State commitment learning: training language models to distinguish computation from memory

# State commitment learning: training language models to distinguish computation from memory
Source: [https://arxiv.org/html/2606.05201](https://arxiv.org/html/2606.05201)
Fei Ding1Yongkang Zhang1Runhao Liu1 Yuhao Liao2Zijian Zeng2Huiming Yang2 1Alibaba Group2Tsinghua University

###### Abstract

Reasoning language models do not distinguish tokens used for*computation*from tokens that constitute*persistent state*: once generated, all hidden thoughts remain in context and influence future predictions\. As a result, downstream reasoning may depend on failed attempts, dead ends, and private scratch work that should not be safely relied on later\. We recast this phenomenon as a new training objective,state commitment learning: training models to explicitly distinguish information that should be committed as persistent state from temporary computation that can be discarded\. We define a counterfactual criterion,*persistent\-state sufficiency*, which makes it trainable and measurable whether an answer remains usable after hidden thoughts are erased\. We then proposeCounterfactual Erasure RL\(CERL\), which evaluates, under the same prefix, both a path that keeps hidden thoughts and a path that erases them, and gives reward only when the erasure path remains correct\. We also introduce the*Erasure Dependence Protocol*and show across mathematics, long\-chain logic, scientific QA, and multi\-turn tool\-use evaluation that CERL substantially reduces answer dependence on hidden thoughts without sacrificing accuracy, consistently outperforming correctness\-only RL and long\-answer SFT baselines\.

Standard reasoningProblemxxHidden thoughtHHAnswerAAFuture contextx\+H\+Ax\+H\+ADownstreampredictionfailed draftsremainState commitment learningProblemxxTemporarycomputationHHCommittedstateAAFuture contextx\+Ax\+ADownstreampredictioneraseHH

Figure 1:Overview of state commitment learning\. Standard reasoning leaves hidden thoughts in context, whereas CERL trains the model to use hidden thoughts as temporary computation, commit only the future\-relevant answer state, and remain correct after erasure\.## 1Introduction

### 1\.1Autoregressive language models do not distinguish computation from memory

Autoregressive language models do not distinguish tokens used for*computation*from tokens used for*memory*: once a reasoning token is generated, it is appended to the context and becomes input to future predictions\. This default makes long reasoning traces both a source of computational burden and a source of spurious dependence\. Downstream predictions may depend on failed attempts, dead ends, or private scratch work that should not enter the model’s persistent state\.

Recent evidence suggests that longer chains of thought are not always beneficial: some reasoning tokens provide no positive contribution to final or downstream correctness and may even interfere with future predictions \(see §[2\.1](https://arxiv.org/html/2606.05201#S2.SS1)\)\. This suggests that retaining every generated reasoning token as context is not a reasonable memory mechanism\.

### 1\.2The state commitment problem and the internalization gap

We recast the above phenomenon as a training\-objective problem that has been less explicitly modeled\. Reasoning models lack a*state commitment*mechanism: after using hidden computation, a model should explicitly commit the information needed for future reasoning into a persistent answer state while discarding the remaining temporary computation\.

Existing mitigations mostly operate at inference time on already generated thought traces, such as eviction, pruning, compression, or selecting shorter traces\. They do not directly train the model itself to decide what is persistent and what is temporary\. Their common blind spot is that they never optimize the property that “answerAAremains a usable state interface after hidden thoughtHHis erased\.” We call this gap the*internalization gap*\.

### 1\.3Method overview

The core distinction from prior work can be summarized as follows:*compression asks how to preserve the past trace; state commitment asks what the future needs*\. Compression treats the reasoning trace as a past object to be compactly preserved\. State commitment treats the visible answer as a future\-facing state interface that must be explicitly committed\.

We instantiate this idea withCounterfactual Erasure RL\(CERL\)\. Under the same problem prefix, CERL evaluates matched continuations in parallel: \(i\) a full\-thought path that keeps hidden thoughts, \(ii\) an erasure path that progressively deletes generated hidden thoughts, and \(iii\) a skip baseline used to audit recomputation\. The main reward is tied to erasure\-path correctness, with auxiliary length and anti\-postponement controls\. This signal directly captures whether answerAAcan support downstream reasoning when hidden thoughtHHis no longer visible\.

Because prior work has not made this property a training objective, naively dropping thoughts at inference time often collapses accuracy\. Training under counterfactual erasure instead lets the answer state itself serve as the persistent interface while preserving accuracy\.

### 1\.4Contributions

Our contributions follow five levels: a new problem, a new objective, a new method, a new protocol, and new empirical findings\.

\(1\) New problem definition\.We propose*State Commitment Learning*, which captures the missing training objective that reasoning models should distinguish temporary computation from persistent state that can be relied on in the future\. We give a concrete setting,*Post\-Answer Hidden\-Thought Erasure*: the model alternates between hidden thoughts and visible answer states; each thought is progressively erased when the next thought begins; the final context retains only visible answers\. We make no a priori assumption about the content relation betweenAAandHH; it emerges from the training objective\.

\(2\) New counterfactual objective: persistent\-state sufficiency\.We define the counterfactual criterion of*persistent\-state sufficiency*: under matched full\-thought / erasure paths, erasing hidden thought should not reduce downstream correctness\. This turns whether an answer has been committed into a future\-reliable state into an optimizable and estimable condition; the paired counterfactual paths are formalized in Definition[1](https://arxiv.org/html/2606.05201#Thmdefinition1)\.

\(3\) New training algorithm: CERL\+HSCO\.We propose Counterfactual Erasure RL as an optimization algorithm for persistent\-state sufficiency, implemented as*Hierarchical State\-Commitment Optimization*\(HSCO\): two GRPO layers train the hidden\-thought policyH1H\_\{1\}and the answer stateA1A\_\{1\}over non\-overlapping token ranges\.

\(4\) New evaluation protocol: erasure dependence\.We propose the*Erasure Dependence Protocol*\. It reports four sufficiency metrics after erasure:*Answer Sufficiency Gap*\(ASG, the accuracy gap between full and erasure paths, lower is better\),*Hidden Thought Dependency Rate*\(HTDR, the fraction of examples where the full path is correct but the erasure path is wrong, lower is better\),*Erasure Success Rate*\(ESR, conditional correctness after erasure given full\-path correctness, higher is better\), and*Answer Interface Sufficiency*\(AIS, the erasure\-path accuracy divided by full\-path accuracy, higher is better\)\. We also report*Marginal State Gain*\(MSG\),

MSG=Acc​\(x\+A1\)−Acc​\(x\+empty\)=Accpe−Accskip\.\\mathrm\{MSG\}\\;=\\;\\mathrm\{Acc\}\(x\+A\_\{1\}\)\\;\-\\;\\mathrm\{Acc\}\(x\+\\texttt\{empty\}\)\\;=\\;\\mathrm\{Acc\}\_\{\\mathrm\{pe\}\}\-\\mathrm\{Acc\}\_\{\\mathrm\{skip\}\}\.MSG rules out the narrow explanation that the erasure path succeeds by ignoringA1A\_\{1\}and recomputing fromxx; it should still be interpreted together with downstream length control and visible\-CoT leakage audits\.

\(5\) Empirical findings\.Our experiments are designed to show that correctness\-only reinforcement learning is insufficient for learning persistent\-state sufficiency, while counterfactual erasure training substantially closes ASG without sacrificing accuracy and improves state management in BFCL\-v3 multi\-turn tool use\. Long\-answer SFT is used as a diagnostic control for the weaker alternative explanation that supervising longer visible answers alone naturally yields the same erasure\-dependence signature; it is not used as a standalone causal attribution\.

## 2Related work

### 2\.1Long\-CoT degradation as motivation

A growing body of work reports that longer chains of thought can be worse\(Hassid et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib4);Wu et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib15);Luo et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib10);Zheng et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib21)\)\. These observations motivate our central question: if many reasoning tokens do not help final or downstream reasoning, should every generated thought token be retained as future context?

