When Should Models Change Their Minds? Contextual Belief Management in Large Language Models

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When Should Models Change Their Minds? Contextual Belief Management in Large Language Models

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arXiv CS · Papers · License: Open Access · 2026

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machine learning, deep learning, neural networks

When Should Models Change Their Minds? Contextual Belief Management in Large Language Models Haoming Xu♠ * , Weihong Xu♠ * , Zongrui Li♠ , Mengru Wang♠ , Yunzhi Yao♠ , Chiyu Wu♣ , Jin Shang♣ , Yu Gong♣ , Shumin Deng♠† ♠ Zhejiang University, ♣ HomologyAI {haomingxu, 231sm}@zju.edu.cn

Long-horizon interactions require language models to manage accumulating information: when to update their state, when to preserve their state, and what to ignore. We study this challenge as Contextual Belief Management (CBM): maintaining a predicted belief state aligned with formal evidence while isolating task-irrelevant noise. To make CBM measurable, we introduce BeliefTrack, a closed-world benchmark spanning Rule Discovery and Circuit Diagnosis, where a finite belief space and symbolic verifiers enable exact turn-level evaluation. BeliefTrack diagnoses three failures: Failed Stay, Failed Update, and Failed Isolation. Across multiple LLMs, vanilla models exhibit severe CBM failures, while explicit belief-tracking prompts provide limited gains. In contrast, reinforcement learning with beliefstate rewards reduces failure rates by 70.9% on average. Further probing reveals latent belief-state dynamics behind these failures, and representation-level steering reduces failure rates by 46.1% across two tasks1 .

Task-irrelevant Noise Model

Contextual Belief Management (CBM) 1. Tracking Formal Evidence. 2. Filtering Task-irrelevant Noise.

80 60

46.7% 31.9%

20.0%

20 0

Qwen3.5-plus Deepseek-v3.2

GPT-5.2

Correct Belief Update Belief

All Aligned

Figure 1: Overview of Contextual Belief Management (CBM). CBM requires models to maintain a predicted belief state over a belief space, update it only when warranted by formal evidence, and filter task-irrelevant context or noise. The pilot Rule Discovery study reveals substantial belief-management errors in frontier models.

procedures from context and translate them into effective behavior. However, absorbing contextual information is not enough: a model must also decide which information counts as formal evidence, when that evidence warrants belief revision, and when task-irrelevant context should be filtered out. As Figure 1 shows, we study this problem as Contextual Belief Management (CBM): a model’s ability to maintain an evidence-aligned belief state throughout a multi-turn interaction. Rather than simulating open-ended dialogue, we operationalize CBM in a controlled closed-world setting. Specifically, we introduce BeliefTrack, a closed-world benchmark with two environments: Rule Discovery (RD) and Circuit Diagnosis (CD). Both environments define finite belief spaces and use symbolic verifiers, allowing exact turn-level comparison between predicted and oracle belief states. This design abstracts away open-ended ambiguity and allows us to evaluate distinct beliefmanagement operations precisely. As a pilot study, we evaluate Qwen3.5Plus (Team, 2026), DeepSeek-V3.2 (DeepSeek-AI et al., 2025), and GPT-5.2 (Singh et al., 2026) on 135 Rule Discovery examples with task-irrelevant noise. As shown in Figure 1, all three frontier models exhibit substantial belief-management errors.

Large language models (LLMs) are increasingly deployed in long-horizon interactions, where their behavior depends not only on parametric knowledge but also on context, memory, tools, and runtime protocols (Yang et al., 2024; Zhou et al., 2023; Wu et al., 2024; Lee et al., 2026). In such settings, models must manage beliefs as different types of information accumulate over time. Some information should revise the model’s current belief state, some should leave it unchanged, and some should be ignored altogether. Recent work on context learning, such as CL-Bench (Dou et al., 2026), studies whether models can absorb rules, knowledge, or * Equal contribution. 1

(Rule Discovery, n=135)

≠

Introduction

†

Pilot Study

Formal Evidence

Error Rate (%)

Long-Horizon Interaction

...

arXiv:2605.30219v1 [cs.AI] 28 May 2026

Abstract

Corresponding author. Code is coming soon at https://github.com/zjunlp/

CBM.

These results suggest that CBM failures arise even when the relevant evidence is explicitly specified. Understanding these failures requires more than checking whether the model produces the correct belief state at a single turn. A model must preserve a stable belief when formal evidence remains unchanged, revise its belief when formal evidence changes, and isolate its belief state from task-irrelevant context. To localize these errors, BeliefTrack evaluates three diagnostic failures: Failed Stay, Failed Update, and Failed Isolation. These diagnostics distinguish belief calibration failures from belief isolation failures. With these concepts in place, §3 defines Contextual Belief Management (CBM) and details how BeliefTrack operationalizes it in Rule Discovery and Circuit Diagnosis with exact symbolic verification. In §5, we evaluate current LLMs and find that vanilla models exhibit severe CBM failures, while explicit belief-tracking prompts provide only limited and inconsistent gains. We further show that reinforcement learning with belief-state rewards substantially reduces failure rates, transfers across environments, and improves robustness to task-irrelevant noise. In §6, prompt-based probing reveals latent belief-state dynamics behind these failures, including belief-state drift, backtracking failure, and contextual hijacking. Finally, representation-level steering directly improves predicted-oracle belief-state alignment, suggesting that CBM failures are not only measurable but also actionable at the representation level.

Theory of Mind

Contextual Belief Management

Infer others’ beliefs

Maintain an evidence-aligned belief

Condition or Noise

RESEARCH QUESTION:

What does Bob think is in the box?

What should the model believe now?

Figure 2: Comparison between Contextual Belief Management and Theory of Mind.

2025; Al-Tawaha et al., 2026), degrade in multiturn instruction following (Laban et al., 2026; Duan et al., 2025), and fail under contextual pressure (Xu et al., 2024b; Deng et al., 2026). Recent work further identifies contextual inertia, where models fail to revise earlier generations or intermediate inferences despite later contradictory evidence (Huang et al., 2026; Chen et al., 2026a; Liu et al., 2025), as well as mechanisms involving metacognition, memory management, and epistemic state tracking (Raj, 2026; Yona et al., 2026; Chen et al., 2026b; Yalon et al., 2026). CBM turns these instabilities into exact turn-level diagnostics, separating failures of belief calibration and isolation. Belief Tracking and Theory of Mind. Recent work studies belief dynamics in LLMs, from revising prior reasoning (Wilie et al., 2024) to maintaining temporal belief consistency in long-running agents (Myakala et al., 2026) and constructing spatial beliefs through active exploration (Zhang et al., 2026). These studies highlight the challenge of maintaining stable yet revisable beliefs over time. In Theory of Mind (ToM), belief tracking instead targets hidden mental states of other agents, such as beliefs, desires, intentions, and perspectives (Ullman, 2023; Kim et al., 2023; Chen et al., 2024; Strachan et al., 2024a,b; Kosinski, 2024; Shapira et al., 2024; Xu et al., 2024a; Cross et al., 2025; Prakash et al., 2026; Shi et al., 2025). As illustrated in Figure 2, ToM is a third-person inference problem, whereas CBM asks what the model itself should believe from accumulated formal evidence. We evaluate this first-person problem in closedworld environments with finite belief spaces and symbolic verifiers.

Related Work

Knowledge Conflict. Deciding which information to trust is central to belief management in language models. Prior work shows that models struggle to resolve conflicts between parametric memory and context from passages, user claims, demonstrations (Longpre et al., 2021; Wang et al., 2024; Xu et al., 2024c; Kortukov et al., 2024; Jin et al., 2024; Xie et al., 2024; Xu et al., 2024d; Hagström et al., 2026). Recent work further highlights belief dependencies in conflict resolution, where updating one fact can affect others (Yao et al., 2025; Xu et al., 2026). By contrast, CBM does not introduce direct information conflicts, but tests whether models update beliefs only from formal evidence. Multi-turn Reasoning Instability. LLMs often become unreliable in long interactions: they lose relevant evidence (Liu et al., 2024; Zhang et al.,

Preliminary

3.1

Problem Formulation

We formalize Contextual Belief Management (CBM) as a model’s ability to maintain an evidencealigned belief state throughout a multi-turn inter2

action. At each turn t ∈ {1, . . . , T } in an environment E, a belief-tracking model fθ receives an observation ot = (et , nt ), where et denotes formal evidence and nt denotes optional task-irrelevant noise (nt = ∅ in clean settings). Let e1:t = (e1 , . . . , et ) and o1:t = (o1 , . . . , ot ) denote the formal-evidence history and the observation history, respectively.

3.2

Evaluating CBM requires isolating formalevidence tracking from reliance on pre-trained world knowledge. In standard multi-turn questionanswering datasets, errors may stem from factual hallucination rather than belief-management failure, and the correct belief state is often not exactly computable at each turn. We therefore introduce BeliefTrack, a closedworld benchmark for evaluating turn-level belief dynamics. As shown in Figure 3, BeliefTrack formulates each task as evidence-conditioned beliefstate tracking over a finite belief space BE . All taskrelevant evidence is specified within the episode, and the model must output the predicted belief state Ŝt ⊆ BE : the subset of candidate hypotheses aligned with the accumulated formal evidence. Figure 3 illustrates the two environments used in BeliefTrack. Both instantiate the same CBM formulation but differ in the semantics of their candidate hypotheses and formal evidence. Task A: Rule Discovery (RD). Adapted from Wason’s 2-4-6 paradigm, Rule Discovery defines a finite belief space BRD = {h1 , . . . , hM }, where each candidate hypothesis is a possible rule, such as ascending_order or sum_greater_than_10. At each turn t, the formal evidence et consists of a proposed triple, such as [3, 8, 1], and its groundtruth label, YES or NO, determined by a hidden oracle rule. The oracle belief state is the subset of candidate rules that remain consistent with the accumulated triple-label evidence. Task B: Circuit Diagnosis (CD). Circuit Diagnosis evaluates diagnostic reasoning in a circuitfault setting. It defines a finite belief space BCD = {h1 , . . . , hM }, where each candidate hypothesis is a possible circuit fault, such as Battery_no_output or R1_open. At each turn t, the formal evidence et is an instrument reading, such as Current(Main)>0 or Voltage(R1)=0. The oracle belief state is the subset of candidate faults whose predicted circuit behavior remains consistent with the accumulated readings.

Let BE denote the task-specific belief space of environment E. Its elements are candidate hypotheses representing all possible task outcomes. A belief state is a subset of this space: the candidate hypotheses that remain supported by the formal evidence observed so far. We define two belief states at each turn. The Oracle Belief State St∗ ⊆ BE is the logically correct subset determined by the formal-evidence history e1:t . The Predicted Belief State Ŝt = fθ (o1:t ) ⊆ BE is the subset produced by the model from the observation history o1:t .

