LAMB：面向长时程大语言模型智能体的隐式多假设信念

# Latent Multi-Hypothesis Beliefsfor Long-Horizon Large Language Agents
Source: [https://arxiv.org/html/2610.26955](https://arxiv.org/html/2610.26955)
LAMB: Latent Multi\-Hypothesis Beliefs for Long\-Horizon Language Agents

Hui Li11footnotemark:1Yuhan Liu††thanks:Corresponding authors\.Jinsong Su22footnotemark:2

###### Abstract

Long\-horizon language agents require compact memories that retain interaction evidence while preserving competing hypotheses about unobserved states\. Although effective, many belief\-state methods encode memory in discrete natural language, where repeated compression can omit evidence or prematurely discard plausible alternatives\. We introduce LAMB, a recurrent framework for latent multi\-hypothesis beliefs\. A Common–Delta factorization combines a shared Common state for interaction evidence with weighted Delta components for competing hypotheses\. Relative evidence from action\-outcome predictions guides recurrent hypothesis updates without generating textual memory\. A context\-conditioned readout exposes hypotheses separately to the policy through a fixed\-token belief prefix in the default configuration\. The belief modules and policy are jointly trained using action imitation and auxiliary state, outcome, and observation\-discrimination objectives\. Across ALFWorld, BabyAI, ScienceWorld, and Crafter, LAMB matches or outperforms PABU, improving ALFWorld SR@30 and ScienceWorld success by 6\.0 and 22\.0 percentage points, respectively, and Crafter achievement progress by 26\.37 points\. On ALFWorld, it also reduces the redundant\-action rate by 17\.70 percentage points\. These results support latent multi\-hypothesis beliefs as compact memory for effective long\-horizon decision making\.

![Refer to caption](https://arxiv.org/html/2610.26955v1/motivation.png)Figure 1:Existing text\-based beliefs \(left\) prematurely drop or misjudge environmental observations, trapping the agent in invalid action loops\. LAMB \(right\) factorizes beliefs into confirmed state facts and candidate spatial distributions, maintaining calibrated uncertainty to resolve the task efficiently in 5 steps\.## 1Introduction

Large language model \(LLM\) agents solve multi\-step tasks through sequences of actions and observations\([Yao et al\., 2023](https://arxiv.org/html/2610.26955#bib.bib46)\), which has wide applications in many fields\([Liu et al\., 2024b](https://arxiv.org/html/2610.26955#bib.bib47);[Liu et al\., 2025b](https://arxiv.org/html/2610.26955#bib.bib48);[Liu et al\., 2025a](https://arxiv.org/html/2610.26955#bib.bib43)\)\. Under partial observability, effective decisions require retaining earlier evidence while reasoning about aspects of the environment that remain unobserved\. Conditioning on the full interaction history provides access to past information, but its growing length increases the context processed at each decision\. Long\-horizon agents therefore need memory that preserves decision\-relevant information without repeatedly processing the complete trajectory\.

Context\-management methods reduce this burden by pruning, masking, or compressing past interactions\([Lindenbauer et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib40);[Xiao et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib41)\)\. Belief\-state methods instead recursively update a compact state for subsequent decisions\. Although simple and effective, many LLM belief\-state methods still represent beliefs in discrete natural language\([Jiang et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib15);[Lidayan et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib14);[Singh et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib16)\)\. Even when augmented with confidence labels, these states must express retained evidence and alternatives within a limited text budget\. Repeated selection and rewriting can omit earlier findings or prematurely discard plausible interpretations\.

For example, Figure[1](https://arxiv.org/html/2610.26955#S0.F1)shows an agent ruling out the dresser while considering the bed, drawer, and shelf as possible laptop locations\. Its memory should preserve the inspection result shared by both alternatives and revise the candidates as new evidence arrives\. Updating textual beliefs through generation also incurs decoding overhead beyond action selection\. These considerations motivate our central question:*how can a fixed\-size recurrent memory retain shared evidence while preserving and revising competing hypotheses without repeatedly verbalizing its state?*

To address this question, we propose LAMB, a latent belief framework that tracks episode information directly in continuous latent space\. Through a Common–Delta factorization, a shared Common component is trained to retain interaction evidence, while weighted Delta components encode hypothesis\-specific information\. Rather than pooling the particles into a single representation, a context\-conditioned readout exposes each Delta to the policy separately, alongside the Common readout\. The default configuration produces a fixed\-token belief prefix, keeping the historical contribution to policy input constant throughout the episode\. To make hypothesis revision responsive to new evidence, LAMB compares each particle’s action\-outcome prediction with the feedback received from the environment\. The resulting relative evidence guides recurrent Delta updates, after which mixture weights are recomputed from the updated Common and Delta states\. We jointly train the belief modules and policy on offline interaction trajectories using action imitation, joint state\-semantic prediction, outcome\-semantic prediction, and observation discrimination for the Common component\. Simulator\-derived state labels are used only during training; online belief updates rely on observed actions and feedback\.

We evaluate LAMB on ALFWorld, BabyAI, ScienceWorld, and Crafter\. Results show that LAMB matches or outperforms the strong baselines across these benchmarks while reducing token consumption on ALFWorld\. Ablations and diagnostic analyses further assess the proposed design and characterize evidence retention, particle readouts, and agent behavior\.

Our main contributions are as follows:

- •We propose LAMB, a fixed\-capacity latent belief framework that supports long\-horizon decisions with a constant\-size policy prefix, without generating textual memory updates\.
- •We develop a Common–Delta representation with action\-outcome prediction, evidence\-guided recurrent updates, and an unpooled readout of weighted hypotheses\.
- •We demonstrate task\-performance gain across four benchmarks and inference cost on ALFWorld, complemented by ablations, redundant\-action analysis, and diagnostics of evidence retention and particle readouts\.

## 2Related Work

### 2\.1Stream\-based Agent Memory

Many agent\-memory systems organize past interactions as records that are retrieved, revised, or compressed for subsequent use\. Generative Agents\([Park et al\., 2023](https://arxiv.org/html/2610.26955#bib.bib1)\)and MemGPT\([Packer et al\., 2024](https://arxiv.org/html/2610.26955#bib.bib2)\)keep the full record in external storage and page relevant entries into context, A\-MEM\([Xu et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib3)\)links and updates memory entries\. Compression methods, including LLMLingua\-2\([Pan et al\., 2024](https://arxiv.org/html/2610.26955#bib.bib4)\), LongLLMLingua\([Jiang et al\., 2024](https://arxiv.org/html/2610.26955#bib.bib5)\), and ACON\([Kang et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib6)\)reduce the context passed to the policy\. Experience\-reuse methods, including Reflexion\([Shinn et al\., 2023](https://arxiv.org/html/2610.26955#bib.bib7)\), ExpeL\([Zhao et al\., 2024](https://arxiv.org/html/2610.26955#bib.bib8)\), AWM\([Wang et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib9)\), and ACE\([Zhang et al\., 2026c](https://arxiv.org/html/2610.26955#bib.bib10)\), distill cross\-episode knowledge into insights, workflows, or playbooks\.

These approaches differ in their retention and access mechanisms; the accessibility of retained evidence can also depend on its placement within the context\([Liu et al\., 2024a](https://arxiv.org/html/2610.26955#bib.bib11)\)\.

Our LAMB focuses on maintaining a fixed\-size state of the current episode through recursive updates, which offers a different interface for carrying interaction evidence into each decision\.

### 2\.2Textual Belief State

Belief\-state approaches maintain a compact representation for decision making under partial observability\([Kaelbling et al\., 1998](https://arxiv.org/html/2610.26955#bib.bib12)\)\.