### 2\.2Inference\-time pruning and compression do not internalize the boundary

Existing methods mostly manage already generated traces through pruning, eviction, compression, selection, or external memory\. These approaches ask how to control or preserve past reasoning traces\. Our question is different: whether the model can be trained so that the visible answer stateAAremains a sufficient downstream interface after hidden thoughtHHis erased\. Detailed positioning is in Appendix[K](https://arxiv.org/html/2606.05201#A11)\.

### 2\.3Long\-answer supervision is not state commitment

Long\-answer SFT is a useful diagnostic control, but it supervises answer form rather than counterfactual sufficiency\. It tests whether longer visible answer states alone can reproduce the erasure\-dependence signature\. In contrast, CERL optimizes whetherAAremains reliable afterHHis erased;AAmay be short or long, and form is not the objective\.

## 3Method

### 3\.1Problem definition: post\-answer hidden\-thought erasure

Given a problemxx, the model may generate hidden thoughtHHto help produce answerAA\. Before the next hidden\-thought segment begins, the previousHHis erased, and only the committedx\+Ax\+Aremains for downstream context\. The central question is whether the model can learn, during training, to makeAAcorrect, sufficient, and future\-reliable afterHHis erased; equivalently, whether it can explicitly distinguish the boundary between computation and state\.

##### Formal definition of progressive erasure\.

```
For t = 1..n:
    H_t = Think(x, A_{1:t-1})
    A_t = Output_State(x, A_{1:t-1}, H_t)
    Before H_{t+1} starts generating: Erase(H_t)
```

The final context isx\+A1\+⋯\+An=x\+A1:nx\+A\_\{1\}\+\\cdots\+A\_\{n\}=x\+A\_\{1:n\}, and all think segments have disappeared\.

A1,…,AnA\_\{1\},\\ldots,A\_\{n\}are all permanently retained answer tokens, and eachAiA\_\{i\}is produced with the help of its corresponding hidden thoughtHiH\_\{i\}\. We do not model whether to enter thought as a separate learning target: if an intermediate step requires no hidden computation, it does not form a separate erasure segment and is treated as part of a neighboring answer state\. Form\-level differences are not prescribed; they are determined by the objective\.

##### Sufficiency target\.

We instantiate persistent\-state sufficiency with two requirements:

- •In single\-turn settings,x\+A1:nx\+A\_\{1:n\}should be sufficient to infer the final correct answer withoutHH\.
- •In multi\-turn settings, the historical visible answersA1:nA\_\{1:n\}should support follow\-up queries after allHHhas been erased\.

### 3\.2Tag semantics

Visible text is the default output channel\. Only`<think\>\.\.\.</think\>`is special and marks temporary hidden\-thought spans\. The generation alternates between hidden\-thought spans and answer states; a full example appears in Appendix[C](https://arxiv.org/html/2606.05201#A3)\.

### 3\.3Training objective: persistent\-state sufficiency

We directly define a future\-facing objective in task\-projected terms\. Given the same generated tuple\(x,H,A\)\(x,H,A\), ifAAhas already served as the persistent\-state interface, then erasingHHshould not reduce downstream correctness:

Prπ⁡\[Cpe=1∣x,H,A\]≥Prπ⁡\[Cfull=1∣x,H,A\]\.\\Pr\_\{\\pi\}\[C\_\{\\mathrm\{pe\}\}=1\\mid x,H,A\]\\;\\geq\\;\\Pr\_\{\\pi\}\[C\_\{\\mathrm\{full\}\}=1\\mid x,H,A\]\.HereCfullC\_\{\\mathrm\{full\}\}andCpeC\_\{\\mathrm\{pe\}\}denote downstream correctness events under the full\-thought and erasure paths, respectively\. The meaning is thatAA, as committed persistent state, should serve as the interface supporting downstream reasoning after hidden thoughtHHis erased\. In this sense,AAcrosses the computation–state boundary and becomes an object that can be relied on in the future\. CERL’s binary correctness reward is a finite\-sample surrogate for this objective\.

We make no content\-level assumption aboutAAandHH:AAmay inherit, rewrite, extend, or be independent ofHH\. Its final form emerges from the training objective rather than from a form constraint\. The commitment is an objective\-level property, persistent\-state sufficiency, not a form\-level constraint\.

### 3\.4Theoretical framework

This section defines the estimable objects used for training and evaluation\. We do not define PSS as equality of full output distributions; instead, we define its task\-projected version over evaluator\-induced binary correctness events\. To avoid notation overload,Pπ​\(y∣⋅\)P\_\{\\pi\}\(y\\mid\\cdot\)denotes only the output distribution of policyπ\\pi;Prπ⁡\[⋅\]\\Pr\_\{\\pi\}\[\\cdot\]denotes event probabilities induced jointly by the policy, data distribution, decoder randomness, and evaluator; andAcc\\mathrm\{Acc\}abbreviates the probability of a correctness event\. The Acc values in experimental tables are finite\-sample estimates\. Specifically,

Accfull​\(π\)=Prπ⁡\[Cfull=1\],Accpe​\(π\)=Prπ⁡\[Cpe=1\],\\mathrm\{Acc\}\_\{\\text\{full\}\}\(\\pi\)=\\Pr\_\{\\pi\}\[C\_\{\\text\{full\}\}=1\],\\qquad\\mathrm\{Acc\}\_\{\\text\{pe\}\}\(\\pi\)=\\Pr\_\{\\pi\}\[C\_\{\\text\{pe\}\}=1\],whereCfullC\_\{\\text\{full\}\}andCpeC\_\{\\text\{pe\}\}denote final\-answer correctness under the full\-thought and progressive\-erasure paths\. The full / erasure comparison is paired counterfactual evaluation: we first fix the same generated tuple\(x,H,A\)\(x,H,A\)and then construct two evaluation contexts

gfull​\(x,H,A\)=\(x,H,A\),gpe​\(x,H,A\)=\(x,A\)\.g\_\{\\mathrm\{full\}\}\(x,H,A\)=\(x,H,A\),\\qquad g\_\{\\mathrm\{pe\}\}\(x,H,A\)=\(x,A\)\.ThusAAis identical in the two paths, and the only intervention is whetherHHremains in the future context\. We compare the effect of this intervention on correctness events; we do not require the two paths to generate identical downstream text or induce identical output distributions\.

###### Definition 1\(Persistent\-state sufficiency\)\.

A policyπ\\pisatisfies*persistent\-state sufficiency*\(PSS\) iff, under the paired context intervention above, erasingHHdoes not reduce downstream correctness:

Prπ⁡\[Cpe=1∣x,H,A\]≥Prπ⁡\[Cfull=1∣x,H,A\]\.\\Pr\_\{\\pi\}\[C^\{\\mathrm\{pe\}\}=1\\mid x,H,A\]\\;\\geq\\;\\Pr\_\{\\pi\}\[C^\{\\mathrm\{full\}\}=1\\mid x,H,A\]\.

###### Definition 2\(Hidden\-thought dependence\)\.

DH​\(π\)=Prπ⁡\[Cfull=1∧Cpe=0\]\.D\_\{H\}\(\\pi\)\\;=\\;\\Pr\_\{\\pi\}\\bigl\[C\_\{\\text\{full\}\}=1\\;\\wedge\\;C\_\{\\text\{pe\}\}=0\\bigr\]\.DHD\_\{H\}is a sample\-level violation rate of PSS:DH=0D\_\{H\}=0means that no sample is correct on the full path and wrong on the erasure path\. LargerDHD\_\{H\}means downstream reasoning relies more on the erased computation side \(HH\) than on the memory side \(AA\)\. Its empirical estimate is HTDR \(§[4\.1](https://arxiv.org/html/2606.05201#S4.SS1)\)\.