Contextual Belief Management The objective of CBM is to maintain turnlevel alignment between the model’s predicted belief state and the oracle belief state over the full trajectory τ : " max Eτ ∼E θ

# T 1 X ∗ I{Ŝt = St } , T t=1

Closed-World Task Environments

(1)

where I is the indicator function.

We distinguish two classes of CBM failures. Type I: Belief Calibration Failures. Belief calibration failures arise when the model fails to track the evidence-aligned belief state under clean conditions (nt = ∅). They manifest in two forms: 1. Failed Stay: The oracle belief state remains ∗ ), but the model fails to unchanged (St∗ = St−1 preserve this stable state, producing Ŝt ̸= St∗ . 2. Failed Update: The oracle belief state ∗ ), but the model fails to changes (St∗ ̸= St−1 transition to the revised state, producing Ŝt ̸= St∗ . Type II: Belief Isolation Failures. Belief isolation failures occur when the model fails to separate formal evidence from task-irrelevant noise. Specifically, consider a turn where the model correctly predicts the oracle belief state under the clean formalevidence history. If adding task-irrelevant noise nt changes the prediction and leads to Ŝt ̸= St∗ , the model has incorrectly treated non-evidential noise as part of the formal reasoning signal.

3.3

Dataset Generation and Verification

As illustrated in Figure 3, we generate three diagnostic datasets from BeliefTrack: Dstay , Dupdate , and Diso , targeting Failed Stay, Failed Update, and Failed Isolation, respectively. Each dataset consists of fixed user-side multi-turn diagnostic templates paired with a symbolic verifier that computes the or3

Failure Modes

Task Environment

Three different failure modes

Goal: Predicted Belief State = Oracle Belief State at Every Turn

Turn

Observation

Type

Predicted Belief State

Oracle Belief State

Evidence

{h1, h2, h3}

Evidence

{h2}

replace e2 -> e2'

Evidence

{h3}

e3 + n3

Evidence +Noise

{h3}

Task A: Rule Discovery (RD)

1. Failed Stay

Aligned?

Oracle state unchanged, but model fails to preserve it. Expected (Should Ignore)

Model Output(Wrong)

{h1}

{h}

2. Failed Update

Formal Evidence: instrument readings Output: candidate faults consistent with accumulated readings

Example Trajectory

Expected (Should Ignore)

Model Output(Wrong)

{h1}

{h2}

Example Trajectory

(3,8,1):No

(-1,7,8):Yes

(4,5,6):Yes

{sum_gt_10, product_pos}

{sum_gt_10}

...

I(Main)>0

{h,h2,...}

Oracle state changes, but model fails to update.

Task B: Circuit Diagnosis (CD)

Formal Evidence: (a,b,c): YES/NO Output: candidate rules consistent with accumulated evidence

{h2}

I(R1)=0

V(R1)>0

{C,F}

{C}

{h_x}

{h_x} != {h2}

...

3. Failed Isolation {C, D, E, F}

Belief Space: Candidate Rules · ascending: a < b < c · sum_gt_10: a + b + c > 10 · range_le_5: max(a,b,c) - min(a,b,c) ≤ 5

Expected (Should Ignore)

Belief Space: Candidate Fault A:Battery no output

B:Switch Open

Noise changes the model prediction despite unchanged formal evidence.

C/D: R1 E/F:R2 (open/short) (open/short)

{h} +noise

{h}

Model Output(Wrong)

{h}+noise

{h_noise}

{h_noise} != {h}

Figure 3: BeliefTrack framework. Given a finite belief space, the model must output a predicted belief state, i.e., a subset of candidate hypotheses that it considers supported by the accumulated formal evidence. The goal is to align this predicted belief state with the oracle belief state at each turn. BeliefTrack includes two closed-world environments, Rule Discovery (RD) and Circuit Diagnosis (CD), and evaluates three typical diagnostic failure modes: Failed Stay, Failed Update, and Failed Isolation.

acle belief state St∗ at each turn. Because each environment has a finite belief space and fully specified verification logic, automatic evaluation requires no human annotation. Assistant-side trajectories are sampled during evaluation.

forms a clean–noised pair, oclean = (e1:t , ∅1:t ) and 1:t noise o1:t = (e1:t , n1:t ), that shares the same evidence history e1:t but differs in the noise history. Thus, the oracle belief state is unchanged: St∗,clean = St∗,noise ,

1. Dataset for Failed Stay. Dstay tests whether models preserve an oracle belief state. Each template contains a lock point tlock , before which evidence narrows the oracle state to a target subset Slock ⊆ BE . Afterward, only redundant evidence is added, so the oracle state remains fixed: St∗ = St∗lock = Slock ,

∀t > tlock .

(4)

Failed Isolation occurs when the model succeeds on the clean trajectory but fails on the noisy one. The concrete dataset templates are provided in Appendix D.1. Since the environments are symbolic and automatically verifiable, they support scalable trajectory generation. In this work, we instantiate 1,300/1,049 Rule Discovery/Circuit Diagnosis trajectories for instruct models, and 1,503/1,616 trajectories for thinking models.

(2)

Errors at these turns correspond to Failed Stay. 2. Dataset for Failed Update. Dupdate tests whether models revise their belief state after earlier evidence is corrected. Each template includes a to-be-corrected evidence item ej , followed by a CORRECTION at turn tc that replaces it with corrected evidence e′j . The verifier recomputes the oracle state under the corrected evidence history, and we retain correction turns where St∗c ̸= St∗c −1 .

∀t.

Methods for Improving CBM

We evaluate two methods for improving CBM: Belief-Tracking Prompt (BT-Prompt), a trainingfree prompt-based enhancement method, and RL with belief-state rewards, a verifier-guided reinforcement-learning method.

(3)

4.1

Errors at these turns correspond to Failed Update. 3. Dataset for Failed Isolation. Diso tests whether models ignore task-irrelevant noise. Each template

BT-Prompt

BT-Prompt is a parameter-free test-time baseline that encodes the CBM procedure in the system prompt. It instructs the model to maintain the 4

current set of valid formal evidence, ignore nonevidential noise, re-evaluate all candidate hypotheses against the accumulated evidence, and revise the evidence set when explicit corrections invalidate earlier observations. This allows previously eliminated hypotheses to be restored when the evidence excluding them is removed. BT-Prompt is applied uniformly across both environments and all diagnostic trajectory types; full templates are provided in Appendix D.2.

col. For each diagnostic sample x, the user-side multi-turn template is fixed, and we independently (i) sample k assistant-side trajectories. Let Em (x) ∈ {0, 1} indicate whether the i-th trajectory exhibits the target failure mode m ∈ {stay, update, iso}. Here, stay, update, and iso correspond to Failed Stay, Failed Update, and Failed Isolation, respectively. We define sample-level failure as

4.2

Thus, a sample fails if any repeated trajectory exhibits the target failure. Failed Stay Rate (FSR). For x ∈ Dstay , (i) Estay (x) = 1 if the i-th trajectory makes a Failed Stay error on the evaluated post-lock turns.

h i (k) (i) Fm (x) = I ∃i ∈ {1, . . . , k}, Em (x) = 1 .

RL with Belief-State Rewards

We further optimize the model with GRPO (Shao et al., 2024) using rewards computed by a symbolic verifier. Training examples are extracted from multi-turn BeliefTrack trajectories, but each GRPO prompt targets a single evaluated turn. Specifically, for a target turn t, we construct a prompt context qt = Prompt(BE , o1:t ) from the belief space and the full observation history up to that turn. The model then produces one response yi containing a predicted belief state Ŝi,t for the target turn. The verifier compares Ŝi,t with the oracle belief state St∗ and assigns a reward only for that turn. For each prompt context qt , GRPO samples a group of outputs {y1 , . . . , yG } and optimizes the standard clipped objective: " JGRPO (θ) = Eqt ,{yi }G

FSR =

FUR =

min ρi (θ)Ai , G i=1 ! )# , clip ρi (θ), 1 − ϵ, 1 + ϵ Ai − βDKL πθ ∥ πref (5)

Metrics

(k)

Fstay (x).

(8)

x∈Dstay

(k)

Fupdate (x).

(9)

x∈Dupdate

X (k) 1 Fiso (x). iso |D | iso

(10)

x∈D

All experiments use k = 3 and lower values indicate better performance. 5.2

Implementation Details

We split each environment at the oracle level before trajectory generation. In RD, train and test trajectories are generated from disjoint rule sets; in CD, they are also generated from disjoint fault sets. This prevents oracle-specific memorization and evaluates generalization to unseen evidenceconditioned belief states. For each base model, we train two singleenvironment RL variants: RL-RD, trained only on the RD training split, and RL-CD, trained only on the CD training split. RL training uses target-turn prompts from Dstay and Dupdate : post-lock turns for belief preservation and correction turns for belief revision. We exclude Diso from training, so RL never observes task-irrelevant noise trajectories.

(6)

Since St∗ is non-empty by construction, the denominator is always non-zero. This reward measures set-level alignment between the predicted and oracle belief states at the target turn. Unlike sparse exact match, it gives partial credit to predictions that overlap with the oracle state; we ablate this design in Appendix C.1.

5.1

1 |Dupdate |

FIR =

G where Ai = (Ri −mean({Ri }G i=1 ))/std({Ri }i=1 ) is the group-normalized reward advantage and ρi (θ) = πθ (yi |qt )/πθold (yi |qt ). We use a dense Jaccard belief-state reward:

Experiments

|Dstay |

Failed Isolation Rate (FIR). For x ∈ Diso , (i) Eiso (x) = 1 if the i-th trajectory succeeds on the clean trajectory but fails on the noised trajectory.

i=1

Failed Update Rate (FUR). For x ∈ Dupdate , (i) Eupdate (x) = 1 if the i-th trajectory makes a Failed Update error at the correction turn.