MEM1\([Zhou et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib13)\)and ABBEL\([Lidayan et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib14)\)learn textual states that consolidate information across interactions, while PABU\([Jiang et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib15)\)tracks task progress together with selectively retained actions and observations\. Agent\-BRACE\([Singh et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib16)\)and BeliefMem\([Liao et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib17)\)further encode uncertainty through confidence labels or candidate records\. Textual beliefs, however, represent uncertainty by*enumeration*: evidence shared across hypotheses is restated for each candidate, so a bounded state must trade evidence retention against hypothesis coverage\. In practice this trade\-off surfaces either as forgotten findings or as premature commitment to a single hypothesis\([Krishnamurthy et al\., 2024](https://arxiv.org/html/2610.26955#bib.bib18)\)\. LAMB keeps hypotheses in latent form on top of a shared common state, so each additional hypothesis costs one prefix token rather than another restatement of the evidence\.

### 2\.3Latent Memory and Filtering

Latent\-memory methods encode context as learned continuous representations, avoiding explicit text generation for the memory itself\. Recurrent Memory Transformer\([Bulatov et al\., 2022](https://arxiv.org/html/2610.26955#bib.bib19)\), AutoCompressors\([Chevalier et al\., 2023](https://arxiv.org/html/2610.26955#bib.bib20)\), Gist tokens\([Mu et al\., 2023](https://arxiv.org/html/2610.26955#bib.bib21)\), Laser\([Wang et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib39)\)and ICAE\([Ge et al\., 2024](https://arxiv.org/html/2610.26955#bib.bib22)\)condense context into a small set of learned vectors\. In agent settings, Mem\-W\([Zhang et al\., 2026b](https://arxiv.org/html/2610.26955#bib.bib23)\)and MemGen\([Zhang et al\., 2026a](https://arxiv.org/html/2610.26955#bib.bib24)\)bring such representations to agents as within\-episode and generative memory\.

However, particle\-based belief models, such as Particle Filter Networks\([Karkus et al\., 2018](https://arxiv.org/html/2610.26955#bib.bib25)\), DVRL\([Igl et al\., 2018](https://arxiv.org/html/2610.26955#bib.bib26)\), and PF\-RNN\([Ma et al\., 2020](https://arxiv.org/html/2610.26955#bib.bib27)\), make hypotheses explicit, but each particle*replicates*the full state, reintroducing the duplication of shared evidence\. In contrast, LAMB factorizes a recurrent latent belief into a common component that stores shared evidence once and hypothesis\-specific deltas that encode only how alternatives differ\.

## 3Method

![Refer to caption](https://arxiv.org/html/2610.26955v1/overview.png)Figure 2:Overview of LAMB\. The recurrent updater incorporates the latest action and observation into a shared Common state and weighted Delta hypotheses\. A context\-conditioned readout produces a fixed 12\-token prefix for action generation\. Simulator facts supervise joint prediction during training only; subsequent interaction feedback enters the next belief update\.### 3\.1Factorizing the Latent Belief

We consider goalgg\-directed interaction under partial observability\. At steptt, the agent receives an observationoto\_\{t\}and selects an actionata\_\{t\}from the admissible set𝒜t\\mathcal\{A\}\_\{t\}\. A history\-conditioned policy acts on:

ht=\(g,o1,a1,…,at−1,ot\),at∼πθ\(⋅∣ht\),h\_\{t\}=\(g,o\_\{1\},a\_\{1\},\\dots,a\_\{t\-1\},o\_\{t\}\),a\_\{t\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid h\_\{t\}\),\(1\)whose input length grows linearly withtt\. Our method LAMB replaces the history before the current step with a fixed\-size beliefBtB\_\{t\}, updated from the latest action and observation:

B0=Init\(g\),Bt=U\(Bt−1,at−1,ot\)\.\\displaystyle B\_\{0\}=\\operatorname\{Init\}\(g\),B\_\{t\}=U\\bigl\(B\_\{t\-1\},a\_\{t\-1\},o\_\{t\}\\bigr\)\.\(2\)The policy conditions on the updated belief and the current contextxt=\(g,ot,𝒜t,at−1\)x\_\{t\}=\(g,o\_\{t\},\\mathcal\{A\}\_\{t\},a\_\{t\-1\}\):

at∼πθ\(⋅∣R\(Bt,xt\),xt\),a\_\{t\}\\sim\\pi\_\{\\theta\}\\bigl\(\\cdot\\mid R\(B\_\{t\},x\_\{t\}\),x\_\{t\}\\bigr\),\(3\)whereRRmaps the belief to a fixed number of continuous prefix tokens\. The historical contribution to policy input therefore remains constant, while the current observation and candidate actions may vary in length\.

Partial observations can leave several world configurations plausible\. To share information across these alternatives while allowing particle\-specific representations, we factorize the belief into a common statectc\_\{t\},KKdelta states\{δti\}i=1K\\\{\\delta\_\{t\}^\{i\}\\\}\_\{i=1\}^\{K\}, and normalized mixture weightswt∈ΔK−1w\_\{t\}\\in\\Delta^\{K\-1\}:

Bt=\(ct,\{δti\}i=1K,wt\),ct,δti∈ℝd\.B\_\{t\}=\\bigl\(c\_\{t\},\\\{\\delta\_\{t\}^\{i\}\\\}\_\{i=1\}^\{K\},w\_\{t\}\\bigr\),\\quad c\_\{t\},\\delta\_\{t\}^\{i\}\\in\\mathbb\{R\}^\{d\}\.\(4\)The common state is intended to retain information useful across hypotheses, while the deltas provide separate representations that can capture unresolved alternatives\. Each pair\(ct,δti\)\(c\_\{t\},\\delta\_\{t\}^\{i\}\)constitutes a learned latent particle\. This factorization encourages information sharing and component\-specific variation without imposing a uniquely identifiable semantic decomposition\.

During training, the belief predicts a set of simulator\-derived fact valuesy1:Qy\_\{1:Q\}through a mixture of conditionally independent distributions:

p\(y1:Q∣Bt\)=∑i=1Kwti∏q=1Qpϕ\(yq∣ct,δti\)\.p\\bigl\(y\_\{1:Q\}\\mid B\_\{t\}\\bigr\)=\\sum\_\{i=1\}^\{K\}w\_\{t\}^\{i\}\\prod\_\{q=1\}^\{Q\}p\_\{\\phi\}\\bigl\(y\_\{q\}\\mid c\_\{t\},\\delta\_\{t\}^\{i\}\\bigr\)\.\(5\)Each component factorizes the fact predictions, while the shared mixture index couples them in the joint distribution\. This construction can capture dependencies such as mutually exclusive object locations\. The resulting consistency depends on the learned component predictions; the mixture form alone does not exclude invalid configurations \(Appendix[E](https://arxiv.org/html/2610.26955#A5)\)\.

Figure[2](https://arxiv.org/html/2610.26955#S3.F2)summarizes the interaction loop\. The updater evaluates the observed action outcome under the previous particles, updates the common and delta states, and recomputes the mixture weights \(Section[3\.2](https://arxiv.org/html/2610.26955#S3.SS2)\)\. A context\-conditioned readout then converts the updated belief into a fixed\-length prefix for action generation \(Section[3\.3](https://arxiv.org/html/2610.26955#S3.SS3)\)\. The policy and belief modules are trained jointly \(Section[3\.4](https://arxiv.org/html/2610.26955#S3.SS4)\)\.

