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AgentCue: Statistical Summaries for Guiding Decentralized Multi-Agent Self-Organization
Guojun Zhu, Xinyou Chen, Kejian Zhang, Sanguo Zhang, Doudou Zhou
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 89%
Last extracted: 10/14/2026, 9:10:29 AM
Summary
The paper introduces AgentCue, a method for guiding decentralized multi-agent systems (MAS) by using statistical summaries derived from a latent space model. It addresses the information explosion problem in large-scale MAS where workers struggle to efficiently select sub-tasks and collaborators from a growing shared board. AgentCue models execution history as two continually updated networks (collaboration and participation) and estimates latent positions for workers and task types. These estimates form a compact summary that guides self-organization without repeated searches.
Entities (7)
Relation Signals (6)
AgentCue → uses → Latent space model
confidence 95% · We then propose AgentCue, a method built on a new latent space model that places workers and task types in a shared latent space.
AgentCue → guides → Self-organization
confidence 93% · workers use the statistical summary to choose sub-tasks and collaborators without repeated searches of the shared board.
Collaboration network → models → Worker collaboration
confidence 92% · The first network is a collaboration network, whose nodes are workers and whose edges count the collaborations between two workers
Participation network → models → Worker task participation
confidence 92% · The second network is a participation network, a worker–task-type bipartite network whose nodes are the workers and the task types
AgentCue → evaluatedon → LongBench v2
confidence 90% · Experiments on LongBench v2 Multi-Doc QA show that AgentCue achieves competitive effectiveness, efficiency, and interpretability.
AgentCue → solves → Information explosion
confidence 90% · In a large-scale MAS, many workers keep adding participation and collaboration records, and the information explodes... The resulting dilemma... is exactly where statistics excels.
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Abstract
Abstract:Multi-agent systems (MAS) combine multiple agents to solve complex tasks, and recent MAS scale to more than a thousand agents. At this scale, decentralized MAS remove the central orchestrator that assigns sub-tasks, and the agents, which we call workers, self-organize through their own choices of sub-tasks and collaborators on a shared board. As workers keep adding participation and collaboration records, however, the information explodes: workers spend more and more effort on reading the records of other workers, and self-organization loses efficiency. In the statistical view, we represent the execution history of an MAS by two continually updated networks, a collaboration network among workers and a participation network between workers and task types. We then propose AgentCue, a method built on a new latent space model that places workers and task types in a shared latent space. Each time the networks are updated, the estimated parameters of the latent space model form a compact and interpretable statistical summary, and workers use the statistical summary to choose sub-tasks and collaborators without repeated searches of the shared board. We establish the identifiability of the latent space model and non-asymptotic error bounds for the estimates under network increments, and we show that a fixed number of gradient updates after each network update suffices for the same error rate. Experiments on LongBench v2 Multi-Doc QA show that AgentCue achieves competitive effectiveness, efficiency, and interpretability.
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- Source: https://arxiv.org/abs/2610.16621v1
- Canonical: https://arxiv.org/abs/2610.16621v1
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AgentCue: Statistical Summaries for Guiding Decentralized Multi-Agent Self-Organization Guojun Zhu Affiliation: School of Mathematical Sciences, University of Chinese Academy of Sciences Affiliation: Department of Statistics and Data Science, National University of Singapore Xinyou Chen Affiliation: Department of Statistics and Data Science, Tsinghua University*Corresponding author: ddzhou@nus.edu.sg Kejian Zhang Affiliation: Department of Statistics and Data Science, National University of Singapore Sanguo Zhang Affiliation: School of Mathematical Sciences, University of Chinese Academy of Sciences Doudou Zhou Affiliation: Department of Statistics and Data Science, National University of Singapore Abstract Multi-agent systems (MAS) combine multiple agents to solve complex tasks, and recent MAS scale to more than a thousand agents. At this scale, decentralized MAS remove the central orchestrator that assigns sub-tasks, and the agents, which we call workers, self-organize through their own choices of sub-tasks and collaborators on a shared board. As workers keep adding participation and collaboration records, however, the information explodes: workers spend more and more effort on reading the records of other workers, and self-organization loses efficiency. In the statistical view, we represent the execution history of an MAS by two continually updated networks, a collaboration network among workers and a participation network between workers and task types. We then propose AgentCue, a method built on a new latent space model that places workers and task types in a shared latent space. Each time the networks are updated, the estimated parameters of the latent space model form a compact and interpretable statistical summary, and workers use the statistical summary to choose sub-tasks and collaborators without repeated searches of the shared board. We establish the identifiability of the latent space model and non-asymptotic error bounds for the estimates under network increments, and we show that a fixed number of gradient updates after each network update suffices for the same error rate. Experiments on LongBench v2 Multi-Doc QA show that AgentCue achieves competitive effectiveness, efficiency, and interpretability. Keywords: Multi-agent systems; Latent space model; Organization learning; Large language model; Self-organization. 1 Introduction Over the past few years, foundation models, that is, large pretrained models such as large language models, have advanced rapidly, and so has the software around these models, which agent developers call a harness. A harness runs a foundation model in a loop: the harness maintains the context of the foundation model, executes the tools that the foundation model calls, and feeds the results back to the foundation model. A foundation model together with a harness forms an agent, and agents are now in wide use (yao2023react). On complex tasks, however, a single agent can only explore step by step: the agent must first decompose the task and then spend a long time on sub-tasks such as coding, inference, and review, one after another (zhan2026agensh). Moreover, on a complex task, a single agent may even lose focus on local details and become highly prone to errors, and the decomposition of the task may itself be flawed (yang2026programbench, huang2026deepswe). Therefore, agent developers equip multiple agents with an auxiliary system for collaboration, and the result is a multi-agent