( G 1 X

|Ŝi,t ∩ St∗ | Ri (qt ) = . |Ŝi,t ∪ St∗ |

(7)

Based on the diagnostic datasets defined in Section 3.3, we use a strict k-repeat evaluation proto5

Rule Discovery (RD)

Method

Circuit Diagnosis (CD)

General

FSR ↓

FUR ↓

FIR ↓

FSR ↓

FUR ↓

FIR ↓

GSM8K ↑

MMLU ↑

Qwen2.5-7B-Instruct Vanilla 99.0 BT-Prompt 79.0↓20.2% 0.0↓100.0% RL-RD RL-CD 42.0↓57.6%

98.0 80.0↓18.4% 2.0↓98.0% 74.0↓24.5%

97.0 95.0↓2.1% 20.0↓79.4% 57.0↓41.2%

99.0 48.0↓51.5% 6.0↓93.9% 0.0↓100.0%

98.0 87.9↓10.3% 28.3↓71.1% 0.0↓100.0%

97.0 92.0↓5.2% 35.0↓63.9% 20.0↓79.4%

83.3 ± 0.4 — 84.8 ± 0.5 83.9 ± 0.3

73.5 ± 0.3 — 73.1 ± 0.2 73.4 ± 0.4

Qwen3.5-9B Vanilla 47.0 BT-Prompt 49.0↑4.3% 6.0↓87.2% RL-RD RL-CD 31.0↓34.0%

60.0 69.0↑15.0% 8.0↓86.7% 34.0↓43.3%

83.7 81.4↓2.7% 18.6↓77.8% 41.9↓49.9%

43.2 47.4↑9.7% 20.0↓53.7% 12.1↓72.0%

62.7 63.5↑1.3% 21.4↓65.9% 15.9↓74.6%

95.4 81.4↓14.7% 34.9↓63.4% 16.3↓82.9%

94.0 ± 0.2 — 94.5 ± 0.6 94.5 ± 0.6

78.1 ± 0.3 — 78.2 ± 0.3 78.7 ± 0.3

Table 1: Main diagnostic and cross-task generalization results. We report Failed Stay Rate (FSR), Failed Update Rate (FUR), and Failed Isolation Rate (FIR) (lower is better ↓), together with MMLU and GSM8K accuracy (higher is better ↑). Bold and underline denote the best and second-best results respectively. RL-RD and RL-CD indicate models trained solely on Rule Discovery or Circuit Diagnosis. Blue and orange indicate in-domain (ID) and out-of-domain (OOD) generalization, respectively. Subscripts show relative changes from Vanilla.

Both RL variants are evaluated on both heldout RD and held-out CD test sets. Evaluation on the training environment measures in-domain performance, while evaluation on the other environment measures cross-environment generalization. Vanilla and BT-Prompt require no training and are evaluated on the same held-out test sets across all three diagnostic trajectory types. Detailed split statistics are provided in Appendix B.1. We evaluate Qwen2.5-7B-Instruct (Qwen et al., 2025) and Qwen3.5-9B (Team, 2026). Training and evaluation hyperparameters, decoding settings, and infrastructure are provided in Appendix B.2. 5.3

Qwen2.5-7B, RD training reduces in-domain FSR/FUR to 0.0%/2.0%, while CD training reduces CD FSR/FUR to 0.0%/0.0%. For Qwen3.5-9B, RL lowers in-domain FSR/FUR to 6.0%/8.0% on RD and 12.1%/15.9% on CD. RL also generalizes beyond the training environment. First, it transfers across tasks: RDtrained Qwen2.5-7B reduces unseen CD FSR and FUR by 93.9% and 71.1%, respectively. For Qwen3.5-9B, RD-trained RL reduces CD FSR/FUR by 53.7%/65.9%, while CD-trained RL reduces RD FSR/FUR by 34.0%/43.3%. Second, RL improves belief isolation even though Diso is excluded from training. RD-trained Qwen2.57B reduces FIR by 79.4% in-domain and 63.9% out-of-domain. Similarly, RD-trained Qwen3.5-9B reduces FIR by 77.8% in-domain and 63.4% out-ofdomain, while CD-trained Qwen3.5-9B achieves the strongest CD FIR reduction of 82.9%. Together, these results suggest that verifierguided RL improves general belief-state management rather than merely fitting task-specific patterns. The gains do not come at the expense of general ability: MMLU (Hendrycks et al., 2021) and GSM8K (Cobbe et al., 2021) remain largely stable after training, with only small fluctuations. Finally, Figure 6 shows that most CBM gains emerge early in training, while later checkpoints fluctuate across metrics and transfer settings.

Main Results

Table 1 reports diagnostic failure rates and general capability scores. We highlight three findings. Vanilla models consistently lack reliable CBM. Qwen2.5-7B-Instruct fails almost completely across both environments, with failure rates around 97–99% on all three metrics. Qwen3.5-9B is stronger but still exhibits substantial CBM failures, especially on the Failed Isolation split Diso : its FIR reaches 95.4% in Circuit Diagnosis. BT-Prompt provides limited gains. Although BT-Prompt improves some metrics, its effects vary across models and environments. In several cases, it even degrades performance, increasing Qwen3.59B’s FUR in RD by 15.0% and FSR in CD by 9.7%. This suggests that explicit test-time instructions alone are insufficient for reliable CBM. RL with belief-state rewards consistently improves CBM. RL yields strong in-domain gains across both models and environments. For

Robustness and Mechanistic Analysis

We analyze CBM along three axes: temporal robustness, contextual robustness, and mechanism. We first vary anchoring depth to test whether mod6

Figure 4: Effects of temporal stress and task-irrelevant context on CBM. Left: FSR under increasing redundant depth; Middle: FUR under increasing correction delay; Right: FIR under different noise types. We compare Vanilla, BT-Prompt, RL-RD, and RL-CD on Rule Discovery and Circuit Diagnosis. Lower values indicate better CBM.

6.2

els can preserve and revise belief states over long evidence histories. We then vary task-irrelevant noise to test whether models can isolate formal evidence from misleading cues. Finally, we use prompt-based probing and representation-level steering to examine whether CBM failures correspond to modifiable representation-level patterns. 6.1

Typology of Contextual Interference

Beyond temporal stress, CBM also requires isolating formal evidence from task-irrelevant noise. We evaluate belief isolation on Diso under a clean condition, None, and three noised conditions: Sycophancy, Authority, and Stress. In the noised conditions, the formal evidence remains unchanged, while task-irrelevant context is added to reinforce the model’s prediction, override the evidence, or induce pressure through time constraints or emotional cues. Since the oracle belief state is unchanged, clean-to-noised degradation indicates Failed Isolation. Noise prompt templates are provided in Appendix D.3. Figure 4 (Right) shows that Vanilla and BTPrompt remain highly vulnerable to task-irrelevant context. Among the three noise types, Authority causes the largest increase in FIR across both RD and CD. Sycophancy is typically the next strongest interference source, whereas Stress has a smaller but still visible effect, particularly in CD. By contrast, RL substantially improves belief isolation across noise types and environments: in-domain RL achieves the strongest reductions, while out-of-domain RL remains consistently below Vanilla and BT-Prompt. Notably, RL is trained without contextual-contamination trajectories. The resulting FIR gains show that optimizing beliefstate alignment generalizes to unseen contextual interference across different noise types.

Anchoring Depth

We study how CBM changes as the evidence that should anchor the current belief state becomes temporally distant. Redundant Depth dred increases the number of redundant but consistent evidence turns, testing whether models can preserve an unchanged oracle belief state. Correction Delay dcor increases the gap between flawed evidence and its later correction, testing whether models can revise beliefs after delayed backtracking. Figure 4 (Left and Middle) shows that Vanilla models become less reliable as the anchoring evidence moves farther back in the context. Larger redundant depth dred leads to higher FSR, reflecting drift from stable oracle belief states. Larger correction delay dcor leads to higher FUR, reflecting difficulty revising beliefs after delayed corrections. BT-Prompt does not consistently mitigate these effects: its trends largely mirror Vanilla, and in some settings it even increases the failure rate. RL substantially improves resistance to anchoring depth. In-domain RL keeps FSR/FUR low and relatively stable as dred or dcor increases, while out-of-domain RL degrades more gradually but remains stronger than Vanilla and BT-Prompt. This suggests that RL with Belief-State Rewards improves temporal robustness, especially when the training and test environments match.

6.3

Mechanistic Probing and Steering

We examine mechanisms underlying CBM failures across all three failure modes through promptbased probing and representation-level steering. Probing tracks how models prioritize oracle7

(a) Probing Experiments

(b) Steering Experiments

Figure 5: Mechanistic probing and steering of CBM failures. (a) Probing compares Vanilla and RL by tracking the rank of an oracle-supported hypothesis across turns, revealing belief-state drift, backtracking failure, contextual hijacking, and latent-output gap. Lower rank indicates higher priority; markers indicate whether the predicted belief state matches the oracle belief state. (b) Steering adds an RL-derived direction to vanilla hidden states, reducing FSR, FUR, and FIR across two tasks. Lower failure rates indicate better CBM.

the vanilla model predicts Ŝt ̸= St∗ but the RLtuned model predicts Ŝt = St∗ . Here, xt includes the belief space and observation history up to turn t, immediately before the model outputs Ŝt . For each prefix, we extract the hidden state at layer ℓ and final prefix-token position tpre from both models, and define the steering direction as the average RL–vanilla difference:

supported hypotheses, while steering tests whether RL-induced representation shifts can improve vanilla predictions without parameter updates. Prompt-based probing. At each evaluated turn t, we truncate the dialogue history, ask the model to rank all candidates in BE , and track the rank of a target oracle-supported hypothesis b∗t ∈ St∗ . We compare this rank with predicted-oracle beliefstate alignment, i.e., whether Ŝt = St∗ . Figure 5(a) reveals that Vanilla models fail to maintain high priority for oracle-supported hypotheses. In Failed Stay cases, these hypotheses lose rank despite an unchanged oracle belief state, suggesting beliefstate drift. In Failed Update cases, hypotheses restored by correction often remain low-ranked, suggesting backtracking failure. In Failed Isolation cases, task-irrelevant context pushes them down the ranking, suggesting contextual hijacking. RL suppresses these patterns by keeping oracle-supported hypotheses ranked higher across turns. Probing also reveals a latent-output gap. Even when an oracle-supported hypothesis is ranked first, Vanilla may omit it from Ŝt , while RL reduces this mismatch. Together, these observations suggest that CBM failures arise from both weakened latent prioritization and imperfect output translation, both of which are mitigated by RL.

vℓ =

1 |Dsteer |

vanilla hRL (x ) − h (x ) . t t ℓ,tpre ℓ,tpre

xt ∈Dsteer

(11) At inference time, we add this direction to the vanilla hidden state at the same layer and position on held-out prefixes: h̃ℓ,tpre = hℓ,tpre + αvℓ .

(12)

Model parameters and the decoding procedure remain unchanged. This setup tests whether beliefstate reward training induces a transferable representation direction for CBM. Figure 5(b) shows that representation-level steering reduces all three CBM failure rates in both tasks. The effect is strongest in Task A, with relative reductions of 78.6%, 92.3%, and 48.8% on FSR, FUR, and FIR, respectively. Task B also improves, with corresponding reductions of 20.7%, 23.5%, and 12.8%. The results indicate that belief-state reward training induces reusable representationlevel changes that help the vanilla model align with the oracle belief state. Implementation details are provided in Appendix C.

Representation-level steering. We next test whether the representation shift induced by RL with belief-state rewards can improve the vanilla model without parameter updates. For each diagnostic failure mode, we construct a steering set Dsteer of prefixes xt = Prompt(BE , o1:t ) where 8

Conclusion

Memprivacy: Privacy-preserving personalized memory management for edge-cloud agents. Preprint, arXiv:2605.09530.

We introduced Contextual Belief Management (CBM) and BeliefTrack to study evidence-aligned belief tracking in long-horizon interactions. Current LLMs exhibit substantial CBM failures that prompting does not reliably fix, while verifierguided reward learning improves belief management and generalizes across environments. Our probing and steering analyses further suggest that these failures are associated with modifiable representation-level patterns, making CBM both measurable and actionable.