### 3\.2Common\-Delta Belief Update

Updating beliefs under partial observability requires resolving a fundamental problem:committing to verified consensus across the episode while preserving distinct, competing hypotheses where ambiguity remains\.To balance shared facts against hypothesis diversity, our update decouples belief evolution into an objective common statectc\_\{t\}and specialized particle deltasδti\\delta\_\{t\}^\{i\}\. At each step, the belief update incorporates the latest transition\(at−1,ot\)\(a\_\{t\-1\},o\_\{t\}\)to evaluate competing hypotheses and evolve the state representations\. We first encode the transition intout=Embed⁡\(at−1,ot\)u\_\{t\}=\\operatorname\{Embed\}\(a\_\{t\-1\},o\_\{t\}\)\. Before modifying the state vectors, LAMB evaluates how accurately each hypothesis anticipated the transition outcomertr\_\{t\}:

ϵti\\displaystyle\\epsilon\_\{t\}^\{i\}=ℓti−1K​∑j=1Kℓtj,\\displaystyle=\\ell\_\{t\}^\{i\}\-\\frac\{1\}\{K\}\\sum\_\{j=1\}^\{K\}\\ell\_\{t\}^\{j\},\(6\)ℓti\\displaystyle\\ell\_\{t\}^\{i\}=log⁡pψ​\(rt∣ct−1,δt−1i,at−1\)\.\\displaystyle=\\log p\_\{\\psi\}\\big\(r\_\{t\}\\mid c\_\{t\-1\},\\delta\_\{t\-1\}^\{i\},a\_\{t\-1\}\\big\)\.Here,ϵti\\epsilon\_\{t\}^\{i\}serves as a relative evidence signal: hypotheses whose predictions better match the observed transition receive positive feedback, while conflicting ones are penalized\.

Next, both components evolve via gated residual updates:

ct\\displaystyle c\_\{t\}=Gatec​\(ct−1,\[ct−1;ut\]\),\\displaystyle=\\mathrm\{Gate\}\_\{c\}\\big\(c\_\{t\-1\},\[c\_\{t\-1\};u\_\{t\}\]\\big\),\(7\)δti\\displaystyle\\delta\_\{t\}^\{i\}=Gateδ​\(δt−1i,\[ct−1;δt−1i;ut;ϵti\]\),\\displaystyle=\\mathrm\{Gate\}\_\{\\delta\}\\big\(\\delta\_\{t\-1\}^\{i\},\[c\_\{t\-1\};\\delta\_\{t\-1\}^\{i\};u\_\{t\};\\epsilon\_\{t\}^\{i\}\]\\big\),\(8\)whereGate⁡\(s,z\)=\(1−σ⁡\(Wz​z\)\)⊙s\+σ⁡\(Wz​z\)⊙MLP⁡\(z\)\\mathrm\{Gate\}\(s,z\)=\(1\-\\sigma\(W\_\{z\}z\)\)\\odot s\+\\sigma\(W\_\{z\}z\)\\odot\\operatorname\{MLP\}\(z\)\. Crucially, the information flow is asymmetric: the common statectc\_\{t\}aggregates objective transition history without consulting individual hypotheses, while each deltaδti\\delta\_\{t\}^\{i\}conditions on bothct−1c\_\{t\-1\}and its specific evidenceϵti\\epsilon\_\{t\}^\{i\}\. We parameterizeGateδ\\mathrm\{Gate\}\_\{\\delta\}with a per\-hypothesis LoRA\([Hu et al\., 2022](https://arxiv.org/html/2610.26955#bib.bib34)\)to encourage specialization across particles\. Finally, a shared scoring headggre\-estimates the belief weights directly from the updated states:

wti=exp⁡\(g⁡\(ct,δti\)\)∑j=1Kexp⁡\(g⁡\(ct,δtj\)\)\.w\_\{t\}^\{i\}=\\frac\{\\exp\\big\(g\(c\_\{t\},\\delta\_\{t\}^\{i\}\)\\big\)\}\{\\sum\_\{j=1\}^\{K\}\\exp\\big\(g\(c\_\{t\},\\delta\_\{t\}^\{j\}\)\\big\)\}\.\(9\)

### 3\.3Unpooled Policy Readout

Directly pooling particle beliefs into a single vector collapses multimodal uncertainty, blending mutually exclusive hypotheses into ambiguous mean states\. To preserve discrete alternatives while anchoring the decision in established facts, we design an asymmetric readout that translates the belief into a structured prefix sequence\.

Specifically, the common statectc\_\{t\}is integrated with the current input representationEmbed⁡\(xt\)\\operatorname\{Embed\}\(x\_\{t\}\)via cross\-attention intoMMcontext tokens, capturing verified environmental constraints\. In parallel, each particleδti\\delta\_\{t\}^\{i\}is projected into an individual hypothesis tokenetie\_\{t\}^\{i\}, modulated by its log\-weight:e~ti=eti\+v​log⁡wti\\tilde\{e\}\_\{t\}^\{i\}=e\_\{t\}^\{i\}\+v\\log w\_\{t\}^\{i\}, wherevvis a learned direction vector\. MaintainingKKseparate hypothesis tokens enables the policy’s self\-attention mechanism to explicitly contrast candidate states, isolating dominant hypotheses from marginal ones without information loss\. Finally, the resultingM\+KM\+Kbelief tokens are prepended as a belief\-conditioned continuous prefix\([Li and Liang, 2021](https://arxiv.org/html/2610.26955#bib.bib35)\)to the embedded inputxtx\_\{t\}, and the policyπθ\\pi\_\{\\theta\}\(fine\-tuned via LoRA\) generates the actionata\_\{t\}\.

### 3\.4Grounded Belief Training

End\-to\-end action imitation risks representation collapse, reducing hypotheses to ungrounded task heuristics\. To enforce faithful state tracking, our method decouples physical world modeling from policy optimization, minimizing a joint objective:

ℒ=ℒact\+λsem​\(ℒstate\+ℒout\)\+λcom​ℒcom,\\mathcal\{L\}=\\mathcal\{L\}\_\{\\text\{act\}\}\+\\lambda\_\{\\text\{sem\}\}\(\\mathcal\{L\}\_\{\\text\{state\}\}\+\\mathcal\{L\}\_\{\\text\{out\}\}\)\+\\lambda\_\{\\text\{com\}\}\\mathcal\{L\}\_\{\\text\{com\}\},\(10\)whereℒact\\mathcal\{L\}\_\{\\text\{act\}\}is the standard imitation loss on expert actionsat∗a\_\{t\}^\{\*\}\.

The auxiliary terms ground the belief in physical dynamics:ℒstate\\mathcal\{L\}\_\{\\text\{state\}\}evaluates Eq\. \([5](https://arxiv.org/html/2610.26955#S3.E5)\) against ground\-truth latent facts to ensure hypotheses span valid world configurations;ℒout\\mathcal\{L\}\_\{\\text\{out\}\}trains the belief to anticipate impending transition feedbackp⁡\(rt\+1∣Bt,a~t\)p\(r\_\{t\+1\}\\mid B\_\{t\},\\tilde\{a\}\_\{t\}\), directly supervising the evidence scoring head in Eq\. \([6](https://arxiv.org/html/2610.26955#S3.E6)\); andℒcom\\mathcal\{L\}\_\{\\text\{com\}\}uses contrastive learning over factually corrupted negatives to enforce semantic invariance inctc\_\{t\}\.

Finally, stop\-gradients on the evidence signal and readout log\-weightslog⁡wt\\log w\_\{t\}insulate the belief weights from downstream policy gradients, preservingwtw\_\{t\}strictly as an environmental likelihood\. At inference time, all privileged state supervision is discarded, and LAMB operates solely on online observations and transition feedback\.