###### Lemma 1\(Relationship between ASG andDHD\_\{H\}\)\.

Let the erasure purification rate beDE​\(π\)=Prπ⁡\[Cfull=0∧Cpe=1\]D\_\{E\}\(\\pi\)=\\Pr\_\{\\pi\}\[C\_\{\\text\{full\}\}=0\\wedge C\_\{\\text\{pe\}\}=1\]\. Then

ASG=Accfull−Accpe=DH−DE,\\mathrm\{ASG\}\\;=\\;\\mathrm\{Acc\}\_\{\\text\{full\}\}\-\\mathrm\{Acc\}\_\{\\text\{pe\}\}\\;=\\;D\_\{H\}\-D\_\{E\},soASG≤DH\\mathrm\{ASG\}\\leq D\_\{H\}, with equality whenDE=0D\_\{E\}=0\.ASG<0\\mathrm\{ASG\}<0meansDE\>DHD\_\{E\}\>D\_\{H\}, i\.e\., the purification effect of erasure exceeds the cost of state\-commitment failure\.

The proof is in Appendix[A](https://arxiv.org/html/2606.05201#A1)\. This decomposition shows that ASG is not a standard accuracy metric; it is the difference between hidden\-thought dependenceDHD\_\{H\}and erasure purificationDED\_\{E\}\. Reporting full\-path accuracy alone cannot determine whether the answer state is sufficient after erasure\. We therefore explicitly includeAccpe\\mathrm\{Acc\}\_\{\\text\{pe\}\}, HTDR, ESR, and MSG in training and evaluation\. In implementation,cfullc\_\{\\text\{full\}\}andcpec\_\{\\text\{pe\}\}are Monte Carlo estimates from repeated rollouts under the two paired contexts above\.

### 3\.5Training signal: CERL and HSCO

We implement erasure\-dependence training at two complementary levels\. CERL optimizes a finite\-sample surrogate of persistent\-state sufficiency through paired full\-thought / erasure dual\-context rollouts\. HSCO uses two GRPO layers to update hidden thoughtH1H\_\{1\}and answer stateA1A\_\{1\}\. Concretely,cpec\_\{\\text\{pe\}\}is a Monte Carlo estimate ofPrπ⁡\[Cpe=1∣x,H,A\]\\Pr\_\{\\pi\}\[C\_\{\\mathrm\{pe\}\}=1\\mid x,H,A\], so the binary correctness reward directly corresponds to the task\-projected PSS objective above, rather than to full output\-distribution matching\. For eachxx, we sample 4 candidates forH1H\_\{1\}; for eachH1H\_\{1\}, we sample 8 candidates forA1A\_\{1\}\. EachA1A\_\{1\}is evaluated under both a full\-thought path that retainsH1H\_\{1\}and an erasure path that deletesH1H\_\{1\}, plus an additional skip baseline\. Full sampling pseudocode and statistics are in Appendix[B](https://arxiv.org/html/2606.05201#A2)\.

problemxxsample44HiH\_\{i\}sample88AiA\_\{i\}full pathx,Hi,Aix,H\_\{i\},A\_\{i\}cfullc\_\{\\rm full\}erasure pathx,Aix,A\_\{i\}cpe,LE,correctc\_\{\\rm pe\},L\_\{E,\\rm correct\}skip baselinex,∅x,\\emptysetLmin,MSGL\_\{\\min\},\\mathrm\{MSG\}RAi:cpeR\_\{A\_\{i\}\}:c\_\{\\rm pe\}−\-\\,length penalty−\-\\,postpone penaltyRHi=avgj=1\.\.8​cpe\(k,j\)R\_\{H\_\{i\}\}=\\mathrm\{avg\}\_\{j=1\.\.8\}c\_\{\\rm pe\}^\{\(k,j\)\}GRPO updatesHiH\_\{i\}andAiA\_\{i\}tokensnon\-overlapping rangesHSCO samplingcounterfactual evaluationtwo\-level reward

Figure 2:CERL/HSCO training flow\. Each candidate answer state is evaluated under matched full, erasure, and skip contexts; the erasure path supplies the main sufficiency reward, while the skip baseline and length terms discourage recomputation and postponed computation\.For eachA1A\_\{1\}, letcfullc\_\{\\text\{full\}\}andcpec\_\{\\text\{pe\}\}denote correctness rates under the full\-thought and erasure paths, and letLEL\_\{E\}be the average downstream length afterA1A\_\{1\}under the erasure path\. LetLF,correctL\_\{F,\\text\{correct\}\},LE,correctL\_\{E,\\text\{correct\}\}, andLS,correctL\_\{S,\\text\{correct\}\}denote the downstream lengths of correct trajectories under the full\-thought, erasure, and skip paths, respectively; if a path has no correct trajectory, its length is set to\+∞\+\\infty\. LetL¯E,correct\(k\)\\overline\{L\}\_\{E,\\text\{correct\}\}^\{\(k\)\}denote the meanLE,correctL\_\{E,\\text\{correct\}\}over multipleA1\(k,i\)A\_\{1\}^\{\(k,i\)\}candidates under the sameH1\(k\)H\_\{1\}^\{\(k\)\}\. We then define the downstream length baseline:

Lmin\(k,i\)=min⁡\(LF,correct,L¯E,correct\(k\),LS,correct\)\.L\_\{\\min\}^\{\(k,i\)\}=\\min\\bigl\(L\_\{F,\\text\{correct\}\},\\;\\overline\{L\}\_\{E,\\text\{correct\}\}^\{\(k\)\},\\;L\_\{S,\\text\{correct\}\}\\bigr\)\.\(1\)LminL\_\{\\min\}is the successful downstream\-length baseline for the current sample and is used to constrainLE,correctL\_\{E,\\text\{correct\}\}; paths with no correct trajectory take length\+∞\+\\infty\.

#### 3\.5\.1A1A\_\{1\}\-level signal

TheA1A\_\{1\}\-level signal uses erasure\-path correctnesscpec\_\{\\text\{pe\}\}as the main reward and adds two form\-control penalties\. For eachA1\(k,i\)A\_\{1\}^\{\(k,i\)\}\(superscripts omitted below\), letm=meani​len​\(A1\(k,i\)\)m=\\mathrm\{mean\}\_\{i\}\\,\\mathrm\{len\}\(A\_\{1\}^\{\(k,i\)\}\)be the average length of the 8A1A\_\{1\}candidates under the sameH1H\_\{1\}\. The reward is

R=\{cpe−α⋅len​\(A1\)m−β⋅max⁡\(0,LE,correctLmin−1\),if​cpe\>0,−ψ,otherwise\.R\\;=\\;\\begin\{cases\}c\_\{\\text\{pe\}\}\\;\-\\;\\alpha\\cdot\\dfrac\{\\mathrm\{len\}\(A\_\{1\}\)\}\{m\}\\;\-\\;\\beta\\cdot\\max\\\!\\Bigl\(0,\\,\\dfrac\{L\_\{E,\\text\{correct\}\}\}\{L\_\{\\min\}\}\-1\\Bigr\),&\\text\{if \}c\_\{\\text\{pe\}\}\>0,\\\\\[5\.69054pt\] \-\\psi,&\\text\{otherwise\}\.\\end\{cases\}\(2\)This reward learns the state\-commitment boundary through erasure\-after\-correctness gating, rather than through a simple linear tradeoff between correctness and length\. First, the main reward comes from erasure\-path correctness: afterHHis deleted, downstream reasoning can rely only onx\+Ax\+A, so ifAAhas not committed information needed later,cpec\_\{\\text\{pe\}\}drops\. This signal pushes future\-relevant information into the visible answer state\. Second, the reward is gated: when the erasure path fails, the sample receives−ψ\-\\psi, so a short but insufficientA1A\_\{1\}cannot benefit\. Only whencpe\>0c\_\{\\text\{pe\}\}\>0do theA1A\_\{1\}length pressure and downstream anti\-postpone pressure act among candidates that are already correct after erasure\. Theα\\alphaterm provides peer\-relative length pressure within the sameH1H\_\{1\}, while theβ\\betaterm discourages postponing critical computation until after erasure or makingA1A\_\{1\}worse than the skip baseline\. In other words, CERL first requiresA1A\_\{1\}to be sufficient for correct post\-erasure reasoning, and only then optimizes the economy of the persistent state: information that remains needed downstream is encouraged to be committed toA1A\_\{1\}, while trial\-and\-error, exploration, and redundant computation are encouraged to remain in temporary<think\>spans that can be erased\. The hard penalty−ψ\-\\psialso closes a reward\-hacking channel in which the model could make the full\-thought path fail as well to avoid penalty\. Design motivation, default hyperparameters, and GRPO implementation details are in Appendix[D](https://arxiv.org/html/2606.05201#A4)\.