system (MAS): several agents work on different sub-tasks at the same time, and one or more agents then integrate their results. Such collaboration greatly improves performance and saves time (zhan2026agensh), and MAS have become a mainstream research paradigm for complex tasks (wu2024autogen, li2026beyondmoreagents). To meet increasingly complex real-world tasks, MAS continue to grow in scale. Well-known MAS such as CAMEL, ChatDev, and LAMBDA use only two to seven agents for role-playing task solving, software development, and data analysis, respectively (li2023camel, qian2024chatdev, sun2026lambda), whereas the reconstruction of a complex codebase within a limited time can now call for as many as 1,024 agents (zhan2026agensh). However, most current MAS rely on a central orchestrator, that is, a main agent that decomposes the task and assigns sub-tasks to the other agents, which we call workers. The capacity of the central orchestrator to allocate sub-tasks and coordinate workers often limits the scalability of the whole MAS (zhan2026agensh). Therefore, decentralized MAS have emerged as a core research paradigm for large-scale MAS (yang2025agentnet, li2026beyondmoreagents). Decentralized MAS replace the central orchestrator with environment components such as a shared workspace for sub-tasks in progress and finished sub-tasks, shared context for reusable findings, a message interface for communication, and a task queue for open sub-tasks (mao2026delm, zhan2026agensh), and we refer to these components collectively as the shared board. Decentralized MAS raise many research questions, and the most important question asks how a decentralized MAS achieves self-organization through the shared board. Here, self-organization means that the division of labor and collaboration arise from the local choices of workers instead of central assignment, where a local choice is a choice of a sub-task or a collaborator that a worker makes on its own. Studies such as DeLM (mao2026delm), Proxifield (tambwekar2026proxifield), and recent analyses of agent systems (kim2025scaling, li2026beyondmoreagents) agree that self-organization without any external design suffers from low efficiency, redundant work, and error propagation, where an external design refers to a mechanism that agent developers add to shape the interactions among workers. Figure 1 illustrates how the difficulties of self-organization grow with the number of workers. There is therefore an urgent need for a suitable external design that helps workers choose their next sub-tasks and collaborators efficiently. Figure 1: Self-organization in a decentralized MAS. Workers read from and publish to the shared board (dashed lines) and collaborate with one another (solid lines), and groups of collaborators form and change over time through local choices. With few workers (upper panel), the shared board holds few records. With many workers (middle panel), records accumulate quickly, and each local choice (lower panel), which sub-task to claim and whom to collaborate with, requires a search through many records. Several recent studies introduce such external designs. AgentNet (yang2025agentnet) equips each agent with a router, which decides whether the agent executes a task, splits the task, or forwards the task to the agent whose capability best matches the task. Decentralized Expertise-Aware Load Serving (DEALS) (huang2026deals) lets each agent forward a task to a neighbor with fewer unfinished tasks or a higher success rate on tasks of the same category. Economy of Minds (EoM) (qi2026economy) lets agents bid in auctions for the right to act and pay one another, and the wealth of each agent then decides whether the agent persists and mutates or goes bankrupt. Proxifield (tambwekar2026proxifield) builds a sparse communication graph in every round through deterministic routing rules. Self-Organizing Agent Teams (SAT) (pappu2026sat) learns a bank of collaboration strategies offline, which specify roles, conversation phases, participation, and information flow, and applies the bank unchanged to new problems. Each design presets a mechanism of interaction, within which the agents still make their own local choices. However, AgentNet, DEALS, EoM, Proxifield, and SAT were developed and evaluated in small-scale MAS, and a new problem takes the stage in large-scale MAS. The new problem lies in the participation and collaboration records: participation records state the task types, that is, pre-specified categories of sub-tasks, in which each worker works, and collaboration records state whose published results, that is, results that workers publish on the shared board, each worker uses or reviews. In a large-scale MAS, many workers keep adding participation and collaboration records, and the information explodes: workers spend more and more effort on reading the records of other workers and choose their sub-tasks more and more slowly, so the efficiency of self-organization declines. Agensh (zhan2026agensh) shows the burden at scale: with 1,024 workers, every new entry in the shared context goes to all other workers, while a direct read of the shared context returns only the 2,000 latest entries. The more entries the workers write, the more time a worker must spend on keyword searches to trace entries beyond the 2,000 latest entries. The gain from more workers also shrinks: in the codebase reconstruction of Agensh, the final test-pass rate rises by 17.05% from 1 to 128 workers but by only 4.12% from 128 to 1,024 workers. The resulting dilemma, an information explosion without a fast and interpretable summary, is exactly where statistics excels. Picture the workers as a large crowd of ‘cowboys’: each cowboy rides a foundation model as a horse and steers the horse through a harness, while a vast list of sub-tasks and potential collaborators awaits selection. What the cowboys need is a statistical estimation procedure11 1 Coincidentally, one of the most influential estimation procedures in statistics shares its name with the cowboy’s rope: the Lasso (tibshirani1996lasso), which shrinks and selects among many candidate variables. that summarizes the information. With such a procedure, the cowboys no longer depend on repeated searches, that is, retrievals of the same unchanged records of the shared board by many workers: the cowboys quickly screen the sub-tasks and potential collaborators, select the sub-tasks and collaborators that fit them best, and reach an efficient self-organization. In the statistical view, we first characterize the data. A preprocessing step condenses the massive execution history of an MAS into two networks. The first network is a collaboration network, whose nodes are workers and whose edges count the collaborations between two workers, that is, the times one worker uses or reviews a published result of the other worker. The second network is a participation network, a worker–task-type bipartite network whose nodes are the workers and the task types and whose edges count the times each worker performs each task type. Latent space models offer a powerful tool for such network data (hoff2002latent, ma2020universal). A latent space model represents each node by a latent position, a low-dimensional vector to be estimated, and nodes with closer latent positions tend to have closer relations. A rich literature has extended latent space