## 4Experiments

### 4\.1Experimental Setup

#### Environments\.

We evaluate LAMB on four interactive benchmarks, reporting SR@30 on ALFWorld\([Xi et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib28)\), task success rate on ScienceWorld\([Wang et al\., 2022](https://arxiv.org/html/2610.26955#bib.bib29)\), AgentBoard subgoal progress on BabyAI\([Ma et al\., 2024](https://arxiv.org/html/2610.26955#bib.bib30)\), and BALROG achievement progress on Crafter\([Paglieri et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib31)\)\. Environment interfaces and evaluation protocols are detailed in Appendix[F](https://arxiv.org/html/2610.26955#A6)\.

#### Baselines\.

We group baselines into three categories: \(1\)*Open\-weight API models*: Qwen3\-8B, Qwen3\-32B\([Yang et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib32)\), and GPT\-OSS\-120B\([Agarwal et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib33)\), using full interaction histories without task\-specific training; \(2\)*Proprietary API models*: GPT\-5\.6\-luna\([OpenAI, 2026](https://arxiv.org/html/2610.26955#bib.bib42)\), Claude\-Haiku\-5\.5\([Anthropic, 2026](https://arxiv.org/html/2610.26955#bib.bib44)\), and Gemini\-3\-Flash\([Google DeepMind, 2025](https://arxiv.org/html/2610.26955#bib.bib45)\), also using full interaction histories; \(3\)*Benchmark\-tuned models*: fine\-tuned Qwen3\-8B\-FH, BeliefMem\([Liao et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib17)\), AgentEvol\([Xi et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib28)\), PABU\([Jiang et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib15)\), and Agent\-BRACE\([Singh et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib16)\)\.

#### Training and inference\.

We jointly train the belief modules and LoRA adapters with a frozen Qwen3\-8B backbone; inference updates only the latent belief\. Training details and hyperparameters are provided in Appendix[D](https://arxiv.org/html/2610.26955#A4)\.

### 4\.2Main Results

RQ1: How well does LAMB perform?

Table[1](https://arxiv.org/html/2610.26955#S4.T1)shows that LAMB achieves the strongest performance across all four environments among Benchmark\-tuned Models, with the largest gains on complex, long\-horizon tasks: \+22\.0 points over PABU on ScienceWorld and \+19\.55 over AgentEvol on Crafter\. Crucially, our 8B agent decisively outperforms full\-history prompting on models up to15×15\\timeslarger \(Qwen3\-32B, GPT\-OSS\-120B\) as well as frontier proprietary APIs, demonstrating that expanding raw context cannot compensate for a structured belief\. Furthermore, the clear advantage over PABU confirms that our method retains decision\-critical details that single natural\-language summaries tend to lose\. See Appendix[H](https://arxiv.org/html/2610.26955#A8)for a qualitative case study\.

ModelALFWorldSR↑\\uparrowScienceWorldSR↑\\uparrowBabyAIProgress↑\\uparrowCrafterProgress↑\\uparrowOpen\-weight API Models zero\-shotQwen3\-8B\-FH\([Yang et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib32)\)24\.01\.562\.160\.00Qwen3\-32B\-FH\([Yang et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib32)\)57\.517\.066\.7312\.73GPT\-OSS\-120B\-FH\([Agarwal et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib33)\)39\.516\.564\.3318\.64Proprietary API ModelsGPT\-5\.6\-luna\([OpenAI, 2026](https://arxiv.org/html/2610.26955#bib.bib42)\)65\.5060\.0073\.3920\.00Claude\-Haiku\-5\.5\([Anthropic, 2026](https://arxiv.org/html/2610.26955#bib.bib44)\)78\.5057\.0081\.3746\.82Gemini\-3\-Flash\([Google DeepMind, 2025](https://arxiv.org/html/2610.26955#bib.bib45)\)97\.5072\.5086\.3250\.91Benchmark\-tuned ModelsQwen3\-8B\-FH\([Yang et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib32)\)62\.5040\.5081\.796\.82BeliefMem\([Liao et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib17)\)6\.01\.557\.467\.27AgentEvol\([Xi et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib28)\)88\.039\.589\.2039\.09PABU\([Jiang et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib15)\)86\.561\.589\.5732\.27Agent\-BRACE\([Singh et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib16)\)30\.52871\.736\.98LAMB \(Ours\)92\.5\(↑\\uparrow4\.5\)83\.5\(↑\\uparrow22\.0\)89\.60\(↑\\uparrow0\.03\)58\.64\(↑\\uparrow19\.55\)Table 1:Overall performance across interactive environments\. ALFWorld reports SR@30 \(success within 30 actions\)\. The three blocks contain open\-model full\-history \(FH\) baselines, proprietary API full\-history baselines, and fine\-tuned baselines\. Best results within the benchmark\-tuned block are inbold\.Redannotations report percentage\-point gains over the strongest benchmark\-tuned baseline in each interactive environment\.RQ2: Does LAMB reduce inference cost?

Table[2](https://arxiv.org/html/2610.26955#S4.T2)reports mean inference cost over all 200 ALFWorld tasks under a 50\-step limit, including both successful and failed episodes\. Relative to zero\-shot Qwen3\-8B\-FH, LAMB reduces cumulative input tokens per task by 98\.65% and mean episode length by 66\.27%\. Compared with PABU, it reduces output tokens per task by 57\.72%, while measured task time is 55\.90% lower\. This efficiency directly reflects our architectural design: previous methods must autoregressively decode an explicit textual belief at every decision step, whereas LAMB updates its hypotheses entirely in latent space, decoding only the final action\.

ModelAvg\.\[\-2pt\]Steps↓\\downarrowInput\[\-2pt\]Tokens↓\\downarrowOutput\[\-2pt\]Tokens↓\\downarrowTask Time\[\-2pt\]\(s/Task\)↓\\downarrowAPI\-based ModelsLlama\-3\.2\-3B\-FH\([Meta, 2024](https://arxiv.org/html/2610.26955#bib.bib36)\)48\.03332,55374149\.10Qwen3\-8B\-FH\([Yang et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib32)\)41\.24297,15465760\.20Qwen3\-32B\-FH\([Yang et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib32)\)31\.12202,09447927\.01GPT\-OSS\-120B\-FH\([Agarwal et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib33)\)37\.25244,2163,927198\.52DeepSeek\-V4\-Flash\-FH\([DeepSeek\-AI et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib38)\)18\.6481,47513,439132\.64Benchmark\-tuned ModelsBeliefMem\([Liao et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib17)\)47\.7881,85482923\.36AgentEvol\([Xi et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib28)\)15\.1519,18535112\.34PABU\([Jiang et al\., 2026](https://arxiv.org/html/2610.26955#bib.bib15)\)13\.215,03749942\.02LAMB \(Ours\)13\.91401521118\.53Table 2:Inference cost on ALFWorld\-200 with a 50\-step limit\. FH denotes full history\. The two blocks contain API\-based models and benchmark\-tuned models; the shaded row highlights LAMB\. Steps, token counts, and task times are averaged over all tasks, including failures\. Best results are in bold\.RQ3: How does policy context scale over time?

Total token volume conflates episode length with per\-step input size\. To isolate how memory representations scale over time, Figure[3](https://arxiv.org/html/2610.26955#S4.F3)tracks the policy context length across successive decision steps on the 45 ALFWorld tasks successfully solved by all four displayed methods\. Input context encompasses all tokens fed to the policy before generating an action, including task instructions, current observations, admissible actions, and history or belief representations\.