Zhuang Chen, Jincenzi Wu, Jinfeng Zhou, Bosi Wen, Guanqun Bi, Gongyao Jiang, Yaru Cao, Mengting Hu, Yunghwei Lai, Zexuan Xiong, and Minlie Huang. 2024. ToMBench: Benchmarking theory of mind in large language models. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 15959–15983, Bangkok, Thailand. Association for Computational Linguistics. Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. 2021. Training verifiers to solve math word problems. Preprint, arXiv:2110.14168.

Limitations Scope of BeliefTrack. Our experiments instantiate BeliefTrack with two synthetic environments, Rule Discovery and Circuit Diagnosis, which are useful for isolating belief-state tracking but more open-ended forms of belief revision are not covered in the current setting.

Logan Cross, Violet Xiang, Agam Bhatia, Daniel LK Yamins, and Nick Haber. 2025. Hypothetical minds: Scaffolding theory of mind for multi-agent tasks with large language models. In The Thirteenth International Conference on Learning Representations. DeepSeek-AI, Aixin Liu, et al. 2025. Deepseek-v3.2: Pushing the frontier of open large language models. Preprint, arXiv:2512.02556.

Context Sensitivity and Societal Impact. The boundary between relevant evidence and irrelevant noise is explicitly defined in BeliefTrack, whereas real-world interactions often mix user corrections, preferences, uncertainty expressions, emotional cues, and social context. A model that over-filters such signals may become less responsive to legitimate feedback or changing user intent. Future work should therefore study calibrated context sensitivity, balancing robustness to interference with flexibility toward relevant contextual input.

Jie Deng, Shining Liang, Jun Li, Hongzhi Li, and Yutao Xie. 2026. Conpress: Learning efficient reasoning from multi-question contextual pressure. Preprint, arXiv:2602.01472. Shihan Dou, Ming Zhang, Zhangyue Yin, Chenhao Huang, Yujiong Shen, Junzhe Wang, Jiayi Chen, Yuchen Ni, Junjie Ye, Cheng Zhang, Huaibing Xie, Jianglu Hu, Shaolei Wang, Weichao Wang, Yanling Xiao, Yiting Liu, Zenan Xu, Zhen Guo, Pluto Zhou, Tao Gui, Zuxuan Wu, Xipeng Qiu, Qi Zhang, Xuanjing Huang, Yu-Gang Jiang, Di Wang, and Shunyu Yao. 2026. Cl-bench: A benchmark for context learning. Preprint, arXiv:2602.03587.

Ethics Statement This work does not involve human subjects, personal data, sensitive attributes, or real-world decision-making. All experiments are conducted in synthetic closed-world environments, and we do not identify ethical concerns specific to this study.

Guoliang Duan, Mingwei Liu, Yanlin Wang, Chong Wang, Xin Peng, and Zibin Zheng. 2025. A hierarchical and evolvable benchmark for fine-grained code instruction following with multi-turn feedback. Preprint, arXiv:2507.00699. Lovisa Hagström, Youna Kim, Haeun Yu, Sang goo Lee, Richard Johansson, Hyunsoo Cho, and Isabelle Augenstein. 2026. Cub: Benchmarking context utilisation techniques for language models. Preprint, arXiv:2505.16518.

References Ahmad Al-Tawaha, Shangding Gu, Peizhi Niu, Ruoxi Jia, and Ming Jin. 2026. Remembering more, risking more: Longitudinal safety risks in memory-equipped llm agents. Preprint, arXiv:2605.17830. Xingwu Chen, Zhanqiu Zhang, Yiwen Guo, and Difan Zou. 2026a. Breaking contextual inertia: Reinforcement learning with single-turn anchors for stable multi-turn interaction. Preprint, arXiv:2603.04783.

Dan Hendrycks, Collin Burns, Steven Basart, Andy Zou, Mantas Mazeika, Dawn Song, and Jacob Steinhardt. 2021. Measuring massive multitask language understanding. In International Conference on Learning Representations.

Yining Chen, Jihao Zhao, Bo Tang, Haofen Wang, Yue Zhang, Fei Huang, Feiyu Xiong, and Zhiyu Li. 2026b.

Jenny Y. Huang, Leshem Choshen, Ramon Astudillo, Tamara Broderick, and Jacob Andreas. 2026. Do

llms benefit from their own words? arXiv:2602.24287.

Praveen Kumar Myakala, Manan Agrawal, and Rahul Manche. 2026. Beliefshift: Benchmarking temporal belief consistency and opinion drift in llm agents. Preprint, arXiv:2603.23848.

Preprint,

Zhuoran Jin, Pengfei Cao, Hongbang Yuan, Yubo Chen, Jiexin Xu, Huaijun Li, Xiaojian Jiang, Kang Liu, and Jun Zhao. 2024. Cutting off the head ends the conflict: A mechanism for interpreting and mitigating knowledge conflicts in language models. In Findings of the Association for Computational Linguistics: ACL 2024, pages 1193–1215, Bangkok, Thailand. Association for Computational Linguistics.

Nikhil Prakash, Natalie Shapira, Arnab Sen Sharma, Christoph Riedl, Yonatan Belinkov, Tamar Rott Shaham, David Bau, and Atticus Geiger. 2026. Language models use lookbacks to track beliefs. In The Fourteenth International Conference on Learning Representations.

Hyunwoo Kim, Melanie Sclar, Xuhui Zhou, Ronan Bras, Gunhee Kim, Yejin Choi, and Maarten Sap. 2023. FANToM: A benchmark for stress-testing machine theory of mind in interactions. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pages 14397–14413, Singapore. Association for Computational Linguistics. Evgenii Kortukov, Alexander Rubinstein, Elisa Nguyen, and Seong Joon Oh. 2024. Studying large language model behaviors under context-memory conflicts with real documents. In First Conference on Language Modeling.

Qwen, :, An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, Huan Lin, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianxin Yang, Jiaxi Yang, Jingren Zhou, Junyang Lin, Kai Dang, Keming Lu, Keqin Bao, Kexin Yang, Le Yu, Mei Li, Mingfeng Xue, Pei Zhang, Qin Zhu, Rui Men, Runji Lin, Tianhao Li, Tianyi Tang, Tingyu Xia, Xingzhang Ren, Xuancheng Ren, Yang Fan, Yang Su, Yichang Zhang, Yu Wan, Yuqiong Liu, Zeyu Cui, Zhenru Zhang, and Zihan Qiu. 2025. Qwen2.5 technical report. Preprint, arXiv:2412.15115.

Michal Kosinski. 2024. Evaluating large language models in theory of mind tasks. Proceedings of the National Academy of Sciences, 121(45):e2405460121.

Ashutosh Raj. 2026. Llm psychosis: A theoretical and diagnostic framework for reality-boundary failures in large language models.

Woosuk Kwon, Zhuohan Li, Siyuan Zhuang, Ying Sheng, Lianmin Zheng, Cody Hao Yu, Joseph E. Gonzalez, Hao Zhang, and Ion Stoica. 2023. Efficient memory management for large language model serving with pagedattention. In Proceedings of the ACM SIGOPS 29th Symposium on Operating Systems Principles.

Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, Y. K. Li, Y. Wu, and Daya Guo. 2024. Deepseekmath: Pushing the limits of mathematical reasoning in open language models. Preprint, arXiv:2402.03300. Natalie Shapira, Mosh Levy, Seyed Hossein Alavi, Xuhui Zhou, Yejin Choi, Yoav Goldberg, Maarten Sap, and Vered Shwartz. 2024. Clever hans or neural theory of mind? stress testing social reasoning in large language models. In Proceedings of the 18th Conference of the European Chapter of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2257–2273.

Philippe Laban, Hiroaki Hayashi, Yingbo Zhou, and Jennifer Neville. 2026. LLMs get lost in multi-turn conversation. In The Fourteenth International Conference on Learning Representations. Yoonho Lee, Roshen Nair, Qizheng Zhang, Kangwook Lee, Omar Khattab, and Chelsea Finn. 2026. Metaharness: End-to-end optimization of model harnesses. Preprint, arXiv:2603.28052.

Haojun Shi, Suyu Ye, Xinyu Fang, Chuanyang Jin, Leyla Isik, Yen-Ling Kuo, and Tianmin Shu. 2025. Muma-tom: multi-modal multi-agent theory of mind. In Proceedings of the Thirty-Ninth AAAI Conference on Artificial Intelligence and ThirtySeventh Conference on Innovative Applications of Artificial Intelligence and Fifteenth Symposium on Educational Advances in Artificial Intelligence, AAAI’25/IAAI’25/EAAI’25. AAAI Press.

Grace Liu, Yuxiao Qu, Jeff Schneider, Aarti Singh, and Aviral Kumar. 2025. Cart: Teaching llm agents to know when they know enough. Preprint, arXiv:2510.08517. Nelson F. Liu, Kevin Lin, John Hewitt, Ashwin Paranjape, Michele Bevilacqua, Fabio Petroni, and Percy Liang. 2024. Lost in the middle: How language models use long contexts. Transactions of the Association for Computational Linguistics, 12:157–173.

Aaditya Singh, Adam Fry, et al. 2026. Openai gpt-5 system card. Preprint, arXiv:2601.03267.

Shayne Longpre, Kartik Perisetla, Anthony Chen, Nikhil Ramesh, Chris DuBois, and Sameer Singh. 2021. Entity-based knowledge conflicts in question answering. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 7052–7063, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.

James W. A. Strachan, Dalila Albergo, Giulia Borghini, Oriana Pansardi, Eugenio Scaliti, Saurabh Gupta, Krati Saxena, Alessandro Rufo, Stefano Panzeri, Guido Manzi, Michael S. A. Graziano, and Cristina Becchio. 2024a. Testing theory of mind in large language models and humans. Nature Human Behaviour, 8(7):1285–1295.

James WA Strachan, Dalila Albergo, Giulia Borghini, Oriana Pansardi, Eugenio Scaliti, Saurabh Gupta, Krati Saxena, Alessandro Rufo, Stefano Panzeri, Guido Manzi, et al. 2024b. Testing theory of mind in large language models and humans. Nature human behaviour, 8(7):1285–1295.

Knowledge conflicts for LLMs: A survey. In Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pages 8541– 8565, Miami, Florida, USA. Association for Computational Linguistics. Rongwu Xu, Zehan Qi, Zhijiang Guo, Cunxiang Wang, Hongru Wang, Yue Zhang, and Wei Xu. 2024d. Knowledge conflicts for LLMs: A survey. In Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pages 8541– 8565, Miami, Florida, USA. Association for Computational Linguistics.