models to various network data. For example, he2025semiparametric analyze longitudinal networks of interaction counts with timestamps, such as bike trips between stations, and fan2024generalized jointly analyze a one-mode network, such as friendships among users, and a two-mode network, such as reviews from users to businesses. For a new form of data, which we call continually updated networks because the edge counts accumulate as the MAS runs, we propose a new latent space model, to our knowledge the first latent space model for such networks, and build our method AgentCue on the latent space model. In AgentCue, at each checkpoint, that is, each time the networks are updated, the MAS estimates the parameters of the latent space model and publishes the estimates as a statistical summary. Workers then locate suitable collaborators and task types in the latent space, and the next stage of self-organization emerges in an orderly and efficient manner. Our work makes a four-fold contribution. (i). For the problem, we identify a new problem for agents: in large-scale decentralized MAS, the information explosion of participation and collaboration records slows down self-organization, and the problem also provides a stage on which statistical methods can be built. (i). For methodology, we propose AgentCue, which combines a new latent space model for continually updated networks with the MAS: the MAS generates the networks, and the statistical summaries of the latent space model guide the MAS. (i). For theory, we establish the identifiability of the latent space model and non-asymptotic error bounds for its estimates when the counts added at each checkpoint depend on the earlier history of the MAS, and we show that a fixed number of gradient updates per checkpoint attains the statistical error rate. (iv). For experiments, we evaluate AgentCue on the Multi-Doc QA subset of LongBench v2 (bai2024longbenchv2), and the results show that AgentCue achieves competitive effectiveness, efficiency, and interpretability. 2 Background and Related Work Large language models support language understanding, reasoning, and code generation. With tools and environmental feedback, agents can plan and act (yao2023react). MAS let agents divide work and exchange information (wu2024autogen). Recent decentralized MAS extend collaboration to many workers, who choose sub-tasks and coordinate through a shared board (zhan2026agensh). Multi-Agent Collaboration. Several systems organize collaboration through role assignment and structured communication. CAMEL pairs a user agent with an assistant agent to solve tasks through role-playing (li2023camel), while AutoGen supports conversations among agents with configurable roles and tools (wu2024autogen). ChatDev and MetaGPT organize software development around specialized roles and workflows (qian2024chatdev, hong2024metagpt). AgentCoder separates programming, test design, and test execution across three agents (huang2023agentcoder). For data analysis, LAMBDA pairs a programmer with an inspector for code correction (sun2026lambda). AgentCoder and LAMBDA illustrate collaboration in single-digit teams. Our concern is how large-scale MAS can use accumulated participation and collaboration records to guide subsequent local choices. Decentralized Multi-Agent Systems. Decentralized MAS allow workers to choose sub-tasks and collaborators through shared context and communication (li2026beyondmoreagents). DeLM combines an asynchronous task queue with verified shared context, so that workers claim sub-tasks and build on earlier findings (mao2026delm). AutoScientists lets workers form teams around hypotheses and share successes and failures (gao2026autoscientists). Agensh, the closest setting to ours, scales to 1,024 workers in the reconstruction of the pandoc codebase (zhan2026agensh). The workers of Agensh claim, execute, verify, and merge work through a shared workspace; reported execution logs illustrate reviewer selection from prior experience. As the number of workers grows and participation and collaboration records accumulate, workers face more potential collaborators and more records to search. The statistical challenge is to jointly summarize participation across task types and collaboration through the use or review of published results, while accounting for overall activity. Self-Organization Learning. Several approaches adapt collaboration through experience. MorphAgent updates textual profiles (lu2024morphagent), while SAT learns strategies for roles, participation, and information flow (pappu2026sat). Auctions and accumulated payments govern the allocation of the right to act, agent mutation, and replacement in EoM (qi2026economy); AgentLance adapts bidding strategies while an allocator selects workers using bids and reputation (liu2026markets). TheBotCompany uses managers to hire, assign, and retire workers (lyu2026thebotcompany); LATTE lets workers propose task-graph changes coordinated by a lead agent (mieczkowski2026latte). In robotics, SwarmCF learns robot and task vectors from outcomes for decentralized allocation (apartsin2026swarmcf). We study the cumulative participation and collaboration records that the local choices of workers produce. Graph-Guided Multi-Agent Collaboration. Graph-based methods select participants or communication links. DyLAN uses contribution scores to select participants (liu2024dylan), G-Designer learns task-dependent communication graphs (zhang2025gdesigner), and SelfOrg rebuilds graphs from evolving answers (tastan2025selforg). AgentNet combines local routing and retrieval-based memory with adaptive connections (yang2025agentnet), and DEALS routes tasks between neighbors by weighing unfinished tasks against success rates (huang2026deals); MeshHeal uses peer verification and routing adjustments to detect degradation and reintegrate recovered agents (chen2026meshheal). DyTopo and Proxifield construct communication links through semantic matching (luo2026dytopo, tambwekar2026proxifield). In our setting, the collaboration network records the use and review of published results. Statistical Studies of Human Collaboration and Latent Space Models. Statistical models of human collaboration analyze collaboration records. Relational event models connect interaction rates to past activity, and relational hyperevent models describe group interactions such as coauthorship (butts2008relational, perry2013point, lerner2023coauthor). Latent space models describe network relationships through low-dimensional latent positions, with methods for large networks (ma2020universal), temporal variation, multiple layers, and hierarchical node categories (durante2014dynamic, loyal2023eigenmodel, pu2026tree). fan2024generalized jointly model one-mode and two-mode networks through shared latent positions and network-specific node effects, that is, node-level parameters for overall activity. We apply the construction of fan2024generalized to the collaboration network and the participation network and account for overall activity in each network. We return the statistical summaries to workers to inform their subsequent local choices. 