As interactions accumulate, full\-history models require increasingly long policy inputs, whereas LAMB maintains an approximately stable context\. At decision step 10, LAMB reduces mean policy input length by approximately 89% relative to Qwen3\-8B\-FH and 88% relative to DeepSeek\-V4\-Flash\-FH\. By step 20, the reduction reaches approximately 94% for both baselines, illustrating the growing savings over longer interactions\.

These dynamics demonstrate that recursive latent belief tracking eliminates cumulative context overhead while preserving decision\-critical state over extended episodes\.

Figure 3:Input context length on the 45 ALFWorld tasks successfully completed by four methods under a 50\-step budget\. Shaded bands denote±1\\pm 1SEM across active tasks at each step; the lower panel reports active task counts\. The inset provides detailed scaling for PABU and LAMB\.

## 5Ablation Studies and Analysis

VariantSR@30\(%\)↑\\uparrowSR@50\(%\)↑\\uparrowΔ50\\Delta\_\{50\}Avg\.Steps↓\\downarrowFull Model92\.595\.50\.013\.91Unstructured Latent46\.551\.5−44\.0\-44\.031\.40w/o Delta86\.090\.5−5\.0\-5\.017\.04w/o Common91\.593\.5−2\.0\-2\.015\.38w/o Belief Input80\.082\.0−13\.5\-13\.519\.13w/oℒstate\\mathcal\{L\}\_\{\\text\{state\}\}86\.592\.0−3\.5\-3\.516\.71w/oℒout\\mathcal\{L\}\_\{\\text\{out\}\}88\.594\.0−1\.5\-1\.516\.17Table 3:Component ablations on 200 ALFWorld tasks under identical training and evaluation protocols\. SR@30 and SR@50 measure success within 30 and 50 actions, respectively\.Δ50\\Delta\_\{50\}denotes the percentage\-point change in SR@50 relative to the full model; average steps include successful and failed tasks under the 50\-step limit\.### 5\.1Component Ablations

Figure 4:Redundant action rates across three environments \(%, lower is better\), with absolute counts in parentheses, evaluated over all executed actions across successful and failed tasks\.Table[3](https://arxiv.org/html/2610.26955#S5.T3)compares six retrained variants\. Replacing the structured Common–Delta latent \(4 Common \+ 8 Delta tokens\) with 12 unstructured latent tokens, while retaining the same policy prefix length and auxiliary losses, reduces SR@30 from 92\.5% to 46\.5% and SR@50 from 95\.5% to 51\.5%, with average steps increasing to 31\.40\. This gap supports the effectiveness of the proposed representation and update structure over the evaluated unstructured alternative\. Removing belief tokens from the policy input \(w/o Belief Input\) reduces SR@50 to 82\.0%\. Within the architecture, removing Delta or Common reduces SR@50 by 5\.0 or 2\.0 points, respectively, supporting their complementary roles\. What’s more, auxiliary supervision also contributes: removing state supervision reduces SR@50 to 92\.0% and increases average steps from 13\.9 to 16\.7\. Removing observation prediction reduces SR@30 from 92\.5% to 88\.5%, indicating its contribution to early task completion\.

### 5\.2Redundant Action Analysis

Evidence degradation manifests in behavioral artifacts before it lowers task success: an agent re\-inspects visited locations, searches for objects it has already found, or retries actions that recently failed\. We utilize redundant actions as an empirical probe to evaluate what the belief retains, defining redundancy by contextual task relevance instead of literal command repetition\. Specifically, an action is deemed redundant if, given past observations and environment rules, it contributes neither to advancing pending subgoals nor to resolving necessary task uncertainties\. We evaluate Qwen3\-8B zero\-shot, PABU, and LAMB across ALFWorld, ScienceWorld, and Crafter \(Figure[4](https://arxiv.org/html/2610.26955#S5.F4)\)\. On ALFWorld and Crafter, LAMB achieves the lowest redundancy rates across all methods \(16\.40% and 0\.94%\), outperforming PABU by 17\.70 and 16\.04 percentage points, and improving over zero\-shot prompting by more than 33 points\. In addition, on ScienceWorld, LAMB cuts zero\-shot redundancy from 35\.55% to 9\.45%\. While PABU records a lower numerical redundancy rate \(4\.66%\), this metric must be contextualized with task completion: PABU succeeds on only 61\.5% of ScienceWorld tasks, terminating prematurely on complex trajectories where redundant behaviors typically accumulate\. In contrast, LAMB achieves 83\.5% task success \(\+22\.0 points over PABU\) while maintaining single\-digit redundancy\. These results demonstrate that structured latent particles systematically curtail aimless interactions across complex long\-horizon environments\.

#### Fine\-grained error breakdown\.

To identify which memory failures our latent belief resolves, we categorize ALFWorld redundant actions into five mutually exclusive failure modes:*re\-observation*,*unnecessary search*,*unnecessary manipulation*,*failed retries*, and*repeated operations*\(formal definitions in Appendix[C](https://arxiv.org/html/2610.26955#A3)\)\. And Figure[5](https://arxiv.org/html/2610.26955#S5.F5)visualizes the distribution of these failure modes across executed actions\. While zero\-shot Qwen3\-8B exhibits high redundancy across all modes, PABU curbs re\-observations but suffers severely from failed retries, which account for 20\.06% of its total actions\. This reveals an inherent limitation of natural\-language summaries: textual memory typically logs that an action was executed, but struggles to retain whether it failed or why\.

In contrast, LAMB completely eliminates failed retries, unnecessary manipulations, and repeated operations \(0% across all three categories\)\. Although our re\-observation rate is slightly higher, eradicating the 20\-point retry failure mode reduces overall redundancy by 17\.70 points\. This decisive gain confirms that our observation\-conditioned particle update effectively propagates failure feedback through negative evidence signals\. Meanwhile, re\-inspecting known information remains the primary residual source of redundancy\.

Figure 5:ALFWorld redundant\-action breakdown across five mutually exclusive categories \(percentage of all executed actions, lower is better\)\. Blue annotations mark LAMB’s rates\.

### 5\.3Observation Retention Analysis

A textual belief selects observation details for its summary, whereas Common maintains a fixed\-capacity latent state through recursive updates\. We evaluate whether previously observed information remains recoverable after these updates through a diagnostic observation\-recall task\.

To isolate memory retention from policy divergence, all methods replay identical ALFWorld action–observation sequences where an agent visits a location and subsequently executes up to eight actions that never mention that location again\. At lags of 0, 1, 2, 4, and 8 steps, we train a separate linear probe for each method and lag to identify the original observation among four candidates \(25% chance baseline\)\. Lag 0 is measured immediately after the target observation has been incorporated into memory\. For this vector\-based diagnostic, textual baselines \(AgentEvol’s raw history and PABU’s summary\) are encoded using frozen Qwen3\-8B embeddings and mean\-pooled into vectors of identical dimensionality before probing, while our method is probed directly from the Common state vectors\.

Method0 steps1 step2 steps4 steps8 stepsAgentEvol78\.1366\.6762\.5052\.6042\.19PABU25\.0025\.0025\.5225\.0025\.00Ours99\.4899\.4898\.9698\.9698\.44Table 4:Observation identification \(%, higher is better\) across subsequent interaction lags on 48 ALFWorld trajectories, evaluated using vector\-based 4\-way classification probes\.Table[4](https://arxiv.org/html/2610.26955#S5.T4)shows that Common maintains high observation\-identification accuracy, decreasing only from 99\.48% to 98\.44% across the tested lags\. PABU remains near chance, while AgentEvol declines from 78\.13% to 42\.19%\. These results measure recovery through the vector probes; we further examine the textual memories using a direct text reader in Appendix[G](https://arxiv.org/html/2610.26955#A7)\. And appendices[A](https://arxiv.org/html/2610.26955#A1)and[B](https://arxiv.org/html/2610.26955#A2)further analyze the diversity of Delta particles and the effect of particle count and token budget\.