#### 3\.5\.2H1H\_\{1\}\-level signal

The 4 sampledH1H\_\{1\}candidates under the samexxform theH1H\_\{1\}\-level GRPO group, with reward

RH1\(k\)=avgi=1\.\.8​cpe\(k,i\)\.R\_\{H\_\{1\}\}^\{\(k\)\}\\;=\\;\\mathrm\{avg\}\_\{i=1\.\.8\}\\,c\_\{\\text\{pe\}\}^\{\(k,i\)\}\.Thus, under the same problem, the preferredH1H\_\{1\}is the one whose 8A1A\_\{1\}descendants are more often correct after erasure\. This signal updates hidden\-thought policies that better support commit\-ready answer states; gradients are applied only toH1H\_\{1\}tokens\.

#### 3\.5\.3Coordination and curriculum activation

The two rewards backpropagate through non\-overlapping token ranges to avoid interference\. Training first activates only theA1A\_\{1\}\-level signal to stabilize the answer\-state interface\. Later end\-to\-end curriculum stages activate theH1H\_\{1\}\-level signal so that hidden\-thought policy and answer state are optimized jointly\. Layer structure and curriculum details are in Appendix[D](https://arxiv.org/html/2606.05201#A4)\.

### 3\.6Training pipeline

##### Stage 1: Skill data production\.

We first construct successful trajectories with a progressive\-erasure timeline: each substep generates a temporary`<think\>`HiH\_\{i\}`</think\>`and then commits a visible stateAiA\_\{i\}; before the next think segment begins, the previousHiH\_\{i\}is deleted\. Each example stores both the full version and the erased version\.Further details are in Appendix[E](https://arxiv.org/html/2606.05201#A5)\.

##### Stage 2: Answer\-state commitment SFT\.

Each trajectory is split into multiple training examples according to the erasure timeline, so that each training context contains at most the current think segment and all previous think segments have already been erased\. SFT therefore teaches the alternating format, answer\-state commitment under hidden\-thought assistance, and continued reasoning when earlier thoughts are not visible\.

##### Stage 3: CERL\.

Starting from the SFT model, we sample full\-thought and progressive\-erasure paths in parallel and optimize erasure\-path correctness with Eq\. \([2](https://arxiv.org/html/2606.05201#S3.E2)\)\. The curriculum gradually expands the erasure range and gradient range: it first stabilizes localA1A\_\{1\}\-level PSS and then enables full\-chain erasure with bothA1A\_\{1\}andH1H\_\{1\}signals\.

## 4Experiments

### 4\.1Erasure dependence protocol

The*Erasure Dependence Protocol*reports both task accuracy and post\-erasure sufficiency\. The basic accuracy metric is Acc, reported in the tables as Avg@8\. The core metrics are

ASG=Accfull−Accpe,HTDR=Prπ⁡\[Cfull=1∧Cpe=0\],\\mathrm\{ASG\}=\\mathrm\{Acc\}\_\{\\text\{full\}\}\-\\mathrm\{Acc\}\_\{\\text\{pe\}\},\\quad\\mathrm\{HTDR\}=\\Pr\_\{\\pi\}\[C\_\{\\text\{full\}\}=1\\wedge C\_\{\\text\{pe\}\}=0\],ESR=Prπ⁡\[Cpe=1∣Cfull=1\],AIS=Accpe/Accfull\.\\mathrm\{ESR\}=\\Pr\_\{\\pi\}\[C\_\{\\text\{pe\}\}=1\\mid C\_\{\\text\{full\}\}=1\],\\quad\\mathrm\{AIS\}=\\mathrm\{Acc\}\_\{\\text\{pe\}\}/\\mathrm\{Acc\}\_\{\\text\{full\}\}\.Lower ASG is better, higher ESR/AIS is better, and HTDR is the empirical estimate of Definition[2](https://arxiv.org/html/2606.05201#Thmdefinition2)\. To rule out the alternative explanation of recomputation after erasure, we also report marginal state gain:

MSG\(Marginal State Gain\)=Acc​\(x\+A1\)−Acc​\(x\+empty\)\.\\textbf\{MSG\}\\text\{ \(Marginal State Gain\)\}\\;=\\;\\mathrm\{Acc\}\(x\+A\_\{1\}\)\\,\-\\,\\mathrm\{Acc\}\(x\+\\texttt\{empty\}\)\.MSG\>0\\mathrm\{MSG\}\>0indicates thatA1A\_\{1\}contributes nontrivial marginal state information beyond the skip\-context recomputation baseline\. Algebraic relations among metrics and form reports \(HTT/RHTT,len¯​\(A\)\\overline\{\\mathrm\{len\}\}\(A\), TTF\) are in Appendix[F](https://arxiv.org/html/2606.05201#A6)\.

### 4\.2Experimental setup

The goal is not to compare who compresses reasoning traces more aggressively, but to test the core claim: hidden thought can serve as temporary computation, and once its conclusion is committed to visible answer state, the original thought trace can be erased without harming downstream solving\. Experiments therefore ask whether \(i\) the progressive\-erasure path maintains high correctness without retaining hidden thought in the final context, and \(ii\) the gain comes from answer\-state commitment rather than ordinary trace compression, long\-answer supervision, test\-time refresh, or correctness\-only RL\. Beyond mathematical, logical, and scientific QA tasks, we include BFCL\-v3 to test whether the state interface is effective in multi\-turn tool use\.

##### Base model and training data\.

The main table uses Qwen3\-8B\(Yang et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib17)\)as the base model; Appendix[J](https://arxiv.org/html/2606.05201#A10)reports same\-protocol Qwen3\-14B/32B results\. Training data uses strictly decontaminated DeepMath\-103K\(He et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib5)\), converted in Stage 1 into answer\-state commitment SFT data with a progressive\-erasure timeline, followed by CERL training using the objective in §[3\.5](https://arxiv.org/html/2606.05201#S3.SS5)\. All experiments use aC=32C=32k context window\.

##### Evaluation sets\.

We evaluate on AIME 2024, AIME 2025, ZebraLogic\(Lin et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib9)\), AutoLogi\(Zhu et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib22)\), BFCL\-v3\(Patil et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib12)\), and GPQA\-Diamond\(Rein et al\.,[2024](https://arxiv.org/html/2606.05201#bib.bib14)\)\. AIME and GPQA\-Diamond test difficult mathematical and scientific reasoning, ZebraLogic and AutoLogi test long\-chain logic, constraint preservation, and multi\-stage state updates, and BFCL\-v3 \(Berkeley Function\-Calling Leaderboard v3\) tests tool selection, parameter maintenance, and cross\-turn state management in multi\-turn function calling\.

##### Decoding and statistics\.

All methods use Temperature=0\.6, TopP=0\.95, TopK=20, and MinP=0\. Each problem is sampled independently 8 times and reported as Avg@8\. Task accuracies are reported over 5 random seeds as mean±\\pm95% bootstrap CI, while erasure\-dependence metrics are aggregate point estimates over the same evaluation set\. Differences from baselines use paired bootstrap tests\. The main tables report task correctness and core erasure\-dependence metrics; HTDR, AIS, retained hidden\-thought tokens,len¯​\(A\)\\overline\{\\mathrm\{len\}\}\(A\), and TTF are protocol extensions defined in Appendix[F](https://arxiv.org/html/2606.05201#A6)\.