3 Methodology 3.1 Network Data Collection We collect network data from the execution history of an Agensh-based MAS (zhan2026agensh). Workers pursue a common goal by choosing their own sub-tasks and collaborators. The shared board makes sub-tasks in progress, published results, and accumulated findings available to all workers. Workers use the shared board to identify useful work, publish their contributions, and continue from their own results or the results of other workers. Self-organization therefore leaves records of who performs each piece of work and how published results are used or reviewed. A piece of published work, which we call a work, is the basic observation unit. The record of a work, which we call a work record, identifies the work, the worker who produced the work, the publication order, and the exact versions of the published results that the work uses or reviews. The work records thus contain the participation and collaboration records of Section 1. Preprocessing removes duplicate work records and assigns task types to each work according to the published results of the work. For network analysis, we pre-specify task types as categories of sub-tasks for the common goal. For example, in cross-document reasoning, we define four task types: extract, compare, inference, and review. A work can belong to multiple task types when the work delivers several independently reusable results. We index the resulting work records by r, the n workers by i,ℓ∈[n]i, ∈[n], and the m task types by k∈[m]k∈[m], where [n]=1,…,n[n]=\1,…,n\. Each work record r produces network increments Δr∈0,1n×n A_r∈\0,1\^n× n and Δr∈0,1n×m B_r∈\0,1\^n× m. The entry (Δr)iℓ( A_r)_i is one when the work of work record r is produced by worker i and uses or reviews a published result of worker ℓ , or vice versa. We aggregate collaboration over worker–worker pairs, so Δr A_r is symmetric with zero diagonal. Each work contributes at most once to a worker–worker pair, even if the work uses several results of the same worker. The entry (Δr)ik( B_r)_ik is one when worker i produced the work of work record r and the work belongs to task type k. Multiple collaborators or task types can therefore update several entries, while each work contributes at most once to each worker–task-type pair. The network increments record participation and collaboration in published work, whether or not the work is correct. A worker acts in sessions, that is, separate activations of the worker. We place checkpoints by the number of completed sessions of the MAS instead of elapsed time, because the elapsed time of a session differs greatly across the host machines that run the MAS. At checkpoint t∈[T]t∈[T], where T is the number of checkpoints, let StS_t be the number of work records published up to checkpoint t. The cumulative network observations are: (t)=∑r=1StΔr,(t)=∑r=1StΔr. A^(t)= _r=1^S_t A_r, B^(t)= _r=1^S_t B_r. Their entries Aiℓ(t)A_i ^(t) and Bik(t)B_ik^(t) count collaboration and participation through checkpoint t. We retain all worker–worker and worker–task-type pairs, including those with zero counts. Successive checkpoints share the accumulated work records, and later local choices can depend on earlier published results. As illustrated in Figure 2, the two networks (t) A^(t) and (t) B^(t) provide complementary observations on the same workers: (t) A^(t) describes whose results support the work of other workers, and (t) B^(t) describes participation across task types. We jointly model these networks below. 3.2 Latent Space Model and Estimation We connect the two networks through shared worker latent positions, as in the one-mode and two-mode construction of fan2024generalized. Let i(t),k(t)∈ℝd z_i^(t), w_k^(t) ^d denote worker and task-type latent positions, where d is the latent dimension. The node effects αi(t) _i^(t) and βi(t) _i^(t) describe the overall collaboration and participation activity of worker i, and the node effect γk(t) _k^(t) describes the overall participation that task type k attracts. The latent space model treats the network counts as Poisson variables with cumulative means θiℓA,(t)>0 _i ^A,(t)>0 and θikB,(t)>0 _ik^B,(t)>0: Aiℓ(t) A_i ^(t) ∼Poisson(θiℓA,(t)), ( _i ^A,(t)), logθiℓA,(t) _i ^A,(t) =αi(t)+αℓ(t)+(i(t))⊤ℓ(t), = _i^(t)+ _ ^(t)+( z_i^(t)) z_ ^(t), (1) Bik(t) B_ik^(t) ∼Poisson(θikB,(t)), ( _ik^B,(t)), logθikB,(t) _ik^B,(t) =βi(t)+γk(t)+(i(t))⊤k(t). = _i^(t)+ _k^(t)+( z_i^(t)) w_k^(t). Node effects αi(t),βi(t),γk(t) _i^(t), _i^(t), _k^(t) capture overall activity, while the inner products (i(t))⊤ℓ(t)( z_i^(t)) z_ ^(t) and (i(t))⊤k(t)( z_i^(t)) w_k^(t) describe pair-specific variation beyond the node effects. Sharing i(t) z_i^(t) lets both networks inform the latent position of each worker. All parameters vary with t and describe cumulative experience at checkpoint t. Define the parameter tuple Θ(t)=((t),(t),(t),(t),(t)) ^(t)=( α^(t), β^(t), γ^(t), Z^(t), W^(t)), where (t)=(αi(t))i=1n α^(t)=( _i^(t))_i=1^n, (t)=(βi(t))i=1n β^(t)=( _i^(t))_i=1^n, and (t)=(γk(t))k=1m γ^(t)=( _k^(t))_k=1^m are column vectors, and (t)=[1(t),…,n(t)]⊤∈ℝn×d Z^(t)=[ z_1^(t),…, z_n^(t)] ^n× d and (t)=[1(t),…,m(t)]⊤∈ℝm×d W^(t)=[ w_1^(t),…, w_m^(t)] ^m× d contain the latent positions as rows. The two Poisson negative log-likelihoods ℒA(t)L_A^(t) and ℒB(t)L_B^(t) omit terms independent of the parameters: ℒA(t)(Θ(t)) _A^(t)( ^(t)) =∑1≤i<ℓ≤nθiℓA,(t)−Aiℓ(t)logθiℓA,(t), = _1≤ i< ≤ n \ _i ^A,(t)-A_i ^(t) _i ^A,(t) \, (2) ℒB(t)(Θ(t)) _B^(t)( ^(t)) =∑i=1n∑k=1mθikB,(t)−Bik(t)logθikB,(t). = _i=1^n _k=1^m \ _ik^B,(t)-B_ik^(t) _ik^B,(t) \. The joint objective ℒ(t)L^(t) is their sum: ℒ(t)(Θ(t))=ℒA(t)(Θ(t))+ℒB(t)(Θ(t)).L^(t)( ^(t))=L_A^(t)( ^(t))+L_B^(t)( ^(t)). (3) Let n 1_n and m 1_m denote vectors of ones. We minimize (3) subject to n⊤(t)=⊤ 1_n Z^(t)= 0 , m⊤(t)=⊤ 1_m W^(t)= 0 , and n⊤(t)=0 1_n β^(t)=0. These centering constraints are imposed for parameter identifiability. When (t) Z^(t) has full column rank, the node effects (t),(t),(t) α^(t), β^(t), γ^(t) are uniquely determined, and the latent positions (t),(t) Z^(t), W^(t) are identifiable up to a common orthogonal transformation. Proposition 1 in Supplementary .A.1 gives the result and proof. For numerical optimization, we follow the projected-gradient implementation of he2025semiparametric and add a small ridge penalty on (t) Z^(t) and (t) W^(t). Supplementary .B.5 gives the penalized objective, the numerical scaling, and the stopping rule. Substituting the numerical estimate Θ^(t) ^(t), that is, the output of the optimization at checkpoint t, into (1) gives fitted means θ^iℓA,(t) θ_i ^A,(t) and θ^ikB,(t) θ_ik^B,(t). At the first checkpoint, we initialize the latent positions (t) Z^(t) and (t) W^(t) with independent standard normal entries and the node effects (t),(t),(t) α^(t), β^(t), γ^(t) from the observed network counts, then apply the centering constraints. At each subsequent checkpoint t, we initialize the optimization with the preceding estimate Θ^(t−1) ^(t-1). 