## 6Conclusion

We introduced LAMB, a recurrent latent belief framework that factorizes memory into a shared state and weighted hypotheses, updated using action\-outcome evidence\. With a fixed token belief prefix, LAMB achieves competitive performance across four interactive benchmarks and reduces token consumption on ALFWorld\. These results support structured latent beliefs as a compact and effective memory representation for long\-horizon language agents\.

We hope this work encourages future memory architectures trained to actively predict the environment instead of passively transcribing the past\.

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Appendix

## Appendix AEffect of Delta Particles

To verify that multi\-hypothesis particles avoid representational collapse, this appendix investigates whether individual Delta tokens yield distinct vocabulary\-space projections when processed by the frozen language model in isolation\.

#### Setup\.

For each evaluated state, we feed each delta token to the backbone individually at the same input position\. We omit task instructions, the Common prefix, and other Delta tokens, ensuring that output differences stem only from the particle embeddings themselves\.

Following the logit lens paradigm\([nostalgebraist, 2020](https://arxiv.org/html/2610.26955#bib.bib37)\), we decode hidden states at every layer through the frozen final normalization and unembedding matrix, recording the ten most probable vocabulary tokens\. LettingSi\(l\)S\_\{i\}^\{\(l\)\}denote this top\-10 set for particleiiat layerll, we compute the mean pairwise overlap across all\(K2\)\\binom\{K\}\{2\}particle pairs:

O\(l\)=1\(K2\)​∑i<j\|Si\(l\)∩Sj\(l\)\|10\.O^\{\(l\)\}=\\frac\{1\}\{\\binom\{K\}\{2\}\}\\sum\_\{i<j\}\\frac\{\|S\_\{i\}^\{\(l\)\}\\cap S\_\{j\}^\{\(l\)\}\|\}\{10\}\.\(11\)Here,O\(l\)=1O^\{\(l\)\}=1indicates identical top\-10 vocabulary selections across particles, whileO\(l\)=0O^\{\(l\)\}=0denotes completely disjoint sets\. Overlap values are averaged over all\(82\)=28\\binom\{8\}\{2\}=28pairs within each state, across 12 representative diagnostic states\.

Figure 6:Layer\-wise top\-10 token overlap across Delta particles, averaged over 28 particle pairs across 12 diagnostic states\. Final\-layer overlap drops to 30\.68%\.
#### Results\.

Figure[6](https://arxiv.org/html/2610.26955#A1.F6)illustrates that token overlap declines from 74\.17% at layer 1 to 30\.68% at the final layer\. While intermediate layers exhibit non\-monotonic fluctuations, particles progressively diverge across the transformer depth\. At the output layer, two particles share only about three of their ten most probable tokens on average, indicating that the Delta particles retain distinct vocabulary\-space projections and successfully mitigate representational collapse\.

## Appendix BParticle Count and Token Budget

We investigate how agent performance scales when the number of Common readout tokens and Delta particles are varied jointly\. We compare three configurations:2\+42\{\+\}4,4\+84\{\+\}8, and8\+168\{\+\}16, corresponding to total prefix budgets of 6, 12, and 24 tokens, with4\+84\{\+\}8serving as the default setting\. Each variant is trained on the raw 7,200/800 train/dev source split without multi\-view trajectory augmentations\. Models are trained independently from the Qwen3\-8B backbone and evaluated on the same 200 ALFWorld test tasks\.

CommonTokensDeltaParticlesTotalPrefixSR@30\(%\)↑\\uparrowSR@50\(%\)↑\\uparrowAvg\.Steps↓\\downarrow24687\.592\.016\.39481289\.094\.015\.788162488\.093\.516\.09Table 5:ALFWorld performance under varying Common and Delta token budgets\. All variants are trained on the unaugmented 7,200/800 source split\. The green\-shaded row denotes our default configuration\. Bold marks the best result per column\.Table[5](https://arxiv.org/html/2610.26955#A2.T5)indicates that policy performance saturates quickly with respect to the belief budget\. Among the three evaluated configurations, the default4\+84\{\+\}8budget achieves the highest SR@30 \(89\.0%\) and SR@50 \(94\.0%\) while requiring the fewest average steps \(15\.78\)\. Constraining capacity to2\+42\{\+\}4lowers SR@50 by 2\.0 points, demonstrating that an undersized particle set lacks sufficient expressive capacity to maintain competing hypotheses\. Conversely, expanding the budget to8\+168\{\+\}16yields no performance benefit \(93\.5% SR@50\) despite doubling the prefix footprint\.

## Appendix CRedundant Action Categories

An action is*redundant*if it \(i\) reveals no new information, \(ii\) causes no task\-relevant state change, and \(iii\) satisfies no precondition for a later step\. Redundancy is judged only from information available before execution: exploring unverified states is not considered redundant merely because a more direct trajectory existed in retrospect\. Under this definition, redundancy encompasses both literal string repetitions and non\-repetitive detours made avoidable by available context\.

We classify redundant actions into five mutually exclusive categories\.*Re\-observation*gathers facts the agent has already observed\.*Unnecessary search*continues exploring after the target location is established\.*Unnecessary manipulation*performs state\-changing actions that are irrelevant or destructive to the task\.*Failed retry*re\-attempts an unachieved objective without addressing the cause of its failure\.*Repeated operation*re\-applies an effect that has already been achieved and still holds\.Figure[5](https://arxiv.org/html/2610.26955#S5.F5)visualizes their rates on ALFWorld\.

#### Category boundaries\.

Three rules resolve actions that could fit two categories:\(1\)*Re\-observation vs\. unnecessary search*depends on whether the target location is known\. Searching a previously eliminated container while the target is still unknown is a re\-observation; searching other containers after the target has been found is an unnecessary search\.\(2\)*Failed retry vs\. repeated operation*depends on whether the intended effect has already been achieved\. Re\-opening a locked door without acquiring the key is a failed retry; opening a door that is already open is a repeated operation\.\(3\)*Unnecessary manipulation vs\. repeated operation*depends on whether the action disrupts progress\. Removing an item already placed in its correct receptacle is an unnecessary manipulation; reheating an object that is already heated is a repeated operation\.

#### Non\-redundant repetition\.

Repetition at the string level does not inherently indicate redundancy\. Multi\-object tasks legitimately require consecutive*pick*actions; combat, mining, or waiting sequences frequently demand successive identical operations; and re\-inspecting an environment following a state\-altering action is often necessary\. Furthermore, an initial failure followed by a retry after resolving the prerequisite condition is strictly classified as valid task execution\.

## Appendix DHyperparameter Details

Table[6](https://arxiv.org/html/2610.26955#A4.T6)summarizes the hyperparameter configuration for the main experiments\.