##### Baselines\.

We group comparisons into four categories\. The first contains base paths without state commitment: Base Qwen3 and Fixed\-ratio erasure \+ free answer, which share the erasure mechanism but have no CERL training\. The second contains alternative training objectives: Long\-answer SFT, Correctness\-only RL, and Length\-penalty RL\. Long\-answer SFT supervises the erased version of the same successful trajectories, i\.e\., inputxxand targetA1:nA\_\{1:n\}only, with noHH; it is a diagnostic control for whether supervising long visible answer states naturally yields similar behavior, not a single\-variable causal comparison\. The third contains trajectory\-retention and test\-time management baselines, including TokenSkip\(Xia et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib16)\), trajectory compression baselines, and Halo\(Li et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib8)\)\. The fourth is CERL full\. ASG/ESR/MSG are defined only for methods with a homologous multi\-segmentHi/AiH\_\{i\}/A\_\{i\}interface that can run both full\-thought and progressive\-erasure paths; ordinary baselines without this interface use “\-” in these columns\.

### 4\.3Experiment 1: main results

The main experiment runs, for each problem, both the full\-thought path and the progressive\-erasure path, compares final\-answer correctness, and reports ASG, ESR, and MSG\. The protocol requires a locatable multi\-segmentHi/AiH\_\{i\}/A\_\{i\}structure\. Methods that produce ordinary answers, single\-segment CoT, compressed traces, or external state management have no erasure points orA1A\_\{1\}skip control aligned with CERL, so these metrics are not computed\.Accskip\\mathrm\{Acc\}\_\{\\text\{skip\}\}and MSG are jointly reported to rule out a narrow recomputation explanation: the erasure path does not succeed by completely ignoringA1A\_\{1\}and recomputing fromxx\. We defineMSG=Accpe−Accskip\\mathrm\{MSG\}=\\mathrm\{Acc\}\_\{\\text\{pe\}\}\-\\mathrm\{Acc\}\_\{\\text\{skip\}\}; positive MSG indicates thatA1A\_\{1\}has nontrivial marginal contribution to downstream reasoning, but does not by itself prove thatA1A\_\{1\}is an abstract state\. We therefore interpret it together with visible\-CoT leakage audits, downstream length control, and form statistics\.

Table 1:Main results \(Qwen3\-8B\)\. Acc is Avg@8; lower ASG is better, higher ESR and MSG are better; “\-” in ASG/ESR/MSG means the metric is not applicable\.The table addresses four alternative explanations\. If CERL were merely ordinary CoT compression or test\-time trace management, TokenSkip / compression baselines and Halo should obtain similar gains\. If correctness optimization were sufficient, Correctness\-only RL should approach CERL full\. Long\-answer SFT improves some task accuracies, but it mainly tests the weaker alternative explanation of long visible\-answer supervision; because it lacks a homologous erasure interface, it does not support ASG/MSG causal attribution\. CERL’s unique signature is zero retained hidden\-thought tokens at final evaluation, low ASG, high ESR, and high MSG, whilelen¯​\(A\)\\overline\{\\mathrm\{len\}\}\(A\)and TTF are only form reports\.

### 4\.4Experiment 2: training\-objective ablation

This experiment provides a protocol\-compatible comparison of erasure dependence: all variants that report ASG/MSG share the homologous multi\-segmentHi/AiH\_\{i\}/A\_\{i\}interface, so differences can be more directly attributed to the training objective and curriculum\. We remove RL, the progressive\-erasure curriculum, and the anti\-postpone penalty; these component removals retain the homologous multi\-segment erasure interface and therefore report ASG/MSG\.

Table 2:Training\-objective ablation \(Qwen3\-8B\)\. Component removals within the CERL family report Avg@8 and core erasure metrics\.Ablations show that removing CERL RL within the same training family substantially degrades ASG and MSG, indicating that SFT initialization alone does not learn answer\-state commitment under erasure\. Removing the progressive\-erasure curriculum preserves an erasure objective but makes credit assignment less stable\. Removing the anti\-postpone penalty increases ASG, suggesting that the model more easily postpones critical computation until after erasure\.

### 4\.5Experiment 3: failure\-mode audit

We audit the main failure modes with two controls; the full rubric and length\-control details are in Appendix[G](https://arxiv.org/html/2606.05201#A7)\. The skip audit comparesAcc​\(x\+A1\)\\mathrm\{Acc\}\(x\+A\_\{1\}\)withAcc​\(x\+empty\)\\mathrm\{Acc\}\(x\+\\texttt\{empty\}\);MSG\>0\\mathrm\{MSG\}\>0indicates that the erasure path does not succeed by completely ignoringA1A\_\{1\}and recomputing fromxx\. The visible\-CoT leakage audit checks whetherAAcontains step\-by\-step trial and error, self\-correction, or scratch\-like derivations\. Downstream length control in the CERL reward further discourages compensating for uncommitted state by extending downstream reasoning after erasure\.

## 5Empirical findings

The experiments support three conclusions\. First, hidden thought can serve as temporary computation: CERL full maintains high Acc while final retained hidden\-thought tokens are zero and ASG is low, showing that once conclusions are committed, the original thought trace can be erased\. Second, correctness training is insufficient for this property: within the same CERL family, removing RL or counterfactual erasure components increases ASG and decreases MSG\. Third, this ability appears not only in mathematical and long\-chain logical tasks, but also in multi\-turn tool use; this suggests that state commitment learning is learning a more reliable downstream state interface, not merely a longer answer form\. Long\-answer SFT only tests the weaker explanation of “longer visible answers,” whereas CERL’s signature is simultaneously high Acc, low ASG, and high MSG\.

## 6Discussion

##### Must long chains of thought be retained?

The current reasoning paradigm assumes that thought traces must remain permanently in context, which conflates computation with memory\. We do not question whether a model should think deeply; we question whether thought conclusions must be explicitly committed to persistent state and whether the original trace must remain available\. The model can still think extensively, but thought is temporary computation\. Once its conclusion is committed into the answer state, the temporary thought is no longer the object that downstream reasoning must retain\.

##### Connection to agent settings\.

Long\-horizon agents already require some form of summarization because context windows cannot hold full trajectories\. Our method internalizes this capability into the model: rather than relying on an external summarizer, the model performs the loop “computation→\\tostate commitment→\\toerasure” during generation\. BFCL\-v3 further indicates that this mechanism can transfer to state maintenance in multi\-turn tool use, rather than being only a formatting trick for single\-turn reasoning problems\. This provides a training\-time approach to context management for agents, rather than an inference\-time post\-processing step\.

## 7Broader impact

If state commitment learning scales, it may shift large\-model reasoning from retaining full chains of thought to using temporary computation plus committed state\. This could reduce long\-context cost, improve agent robustness, reduce unnecessary retention of private intermediate reasoning, and make committed state easier to audit, offering substantial economic value in the current AI coding boom\.

## 8Limitations

CERL learns the allocation boundary between temporary computation and visible state under an explicit<think\>/ answer interface; it is a restricted instance of state\-management training, not a solution to general memory writing or arbitrary state management\.