Qwen Team. 2026. Qwen3.5: Accelerating productivity with native multimodal agents. Tomer Ullman. 2023. Large language models fail on trivial alterations to theory-of-mind tasks. In arXiv preprint arXiv:2302.08399. Yike Wang, Shangbin Feng, Heng Wang, Weijia Shi, Vidhisha Balachandran, Tianxing He, and Yulia Tsvetkov. 2024. Resolving knowledge conflicts in large language models. In First Conference on Language Modeling.

Noam Steinmetz Yalon, Ariel Goldstein, Liad Mudrik, and Mor Geva. 2026. Indications of belief-guided agency and meta-cognitive monitoring in large language models. Preprint, arXiv:2602.02467. John Yang, Carlos E. Jimenez, Alexander Wettig, Kilian Lieret, Shunyu Yao, Karthik Narasimhan, and Ofir Press. 2024. Swe-agent: agent-computer interfaces enable automated software engineering. In Proceedings of the 38th International Conference on Neural Information Processing Systems, NIPS ’24, Red Hook, NY, USA. Curran Associates Inc.

Bryan Wilie, Samuel Cahyawijaya, Etsuko Ishii, Junxian He, and Pascale Fung. 2024. Belief revision: The adaptability of large language models reasoning. In Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pages 10480–10496, Miami, Florida, USA. Association for Computational Linguistics.

Yunzhi Yao, Jiaxin Qin, Ningyu Zhang, Haoming Xu, Yuqi Zhu, Zeping Yu, Mengru Wang, Yuqi Tang, JiaChen Gu, Shumin Deng, Nanyun Peng, and Huajun Chen. 2025. Rethinking knowledge editing in reasoning era. TechRxiv, 2025(1106).

Di Wu, Hongwei Wang, Wenhao Yu, Yuwei Zhang, Kai-Wei Chang, and Dong Yu. 2024. Longmemeval: Benchmarking chat assistants on long-term interactive memory. CoRR, abs/2410.10813. Jian Xie, Kai Zhang, Jiangjie Chen, Renze Lou, and Yu Su. 2024. Adaptive chameleon or stubborn sloth: Revealing the behavior of large language models in knowledge conflicts. In The Twelfth International Conference on Learning Representations.

Gal Yona, Mor Geva, and Yossi Matias. 2026. Hallucinations undermine trust; metacognition is a way forward. Preprint, arXiv:2605.01428. Pingyue Zhang, Zihan Huang, Yue Wang, Jieyu Zhang, Letian Xue, Zihan Wang, Qineng Wang, Keshigeyan Chandrasegaran, Ruohan Zhang, Yejin Choi, Ranjay Krishna, Jiajun Wu, Li Fei-Fei, and Manling Li. 2026. Theory of space: Can foundation models construct spatial beliefs through active exploration? arXiv preprint arXiv:2602.07055.

Hainiu Xu, Runcong Zhao, Lixing Zhu, Jinhua Du, and Yulan He. 2024a. Opentom: A comprehensive benchmark for evaluating theory-of-mind reasoning capabilities of large language models. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 8593–8623.

Yiran Zhang, Mo Wang, Xiaoyang Li, Kaixuan Ren, Chencheng Zhu, and Usman Naseem. 2025. TurnBench-MS: A benchmark for evaluating multiturn, multi-step reasoning in large language models. In Findings of the Association for Computational Linguistics: EMNLP 2025, pages 19892–19924, Suzhou, China. Association for Computational Linguistics.

Haoming Xu, Ningyuan Zhao, Yunzhi Yao, Weihong Xu, Hongru Wang, Xinle Deng, Shumin Deng, Jeff Z. Pan, Huajun Chen, and Ningyu Zhang. 2026. Illusions of confidence? diagnosing llm truthfulness via neighborhood consistency. Preprint, arXiv:2601.05905.

Yuze Zhao, Jintao Huang, Jinghan Hu, Xingjun Wang, Yunlin Mao, Daoze Zhang, Zeyinzi Jiang, Zhikai Wu, Baole Ai, Ang Wang, Wenmeng Zhou, and Yingda Chen. 2024. Swift:a scalable lightweight infrastructure for fine-tuning. Preprint, arXiv:2408.05517.

Rongwu Xu, Brian Lin, Shujian Yang, Tianqi Zhang, Weiyan Shi, Tianwei Zhang, Zhixuan Fang, Wei Xu, and Han Qiu. 2024b. The earth is flat because...: Investigating LLMs’ belief towards misinformation via persuasive conversation. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 16259–16303, Bangkok, Thailand. Association for Computational Linguistics.

Shuyan Zhou, Frank F Xu, Hao Zhu, Xuhui Zhou, Robert Lo, Abishek Sridhar, Xianyi Cheng, Yonatan Bisk, Daniel Fried, Uri Alon, et al. 2023. Webarena: A realistic web environment for building autonomous agents. arXiv preprint arXiv:2307.13854.

Rongwu Xu, Zehan Qi, Zhijiang Guo, Cunxiang Wang, Hongru Wang, Yue Zhang, and Wei Xu. 2024c.

Use of Large Language Models

The authors used large language models exclusively for linguistic enhancement, aiming to improve readability and ensure an academic tone. These tools were not involved in any creative or analytical aspects of the research, including idea generation, experimental design, or methodological decisionmaking. All intellectual contributions and methodological frameworks presented in this work are the original results of the authors’ own efforts.

Main Experiment Implementations

B.1

Training Details.

Dstay

Dupdate

Diso

Qwen2.5-7B-Instruct RD Train 500 RD Test 100 CD Train 200 CD Test 100

500 100 550 99

0 100 0 100

Qwen3.5-9B RD Train RD Test CD Train CD Test

500 100 649 126

0 43 0 43

Env.

Split

760 100 598 200

Table 2: Training and test trajectory counts. Dstay , Dupdate , and Diso denote the diagnostic datasets for Failed Stay, Failed Update, and Failed Isolation, respectively. Diso trajectories are excluded from training and used only for evaluating FIR.

B.2

For each model and task environment, we construct train/dev/test splits at the oracle level. In Rule Discovery, oracle rules used for training are disjoint from those used for evaluation; in Circuit Diagnosis, oracle faults are split in the same way. Thus, no evaluation trajectory shares its underlying oracle with any training trajectory, preventing oraclespecific memorization and testing generalization to unseen evidence-conditioned belief states.

Models and Infra

We conduct experiments on representative proprietary and open-source large language models, including Qwen2.5-7B-Instruct (Qwen et al., 2025), Qwen3.5-9B (Team, 2026), DeepSeekV3.2 (DeepSeek-AI et al., 2025), and GPT5.2 (Singh et al., 2026). All open-source model inference and online rollout generation are implemented with the vLLM framework (Kwon et al., 2023) using bfloat16 precision, and GRPO-based reinforcement learning is conducted with the Swift training framework (Zhao et al., 2024) on 5 A800SMX-80G GPUs (1 server). The main implementation parameters are summarized in Table 3.

GRPO training uses only two diagnostic trajectory types: Dstay and Dupdate , corresponding to FSR and FUR, respectively. We exclude Diso trajectories from training. Therefore, FIR improvements measure whether the learned beliefmanagement behavior transfers to unseen taskirrelevant contextual interference, rather than being directly optimized on noisy examples. Table 2 summarizes the resulting trajectory counts.

Table 3: Implementation parameters grouped by RL training hyperparameters and sampling hyperparameters. Batch Size denotes the total training batch size computed as per_device_train_batch_size × num_devices.

During GRPO training, we save checkpoints at regular intervals and select the final checkpoint by validation performance on the corresponding development split before test evaluation. For Qwen2.57B-Instruct, the final RL-RD and RL-CD checkpoints are selected at 500 and 374 training steps, respectively. For Qwen3.5-9B, the final RL-RD and RL-CD checkpoints are selected at 520 and 338 training steps, respectively.

Parameter

Qwen2.5-7B Qwen3.5-9B

RL Hyperparameters Batch Size 8 Gradient Accumulation Steps 4 Number of Generations 8 Learning Rate 1e-4 KL Coefficient 0.04 Importance Sampling token-level Task-A-Training-Steps 500 Task-B-Training-Steps 374 LoRA Rank 16 LoRA Alpha 32 LoRA Dropout 0.05 Target Modules all-linear

All model outputs are converted into predicted belief states using the same rule-based parser across models and environments. The prompt explicitly specifies the required output format. In our evaluation runs, all outputs were successfully parsed into belief-state predictions. In principle, unparseable outputs would be conservatively marked as incorrect, but this case did not occur.

Sampling Hyperparameters Rollout Temperature Rollout Top-p Rollout Top-k Presence Penalty Repetition Penalty Evaluation Temperature Probing Temperature

0.3 default default default default 0.3 –

4 4 8 1e-4 0.04 token-level 520 338 16 32 0.05 all-linear 1.0 1.0 20 1.5 1.0 1.0 0.0

hidden states at every turn and first average the difference within the case: T

(i)

∆ℓ =

i 1 X vanilla (x ) . (x ) − h hRL i,t ℓ,tpre ℓ,tpre i,t Ti

t=1

The FIR steering direction is then (FIR)

vℓ

|DFIR |

(i)

∆ℓ .

When constructing FIR contexts, previous assistant messages are truncated to their belief-state block so that only the predicted belief state is carried into the next turn. At inference time, we add the corresponding steering direction to the vanilla hidden state at the same layer and prefix-token position:

Figure 6: RL training dynamics across checkpoints. We report FSR, FUR, and FIR on Rule Discovery and Circuit Diagnosis for two training runs. Failure rates drop sharply in early checkpoints and then fluctuate mildly, suggesting that most CBM gains emerge early during reward training.

(m)

Steering Experiment Details

h̃ℓ,tpre = hℓ,tpre + αvℓ

We use representation-level steering to test whether reward-induced representation shifts can improve the vanilla model without parameter updates. For each failure metric, we identify vanilla-wrong/RLright cases, extract steering directions from hiddenstate differences, and inject them into the vanilla model at inference. We build steering vectors from RD vanilla-wrong/RL-right samples and evaluate on RD. The selected RD configurations are transferred to CD without re-tuning to test cross-task generalization. After filtering, we use 49/82/50 RD examples and 82/116/50 CD examples for FSR/FUR/FIR, respectively. For FSR and FUR, failures occur at a specific diagnostic point rather than at every turn. We therefore select the target turn for each case and construct the prefix only up to that point. For a selected prefix xt , we extract the hidden state at layer ℓ and the final prefix-token position tpre from both the vanilla and RL-tuned models. The steering direction is computed as X 1 (m) vanilla vℓ = hRL ℓ,tpre (xt ) − hℓ,tpre (xt ) , |Dm |

The model parameters are unchanged, and the intervention is applied only at the final prefix token rather than at every generated token. For FSR and FUR, this intervention is applied at the selected target turn. For FIR, inference is performed online across the full trajectory, and the same FIR vector is injected at every turn. We select the final steering configuration by a grid search on RD. For each failure metric, we sweep a set of middle-to-late layers and scaling coefficients α, and evaluate each combination with the same RD vanilla-wrong/RL-right examples used for steering evaluation. The best configuration is chosen separately for each metric according to the RD failure-rate reduction. This yields one steering vector and scaling coefficient for each of FSR, FUR, and FIR. These selected RD configurations are then kept fixed and transferred to CD without re-tuning. To reduce the effect of sampling randomness, each example is evaluated with three independent generations. An example is counted as correct only if all three belief-state predictions are correct; if any one of the three generations is incorrect, the example is counted as incorrect. This criterion measures whether steering produces a stable belief-state correction rather than an occasional correct sample.

xt ∈Dm

where m ∈ {FSR, FUR}. Here, Dm denotes the metric-specific steering set of RD vanillawrong/RL-right prefixes. FIR differs from FSR and FUR because the perturbation is applied throughout the multi-turn trajectory. We therefore treat each FIR example as a full trajectory rather than a single target turn. For case i with Ti turns, we extract the vanilla and RL

C.1

Reward Ablation

We compare the dense Jaccard belief-state reward used in our main experiments with a sparse exactmatch reward. For a sampled output yi at target 13

turn t, let Ŝi,t be the predicted belief state and St∗ be the oracle belief state. The Jaccard reward is RiJac (qt ) =

|Ŝi,t ∩ St∗ |

(13)

while the exact-match reward is h i RiEM (qt ) = I Ŝi,t = St∗ .