HyperparameterValueBackbone & AdaptationPolicy modelQwen3\-8B \(frozen\)LoRA rankrr/α\\alpha/ dropout32 / 64 / 0\.05Max sequence length2,048 \(\+12 prefix\)Backbone precisionBF16Belief RepresentationCommon / Delta state dimdd4,096Delta particle countKK8Common / Delta prefix tokens4 / 8 \(12 total\)Initialization noise scaleσ\\sigma0\.01Particle adapter bottleneck64 \(scale 1\.0\)Mixture\-gate hidden dim512Belief module precisionFP32OptimizationOptimizerAdamWAdamWβ1\\beta\_\{1\}/β2\\beta\_\{2\}0\.9 / 0\.999AdamWβ1\\beta\_\{1\}/β2\\beta\_\{2\}0\.9 / 0\.999AdamWϵ\\epsilon/ weight decay10−810^\{\-8\}/ 0\.01Learning rate / schedule2×10−52\\times 10^\{\-5\}/ linearWarmup ratio / epochs0\.03 / 3Effective batch size16 episodesGradient clipping norm1\.0BPTT truncation window1 env\. stepAuxiliary ObjectivesState loss weightλsem\\lambda\_\{\\mathrm\{sem\}\}0\.2Contrastive loss weightλcom\\lambda\_\{\\mathrm\{com\}\}0\.1Max state queries per step6Contrastive samples1 pos\. / 31 neg\.Contrastive temperature1\.0Table 6:Hyperparameter settings for the main experiments\.
## Appendix EJoint Prediction of Latent Facts

![Refer to caption](https://arxiv.org/html/2610.26955v1/fig_joint_paper.png)Figure 7:Joint vs\. per\-fact prediction \(Eq\.[5](https://arxiv.org/html/2610.26955#S3.E5)\)\.With countertop 1 known to be empty, the mug is in either cabinet 1 or sink 1\.Top:per\-hypothesis predictionspϕ​\(yq∣ct,δti\)p\_\{\\phi\}\(y\_\{q\}\\mid c\_\{t\},\\delta\_\{t\}^\{i\}\)for \(a\) uncertain and \(b\) competing hypotheses\.Bottom:the resulting joint distributions\. Both cases share the same per\-fact marginals \(0\.5\), but only \(b\) assigns zero probability to the mug being in both or neither location\. The agreed fact \(gray\) factors out as a shared multiplier\.In Section[3\.1](https://arxiv.org/html/2610.26955#S3.SS1), we formulate the prediction ofQQlatent facts as a mixture of factorized distributions \(Eq\.[5](https://arxiv.org/html/2610.26955#S3.E5)\):

p\(y1:Q∣Bt\)=∑i=1Kwti∏q=1Qpϕ\(yq∣ct,δti\)\.p\(y\_\{1:Q\}\\mid B\_\{t\}\)=\\sum\_\{i=1\}^\{K\}w\_\{t\}^\{i\}\\prod\_\{q=1\}^\{Q\}p\_\{\\phi\}\(y\_\{q\}\\mid c\_\{t\},\\delta\_\{t\}^\{i\}\)\.\(12\)We further explain why placing the product inside the mixture couples the latent facts and how this motivates the Common–Delta split\.

### E\.1Coupled Joint Prediction vs\. Independent Factorization

Predicting each fact independently instead moves the product outside the mixture:

pindep\(y1:Q∣Bt\)=∏q=1Q\[∑i=1Kwtipϕ\(yq∣ct,δti\)\]\.p\_\{\\text\{indep\}\}\(y\_\{1:Q\}\\mid B\_\{t\}\)=\\prod\_\{q=1\}^\{Q\}\\left\[\\sum\_\{i=1\}^\{K\}w\_\{t\}^\{i\}p\_\{\\phi\}\(y\_\{q\}\\mid c\_\{t\},\\delta\_\{t\}^\{i\}\)\\right\]\.\(13\)A marginal of0\.50\.5can arise from hypotheses that are each uncertain, or from hypotheses that are each certain but disagree \(Figure[7](https://arxiv.org/html/2610.26955#A5.F7)\(a,b\)\)\. Eq\. \([13](https://arxiv.org/html/2610.26955#A5.E13)\) assigns both cases the same joint distribution and thus ignores mutual exclusivity between facts\.

### E\.2Numerical Walkthrough of the Toy Example

Consider a mug that is in either the cabinet \(ycaby\_\{\\text\{cab\}\}\) or the sink \(ysinky\_\{\\text\{sink\}\}\), with two equally weighted hypotheses \(w1=w2=0\.5w^\{1\}=w^\{2\}=0\.5\)\. We write joint states as pairs\(ycab,ysink\)\(y\_\{\\text\{cab\}\},y\_\{\\text\{sink\}\}\)\.

#### Case \(a\): Uncertain hypotheses\.

Both hypotheses predictp⁡\(ycab=1\)=p⁡\(ysink=1\)=0\.5p\(y\_\{\\text\{cab\}\}\{=\}1\)=p\(y\_\{\\text\{sink\}\}\{=\}1\)=0\.5, so Eq\. \([12](https://arxiv.org/html/2610.26955#A5.E12)\) gives

p⁡\(1,1\)=p⁡\(0,0\)=0\.5​\(0\.52\)\+0\.5​\(0\.52\)=0\.25,p\(1,1\)=p\(0,0\)=0\.5\(0\.5^\{2\}\)\+0\.5\(0\.5^\{2\}\)=0\.25,\(14\)assigning 25% to the mug being in both places and 25% to it being in neither\.

#### Case \(b\): Competing hypotheses \(LAMB\)\.

Each particle commits to one configuration:h1h\_\{1\}places the mug in the cabinet\(1,0\)\(1,0\)andh2h\_\{2\}in the sink\(0,1\)\(0,1\)\. Eq\. \([12](https://arxiv.org/html/2610.26955#A5.E12)\) then gives

p⁡\(1,1\)=p⁡\(0,0\)\\displaystyle p\(1,1\)=p\(0,0\)=0\.5​\(1⋅0\)\+0\.5​\(0⋅1\)=0\.00,\\displaystyle=0\.5\(1\{\\cdot\}0\)\+0\.5\(0\{\\cdot\}1\)=\\mathbf\{0\.00\},\(15\)p⁡\(1,0\)=p⁡\(0,1\)\\displaystyle p\(1,0\)=p\(0,1\)=0\.5​\(1⋅1\)\+0\.5​\(0⋅0\)=0\.50,\\displaystyle=0\.5\(1\{\\cdot\}1\)\+0\.5\(0\{\\cdot\}0\)=\\mathbf\{0\.50\},\(16\)ruling out both impossible states\. Eq\. \([13](https://arxiv.org/html/2610.26955#A5.E13)\) would instead average the marginals to0\.50\.5and again assign0\.250\.25to each impossible state\.

### E\.3Algebraic Separation of Agreed and Contested Facts

Suppose the facts split into agreed facts𝒜\\mathcal\{A\}, on which all hypotheses agree,pϕ​\(ya∣ct,δti\)=pϕ​\(ya∣ct\)p\_\{\\phi\}\(y\_\{a\}\\mid c\_\{t\},\\delta\_\{t\}^\{i\}\)=p\_\{\\phi\}\(y\_\{a\}\\mid c\_\{t\}\), and contested facts𝒞\\mathcal\{C\}\. The agreed terms then factor out of the mixture:

p\(y1:Q∣Bt\)\\displaystyle p\(y\_\{1:Q\}\\mid B\_\{t\}\)=\[∏a∈𝒜pϕ​\(ya∣ct\)\]\\displaystyle=\\left\[\\prod\_\{a\\in\\mathcal\{A\}\}p\_\{\\phi\}\(y\_\{a\}\\mid c\_\{t\}\)\\right\]\(17\)×∑i=1Kwti​∏c∈𝒞pϕ​\(yc∣ct,δti\)\.\\displaystyle\\times\\sum\_\{i=1\}^\{K\}w\_\{t\}^\{i\}\\prod\_\{c\\in\\mathcal\{C\}\}p\_\{\\phi\}\(y\_\{c\}\\mid c\_\{t\},\\delta\_\{t\}^\{i\}\)\.The consensus becomes a shared multiplier \(gray in Figure[7](https://arxiv.org/html/2610.26955#A5.F7)\), and uncertainty is confined to the mixture over contested facts \(orange\)\. This motivates storing confirmed evidence in the common statectc\_\{t\}and reserving the deltasδti\\delta\_\{t\}^\{i\}for alternatives that still diverge\.