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## Appendix ASupplementary theory proof

###### Proof of Lemma[1](https://arxiv.org/html/2606.05201#Thmlemma1)\.

Decompose the event\{Cfull=1\}\\\{C\_\{\\text\{full\}\}=1\\\}according to whether\{Cpe=1\}\\\{C\_\{\\text\{pe\}\}=1\\\}holds:

Prπ⁡\[Cfull=1\]=Prπ⁡\[Cfull=1∧Cpe=1\]\+DH\.\\Pr\_\{\\pi\}\[C\_\{\\text\{full\}\}=1\]=\\Pr\_\{\\pi\}\[C\_\{\\text\{full\}\}=1\\wedge C\_\{\\text\{pe\}\}=1\]\+D\_\{H\}\.Similarly,

Prπ⁡\[Cpe=1\]=Prπ⁡\[Cfull=1∧Cpe=1\]\+DE\.\\Pr\_\{\\pi\}\[C\_\{\\text\{pe\}\}=1\]=\\Pr\_\{\\pi\}\[C\_\{\\text\{full\}\}=1\\wedge C\_\{\\text\{pe\}\}=1\]\+D\_\{E\}\.Subtracting the two equations givesASG=DH−DE\\mathrm\{ASG\}=D\_\{H\}\-D\_\{E\}\. ∎

## Appendix BCERL sampling and statistics

For curriculum step one, where onlyH1H\_\{1\}is erased and tokens afterH2H\_\{2\}are neither erased nor updated, sampling proceeds as follows:

```
Input: x

Sample 4 H_1:
    H_1^{(1)}, H_1^{(2)}, H_1^{(3)}, H_1^{(4)}

For each H_1^{(k)}, sample 8 A_1:
    A_1^{(k,1)}, ..., A_1^{(k,8)}

For each A_1^{(k,i)}, under two conditions:
    Full-thought (keep H_1^{(k)}):
        ctx = x + H_1^{(k)} + A_1^{(k,i)}
        Generate 4 trajectories: F^{(k,i,1..4)}
    Erasure (delete H_1^{(k)}):
        ctx = x + A_1^{(k,i)}
        Generate 4 trajectories: E^{(k,i,1..4)}

Additionally sample Skip (per x):
    ctx = x + <think></think>
    Generate 4 trajectories: S^{(1..4)}

Total: 4 * 8 * (4+4) + 4 = 260 trajectories.
```

For eachA1\(k,i\)A\_\{1\}^\{\(k,i\)\}, we recordcfullc\_\{\\text\{full\}\},cpec\_\{\\text\{pe\}\},LEL\_\{E\},LF,correctL\_\{F,\\text\{correct\}\}, andLE,correctL\_\{E,\\text\{correct\}\}\. For eachxx, we record shared skip statisticscskipc\_\{\\text\{skip\}\}andLS,correctL\_\{S,\\text\{correct\}\}\. A path with no correct trajectory takes length\+∞\+\\inftyand automatically drops out of themin\\minin Eq\. \([1](https://arxiv.org/html/2606.05201#S3.E1)\)\.

## Appendix CProgressive erasure example

Visible text is the default output channel\. Only`<think\>\.\.\.</think\>`marks hidden\-thought spans\. The model alternates between hidden\-thought spans and answer statesAiA\_\{i\}; eachHHis erased before the nextHHbegins, so the context contains at most one think span at any time\.

##### Example\.

```
<think>H_1: Analyze the variable relation...</think>   # temporary span
A_1: Let the ball price be x
<think>H_2: Set up the equation...</think>             # H_1 is erased
A_2: x+(x+1)=1.10, so x=0.05
A_3 (final): The ball costs 5 cents.
```

The process has three key properties: \(i\) the model alternates between temporary hidden thought and visible answer state; \(ii\) the context retains only the current think span, and previous thoughts are erased when a new span starts; and \(iii\) if a local step needs no hidden computation, it does not form a separate erasure segment and is merged into a neighboring answer state\.

## Appendix DCERL reward\-design details

##### Why CERL does not directly optimize the signed PSS gap\.

Definition[1](https://arxiv.org/html/2606.05201#Thmdefinition1)gives a counterfactual sufficiency criterion, not a scalar reward that should be optimized literally\. Directly optimizing the signed gapcpe−cfullc\_\{\\text\{pe\}\}\-c\_\{\\text\{full\}\}would create an unsafe incentive: the policy could increase this objective either by improving post\-erasure correctness, which is desired, or by reducing full\-thought correctness, which is a degenerate solution\. CERL therefore uses a guarded one\-sided surrogate\. The main positive signal iscpec\_\{\\text\{pe\}\}, because correctness after erasure is the intended direction of improvement; the full\-thought path is retained for paired counterfactual evaluation, online monitoring, and successful\-length baselines\. At the same time, the hard failure penalty, skip baseline, and downstream\-length penalty jointly discourage three reward\-hacking modes: degrading the full\-thought path, ignoringA1A\_\{1\}and recomputing fromxx, and postponing critical computation until after erasure\. Thus the gap between Definition[1](https://arxiv.org/html/2606.05201#Thmdefinition1)and Eq\. \([2](https://arxiv.org/html/2606.05201#S3.E2)\) is intentional: PSS is the population\-level criterion, while CERL is a practical finite\-sample surrogate designed to encourage PSS without rewarding reductions in full\-thought competence\.

##### Whycpe=0c\_\{\\text\{pe\}\}=0always receives−ψ\-\\psi\.

A natural alternative is to skip an example whencfull=0c\_\{\\text\{full\}\}=0, settingR=0R=0, to avoid penalizingA1A\_\{1\}on hard tasks\. But this introduces a reward\-hacking channel: whencpe=0c\_\{\\text\{pe\}\}=0, the model can makeA1A\_\{1\}mislead downstream reasoning so thatcfullc\_\{\\text\{full\}\}also fails, raising reward from−ψ\-\\psito0\. Eq\. \([2](https://arxiv.org/html/2606.05201#S3.E2)\) closes this channel:cpe=0c\_\{\\text\{pe\}\}=0always receives−ψ\-\\psi, independent ofcfullc\_\{\\text\{full\}\}\. When an entire hard group receives the same−ψ\-\\psi, group variance is zero and GRPO advantage is zero, functionally equivalent to skipping but without the marginal benefit of hacking\.

##### Motivation for the three penalties\.

Theα\\alphaterm uses the mean length of the 8A1A\_\{1\}candidates under the same\(x,H1\)\(x,H\_\{1\}\)as a peer\-relative efficiency baseline\. Theβ\\betaterm constrains two failure modes: \(i\)A1A\_\{1\}postpones state that should have been committed until downstream post\-erasure thought, causingLE,correct\>LminL\_\{E,\\text\{correct\}\}\>L\_\{\\min\}; and \(ii\)A1A\_\{1\}misleads downstream reasoning and becomes worse than the no\-A1A\_\{1\}skip baseline\. The asymmetricmax⁡\(⋅,0\)\\max\(\\cdot,0\)imposes no penalty whenLE,correct<LminL\_\{E,\\text\{correct\}\}<L\_\{\\min\}, which may indicate noise purification or improved efficiency\. Theψ\\psiterm treats complete erasure\-path failure as PSS failure\.

##### WhyLminL\_\{\\min\}comes from three paths\.

LminL\_\{\\min\}is a compact known\-success length baseline:LF,correctL\_\{F,\\text\{correct\}\}gives the natural length whenHHis present;L¯E,correct\(k\)\\overline\{L\}\_\{E,\\text\{correct\}\}^\{\(k\)\}gives the within\-group typical length when the answer interface succeeds under the sameH1\(k\)H\_\{1\}^\{\(k\)\}; andLS,correctL\_\{S,\\text\{correct\}\}gives the lower bound withoutA1A\_\{1\}\. Taking the minimum imposes the strictest pressure\. Paths with no correct trajectory take\+∞\+\\inftyand need no extra indicator\. The full\-thought path cannot be replaced by within\-group statistics, because the natural length whenHHis present is not recoverable from the 8A1A\_\{1\}candidates alone\. The skip path is independent ofH1H\_\{1\}andA1A\_\{1\}, can be shared perxx, and costs only 4 additional trajectories\.

