varepsilon\_\{k\}\\equiv\\varepsilonfor allkk:

‖Qk1−Qk1,∗‖∞\\displaystyle\\\|Q^\{k\_\{1\}\}\-Q^\{k\_\{1\},\*\}\\\|\_\{\\infty\}≤γk1γk2​‖Qk2−Qk2,∗‖∞\+ε⋅1−γk1/γk21−γk1\.\\displaystyle\\leq\\frac\{\\gamma^\{k\_\{1\}\}\}\{\\gamma^\{k\_\{2\}\}\}\\\|Q^\{k\_\{2\}\}\-Q^\{k\_\{2\},\*\}\\\|\_\{\\infty\}\+\\varepsilon\\cdot\\frac\{1\-\\gamma^\{k\_\{1\}\}/\\gamma^\{k\_\{2\}\}\}\{1\-\\gamma^\{k\_\{1\}\}\}\.\(96\)
Since1−γk1/γk2\>01\-\\gamma^\{k\_\{1\}\}/\\gamma^\{k\_\{2\}\}\>0, the second term is positive, meaning the shorter\-horizon critic’s error is strictly bounded by a fraction of the longer\-horizon critic’s error plus a small correction\.

### I\.10Proof of[Theorem˜H\.18](https://arxiv.org/html/2605.05544#A8.Thmtheorem18)

DQC’s policy uses a fixed partial chunk sizehah\_\{a\}\. By[Theorem˜H\.8](https://arxiv.org/html/2605.05544#A8.Thmtheorem8)withk=hak=h\_\{a\}:

VAQC​\(s\)−VDQC​\(s\)\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{\\mathrm\{DQC\}\}\(s\)≥γkmin​\(1−2​ε¯/\(γkmin​Δ\)\)1−γ​𝔼s′∼dAQC​\[A¯k†,∗​\(s′\)−A¯ha,∗​\(s′\)\]\.\\displaystyle\\geq\\frac\{\\gamma^\{k\_\{\\min\}\}\(1\-2\\bar\{\\varepsilon\}/\(\\gamma^\{k\_\{\\min\}\}\\Delta\)\)\}\{1\-\\gamma\}\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\\\!\\left\[\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\-\\bar\{A\}^\{h\_\{a\},\*\}\(s^\{\\prime\}\)\\right\]\.\(97\)
Decompose the expectation over𝒮adapt\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}and its complement:

𝔼s′∼dAQC​\[A¯k†,∗​\(s′\)−A¯ha,∗​\(s′\)\]\\displaystyle\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\\\!\\left\[\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\-\\bar\{A\}^\{h\_\{a\},\*\}\(s^\{\\prime\}\)\\right\]\(98\)=dAQC\(𝒮adapt\)⋅𝔼\[⋅∣s′∈𝒮adapt\]\+\(1−dAQC\(𝒮adapt\)\)⋅𝔼\[⋅∣s′∉𝒮adapt\]\\displaystyle\\quad=d^\{\\mathrm\{AQC\}\}\(\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}\)\\cdot\\mathbb\{E\}\[\\cdot\\mid s^\{\\prime\}\\in\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}\]\+\(1\-d^\{\\mathrm\{AQC\}\}\(\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}\)\)\\cdot\\mathbb\{E\}\[\\cdot\\mid s^\{\\prime\}\\notin\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}\]\(99\)≥ρ⋅𝔼s′∼dAQC​\[δadapt​\(s′\)∣s′∈𝒮adapt\],\\displaystyle\\quad\\geq\\rho\\cdot\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\[\\delta\_\{\\mathrm\{adapt\}\}\(s^\{\\prime\}\)\\mid s^\{\\prime\}\\in\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}\],\(100\)sinceA¯k†,∗​\(s′\)≥A¯ha,∗​\(s′\)\\bar\{A\}^\{k^\{\\dagger\},\*\}\(s^\{\\prime\}\)\\geq\\bar\{A\}^\{h\_\{a\},\*\}\(s^\{\\prime\}\)for alls′s^\{\\prime\}and≥δadapt​\(s′\)\\geq\\delta\_\{\\mathrm\{adapt\}\}\(s^\{\\prime\}\)on𝒮adapt\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}\.