3.3 Summary Guiding in Self-Organization In Agensh, workers search the work records on the shared board before they choose sub-tasks and collaborators (zhan2026agensh). Each session of a worker has a fixed budget of actions on the shared board, and every read of the work records of another worker costs one action. Many workers also retrieve the same unchanged work records to compare the same sub-tasks and collaborators, a pattern that Section 1 calls repeated searches. In AgentCue, the MAS instead estimates, once per checkpoint, the parameters of the latent space model (1) from the work records on behalf of all workers and publishes the estimates ^(t),^(t),^(t),^(t),^(t) α^(t), β^(t), γ^(t), Z^(t), W^(t), which we call the statistical summary at checkpoint t, together with a summary version that identifies the network observations behind the estimates. When a session first requests the statistical summary, the worker reads the latest statistical summary with one action and keeps the same summary version for the rest of the session. Every later local comparison, that is, a calculation of fitted means in (1) from the statistical summary that the worker holds, costs no action on the shared board. Figure 2 gives an overview of AgentCue. Figure 2: Overview of AgentCue. Upper and middle bands: once per checkpoint, the MAS publishes the statistical summary, and worker i reads the statistical summary once. Lower band: the local comparison, enlarged. Bars are log fitted means split as in (4); the struck-through term cancels, and hatched segments are negative affinity terms. The latent-space map shows the projections onto the line of sight of worker i. Small squares scale with the fitted participation mean of each candidate in the task type of the inputs that the sub-task needs; the dashed box and the rings mark the shortlist; the magnifier marks a work-record search. Superscripts (t)(t) are omitted. Each fitted mean comes with an explanation, because the logarithm of each fitted mean in (1) splits into two kinds of terms: logθ^ikB,(t)=β^i(t)+γ^k(t)⏟activity+(^i(t))⊤^k(t)⏟affinity,logθ^iℓA,(t)=α^i(t)+α^ℓ(t)⏟activity+(^i(t))⊤^ℓ(t)⏟affinity. θ_ik^B,(t)= β_i^(t)+ γ_k^(t)_activity+ ( z_i^(t)) w_k^(t)_affinity, θ_i ^A,(t)= α_i^(t)+ α_ ^(t)_activity+ ( z_i^(t)) z_ ^(t)_affinity. (4) We call θ^ikB,(t) θ_ik^B,(t) the fitted participation mean of worker i in task type k and θ^iℓA,(t) θ_i ^A,(t) the fitted collaboration mean between workers i and ℓ . We call the node effects activity terms and the inner products of latent positions affinity terms. The affinity terms measure whether worker i leans toward task type k and whether workers i and ℓ collaborate more often than their activity terms imply. A large network count alone cannot tell an active worker from a close collaborator, and the split in (4) separates the two. If all affinity terms (^i(t))⊤^k(t)( z_i^(t)) w_k^(t) vanish, every worker spreads participation over the task types in the same proportions, so the affinity terms describe the division of labor. A local comparison of worker i involves two fitted means: two fitted participation means of worker i, or two fitted collaboration means between worker i and two candidates, that is, two other workers whom worker i considers as collaborators. In both ratios, the activity term of worker i cancels: θ^ikB,(t)θ^ik′B,(t) θ_ik^B,(t) θ_ik ^B,(t) =exp(γ^k(t)−γ^k′(t))⏟task typesexp(^i(t))⊤(^k(t)−^k′(t))⏟worker i, = ( γ_k^(t)- γ_k ^(t) )_task types\, \( z_i^(t)) ( w_k^(t)- w_k ^(t)) \_worker i, (5) θ^iℓA,(t)θ^iℓ′A,(t) θ_i ^A,(t) θ_i ^A,(t) =exp(α^ℓ(t)−α^ℓ′(t))⏟candidatesexp(^i(t))⊤(^ℓ(t)−^ℓ′(t))⏟pair. = ( α_ ^(t)- α_ ^(t) )_candidates\, \( z_i^(t)) ( z_ ^(t)- z_ ^(t)) \_pair. The first factors in the two rows of (5), which we call the task-type factor and the candidate factor, are common to all workers and compare how much participation task types k and k′k attract and how active candidates ℓ and ℓ′ are. The second factors, which we call the worker factor and the pair factor, belong to worker i alone. Because the centering constraint n⊤^(t)=⊤ 1_n Z^(t)= 0 makes the affinity terms of all workers with a task type sum to zero, a worker factor close to one means that worker i resembles the average worker on task types k and k′k , and a pair factor far from one separates even two equally active candidates. The affinity terms have a geometric reading. For any ∈ℝd x ^d, the inner product (^i(t))⊤( z_i^(t)) x equals the length of ^i(t) z_i^(t) times the signed length of the projection of x onto the line through the origin along ^i(t) z_i^(t), which we call the line of sight of worker i. The projections of the task-type latent positions and of the latent positions of the other workers onto the line of sight order the task types and the other workers by affinity with worker i, so one latent position serves both the sub-task choice and the collaborator choice of worker i. Under the ridge penalty of Section 3.2, a worker without work records stays near the origin, so the small fitted means of the worker come from the activity terms and reflect missing work records, not missing ability. The lower band of Figure 2 enlarges the local comparisons of worker i. To choose a sub-task, worker i compares the fitted participation means of worker i across the task types of the open sub-tasks. To choose a collaborator for a sub-task, worker i compares candidates by two fitted means, namely the fitted participation mean of each candidate in the task type of the inputs that the sub-task needs, for example extract for a sub-task of task type inference, and the fitted collaboration mean between worker i and each candidate. The local comparisons leave a shortlist of sub-tasks and candidates. Worker i then spends actions only to check the current needs, which the statistical summary cannot settle: whether a sub-task remains open, whether a candidate is occupied, and whether a published result supplies the required evidence. The statistical summary answers questions about the past, and the shared board answers questions about the present. The statistical summary proposes, and the worker chooses. When the statistical summary gives weak support, for example for a candidate without work records, when candidates tie, when a sub-task is new, or when the current needs contradict the statistical summary, worker i searches only the work records that the choice requires and states a concrete reason for the choice. Each choice leaves a Choice Record, which stores the summary version, whether the statistical summary informed the choice, and the reason for the choice. The statistical summary guides the choices, the choices create new work records, and the new work records revise the statistical summary at the next checkpoint; along the loop, the Choice Records trace every choice to a summary version and a reason. 3.4 Feedback of the Statistical Summary In the usual analysis with latent space models, the network does not respond to the estimates. In AgentCue, the statistical summary guides the choices that produce the next work records, so later estimates partly describe the effect of earlier estimates. This section analyzes the feedback and asks what the statistical summary describes in the long run. We follow one worker i through four steps of the loop in the participation network. Step 1: the statistical summary reports past shares. The statistical summary gives worker i a fitted participation share for each task type k: π^ik(t)=θ^ikB,(t)∑k′=1mθ^ik′B,(t)=expγ^k(t)+(^i(t))⊤^k(t)∑k′=1mexpγ^k′(t)+(^i(t))⊤^k′(t). π_ik^(t)= θ_ik^B,(t) _k =1^m θ_ik ^B,(t)= \ γ_k^(t)+( z_i^(t)) w_k^(t)\ _k =1^m \ γ_k ^(t)+( z_i^(t)) w_k ^(t)\. (6) The activity term β^i(t) β_i^(t) of worker i cancels in (6), as in (5). The observed participation share pik(t)=Bik(t)/∑k′=1mBik′(t)p_ik^(t)=B_ik^(t)/ _k =1^mB_ik ^(t) is the fraction of the participation counts of worker i that falls on task type k up to checkpoint t; let ^i(t) π_i^(t) and i(t) p_i^(t) denote the vectors of the two shares over