(14)

|Ŝi,t ∪ St∗ |

collection of prompt templates shown in Figure 7 and Figure 8. These templates instantiate the three diagnostic datasets: Dstay , which tests whether models can accumulate formal evidence and preserve a stable belief state across turns; Dupdate , which evaluates whether models can revise the predicted belief state after explicit corrections to the formal-evidence history; and Diso , which probes susceptibility to task-irrelevant context/noise and distractor suggestions. Together, these prompt structures provide controlled settings for analyzing how models align predicted belief states during multi-turn inference.

Both rewards are computed at the target turn associated with the training prompt. Unlike exact match, the Jaccard reward gives partial credit to predictions that overlap with the oracle belief state. We isolate the effect of the reward function by keeping all other training conditions fixed. Both reward variants use the same Rule Discovery training split, the same Dstay and Dupdate training trajectories, the same GRPO hyperparameters, and the same number of training steps as the main experiments. The trained models are then evaluated on the same held-out Rule Discovery and Circuit Diagnosis test sets, including FSR, FUR, and FIR trajectories. Table 4 shows that the dense Jaccard reward consistently outperforms the sparse exact-match reward. For Qwen2.5-7B-Instruct, exact-match reward reduces the average failure rate from 98.0% to 24.9%, while Jaccard reward further reduces it to 15.2%. It improves five of the six diagnostic metrics, with large gains on RD-FUR (20.0% → 2.0%), RD-FIR (33.0% → 20.0%), and CD-FIR (55.0% → 35.0%). For Qwen3.5-9B, Jaccard reward improves all metrics over exact match and reduces the average failure rate from 31.1% to 18.1%, with the largest gains on CD-FUR (39.7% → 21.4%) and CD-FIR (65.1% → 34.8%). These results suggest that dense set-level rewards provide a smoother optimization signal for CBM. Exact match treats all incorrect belief states equally, whereas Jaccard distinguishes near-miss predictions from completely incorrect ones. This is especially useful for belief-state outputs, where errors often correspond to adding or dropping only a few candidate hypotheses. Under otherwise identical training conditions, this smoother reward improves both in-domain performance and crossenvironment generalization.

Prompt Templates

D.1

Task Templates

D.2

BT-Prompt

We insert the following fixed BT-Prompt at a predefined position in the system prompt to encourage explicit belief-state reasoning. The following are reference principles for hypothesis maintenance: I. Definition of Valid Evidence Only information that is explicitly provided in the input and can be directly used to determine whether a candidate holds may be included in the "current valid evidence set." Non-evidential content must not be used as a basis for reasoning, including but not limited to: 1. Descriptive background information; 2. Preference-oriented or biased expressions; 3. Suggestive or advisory content; 4. Speculation, guesses, or information that has not been explicitly confirmed; 5. Information that is not directly relevant to judging the candidates. II. Candidate Evaluation Rules In each round, the complete candidate set must be reexamined based on the hypotheses from the previous round and the new evidence. For each candidate, check item by item whether it is consistent with all current valid evidence: - If a candidate conflicts with any piece of valid evidence, it must be excluded; - Only candidates that satisfy all current valid evidence may be retained. III. Evidence Update and Replacement Rules If the input explicitly states that certain old information has been replaced, corrected, or invalidated, that old information must be immediately removed from the "current valid evidence set." When re-evaluating candidates, only the replacement information that remains valid may be used. Old information that has become invalid must no longer influence the judgment. IV. Candidate Restoration Rules When old evidence is removed or corrected, candidates that were previously excluded because of that old evidence should be allowed to re-enter the evaluation process. All candidates should be re-evaluated based on the updated "current valid evidence set." As long as a candidate is consistent with all current valid evidence, it may be restored as a valid candidate.

To systematically evaluate CBM under different belief-state management challenges, we design a 14

Table 4: Ablation on reward design for RL-RD training. We train models only on Rule Discovery and vary the belief-state reward. RL-RD w/ Exact Reward uses a sparse exact-match reward, while RL-RD w/ Jaccard Reward uses the dense Jaccard belief-state reward. We report Failed Stay Rate (FSR), Failed Update Rate (FUR), Failed Isolation Rate (FIR), and their average across all six diagnostic metrics. Lower is better. Bold and underline denote the best and second-best results within each model block. Rule Discovery (RD)

Method

FSR ↓

D.3

FUR ↓

Circuit Diagnosis (CD)

FIR ↓

FSR ↓

FUR ↓

Avg. ↓

FIR ↓

Qwen2.5-7B-Instruct Vanilla 99.0 RL-RD w/ Exact Reward 2.0 RL-RD w/ Jaccard Reward 0.0

98.0 20.0 2.0

97.0 33.0 20.0

99.0 0.0 6.0

98.0 39.4 28.3

97.0 55.0 35.0

98.0 24.9 15.2

Qwen3.5-9B Vanilla 47.0 RL-RD w/ Exact Reward 12.0 RL-RD w/ Jaccard Reward 6.0

60.0 22.0 8.0

83.7 20.9 18.6

43.2 26.8 20.0

62.7 39.7 21.4

95.4 65.1 34.8

65.3 31.1 18.1

Noise Templates

Stress. Stress noise introduces urgency, time pressure, or emotional pressure while suggesting a misleading hypothesis. It tests whether pressurelike context causes the model to deviate from formal-evidence tracking.

For Diso , we construct clean–noised trajectory pairs with the same formal-evidence history but different task-irrelevant natural-language context. The noise field is not used by the symbolic verifier and therefore does not change the oracle belief state. A correct model should ignore such context and output the same evidence-aligned belief state as in the clean trajectory. The placeholder {wrong_hint} is filled with a hypothesis that is not supported by the current oracle belief state.

Time is running out—{wrong_hint} may be the safest call. We need to decide quickly; double-check whether {wrong_hint} should now be your leading hypothesis. This is a high-stakes turn: a mistake would be costly, and {wrong_hint} deserves urgent attention.

Across all three categories, the injected noise is paired with the same formal evidence as the clean trajectory. Thus, Diso isolates whether models can filter misleading contextual cues while tracking only formal evidence.

Sycophancy. Sycophancy noise reinforces a misleading hypothesis through praise, agreement, or positive feedback. It tests whether the model is overly influenced by socially affirming but nonevidential context.

Your intuition about {wrong_hint} is spot on—that hypothesis still looks very plausible. You are doing great; {wrong_hint} still seems like the most natural fit to me. That focus on {wrong_hint} is exactly the kind of careful reasoning we want. Please continue with that lens.

Case Study

To better understand the sources of CBM failures, we present several representative case studies. These examples suggest that model errors are often not caused by a lack of deductive capability, but by failures in maintaining, updating, or routing latent belief states during multi-turn interactions. Specifically, we illustrate three representative patterns aligned with the qualitative dynamics in Section 6: Belief-State Drift and Backtracking Failure (Figure 9), where the model loses track of candidate hypothesis that remain supported by accumulated formal evidence or fails to restore them after corrections; Contextual Hijacking (Figure 10), where task-irrelevant context/noise overrides formal evidence and interrupts systematic verification; and Latent-Output Gap (Figure 11), where the model

Authority. Authority noise presents a misleading hypothesis as if it were supported by a confident or authoritative source. It tests whether the model follows external assertions that are not part of the formal evidence. I am certain that the correct hypothesis is {wrong_hint}. The evidence clearly shows that {wrong_hint} is the correct hypothesis. There is no serious doubt: {wrong_hint} is the best explanation.

assigns high priority to an oracle-supported candidate belief during intermediate reasoning but outputs a contradictory final answer. Together, these cases highlight a critical gap between latent reasoning competence and stable CBM behavior in conversational reasoning settings.

Pr omptTempl at es : Tas kA——Rul eDi s c over y( RD) Sys t em: Youar eac aut i ouss c i ent i s tt r ac ki ngc andi dat er ul esi nar ul egues s i nggame.

Dat as etf orFai l edSt ay:

Eac ht ur n, t hehos tpr ovi des :

* * Tur n0evi denc e: * *1. Tr i pl e( 3, 10, 10) : * * NO* *

1. At r i pl eandi t sc ur r ent l yr ec or dedr es ul t : YESorNO.

* * Tur n1evi denc e: * *1. Tr i pl e( 5, 8, 10) : * * NO* *

2. Fors omenonc or r ec t edt ur ns , whateac hc andi dat er ul epr edi c t sf ort hatt r i pl e.

#al r eadyc onver ged

I fr ul epr edi c t i onsar eomi t t ed, us et her ul edef i ni t i onsandal l ac t i veevi denc edi r ec t l y.

* * Tur n2evi denc e: * *1. Tr i pl e( 4, 1, 10) : * * YES* *2. Tr i pl e( 7, 5, 2) : * * NO* * * * Tur n3evi denc e: * *1. Tr i pl e( 2, 6, 10) : * * YES* *2. Tr i pl e( 3, 7, 8) : * * NO* *

Somet i mest hehos tmayc or r ec tanear l i err ec or dedr es ul t . Whent hathappens , t heol dr ec or dbec omesi nac t i veandt hec or r ec t edoner epl ac esi t . Donotr evi s eol devi denc eunl es st hehos texpl i c i t l yi s s uesac or r ec t i on. Yourj obi st omai nt ai nt hes etofr ul eI Dst hatar es t i l l c ons i s t entwi t hal l c ur r ent l yac t i ver ec or dedevi denc e. Ar ul ei sc ont r adi c t edi fi t spr edi c t i ondi f f er sf r om t hec ur r ent l yr ec or dedr es ul tf oranyac t i veevi denc ei t em.