Accounting for the selector error probability:

VAQC​\(s\)−VDQC​\(s\)\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{\\mathrm\{DQC\}\}\(s\)≥γkmin​ρ1−γ⋅𝔼s′∼dAQC​\[δadapt​\(s′\)∣s′∈𝒮adapt\]−2​ε¯𝒦\(1−γ\)​γkmin​Δ⋅maxk⁡R¯kγk\.\\displaystyle\\geq\\frac\{\\gamma^\{k\_\{\\min\}\}\\rho\}\{1\-\\gamma\}\\cdot\\mathbb\{E\}\_\{s^\{\\prime\}\\sim d^\{\\mathrm\{AQC\}\}\}\[\\delta\_\{\\mathrm\{adapt\}\}\(s^\{\\prime\}\)\\mid s^\{\\prime\}\\in\\mathcal\{S\}\_\{\\mathrm\{adapt\}\}\]\-\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(101\)

### I\.11Proof of[Theorem˜H\.19](https://arxiv.org/html/2605.05544#A8.Thmtheorem19)

By DQC’s comparison theorem\[li2026decoupled\], the chunked critic’s nominal value satisfies:

V^ac\+​\(s\)−V^n\+​\(s\)\\displaystyle\\hat\{V\}^\{\+\}\_\{\\mathrm\{ac\}\}\(s\)\-\\hat\{V\}^\{\+\}\_\{n\}\(s\)≥δn​H¯n−3​ε𝒦​H​H¯kmin\.\\displaystyle\\geq\\delta\_\{n\}\\bar\{H\}\_\{n\}\-3\\varepsilon\_\{\\mathcal\{K\}\}H\\bar\{H\}\_\{k\_\{\\min\}\}\.\(102\)
AQC’s value is at least as large as the best fixed\-chunk policy’s value \(by[Theorem˜H\.8](https://arxiv.org/html/2605.05544#A8.Thmtheorem8)\), minus the selector regret:

VAQC​\(s\)\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)≥V^ac\+​\(s\)−2​ε¯𝒦\(1−γ\)​γkmin​Δ⋅maxk⁡R¯kγk\.\\displaystyle\\geq\\hat\{V\}^\{\+\}\_\{\\mathrm\{ac\}\}\(s\)\-\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(103\)
Combining:

VAQC​\(s\)−Vn​\(s\)\\displaystyle V^\{\\mathrm\{AQC\}\}\(s\)\-V^\{n\}\(s\)≥δn​H¯n−3​ε𝒦​H​H¯kmin−2​ε¯𝒦\(1−γ\)​γkmin​Δ⋅maxk⁡R¯kγk\.\\displaystyle\\geq\\delta\_\{n\}\\bar\{H\}\_\{n\}\-3\\varepsilon\_\{\\mathcal\{K\}\}H\\bar\{H\}\_\{k\_\{\\min\}\}\-\\frac\{2\\bar\{\\varepsilon\}\_\{\\mathcal\{K\}\}\}\{\(1\-\\gamma\)\\gamma^\{k\_\{\\min\}\}\\Delta\}\\cdot\\max\_\{k\}\\frac\{\\bar\{R\}\_\{k\}\}\{\\gamma^\{k\}\}\.\(104\)

### I\.12Proof of Proposition[5\.1](https://arxiv.org/html/2605.05544#S5.Thmtheorem1)

Under the sparse\-reward approximation,Qk​\(st,𝐚t:t\+k\)≈γk​Vh​\(st\+ka\)Q^\{k\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\\approx\\gamma^\{k\}V^\{h\}\(s\_\{t\+k\}^\{a\}\)andVk​\(st\)≈𝔼πβ​\[γk​Vh​\(st\+k\)∣st\]V^\{k\}\(s\_\{t\}\)\\approx\\mathbb\{E\}\_\{\\pi\_\{\\beta\}\}\[\\gamma^\{k\}V^\{h\}\(s\_\{t\+k\}\)\\mid s\_\{t\}\]\. WhenVh​\(s\)≤ϵV^\{h\}\(s\)\\leq\\epsilonfor all reachable states, both terms are bounded\. Including function approximation errors bounded byσ\\sigma:

\|Qk​\(st,𝐚t:t\+k\)γk\|\\displaystyle\\left\|\\frac\{Q^\{k\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\}\{\\gamma^\{k\}\}\\right\|≤\|Vh​\(st\+ka\)\|\+σ≤ϵ\+σ,\\displaystyle\\leq\\left\|V^\{h\}\(s\_\{t\+k\}^\{a\}\)\\right\|\+\\sigma\\leq\\epsilon\+\\sigma,\(105\)\|Vk​\(st\)γk\|\\displaystyle\\left\|\\frac\{V^\{k\}\(s\_\{t\}\)\}\{\\gamma^\{k\}\}\\right\|≤\|𝔼πβ\[Vh\(st\+k\)∣st\]\|\+σ≤ϵ\+σ\.\\displaystyle\\leq\\left\|\\mathbb\{E\}\_\{\\pi\_\{\\beta\}\}\\\!\\left\[V^\{h\}\(s\_\{t\+k\}\)\\mid s\_\{t\}\\right\]\\right\|\+\\sigma\\leq\\epsilon\+\\sigma\.\(106\)By the triangle inequality:

\|δk​\(s\)\|\\displaystyle\|\\delta\_\{k\}\(s\)\|=\|Qk​\(st,𝐚t:t\+k\)γk−Vk​\(st\)γk\|\\displaystyle=\\left\|\\frac\{Q^\{k\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\}\{\\gamma^\{k\}\}\-\\frac\{V^\{k\}\(s\_\{t\}\)\}\{\\gamma^\{k\}\}\\right\|\(107\)≤\|Qk​\(st,𝐚t:t\+k\)γk\|\+\|Vk​\(st\)γk\|\\displaystyle\\leq\\left\|\\frac\{Q^\{k\}\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\)\}\{\\gamma^\{k\}\}\\right\|\+\\left\|\\frac\{V^\{k\}\(s\_\{t\}\)\}\{\\gamma^\{k\}\}\\right\|\(108\)≤ϵ\+2​σ\.\\displaystyle\\leq\\epsilon\+2\\sigma\.\(109\)Whenϵ≪σ\\epsilon\\ll\\sigma\(far from rewards, the value signal is smaller than the noise floor\), the advantage is dominated by approximation noise and allkkscore near\-zero\.

By contrast, for the uncorrected selectorarg​maxk⁡Qk/γk\\operatorname\*\{arg\\,max\}\_\{k\}Q^\{k\}/\\gamma^\{k\}, when allQk/γk≈ϵ\+σkQ^\{k\}/\\gamma^\{k\}\\approx\\epsilon\+\\sigma\_\{k\}, the argmax picks the scale with the largest positive noise realizationσk\\sigma\_\{k\}\. Since the maximum of finitely many noisy estimates is biased upward, this produces a systematically biased choice that does not average out over repeated queries\. The advantage selector centers each score around zero via subtraction ofVk/γkV^\{k\}/\\gamma^\{k\}, removing this bias\.

### I\.13Proof of Proposition[H\.21](https://arxiv.org/html/2605.05544#A8.Thmtheorem21)

Let𝒯hk\\mathcal\{T\}^\{k\}\_\{h\}denote thekk\-step Bellman operator usingV¯ξh\\bar\{V\}^\{h\}\_\{\\xi\}as bootstrap:

\(𝒯hk​Q\)​\(st,𝐚t:t\+k\)\\displaystyle\(\\mathcal\{T\}^\{k\}\_\{h\}Q\)\(s\_\{t\},\\mathbf\{a\}\_\{t:t\+k\}\):=∑j=0k−1γj​rt\+j\+γk​V¯ξh​\(st\+k\)\.\\displaystyle:=\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\+\\gamma^\{k\}\\bar\{V\}^\{h\}\_\{\\xi\}\(s\_\{t\+k\}\)\.\(110\)This operator is aγk\\gamma^\{k\}\-contraction in the sup\-norm\. LetQfpkQ^\{k\}\_\{\\mathrm\{fp\}\}be its unique fixed point:Qfpk=𝒯hk​QfpkQ^\{k\}\_\{\\mathrm\{fp\}\}=\\mathcal\{T\}^\{k\}\_\{h\}Q^\{k\}\_\{\\mathrm\{fp\}\}\.

For the optimalkk\-step action\-valueQk,∗Q^\{k,\*\}:

\|Qk,∗​\(s,a\)−Qfpk​\(s,a\)\|\\displaystyle\|Q^\{k,\*\}\(s,a\)\-Q^\{k\}\_\{\\mathrm\{fp\}\}\(s,a\)\|=\|𝒯k​Qk,∗​\(s,a\)−𝒯hk​Qfpk​\(s,a\)\|\\displaystyle=\|\\mathcal\{T\}^\{k\}Q^\{k,\*\}\(s,a\)\-\\mathcal\{T\}^\{k\}\_\{h\}Q^\{k\}\_\{\\mathrm\{fp\}\}\(s,a\)\|\(111\)=\|γk​V∗​\(st\+k\)−γk​V¯ξh​\(st\+k\)\|\\displaystyle=\\left\|\\gamma^\{k\}V^\{\*\}\(s\_\{t\+k\}\)\-\\gamma^\{k\}\\bar\{V\}^\{h\}\_\{\\xi\}\(s\_\{t\+k\}\)\\right\|\(112\)≤γk​‖V∗​\(st\+k\)−V¯ξh​\(st\+k\)‖∞\\displaystyle\\leq\\gamma^\{k\}\\\|V^\{\*\}\(s\_\{t\+k\}\)\-\\bar\{V\}^\{h\}\_\{\\xi\}\(s\_\{t\+k\}\)\\\|\_\{\\infty\}\(113\)≤γk​εh\.\\displaystyle\\leq\\gamma^\{k\}\\varepsilon\_\{h\}\.\(114\)
Since𝒯hk\\mathcal\{T\}^\{k\}\_\{h\}is aγk\\gamma^\{k\}\-contraction, repeated application gives:

‖Qψk−Qfpk‖∞≤εk1−γk\.\\displaystyle\\bigl\\\|Q^\{k\}\_\{\\psi\}\-Q^\{k\}\_\{\\mathrm\{fp\}\}\\bigr\\\|\_\{\\infty\}\\leq\\frac\{\\varepsilon\_\{k\}\}\{1\-\\gamma^\{k\}\}\.\(115\)
By the triangle inequality:

‖Qψk−Qk,∗‖∞\\displaystyle\\bigl\\\|Q^\{k\}\_\{\\psi\}\-Q^\{k,\*\}\\bigr\\\|\_\{\\infty\}≤‖Qψk−Qfpk‖∞\+‖Qfpk−Qk,∗‖∞\\displaystyle\\leq\\bigl\\\|Q^\{k\}\_\{\\psi\}\-Q^\{k\}\_\{\\mathrm\{fp\}\}\\bigr\\\|\_\{\\infty\}\+\\bigl\\\|Q^\{k\}\_\{\\mathrm\{fp\}\}\-Q^\{k,\*\}\\bigr\\\|\_\{\\infty\}\(116\)≤εk1−γk\+γk1−γk​εh,\\displaystyle\\leq\\frac\{\\varepsilon\_\{k\}\}\{1\-\\gamma^\{k\}\}\+\\frac\{\\gamma^\{k\}\}\{1\-\\gamma^\{k\}\}\\,\\varepsilon\_\{h\},\(117\)yielding the first bound\.

For the 1\-step bootstrap comparison, a 1\-step value functionV1V^\{1\}must be constructed viaO​\(H\)=O​\(1/\(1−γ\)\)O\(H\)=O\(1/\(1\-\\gamma\)\)TD updates from the per\-step reward, each step contributing errorε1\\varepsilon\_\{1\}\. By the standardnn\-step return bias analysis\[fedus2020revisiting,kozuno2021revisiting\], the resulting error in the bootstrap target isO​\(H​ε1\)O\(H\\,\\varepsilon\_\{1\}\), which replacesεh\\varepsilon\_\{h\}in the bound above to give the second result\.

### I\.14Proof of Proposition[H\.22](https://arxiv.org/html/2605.05544#A8.Thmtheorem22)

At the fixed point of the TD loss[Equation˜14](https://arxiv.org/html/2605.05544#S5.E14),AQC’s partial critic satisfies

Qψk​\(st,at:t\+k\)=∑j=0k−1γj​rt\+j\+γk​V¯ξh​\(st\+k\)\.\\displaystyle Q^\{k\}\_\{\\psi\}\(s\_\{t\},a\_\{t:t\+k\}\)=\\sum\_\{j=0\}^\{k\-1\}\\gamma^\{j\}r\_\{t\+j\}\+\\gamma^\{k\}\\bar\{V\}^\{h\}\_\{\\xi\}\(s\_\{t\+k\}\)\.\(118\)This differs fromQψPQ^\{P\}\_\{\\psi\}only in the bootstrap target:V∗V^\{\*\}in DQC versusVξhV^\{h\}\_\{\\xi\}in AQC\. Since‖Vξh−V∗‖∞≤εh\\\|V^\{h\}\_\{\\xi\}\-V^\{\*\}\\\|\_\{\\infty\}\\leq\\varepsilon\_\{h\}by definition, the gap between the two fixed points is:

‖Qψk−QψP‖∞\\displaystyle\\bigl\\\|Q^\{k\}\_\{\\psi\}\-Q^\{P\}\_\{\\psi\}\\bigr\\\|\_\{\\infty\}≤γk​‖V¯ξh−V∗‖∞\\displaystyle\\leq\\gamma^\{k\}\\\|\\bar\{V\}^\{h\}\_\{\\xi\}\-V^\{\*\}\\\|\_\{\\infty\}\(119\)≤γk​εh\.\\displaystyle\\leq\\gamma^\{k\}\\varepsilon\_\{h\}\.\(120\)This isO​\(εh\)O\(\\varepsilon\_\{h\}\)for boundedkkandγ<1\\gamma<1\.

The implication:AQCrealizes DQC’s objective through a simpler mechanism\. DQC requires goal\-conditioned hindsight relabeling and a distillation hyperparameterκd\\kappa\_\{d\}to constructV∗V^\{\*\}\. AQC reusesVhV^\{h\}, which is already trained for thek=hk=hbaseline, requiring neither\.