task types. At a minimizer of the joint objective (3), the fitted participation means of each worker add up to the participation count of the worker, so ^i(t) π_i^(t) differs from i(t) p_i^(t) only through the low-rank latent space and the latent positions shared with the collaboration network. Step 3 therefore takes ^i(t)=i(t) π_i^(t)= p_i^(t). Step 2: workers act on the shares. How a worker uses the statistical summary depends on the foundation model and the prompt, so we adopt a stylized choice rule: worker i chooses task type k with probability ϕik(^i(t))=(π^ik(t))ωexp(uik)∑k′=1m(π^ik′(t))ωexp(uik′). _ik( π_i^(t))= ( π_ik^(t))^ω (u_ik) _k =1^m( π_ik ^(t))^ω (u_ik ). (7) The baseline preference uiku_ik is the log-propensity of worker i for task type k without the statistical summary and combines the suitability of worker i for task type k with the current needs. The reliance ω≥0ω≥ 0, common to all workers, measures how strongly workers follow the statistical summary: in the odds of two task types, the reliance raises the task-type and worker factors of (5) to the power ω. With ω=0ω=0, a worker ignores the statistical summary; with ω=1ω=1, a worker treats the fitted participation share as a prior, as in Bayes’ rule. Figure 3: Simulation of (8) in the setting of Section 3.4; each line is one run. Dotted lines: fixed point without the statistical summary (ω=0ω=0). Dashed lines: interior fixed point, stable for ω=0.5ω=0.5 and unstable for ω=2ω=2; for ω=1ω=1, no interior fixed point exists. Step 3: cumulative counts feed the choices back. We hold uiku_ik fixed, take ^i(t)=i(t) π_i^(t)= p_i^(t), and let worker i add the same number of participation counts before the first checkpoint and between consecutive checkpoints. Each checkpoint then moves the observed participation share a fraction 1/(t+1)1/(t+1) of the way toward the choice probabilities ϕi(i(t)) φ_i( p_i^(t)): i(t+1)=i(t)+1t+1ϕi(i(t))−i(t)+i(t+1), p_i^(t+1)= p_i^(t)+ 1t+1 \ φ_i( p_i^(t))- p_i^(t)+ ξ_i^(t+1) \, (8) where ϕi=(ϕi1,…,ϕim)⊤ φ_i=( _i1,…, _im) , and the noise i(t+1) ξ_i^(t+1), the share of the added participation counts minus ϕi(i(t)) φ_i( p_i^(t)), has mean zero given the history. The recursion (8) is a stochastic approximation, so the observed participation share can settle only at a stable fixed point, that is, a share i∘=ϕi(i∘) p_i = φ_i( p_i ) that attracts nearby shares (borkar2023stochastic). At any fixed point, stable or not, the statistical summary describes the very choices that the statistical summary induces, a property known as performative stability (perdomo2020performative, mendlerdunner2020stochastic). Figure 3 simulates (8) in an illustrative setting: one worker, two task types with ui1−ui2=0.4u_i1-u_i2=0.4, and ω∈0.5,1,2ω∈\0.5,1,2\. Each of 30 independent runs of (8) starts from 32 participation counts split with probability 1/21/2 and adds 32 participation counts per checkpoint for 100 checkpoints. Step 4: the fixed point in the latent space. A fixed point is interior when all shares are positive; for ω>0ω>0, each vertex, where worker i performs one task type only, is also a fixed point, and a run locks in when the run converges to a vertex. For ω≠1ω≠ 1, the interior fixed point satisfies logpik∘=uik/(1−ω)+ci p_ik =u_ik/(1-ω)+c_i, where cic_i does not depend on k. Let u¯k u_k denote the average of uiku_ik over workers, and let u~ik u_ik denote uiku_ik centered over workers and over task types. If the observed participation shares of all workers sit at the interior fixed points and the matrix (u~ik)( u_ik) has rank at most d, the centering constraints give γ^k(t)=u¯k1−ω+c(t),(^i(t))⊤^k(t)=u~ik1−ω, γ_k^(t)= u_k1-ω+c^(t), ( z_i^(t)) w_k^(t)= u_ik1-ω, (9) where c(t)c^(t) absorbs the growth of the counts and does not depend on k. For 0≤ω<10≤ω<1, the interior fixed point is stable and the vertices are unstable, and (9) magnifies the baseline preferences by 1/(1−ω)≥11/(1-ω)≥ 1 and keeps the order and the signs of u~ik u_ik: the statistical summary sharpens the division of labor. The sharper division of labor forms more slowly, since near the fixed point each checkpoint closes a fraction (1−ω)/(t+1)(1-ω)/(t+1) of the gap instead of 1/(t+1)1/(t+1); in the left panel of Figure 3, most runs remain below the interior fixed point at t=100t=100. For ω=1ω=1, a run drifts slowly toward the vertex of the task type with the largest baseline preference. For ω>1ω>1, the factor 1/(1−ω)1/(1-ω) is negative, so the interior fixed point reverses the order of the baseline preferences and is unstable: the early observed participation share decides the vertex at which a run locks in. In the right panel of Figure 3, 4 of 30 runs head to the task type with the smaller baseline preference. For ω=0ω=0, (9) recovers the baseline preferences, as in the usual analysis with latent space models. For 0≤ω<10≤ω<1, the feedback only rescales γ^k(t) γ_k^(t) and the affinity terms by 1/(1−ω)1/(1-ω), so the order and the signs of the affinity terms keep the usual interpretation. In the MAS, the current needs counteract lock-in, because the open sub-tasks of a crowded task type run out and the baseline preferences for the crowded task type fall; Section 5 examines lock-in. 4 Theory We study estimation of cumulative means and the accuracy of warm-started gradient updates, which start each checkpoint from the preceding estimate. The statistical results control error relative to the current estimation target, defined below. The optimization results control error relative to the current exact minimizer, that is, a minimizer of the joint objective (3) over the feasible parameter set in Assumption 2. Throughout, n≥3n≥ 3, m≥1m≥ 1, and d≥1d≥ 1 is fixed. For the work records indexed by r in Section 3.1, let ℱrF_r denote the information available up to and including work record r. With increments Δr,Δr A_r, B_r and the cumulative number StS_t of work records, we have (t)=∑r=1StΔr A^(t)= _r=1^S_t A_r and (t)=∑r=1StΔr B^(t)= _r=1^S_t B_r. The increments are ℱrF_r-measurable binary matrices. The finite random counts StS_t are nondecreasing in t. We define the estimation targets by summing the expected network increments conditional on the preceding history: θ⋆,iℓA,(t)=∑r=1St(Δr)iℓ∣ℱr−1,θ⋆,ikB,(t)=∑r=1St(Δr)ik∣ℱr−1. _ ,i ^A,(t)= _r=1^S_tE\( A_r)_i _r-1\, _ ,ik^B,(t)= _r=1^S_tE\( B_r)_ik _r-1\. The estimation targets can depend on the realized history and generally differ from the unconditional means (t),(t)E A^(t),E B^(t). Write the true parameter tuple as Θ⋆(t)=(⋆(t),⋆(t),⋆(t),⋆(t),⋆(t)) _ ^(t)=( α_ ^(t), β_ ^(t), γ_ ^(t), Z_ ^(t), W_ ^(t)), where logθ⋆,iℓA,(t)=α⋆,i(t)+α⋆,ℓ(t)+(⋆,i(t))⊤⋆,ℓ(t) _ ,i ^A,(t)= _ ,i^(t)+ _ , ^(t)+( z_ ,i^(t)) z_ , ^(t) and logθ⋆,ikB,(t)=β⋆,i(t)+γ⋆,k(t)+(⋆,i(t))⊤⋆,k(t) _ ,ik^B,(t)= _ ,i^(t)+ _ ,k^(t)+( z_ ,i^(t)) w_ ,k^(t). The parameters satisfy the centering constraints n⊤⋆(t)=⊤ 1_n Z_ ^(t)= 0 , m⊤⋆(t)=⊤ 1_m W_ ^(t)= 0 , and n⊤⋆(t)=0 1_n β_ ^(t)=0. Supplementary .A.2 shows that the parameters can be centered without changing the estimation targets θ⋆,iℓA,(t) _ ,i ^A,(t) and θ⋆,ikB,(t) _ ,ik^B,(t). Assumption 1. There is a constant K>0K>0, independent of n,m,Tn,m,T, such that, for every work record r, ∑i<ℓ(Δr)iℓ+∑i,k(Δr)ik≤K _i< ( A_r)_i + _i,k( B_r)_ik≤ K. Assumption 1 bounds the contribution of each work record to the network increments Δr A_r and Δr B_r. A work record may involve multiple collaborators and task