##### Why the denominator usesmean​\(A1\)\\mathrm\{mean\}\(A\_\{1\}\)rather thanlen​\(H1\)\\mathrm\{len\}\(H\_\{1\}\)\.

There are three reasons to choose the mean length ofA1A\_\{1\}under the sameH1H\_\{1\}rather thanlen​\(H1\)\\mathrm\{len\}\(H\_\{1\}\):

1. 1\.No compression assumption\.The ratiolen​\(A1\)/len​\(H1\)<1\\mathrm\{len\}\(A\_\{1\}\)/\\mathrm\{len\}\(H\_\{1\}\)<1implies thatA1A\_\{1\}should be a compressed version ofH1H\_\{1\}, i\.e\.,A≈Compress​\(H\)A\\approx\\mathrm\{Compress\}\(H\)\. §[3\.3](https://arxiv.org/html/2606.05201#S3.SS3)explicitly rejects this view: the objective is PSS, not reconstructingHH\. Usingmean​\(A1\)\\mathrm\{mean\}\(A\_\{1\}\)removesH1H\_\{1\}from the denominator and keeps the reward aligned with the objective\.
2. 2\.Task difficulty adapts without routing throughH1H\_\{1\}\.Easy tasks naturally produce shorterA1A\_\{1\}and a smallermm, creating stronger pressure for short states; hard tasks produce largermmand weaker length pressure\. The policy’s own answer\-length distribution is a direct implicit baseline for task difficulty\.
3. 3\.Form emerges from the objective\.Under themean​\(A1\)\\mathrm\{mean\}\(A\_\{1\}\)baseline,A1A\_\{1\}can be shorter or longer thanH1H\_\{1\}, depending on the task and answer strategy\.A1<H1A\_\{1\}<H\_\{1\},A1≈H1A\_\{1\}\\approx H\_\{1\}, andA1\>H1A\_\{1\}\>H\_\{1\}are all valid if PSS holds\.

##### Two roles of the full\-thought path\.

The full\-thought path providesLF,correctL\_\{F,\\text\{correct\}\}and also monitors training dynamics online\.

Role 1: vertical scale for reward\.Different problems have different natural downstream lengths; penalizing absoluteLE,correctL\_\{E,\\text\{correct\}\}would over\-penalize hard problems and under\-penalize easy ones\.LF,correctL\_\{F,\\text\{correct\}\}provides a per\-instance scale for how long downstream solving naturally is whenHHis available\. Theα\\alphaterm gives a horizontal scale across the 8A1A\_\{1\}candidates under the sameH1H\_\{1\}, while theβ\\betaterm compares the current erasure\-success lengthLE,correctL\_\{E,\\text\{correct\}\}against the achievable baselineLminL\_\{\\min\}; these two scales are orthogonal and not interchangeable\.LminL\_\{\\min\}further tightens the baseline to a compact known\-success length and provides stricter training pressure\.

Role 2: online monitoring\.The full\-thought path producescfullc\_\{\\text\{full\}\}, enabling core metrics to be computed within each training batch:

ASG\\displaystyle\\mathrm\{ASG\}=cfull−cpe,\\displaystyle=c\_\{\\text\{full\}\}\-c\_\{\\text\{pe\}\},\(sufficiency gap; lower is better\)AIS\\displaystyle\\mathrm\{AIS\}=cpe/cfull,\\displaystyle=c\_\{\\text\{pe\}\}/c\_\{\\text\{full\}\},\(interface\-sufficiency ratio\)ESR\\displaystyle\\mathrm\{ESR\}=Prπ⁡\[Cpe=1∣Cfull=1\]\.\\displaystyle=\\Pr\_\{\\pi\}\\bigl\[C\_\{\\text\{pe\}\}=1\\mid C\_\{\\text\{full\}\}=1\\bigr\]\.\(erasure success rate\)Without full\-thought paths, these quantities would require a separate offline evaluation pass and would lose fine\-grained training dynamics\.

##### No length penalty onH1H\_\{1\}\.

H1H\_\{1\}is transient: it is erased before the next thought segment and does not enter permanent context\. Its length only incurs one\-time generation cost and is not part of the paper’s efficiency claim\. If a longerH1H\_\{1\}does not improvecpec\_\{\\text\{pe\}\}, policy gradients will stop extending it; an explicit length penalty could harm thought quality\.

##### Curriculum activation details\.

Table 3:HSCO two\-level signals: reward, group size, gradient tokens, and length penalty\.- •Early curriculum: activate only theA1A\_\{1\}\-level signal; gradients onH1H\_\{1\}and tokens fromH2H\_\{2\}onward are masked\. BecauseA1A\_\{1\}directly corresponds to the PSS signal, it should stabilize before upstreamH1H\_\{1\}changes\.
- •End\-to\-end curriculum: activate bothH1H\_\{1\}\-level andA1A\_\{1\}\-level signals; bothH1H\_\{1\}andA1A\_\{1\}tokens receive gradients, jointly optimizing hidden\-computation quality and state\-commitment quality\.

##### Default hyperparameters\.

We useα≈0\.3\\alpha\\approx 0\.3for peer\-relativeA1A\_\{1\}length pressure,β≈0\.5\\beta\\approx 0\.5for theLE,correct/LminL\_\{E,\\text\{correct\}\}/L\_\{\\min\}penalty, andψ≈0\.5\\psi\\approx 0\.5for hard answer\-interface failure\. In practice, we first tuneα\\alphato place theA1A\_\{1\}length distribution in a reasonable range, then tuneβ\\betato prevent postponement, and finally tuneψ\\psito balance exploration and stability\.

## Appendix ETraining data and curriculum details

##### Skill construction and data production\.

We first train a skill that answers problems and keep trajectories whose final answers are correct\. The skill plans the problem into multiple steps\. For each step, a subagent thinks and then answers; its thought and answer are concatenated back into the main\-agent prefix, and the previous thought is deleted before dispatching the next subagent to think\. This ensures that data generation itself performs progressive erasure, follows the format`<think\>`H1H\_\{1\}`</think\>`→A1→\\to A\_\{1\}\\toeraseH1H\_\{1\}→\\to`<think\>`H2H\_\{2\}`</think\>`→A2→⋯\\to A\_\{2\}\\to\\cdots, and stores both full and erased versions\.

##### SFT sample splitting\.

For a trajectory with three think segments:

Each training example contains at most the current think segment in context, with all earlier thoughts erased\. This teaches the alternating format, answer\-state commitment under hidden\-thought assistance, and continued reasoning when previous thoughts are invisible\.

##### Curriculum learning\.

We progressively expand the erasure and gradient ranges\. Step one erases onlyH1H\_\{1\}and masks tokens fromH2H\_\{2\}onward\. Step two erasesH1,H2H\_\{1\},H\_\{2\}and masks tokens fromH3H\_\{3\}onward\. Later steps expand erasure points and gradient ranges\. The end\-to\-end step enables full\-chain erasure and bothA1,H1A\_\{1\},H\_\{1\}signals\. This makes early credit assignment controlled and later extends training to the full policy\.

## Appendix FEvaluation metric details

The four erasure metrics are not fully independent\. AIS and ASG are algebraic transformations of the two\-dimensional state\(cfull,cpe\)\(c\_\{\\text\{full\}\},c\_\{\\text\{pe\}\}\), withASG=cfull​\(1−AIS\)\\mathrm\{ASG\}=c\_\{\\text\{full\}\}\(1\-\\mathrm\{AIS\}\)\. HTDR and ESR imply one another throughHTDR=cfull​\(1−ESR\)\\mathrm\{HTDR\}=c\_\{\\text\{full\}\}\(1\-\\mathrm\{ESR\}\)\. The two independent axes are ASG \(net effect, including negative values\) and ESR \(conditional retention\)\. We still report all four to support comparison under different narratives\.