## Appendix FBenchmark Details

ALFWorld\([Xi et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib28)\)is a text\-based household environment in which agents locate and manipulate objects across rooms, often through multi\-stage operations such as cleaning, heating, and cooling\. The AgentGym evaluation setting includes 200 test tasks with a 50\-action limit\. Performance is measured by SR@30 and SR@50, the success rates within the first 30 and 50 actions, respectively\.

BabyAI\([Ma et al\., 2024](https://arxiv.org/html/2610.26955#bib.bib30)\)tests grounded instruction following in partially observable grid worlds, requiring agents to track spatial layout, inventory, and prerequisites such as finding a key before opening a locked door\. The AgentBoard evaluation setting includes 112 test instances across 28 levels, with a 30\-action limit\. Performance is measured by mean subgoal progress\.

Crafter\([Paglieri et al\., 2025](https://arxiv.org/html/2610.26955#bib.bib31)\)is a procedurally generated open\-world survival game with 22 achievements covering resource gathering, crafting, and combat, requiring agents to track vital stats, inventory, and prerequisite chains\. Evaluation uses the BALROG text interface and evaluator on ten held\-out seeds, with up to 2,000 steps per episode\. Performance is measured by achievement progress, the fraction of achievements unlocked averaged over seeds\.

ScienceWorld\([Wang et al\., 2022](https://arxiv.org/html/2610.26955#bib.bib29)\)is a text environment for multi\-step scientific experiments, such as measuring melting points, building circuits, and classifying organisms\. Because standard subsets are dominated by short\-horizon tasks, we evaluate on 200 longer\-horizon variations from 19 task categories of the official unseen test split, with a 30\-decision limit\. We use the official easy setting, which removes navigation and door\- or container\-opening steps but leaves the experimental procedure unchanged, so agents must still retain observations across multiple steps to complete each task\. A task succeeds only if it reaches the maximum score of 100\.

## Appendix GMemory Recovery with Direct Text Reading

While vector probes may not fully reflect the recoverability of textual memories, we use a frozen Qwen3\-8B text reader for Full history and PABU\. All methods receive the same four complete observation candidates on 24 paired branches that differ in target\-object presence and share identical subsequent updates\. Observation identification and target\-object presence derived from the selected observation are scored separately\.

MethodObs\., lag 0Obs\., lag 8Fact, lag 8Full history100\.00100\.00100\.00PABU41\.6733\.3350\.00Ours100\.00100\.00100\.00Table 7:Candidate\-assisted recovery \(%\) on 24 paired observation branches from 12 base trajectories\. Obs\.: identification among four complete observations\. Fact: target\-object presence derived from the selected observation\. The shaded row highlights LAMB; best results \(including ties\) are in bold\.Table[7](https://arxiv.org/html/2610.26955#A7.T7)shows that Full history and Common achieve 100% observation identification and fact accuracy after eight updates, whereas PABU achieves 33\.33% and 50\.00%, respectively\. At lag 8, the text reader selects the same candidate for both members of every PABU observation pair, failing to distinguish whether the target object was previously seen\. These results show that target facts remain recoverable from Common after eight recursive updates under candidate\-assisted reading, matching Full history and outperforming PABU\.

## Appendix HCase Study

Figures[8](https://arxiv.org/html/2610.26955#A8.F8)and[9](https://arxiv.org/html/2610.26955#A8.F9)present the complete 15\-step ALFWorld episode of LAMB, with detailed Common–Delta state descriptions at three key decisions\.

Recursive Latent Belief Update: A Case Study \(1/2\)Task:examine the bowl with the desklamp\.Environment:ALFWorldBackbone:Qwen3\-8BMethod:LAMBInitial observation:You are in the middle of a room\.Step 1Action:go to shelf 1Observation:On the shelf 1, you see a cd 1, a cellphone 1, a creditcard 1, a keychain 1, a mug 1, and a pen 1\.Step 2Action:go to garbagecan 1Observation:On the garbagecan 1, you see nothing\.Step 3Action:go to dresser 1Observation:On the dresser 1, you see a desklamp 1, a pen 3, a pen 2, and a pencil 3\.Step 4: Search under partial observationAction:go to drawer 1Observation:The drawer 1 is closed\.Common component• The lamp is on dresser 1\.• Drawer 1 is closed\.Delta componentsδ1\\delta\_\{1\}bowl may be in drawer 1δ2\\delta\_\{2\}bowl may be in drawer 2δ3\\delta\_\{3\}bowl may be on desk 1Next action:open drawer 1Step 5Action:open drawer 1Observation:You open the drawer 1\. The drawer 1 is open\. In it, you see nothing\.Step 6Action:go to drawer 2Observation:The drawer 2 is closed\.Step 7: Revise candidates after negative evidenceAction:open drawer 2Observation:You open the drawer 2\. The drawer 2 is open\. In it, you see nothing\.Common component• The lamp is on dresser 1\.• Drawer 1 is empty\.• Drawer 2 is empty\.Delta componentsδ1\\delta\_\{1\}bowl may be in drawer 3δ2\\delta\_\{2\}bowl may be in drawer 5δ3\\delta\_\{3\}bowl may be on desk 1Next action:go to drawer 3Figure 8:In this ALFWorld case\-study, the Common component retains the lamp location on dresser 1, while Delta hypotheses shift away from drawers confirmed empty toward uninspected locations\. Once the bowl is found on desk 1, the agent returns to the lamp and uses it to examine the bowl, ultimately completing the task successfully\. Cross\-hatched boxes denote latent belief components\.Recursive Latent Belief Update: A Case Study \(2/2\)Steps 8–15:continued search and task completion\.Step 8Action:go to drawer 3Observation:The drawer 3 is closed\.Step 9Action:open drawer 3Observation:You open the drawer 3\. The drawer 3 is open\. In it, you see a creditcard 2\.Step 10Action:go to drawer 4Observation:On the drawer 4, you see nothing\.Step 11Action:lookObservation:You are facing the drawer 5, and drawer 4\. Next to it, you see nothing\.Step 12: Locate the bowl and reuse the lamp evidenceAction:go to desk 1Observation:On the desk 1, you see a alarmclock 1, a book 1, a bowl 3, a bowl 2, a bowl 1, a keychain 3, a keychain 2, a pencil 2, and a pencil 1\.Common component• The lamp is on dresser 1\.• Drawers 1 and 2 are empty\.• Bowls are on desk 1\.Delta componentsδ1\\delta\_\{1\}bowl is on desk 1δ2\\delta\_\{2\}bowl is on desk 1δ3\\delta\_\{3\}bowl is on desk 1Next action:take bowl 3 from desk 1Step 13Action:take bowl 3 from desk 1Observation:You pick up the bowl 3 from the desk 1\.Step 14Action:go to dresser 1Observation:On the dresser 1, you see a desklamp 1, a pen 3, a pen 2, and a pencil 3\.Step 15Action:use desklamp 1Observation:You turn on the desklamp 1\.Final outcome:Success in 15 actions\.Figure 9:Continuation of Figure[8](https://arxiv.org/html/2610.26955#A8.F8)\.