types, but each work record updates at most K worker–worker and worker–task-type pairs in total. Assumption 2. There are constants Rz,Rw,κ>0R_z,R_w,κ>0, independent of n,m,Tn,m,T, such that, for every t∈[T]t∈[T], the centered true parameters satisfy maxi∥⋆,i(t)∥2≤Rz _i z_ ,i^(t) _2≤ R_z, maxk∥⋆,k(t)∥2≤Rw _k w_ ,k^(t) _2≤ R_w, and λmin(⋆(t))⊤⋆(t)/n≥κ _ \( Z_ ^(t)) Z_ ^(t)/n\≥κ. The feasible parameter set consists of tuples Θ~ with finite node effects that satisfy the same centering constraints and maxi∥~i∥2≤2Rz _i z_i _2≤ 2R_z, maxk∥~k∥2≤2Rw _k w_k _2≤ 2R_w. Assumption 2 bounds the norms of the worker and task-type latent positions ⋆,i(t) z_ ,i^(t) and ⋆,k(t) w_ ,k^(t). The lower bound on λmin(⋆(t))⊤⋆(t)/n _ \( Z_ ^(t)) Z_ ^(t)/n\ ensures that worker latent positions vary sufficiently in every latent dimension. The larger radii 2Rz2R_z and 2Rw2R_w define the radius constraints, that is, the norm bounds of the feasible parameter set, and keep the true latent positions strictly inside the radius constraints. For a feasible parameter tuple Θ~ , let θiℓA(Θ~) _i ^A( ) and θikB(Θ~) _ik^B( ) denote the cumulative means in (1) evaluated at Θ~ . We consider feasible estimates Θ^(t) ^(t) satisfying: ℒ(t)(Θ^(t))≤infΘ~feasibleℒ(t)(Θ~)+εt,0≤εt≤ε0,L^(t)( ^(t))≤ _ \ feasibleL^(t)( )+ _t, 0≤ _t≤ _0, (10) where εt _t is the optimization error allowed at checkpoint t and ε0 _0 is a fixed constant. For Θ=(,,,,) =( α, β, γ, Z, W) and Θ′=(′,…,′) =( α ,…, W ), define the normalized distance: distn,m2(Θ,Θ′) _n,m^2( , ) =∥−′∥22+∥−′∥22n+∥−′∥22m = α- α _2^2+ β- β _2^2n+ γ- γ _2^2m (11) +min⊤=d∥−′∥F2n+∥−′∥F2m, + _ Q Q= I_d \ Z- Z Q _F^2n+ W- W Q _F^2m \, where ∥⋅∥F · _F is the Frobenius norm and d I_d is the d×d× d identity matrix. The square root distn,mdist_n,m is a metric on equivalence classes under a common orthogonal transformation. The same orthogonal d×d× d matrix Q acts on , Z, W, and may differ across checkpoints. We write Θ=(,,,,) Q=( α, β, γ, Z Q, W Q). Assumption 3. There are constants 0<c0≤C0<∞0<c_0≤ C_0<∞, independent of n,m,Tn,m,T, such that, for every t∈[T]t∈[T] and all workers i and task types k: c0t≤∑ℓ≠iθ⋆,iℓA,(t)≤C0t,c0t≤∑kθ⋆,ikB,(t)≤C0t,c0nt/m≤∑iθ⋆,ikB,(t)≤C0nt/m.c_0t≤ _ ≠ i _ ,i ^A,(t)≤ C_0t, c_0t≤ _k _ ,ik^B,(t)≤ C_0t, c_0nt/m≤ _i _ ,ik^B,(t)≤ C_0nt/m. (12) Assumption 3 specifies the scale of cumulative network activity. For each worker, the sums ∑ℓ≠iθ⋆,iℓA,(t) _ ≠ i _ ,i ^A,(t) and ∑kθ⋆,ikB,(t) _k _ ,ik^B,(t) of estimation targets for collaboration and participation are of order t. The sum ∑i,kθ⋆,ikB,(t) _i,k _ ,ik^B,(t) is therefore of order ntnt, and each task type is required to have a sum of estimation targets ∑iθ⋆,ikB,(t) _i _ ,ik^B,(t) of order nt/mnt/m. In the results below and Supplementary .A, C,c,n0C,c,n_0 denote positive constants that may depend only on d,K,c0,C0,Rz,Rw,κ,ε0d,K,c_0,C_0,R_z,R_w,κ, _0. Under Assumption 2, Lemma S1 shows that Assumption 3 implies the following bounds for n≥n0n≥ n_0: ctn≤θ⋆,iℓA,(t)≤Ctn(i<ℓ),ctm≤θ⋆,ikB,(t)≤Ctm(i,k).c tn≤ _ ,i ^A,(t)≤ C tn (i< ), c tm≤ _ ,ik^B,(t)≤ C tm (i,k). (13) Assumptions 1 and 3 imply that the cumulative number of work records satisfies St≍ntS_t nt, that is, cnt≤St≤Cntcnt≤ S_t≤ Cnt. The proof of St≍ntS_t nt is given at the beginning of Supplementary .A.3. Theorem 1. Suppose Assumptions 1–3 hold and n≥n0n≥ n_0. For every δ∈(0,1)δ∈(0,1), there is an event of probability at least 1−δ1-δ on which all checkpoints with t≥C(1+m/n)log((2n+m)/δ)+loglog(e+t)t≥ C(1+m/n)\ ((2n+m)/δ)+ (e^e+t)\ admit exact minimizers, and every feasible estimate with (10) obeys: distn,m2(Θ^(t),Θ⋆(t))≤Cn+mntlog2n+mδ+loglog(e+t).dist_n,m^2( ^(t), _ ^(t))≤ C n+mnt \ 2n+mδ+ (e^e+t) \. (14) 1n2∑i<ℓlogθiℓA(Θ^(t))θ⋆,iℓA,(t)2+1nm∑i,klogθikB(Θ^(t))θ⋆,ikB,(t)2≤Cn+mntlog2n+mδ+loglog(e+t). 1n^2 _i< \ _i ^A( ^(t)) _ ,i ^A,(t) \^2+ 1nm _i,k \ _ik^B( ^(t)) _ ,ik^B,(t) \^2≤ C n+mnt \ 2n+mδ+ (e^e+t) \. (15) For a fixed a∈(0,1)a∈(0,1), let CaC_a denote a positive constant that may additionally depend on a. For any fixed T satisfying T≥Ca(1+m/n)log(2n+m)/δT≥ C_a(1+m/n) \(2n+m)/δ\, there is an event of probability at least 1−δ1-δ on which the maxima of distn,m2(Θ^(t),Θ⋆(t))dist_n,m^2( ^(t), _ ^(t)) and the left-hand side of (15) over ⌈aT⌉≤t≤T aT ≤ t≤ T are both at most Ca(n+m)log(2n+m)/δ/(nT)C_a(n+m) \(2n+m)/δ\/(nT). Theorem 1 quantifies the accuracy of the statistical summaries estimated from the cumulative networks. Bound (14) controls the average squared errors of the estimated node effects ^(t),^(t),^(t) α^(t), β^(t), γ^(t) and latent positions ^(t),^(t) Z^(t), W^(t), with the latent positions compared up to a common orthogonal transformation. Bound (15) controls the average squared errors of the corresponding log-means, that is, the logarithms of the cumulative means, in both networks. Together, Bounds (14) and (15) provide statistical guarantees for the estimated parameters and for the participation and collaboration patterns that the estimated parameters describe. Apart from logarithmic factors, both error bounds scale as (n+m)/(nt)=1/t+m/(nt)(n+m)/(nt)=1/t+m/(nt). Under Assumption 3, each worker accumulates activity of order t, while each task type accumulates participation of order nt/mnt/m. The two terms in the rate are the reciprocals of the two activity scales. When m=O(n)m=O(n), the rate is of order 1/t1/t. When m≫nm n, the term m/(nt)m/(nt) dominates, because less participation data is available for each task type. Thus, a larger number m of task types requires more work records to obtain the same error guarantee. For fixed n and m, both bounds tend to zero as t increases. Remark 1. The first conclusion of Theorem 1 holds simultaneously at all checkpoints satisfying the stated lower bound on t, which we call the information threshold, and the simultaneous control adds the loglog(e+t) (e^e+t) term. For a fixed T, the second conclusion bounds the largest error over ⌈aT⌉≤t≤T aT ≤ t≤ T without the loglog(e+t) (e^e+t) term and therefore gives a bound of order (n+m)/(nT)(n+m)/(nT), up to logarithmic factors, uniformly over ⌈aT⌉≤t≤T aT ≤ t≤ T. Let Θ^⋆(t) _ ^(t) denote an exact minimizer of ℒ(t)L^(t) over the feasible parameter set, whose existence at the checkpoints covered by Theorem 1 is established by Lemma S3. We analyze the optimization updates from a checkpoint t0t_0 at which Θ^(t0) ^(t_0) satisfies (10). The required lower bound on t0t_0 is specified in Theorem 2. For t≥2t≥ 2, define the checkpoint increments Δ(t)=(t)−(t−1) A^(t)= A^(t)- A^(t-1) and Δ(t)=(t)−(t−1) B^(t)= B^(t)- B^(t-1). The checkpoint increments aggregate the contributions of the work records added between checkpoints t−1t-1 and t. Assumption 4. There is a constant K1>0K_1>0, independent of n,m,Tn,m,T, such that, for every checkpoint t=t0+1,…,Tt=t_0+1,…,T: 1n∑i∑ℓ≠iΔAiℓ(t)+∑kΔBik(t)2+mn2∑k∑iΔBik(t)2≤K12(1+mn). 