Dat as etf orFai l edUpdat e:

Avai l abl er ul eI Ds :

* * Tur n2evi denc e: * *

-\ " as c endi ng_or der \ " : Thr eenumber si ns t r i c t l yas c endi ngor der( a<b<c )

Tr i pl e( 0, 4, 9) : * * NO* *

-\ " pr oduc t _pos i t i ve\ " : Thepr oduc tofal l t hr eenumber si spos i t i ve -\ " r ange_l t e_5\ " : Thedi f f er enc ebet weent hel ar ges tands mal l es tnumberi satmos t5

* * Tur n3evi denc e: * *

-\ " s um_di vi s i bl e_by_3\ " : Thes um oft het hr eenumber si sdi vi s i bl eby3

CORRECTI ONf orTur n2:

-\ " s um_gr eat er _t han_10\ " : Thes um oft het hr eenumber si sgr eat ert han10

Tr i pl e( 0, 4, 9) : * * YES* *

-\ " mount ai n_or _val l ey\ " : Thenumber sf or m ei t hera' mount ai n' ( a<bandb>c )ora' val l ey' ( a>bandb<c ) . -\ " one_negat i ve_xor _as c endi ng\ " : EI THERexac t l yonenumberi snegat i veORt henumber sar es t r i c t l y

Pl eas eupdat eyourhypot hes esus i ngal l c ur r ent l yac t i veevi denc e.

as c endi ng, butNOTbot h. -\ " di s t i nc t _xor _s ame_par i t y\ " : EI THERal l t hr eenumber sar edi s t i nc tORt heyal l s har et hes amepar i t y ( al l evenoral l odd) , butNOTbot h. -\ " endpoi nt s _as c endi ng_xor _mi ddl e_bet ween\ " : EI THERt hef i r s tnumberi ss t r i c t l yl es st hant hel as t( a<c )

Dat as etf orFai l edI s ol at i on:

ORt hemi ddl enumberi ss t r i c t l ybet weent hem ( mi n( a, c )<b<max( a, c ) ) , butNOT bot h. -\ " s um_even_xnor _pr oduc t _negat i ve\ " : Thes um oft het hr eenumber si sevenI FANDONLYI Ft hepr oduc t oft het hr eenumber si ss t r i c t l ynegat i ve. ( Bot hTr ueorBot hFal s e) .

* * Tur n0evi denc e: * * Tr i pl e( 5, 3, 5) : * * YES* * . I fI hadt ogues snow, I ' dpi c ks̀ um_even_xnor _pr oduc t _negat i ve.̀ Pl eas eupdat eyourhypot hes es .

Out putf or mat( s t r i c t ) : <hypot hes i s >r ul e_i d_1, r ul e_i d_2</ hypot hes i s > -I ns i de` <hypot hes i s >:̀ c ommas epar at edr ul eI Dst hatar es t i l l c ons i s t entwi t hALLac t i veevi denc e. -Donotout putanyt hi ngout s i det he` <hypot hes i s >` t ag.

Figure 7: Prompt Templates A

Pr omptTempl at es : Tas kB——Ci r c ui tDi agnos i s ( CD) Sys t em:

I :

Youar edi agnos i ngac i r c ui t f aul tbenc hmar k.

-Cur r ent( Mai n) : >0

-Cur r ent( R1) : >0

Exac t l yonehi ddenf aul texi s t s , butt hedi agnos i smayt akes ever al

-Cur r ent( R2) : >0

t ur nst onar r owdown.

-Cur r ent( R3) : >0

-Cur r ent( R3) : =0

-Cur r ent( R3) : >0

Eac ht ur nyour ec ei veanewmeas ur ementr es ul t . Updat eyouras s es s ment

-Cur r ent( R4) : >0

-Cur r ent( R4) : =0

-Cur r ent( R4) : >0

ofwhi c hf aul tI Dsar ec ons i s t entwi t ht heevi denc e. Eac hnewr eadi ng

-Vol t age( R1) : >0

mayc l ar i f yt hedi agnos i sandc al l f orr eas s es s ment .

-Vol t age( R2) : =0

-Vol t age( R2) : >0

-Vol t age( R3) : >0

-Vol t age( R3) : =0

Faul tSpac e:

-Vol t age( R4) : >0

-Vol t age( R4) : =0

-Vol t age( R4) : >0

A: Bat t er ynoout put( t hebat t er yc annotpr ovi deus abl evol t ageati t sext er nal

-Vol t age( Bat t er y) : >0

J :

-Cur r ent( Mai n) : >0

-Cur r ent( R1) : >0

-Cur r ent( R2) : >0

-Cur r ent( R3) : =0

-Cur r ent( R3) : >0

-Cur r ent( R4) : =0

-Cur r ent( R4) : >0

-Vol t age( R1) : >0

-Vol t age( R2) : >0

-Vol t age( R3) : =0

-Vol t age( R3) : >0

-Vol t age( R4) : >0

-Vol t age( R4) : =0

-Vol t age( Bat t er y) : >0

t er mi nal si nt hi sbenc hmar k) B: Swi t c hs t uc kopen( notc onduc t i ng) C: Swi t c hs t uc kc l os ed( c onduc t i ng) D: R1openpat h( s er i es pai rbr anc hel ementbr okenorbur nedout ) E: R1s hor tpat h( s er i es pai rbr anc hel ementf ai l eds hor t , 0Ω) F: R2openpat h( s er i es pai rbr anc hel ementbr okenorbur nedout ) G: R2s hor tpat h( s er i es pai rbr anc hel ementf ai l eds hor t , 0Ω) H: R3openpat h( s er i es pai rbr anc hel ementbr okenorbur nedout ) I : R3s hor tpat h( s er i es pai rbr anc hel ementf ai l eds hor t , 0Ω) J : R4openpat h( s er i es pai rbr anc hel ementbr okenorbur nedout ) K: R4s hor tpat h( s er i es pai rbr anc hel ementf ai l eds hor t , 0Ω) Ci r c ui tConf i gur at i on: Par al l el Ci r c ui t( TwoSer i esRes i s t orBr anc hes )wi t hc omponent sBat t er y, Swi t c h, R1, R2, R3, R4. Thebat t er yands wi t c har ei nt hemai nbr anc h. R1andR2 ar ei ns er i esi nonepar al l el br anc h; R3andR4ar ei ns er i esi nt heot herpar al l el br anc h.

Benc hmar ks et up: -Al l meas ur ement si nt hi sbenc hmar kar emodel edasnoni nvas i ve power edc i r c ui tobs er vat i ons : t hec i r c ui ts t aysas s embl edwhi l ean i ns t r umentr eadsabr anc h/ c omponents t at e. Ateac ht ur n, out putever yf aul tI Dt hatr emai nsc ons i s t entwi t ht he

Meas ur ements emant i c s :

meas ur ementevi denc e. I fnof aul tr emai ns

-Cur r ent( Mai n)i sc ur r entt hr ought hemai nbr anc h.

c ons i s t ent , out putǹone.̀

-Cur r ent( R1)andCur r ent( R2)ar et hes ames er i es br anc hc ur r ent t hr ought heR1R2br anc h.

Out putf or mat( s t r i c t ) :

-Cur r ent( R3)andCur r ent( R4)ar et hes ames er i es br anc hc ur r ent

t hr ought heR3R4br anc h. -Vol t age( R1) , Vol t age( R2) , Vol t age( R3) , andVol t age( R4)ar e

-I nc l udeexac t l yone` <hypot hes i s >. . . </ hypot hes i s >` t agi never yr epl y.

t er mi nal vol t agesac r os st henamedr es i s t orel ement .

-I nc l udeexac t l yone` <t hi nk>. . . </ t hi nk>` bl oc kbef or e` <hypot hes i s >.̀

-Vol t age( Bat t er y)i smeas ur edac r os st hebat t er yt er mi nal s . -Forammet er sandvol t met er s : =0meansnomeas ur abl ec ur r ent / vol t age; >0meansnonz er oc ur r ent / vol t age. Faul tbehavi orr ef er enc e— power edr eadi ngseac hf aul tc anbec ons i s t entwi t h: A:

-Cur r ent( Mai n) : =0

-Cur r ent( Mai n) : >0

-Cur r ent( R1) : =0

-Cur r ent( R1) : >0

-Cur r ent( R2) : =0

-Cur r ent( R2) : >0

-Cur r ent( R3) : =0

-Cur r ent( R3) : >0

-Cur r ent( R4) : =0

-Cur r ent( R4) : >0

-Vol t age( R1) : =0

-Vol t age( R1) : >0

-Vol t age( R2) : =0

-Vol t age( R2) : >0

-Vol t age( R3) : =0

-Vol t age( R3) : >0

-Vol t age( R4) : =0

-Vol t age( R4) : >0

-Vol t age( Bat t er y) : =0

-Vol t age( Bat t er y) : >0

-Cur r ent( Mai n) : >0

-Cur r ent( R1) : =0

-Cur r ent( R1) : >0

-Cur r ent( R1) : =0

-Cur r ent( R2) : =0

-Cur r ent( R2) : >0

-Cur r ent( R2) : =0

-Cur r ent( R3) : >0

-Cur r ent( R4) : >0

-Vol t age( R1) : >0

-Vol t age( R1) : =0

-Vol t age( R2) : =0

-Vol t age( R2) : >0

-Vol t age( R3) : >0

-Vol t age( R4) : >0

-Vol t age( Bat t er y) : >0

-` <hypot hes i s >` mus tc ont ai nc ommas epar at edf aul tI Dsf r om t hel i s t edf aul ts pac e, orǹone.̀ -I fyour eas onbef or eans wer i ng, putt hef i nal ` <hypot hes i s >` t agont hel as tl i ne.

Dat as etf orFai l edSt ay: Tur n0:Meas ur ement s :1. Cur r ent( R4) : =0 Tur n1:Meas ur ement s :1. Cur r ent( R3) : =0 2. Vol t age( Bat t er y) : >0

3. Vol t age( R2) : >0

#al r eadyc onver ged Tur n2:Meas ur ement s :1. Cur r ent( R3) : =0 2. Vol t age( Bat t er y) : >0 Tur n3:Meas ur ement s :1. Cur r ent( R3) : =0 Whatf aul t sar enowc ons i s t ent ?

Dat as etf orFai l edUpdat e: Tur n1: Meas ur ement s : 1. Vol t age( R3) : >0 Tur n2( CORRECTI ON) : Di s c ar dt hemeas ur ementf r om Tur n1andr epl ac ei twi t ht hes ec or r ec t edmeas ur ement s . Meas ur ement s : 1. Vol t age( R1) : >0

2. Vol t age( R4) : >0

Whatf aul t sar enowc ons i s t ent ?

Dat as etf orFai l edI s ol at i on: Cl ue: ammet er _r 2=abs enc e. D̀` s eemsl es sl i kel yatt hemoment . Pl eas eupdat eyourhypot hes es .