Form reports includelen¯​\(A\)\\overline\{\\mathrm\{len\}\}\(A\)andTTF\(Transient Token Fraction\), defined asHTT/\(HTT\+len¯​\(A\)\)\\mathrm\{HTT\}/\(\\mathrm\{HTT\}\+\\overline\{\\mathrm\{len\}\}\(A\)\)\. We avoid subtraction forms such as\(H−A\)/H\(H\-A\)/Hbecause they imply thatAAis a compression ofHH, contrary to the training objective\.len¯​\(A\)\\overline\{\\mathrm\{len\}\}\(A\)and TTF are reported only as emergent properties, not as training signals or win conditions\.

## Appendix GFailure\-mode audit details

##### Skip audit\.

The skip audit targets one narrow recomputation explanation: whether the erasure path completely ignoresA1A\_\{1\}and recomputes fromxx\. We compareAcc​\(x\+A1\)\\mathrm\{Acc\}\(x\+A\_\{1\}\)withAcc​\(x\+empty\)\\mathrm\{Acc\}\(x\+\\texttt\{empty\}\)under the same downstream decoding budget and reportMSG=Accpe−Accskip\\mathrm\{MSG\}=\\mathrm\{Acc\}\_\{\\text\{pe\}\}\-\\mathrm\{Acc\}\_\{\\text\{skip\}\}\. Positive MSG indicates thatA1A\_\{1\}has marginal contribution to downstream reasoning, but does not by itself prove thatA1A\_\{1\}is an abstract state\.

##### Visible\-CoT leakage audit\.

We inspect sampled answer statesAAusing a fixed rubric: ifAAcontains step\-by\-step trial and error, self\-correction, explicit search branches, scratch\-like derivations, or other process text resembling CoT, it is flagged as visible\-CoT leakage\. This audit is not a training objective or main\-table win condition; it checks whether long\-answer supervision degenerates into copying hidden thought into visible state\.

##### Downstream length control\.

The length term in the CERL reward constrains downstream length on the erasure path, discouraging the model from compensating for uncommitted state by extending downstream reasoning after erasure\. Length control alone cannot rule out visible\-CoT leakage; we therefore interpret MSG, visible\-CoT leakage audits, downstream length control, and form statistics jointly\.

## Appendix HExperimental scope and extrapolation boundary

Current experiments cover single\-turn benchmarks, offline counterfactual erasure audits, and multi\-turn tool use; AI coding tasks, open\-ended tool environments, and multi\-document reasoning require further validation\.

## Appendix IBFCL\-v3 public reference baselines

BFCL\-v3 \(Berkeley Function\-Calling Leaderboard v3\) evaluates function calling and agent\-style tool use; its official leaderboard and code/data page is[https://gorilla\.cs\.berkeley\.edu/leaderboard](https://gorilla.cs.berkeley.edu/leaderboard)\. The Qwen3 technical report evaluates BFCL\-v3 in FC format and uses a 64k context length for multi\-turn evaluation\(Yang et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib17)\)\. Table[4](https://arxiv.org/html/2606.05201#A9.T4)reports the public Qwen3 reference scores most relevant to our model scales; these are copied from the Qwen3 report and are not CERL erasure\-dependence measurements\.

Table 4:Public BFCL\-v3 reference scores from the Qwen3 technical report\.
## Appendix JLarger\-model supplementary results

Tables[5](https://arxiv.org/html/2606.05201#A10.T5)and[6](https://arxiv.org/html/2606.05201#A10.T6)report supplementary results for Qwen3\-14B and Qwen3\-32B under the same evaluation protocol\. Acc is Avg@8; task\-accuracy columns report the mean and 95% bootstrap confidence interval over 5 random seeds\. ASG, ESR, and MSG are aggregated under the Erasure Dependence Protocol to measure post\-erasure state sufficiency and marginal state gain\.

##### Interpreting negative ASG\.

By Lemma[1](https://arxiv.org/html/2606.05201#Thmlemma1),ASG<0\\mathrm\{ASG\}<0means that the erasure\-purification termDED\_\{E\}exceeds the hidden\-thought dependence termDHD\_\{H\}\. We do not interpret negative ASG as evidence that erasure intrinsically improves reasoning\. Instead, we interpret it as indicating that in some examples, retained hidden thoughts may introduce noise, erroneous branches, or redundant context, so the model can reason more reliably from the committed answer state after erasure\. This is consistent with recent findings on long\-CoT degradation and overthinking: longer or retained chains of thought do not always yield positive gains and may become a source of interference\(Hassid et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib4);Wu et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib15);Luo et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib10);Zheng et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib21)\)\. We therefore interpret low ASG as evidence for downstream\-reliable answer state only when it co\-occurs with high ESR and positive MSG\.

Table 5:Qwen3\-14B supplementary experimental results\. Acc is Avg@8; task\-accuracy columns report mean and 95% bootstrap CI over 5 random seeds; ASG/ESR/MSG are aggregate erasure\-dependence metrics for CERL full\.Table 6:Qwen3\-32B supplementary experimental results\. Acc is Avg@8; task\-accuracy columns report mean and 95% bootstrap CI over 5 random seeds; ASG/ESR/MSG are aggregate erasure\-dependence metrics for CERL full\.

## Appendix KSupplementary positioning of related directions

Long\-CoT degradation provides the empirical motivation for state commitment learning\.Don’t Overthink Itobserves that short chains can outperform long chains by up to 34\.5% across reasoning tasks\(Hassid et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib4)\)\.When More is Lesscharacterizes a non\-monotonic relationship between accuracy and chain length and estimates scaling laws for an optimal length\(Wu et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib15)\)\.Through the Valleynames this phenomenon*Long CoT Degradation*\(Luo et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib10)\)\.The Curse of CoTreports settings where adding chains of thought harms performance\(Zheng et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib21)\)\. Together, these results suggest that some reasoning tokens can become distractors rather than useful persistent state\.

Inference\-time methods mainly manage traces after they have already been generated\. KV\-cache eviction methods such as H2O, SnapKV, and LazyEviction remove tokens from attention caches\(Zhang et al\.,[2023](https://arxiv.org/html/2606.05201#bib.bib20);Li et al\.,[2024](https://arxiv.org/html/2606.05201#bib.bib7);Zhang et al\.,[2025a](https://arxiv.org/html/2606.05201#bib.bib18)\); token\-pruning methods such as Think Clearly delete tokens by importance\(Choi et al\.,[2025](https://arxiv.org/html/2606.05201#bib.bib1)\); thought\-aware compression methods such as ThinKV, KaVa, and LightThinker compress thought representations\(Ramachandran et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib13);Kuzina et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib6);Zhang et al\.,[2025b](https://arxiv.org/html/2606.05201#bib.bib19)\); short\-chain selection chooses shorter solutions from multiple samples\(Hassid et al\.,[2026](https://arxiv.org/html/2606.05201#bib.bib4)\); gist tokens\(Mu et al\.,[2023](https://arxiv.org/html/2606.05201#bib.bib11)\)and memory\-augmented architectures such as NTM and DNC store or read information in more compact forms\(Graves et al\.,[2014](https://arxiv.org/html/2606.05201#bib.bib2),[2016](https://arxiv.org/html/2606.05201#bib.bib3)\)\. Process\-level supervision, step verification, and trace editing/intervention also study the controllability of intermediate reasoning\. The distinction is that prior work mainly controls intermediate reasoning traces, whereas this paper trains whether the visible answer stateAAcan serve as the downstream reasoning interface after hidden thoughtHHis erased\.

A superficially similar alternative is Long\-answer SFT: supervise only the erased visible answer sequenceA1:nA\_\{1:n\}, without hidden thoughtsHHand without full / erasure / skip counterfactual comparisons\. This baseline is a diagnostic control rather than a strict causal ablation, because it changes both the supervision target and the optimization procedure\. Its role is to test a weaker alternative explanation: whether supervising longer visible answer states alone naturally yields the same erasure\-dependence signature\. The key distinction is*supervising answer form vs\. training counterfactual sufficiency*, or*what is written now vs\. what can still be relied on later*\.

Table 7:Positioning of chain\-of\-thought related directions\.