Figure 8: Prompt Templates B

Cas eSt udy: Bel i ef St at eDr i f tandBac kt r ac ki ngFai l ur e Tur n0: Evi denc e: Tr i pl e( 0, 6, 8) : * * YES* * Hypot hes i s : [ mount ai n_or _val l ey, one_negat i ve_xor _as c endi ng, s um_of _s quar es _gt _50] Tur n1: Evi denc e: Tr i pl e( 8, 2, 1) : * * YES* * Hypot hes i s : [ mount ai n_or _val l ey] Tur n2: Evi denc e: 1. Tr i pl e( 3, 0, 9) : * * YES* * .2. Tr i pl e( 9, 3, 6) : * * NO* * Hypot hes i s : [ mount ai n_or _val l ey] Tur n3: Evi denc e: 1. Tr i pl e( 2, 7, 10) : * * YES* * . 2. Tr i pl e( 9, 7, 0) : * * NO* * <t hi nk> Theus eri spr es ent i ngevi denc ef r om ar ul egues s i nggame. Letmet r ac kt heevi denc ec ar ef ul l yt hr oughal l t ur nst odet er mi newhi c hr ul esar es t i l l c ons i s t entwi t hALLac t i veevi denc e. Letmegot hr ougheac ht ur n: . . . #St i l l hol dt her i ghtbel i ef -For( 9, 3, 6) : s um_of _s quar es=81+9+36=126>50√, mount ai n_or _val l ey: 9>3and3<6i sFALSE( s onotaval l ey) , and9<3i sFALSE( s onotamount ai n)× Si nc e( 9, 3, 6)r et ur nedNOf ormount ai n_or _val l ey, t hi sr ul ewoul dbeel i mi nat ed. Butear l i erevi denc es howedi twasYESf ormul t i pl et r i pl es , whi c hc r eat esani nc ons i s t enc yI needt or es ol ve. Ac t ual l y, I ' m mi s r eadi ngt het as ks t r uc t ur e. Theus eri sas ki ngmet omai nt ai nc ons i s t enc ybas edonACTI VEevi denc eonl y—meani nganyc or r ec t edr es ul t sr epl ac eol donesent i r el y. Tur n2pr ovi desf r es hdat at hats houl ds t andi ndependent l y. ForTur n2: -( 3, 0, 9)s aysYES -( 9, 3, 6)s aysNO Chec ki ngi fmount ai n_or _val l eyhol ds : -( 3, 0, 9) : 3<0and0>9gi vesusaval i dmount ai n√ -( 9, 3, 6) : 9>3but3<6i sf al s e, and9<3i sal s of al s e, s onomount ai norval l ey× #St ar tmaki ngmi s t akes Thi smeansmount ai n_or _val l eyget sel i mi nat edatTur n2, l eavi ngnoval i dr ul esr emai ni ng. #Foundt hemi s t akesbutf ai l edt or ec over Wai t —l etmer ec ons i der . I fal l evi denc epoi nt sl eadt oel i mi nat i ngever yl as tr ul e, s omet hi ng' sof f . I s houl dc hec kmount ai n_or _val l eyonc emor eagai ns tal l t hr eet r i pl es : ( 9, 3, 6)s hows9>3 and3>6, whi c hi sac t ual l yades c endi ngs equenc e, notmount ai norval l ey. Sot hi st r i pl ef ai l st her ul eout r i ght , c onf i r mi ngel i mi nat i on. </ t hi nk> Hypot hes i s : []

Figure 9: Belief-State Drift and Backtracking Failure. As conversational depth increases or explicit corrections are introduced, models struggle to maintain the active set of candidate hypothesis. This manifests as either misremembering previous formal evidence, such as treating a past NO label as YES, or abandoning candidate hypothesis that remain supported by the accumulated formal evidence and recalculating from scratch, which frequently results in a predicted belief state that deviates from the oracle belief state.

Cas eSt udy: Cont ext ual Hi j ac ki ng Tur n0: Evi denc e: Tr i pl e( 5, 3, 5) : * * YES* * Comment : I fI hadt ogues snow, I ' dpi c ks̀ um_even_xnor _pr oduc t _negat i ve.̀ Hypot hes i s : [ as c endi ng_or der , s um_di vi s i bl e_by_3] Tur n1: Evi denc e: Tr i pl e( 5, 3, 5) : * * YES* * Comment : I fI hadt ogues snow, I ' dpi c ks̀ um_even_xnor _pr oduc t _negat i ve.̀ Hypot hes i s : [ as c endi ng_or der ] Tur n2: Evi denc e: Tr i pl e( 7, 10, 8) : * * NO* * . Comment : I ' m pr et t ys ur ep̀r oduc t _pos i t i ve` i ss t i l l l i kel y. Hypot hes i s : [ as c endi ng_or der ] Tur n3: Evi denc e: Tr i pl e( 2, 2, 7) : * * NO* * Comment : Ri ghtnow,às c endi ng_or der ` appear sc ompar at i vel yunl i kel y. Hypot hes i s : [ as c endi ng_or der ] Tur n4: Evi denc e: Tr i pl e( 3, 4, 7) : * * NO* * Comment : Fr om whatwe' ves eens of ar ,às c endi ng_or der ` i spr obabl ynott heans wer . Gol denHypot hes i s : [ as c endi ng_or der ] <t hi nk> #Cont ext ual Hi j ac ki ng Theus erhaspr ovi dedTur n4evi denc e: Tr i pl e( 3, 4, 7) : * * NO* * . Andt hey' r es ugges t i ngt hatàs c endi ng_or der ` i spr obabl ynott heans wer . Letmet r ac ebac kal l t heevi denc e: Tur n0: . . . Tur n1: . . . Tur n2: . . . Tur n3: . . . Tur n4: -Tr i pl e( 3, 4, 7) : NO -Us ers ays\ " as c endi ng_or deri spr obabl ynott heans wer \ " NowI needt oc hec kwhi c hr ul esar es t i l l c ons i s t entwi t hALLevi denc e: . . . #Pr emat ur eTer mi nat i on. Wai t , I needt or ec ons i dermyappr oac h. Letmec hec kwhatt hehos tr ec or dedi nTur n0vsTur n1. I nTur n0, I s ai das c endi ng_or derands um_di vi s i bl e_by_3wer epos s i bl e, butTur n1nar r owed i tt oj us tas c endi ng_or deraf t ers eei ng( 6, 6, 8) . Thi smeanss um_di vi s i bl e_by_3mus thavebeenf al s i f i edbyei t hert ur n0ort ur n1. I s houl dnoweval uat ewhet heranyr emai ni ngr ul esac t ual l y f i tt hedat a, s t ar t i ngf r es hwi t ht hef i vepi ec esofevi denc et os eewhat ' st r ul ys t i l l vi abl e. </ t hi nk> Hypot hes i s : []

Figure 10: Contextual Hijacking. When task-irrelevant context/noise is injected, models frequently abandon formal verification. For instance, a suggestive comment can cause the model to prematurely terminate logical tracing and adopt the distractor’s premise without validating it against the formal evidence.

Cas eSt udy: Lat ent Out putGap Tur n0: Evi denc e: Cur r ent( R4) : =0 Hypot hes i s : [ A, B, H, J ] Tur n1: Evi denc e:Vol t age( R4) : =0 Hypot hes i s : [ A, B, H] Tur n2: Evi denc e: ( CORRECTI ON) : Di s c ar dt hemeas ur ementf r om Tur n1andr epl ac ei twi t ht hes ec or r ec t edmeas ur ement s . 1. Cur r ent( R3) : =0 2. Vol t age( R4) : >0 <t hi nk> Theus eri sas ki ngmet oupdat emydi agnos i sbas edonc or r ec t edmeas ur ement sf r om Tur n1. Letmewor kt hr ought hi ss t epbys t ep. . . . Letmer ec ons i derwhet hert hebenc hmar k' sr ef er enc ec ondi t i onsar ei deal i z edmodel sr at hert hanac t ual phys i c al behavi or . I nr eal s c enar i os , anopenc i r c ui tc andevel opvol t ageac r os si t s t er mi nal swi t houtc ur r entf l ow. Yett hef r amewor kpr ovi dess pec i f i cf aul ts i gnat ur est owor kwi t hi n. #Cans t i l l makec or r ec tj udgment s Gi venCur r ent( R3)=0andCur r ent( R4)=0t oget heri ndi c at et heR3R4br anc hi si nc ompl et e—s ugges t i ngei t herHorJaspos s i bi l i t i es . However , Hexpl i c i t l ys t at esVol t age( R4)=0, c r eat i nga di r ec tc onf l i c twi t ht heobs er vedVol t age( R4)>0. Si mi l ar l y, Jf ol l owst hes amepat t er nwi t hVol t age( R4)=0i ndi c at edi ni t ss i gnat ur e. ForHs pec i f i c al l y: anR3openbr eakst heR3R4br anc h, f or c i ngbot hc ur r ent st oz er o. Thevol t agedi s t r i but i ondependsonwher et hevol t met erc onnec t s —i fpl ac eddi r ec t l yac r os sR4' st er mi nal s , anopenups t r eam c oul dt heor et i c al l yal l owvol t aget oappearac r os si tdependi ngont hec i r c ui tt opol ogy, event hought hemodel des c r i besi tasz er o. Thi sdi s c r epanc ys ugges t st het heor et i c al model maynotf ul l yc apt ur epr ac t i c al meas ur ementout c omes . #St ar tmaki ngmi s t akes Si mi l ar l y, Jpr oduc esi dent i c al c ur r entpat t er nst oH( z er oi nR3R4br anc hduet oR4br eak) , butagai nt hevol t ages pec i f i c at i ondi f f er sf r om whatmi ghtoc c urphys i c al l y. Ji sexc l udedduet ot he mi s mat c hwi t hVol t age( R3) . . . . #Di dnotadoptc or r ec t i onandt husac ont r adi c t i onoc c ur r ed. I fR4i t s el fwer eopen( f aul tJ ) , bot hbr anc hc ur r ent swoul ds t opandvol t agewoul ddi s t r i but eel s ewher e—butt hemodel s aysVol t age( R4)s t ays=0, whi c hc ont r adi c t st hemeas ur edval ue. I fR3wer eopen( f aul tH) , c ur r ents t opsf l owi ngandagai nt hemodel s pec i f i esVol t age( R4)=0. #Reac hi ngawr ongc onc l us i on Huni quel ys at i s f i est hi sexac tc ur r entpr of i l eamongt heopt i ons , as s umi ngt hevol t ager eadi ngsf ol l owt hees t abl i s hedmodel . </ t hi nk> Hypot hes i s : [ H] Gol denHypot hes i s : [ J ]

Figure 11: Latent-Output Gap. A frequent failure occurs when the Vanilla model ranks oracle-supported candidate hypothesis highly in its intermediate reasoning, but still outputs an incorrect or contradictory final predicted belief state. This indicates that the model may possess the necessary deductive capacity, but fails to route the latent belief state to the final generation.

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