1n _i \ _ ≠ i A_i ^(t)+ _k B_ik^(t) \^\!2+ mn^2 _k \ _i B_ik^(t) \^\!2≤ K_1^2 (1+ mn ). (16) Assumption 1 bounds each work record’s contribution, whereas Assumption 4 controls the checkpoint increments Δ(t) A^(t) and Δ(t) B^(t). The two terms in (16) average squared increments over workers and task types, with the task-type term scaled by m/nm/n to reflect the count scale nt/mnt/m of each task type under Assumption 3. The factor 1+m/n1+m/n accommodates sparse integer counts: when m≥nm≥ n, one new participation per worker, each assigned to a distinct task type, with no new collaborations gives the two terms 11 and m/nm/n. Following he2025semiparametric, we prescribe parameter-specific step sizes for the theoretical analysis, scaled according to the cumulative count scales in Assumption 3. Let P denote the orthogonal projection enforcing the centering constraints: it subtracts the column means from Z and W and the mean from β, while leaving α and γ unchanged. Starting from a feasible estimate Θ^(t0) ^(t_0) satisfying (10), initialize each subsequent checkpoint at Θ^(t,0)=Θ^(t−1) ^(t,0)= ^(t-1). For a step-size constant η>0η>0 and an iteration count J, both independent of n,m,Tn,m,T, perform the following updates for j=0,…,J−1j=0,…,J-1, with all gradients evaluated at Θ^(t,j) ^(t,j): Θ^(t,j+1)=[Θ^(t,j)−ηt(∇ℒ(t),∇ℒ(t),mn∇ℒ(t),∇ℒ(t),mn∇ℒ(t))⊤]. ^(t,j+1)= P\! [ ^(t,j)- ηt ( _ αL^(t), _ βL^(t), mn _ γL^(t), _ ZL^(t), mn _ WL^(t) ) ]. (17) Remark 2. Following the numerical implementation of he2025semiparametric, we use adaptive Barzilai–Borwein step sizes in numerical optimization. Our implementation in Supplementary .B.5 uses an adaptive scalar step size, obtained from Barzilai–Borwein step sizes and Armijo backtracking, to minimize the penalized objective ( .B.1). Set Θ^(t)=Θ^(t,J) ^(t)= ^(t,J), which we call the computed estimate at checkpoint t. Theorem 2 analyzes the step sizes and fixed iteration count in (17). Theorem 2. Suppose Assumptions 1–4 hold and n≥n0n≥ n_0. For fixed δ∈(0,1)δ∈(0,1), choose a deterministic integer t0<Tt_0<T satisfying t0≥C(n+m)2log(2n+m)/δ/nt_0≥ C(n+m)^2 \(2n+m)/δ\/n, and let Θ^(t0) ^(t_0) be a feasible estimate satisfying (10). There exist η>0η>0, an integer J≥1J≥ 1, and q∈(0,1)q∈(0,1), independent of n,m,Tn,m,T, such that, with probability at least 1−δ1-δ, all iterates of (17) remain feasible and the following bounds hold simultaneously for t0≤t≤Tt_0≤ t≤ T: distn,m2(Θ^(t),Θ^⋆(t))≤Cε0nt0q2(t−t0)+Cn+mnt2,dist_n,m^2( ^(t), _ ^(t))≤ C _0nt_0q^2(t-t_0)+C n+mnt^2, (18) 0≤ℒ(t)(Θ^(t))−ℒ(t)(Θ^⋆(t))≤C(1+ε0),0 ^(t)( ^(t))-L^(t)( _ ^(t))≤ C(1+ _0), (19) where the constants may additionally depend on K1K_1. Theorem 2 shows that a fixed number J of updates per checkpoint suffices to maintain a bounded optimization error. In (18), the contribution of the initial error decays geometrically, while the term (n+m)/(nt2)(n+m)/(nt^2) accounts for changes in the exact minimizer as new work records are added. Bound (19) ensures that the computed estimates satisfy a fixed optimization-error bound throughout t0≤t≤Tt_0≤ t≤ T. Combining Bound (19) with Theorem 1 gives the following corollary, which bounds the statistical error of the computed estimates relative to the true parameters Θ⋆(t) _ ^(t). Corollary 1. Fix δ∈(0,1)δ∈(0,1) and a∈(0,1)a∈(0,1). Suppose the conditions of Theorem 2 hold and ⌈aT⌉≥t0 aT ≥ t_0. Then the estimates Θ^(t) ^(t) produced by (17), with η and J chosen as in Theorem 2, satisfy, with probability at least 1−δ1-δ: max⌈aT⌉≤t≤Tdistn,m2(Θ^(t),Θ⋆(t))≤Can+mnTlog2n+mδ. _ aT ≤ t≤ Tdist_n,m^2( ^(t), _ ^(t))≤ C_a n+mnT 2n+mδ. (20) The same bound holds for the maximum of the normalized squared log-mean error on the left-hand side of (15) over these checkpoints. Corollary 1 shows that the estimates obtained with a fixed number of updates per checkpoint attain the statistical error rate in Theorem 1. Under Assumptions 1–4 and condition (10) at t0t_0, with constants uniform along a sequence of (n,m,T,t0)(n,m,T,t_0), a sufficient condition for uniform consistency in probability over ⌈aT⌉≤t≤T aT ≤ t≤ T is (n+m)2log(2n+m)/(nt0)→0\(n+m)^2 (2n+m)\/(nt_0)→ 0 with t0≤⌈aT⌉t_0≤ aT . Supplementary .A.5 gives the derivation. Each gradient evaluation in (17) costs On(n+m)dO\n(n+m)d\ operations when all cumulative means θiℓA _i ^A and θikB _ik^B are computed directly. The J updates at each of the T−t0T-t_0 subsequent checkpoints therefore cost OJ(T−t0)n(n+m)dO\J(T-t_0)n(n+m)d\ operations in total, excluding initialization. 5 Experiments 5.1 LongBench v2 Multi-Doc QA LongBench v2 evaluates understanding and reasoning over realistic long contexts (bai2024longbenchv2). We use all 125 questions in its Multi-Doc QA subset, covering academic, financial, governmental, legal, and multi-news documents. DeLM also evaluates decentralized evidence aggregation on this subset (mao2026delm). Each question requires the combination of evidence from multiple documents, so both the division of labor and the reuse of published results become observable. We use the four task types in Section 3.1: extract for recording source-grounded evidence, compare for relating objects along specified dimensions, inference for calculations, deductions, and answer synthesis, and review for checking the exact version of another worker’s published result. A work may deliver several independently reusable results and contribute to several task types. All workers may perform any task type and choose their own sub-tasks and collaborators. We compare two methods: AgentCue, in which workers receive the statistical summary, and History only, in which workers receive no statistical summary. A replicate is one realization of the following paired design. The replicate begins with 32 warm-up sessions, which follow the History-only rules, and then copies the resulting shared board and the private memories, that is, the notes that each worker keeps across sessions, into two branches, the AgentCue branch and the History-only branch. Each branch adds 224 sessions. A trajectory consists of the shared warm-up and one branch, and we call the 256 sessions of a trajectory the effective sessions of the trajectory; a replicate runs 480 actual sessions because the two trajectories share the warm-up. The MAS uses 32 persistent workers with GPT-5.6 Sol at high reasoning effort and at most 32 concurrent sessions. Every completed session counts, including sessions without published work or a correct answer. Both methods expose the same sources, published results, work records, review outcomes, and current workload. A worker designates a published answer as the final answer to a question. The first designated answer to a question must use a relevant published result of another worker, and designation does not require a PASS verdict, that is, an approval in a review. Within these common rules, workers first prioritize uncovered questions, answer-critical gaps, and the continuation of useful existing works. Workers in the AgentCue branch additionally use the statistical summary to narrow local choices, as in Section 3.3, and search the work records when the statistical summary gives weak support or the current needs contradict the statistical summary. Each new session reads the latest statistical summary once and keeps the same summary version throughout the session. In the AgentCue branch, the MAS estimates the parameters of the latent space model with d=2d=2 at eight checkpoints, at 32,64,…,25632,64,…,256 completed effective sessions; estimation at intermediate checkpoints runs in the background while sessions continue. The three replicates, R1–R3, each evaluate the AgentCue branch followed by the History-only branch from their shared warm-up. Supplementary .B.1– .B.5 give the design, prompts, and estimation procedure. Table 1: Final results over three paired replicates (mean ± sample standard deviation), with 256 effective sessions per trajectory and 125 questions. Measure AgentCue History only Correct answers 76.0±1.076.0± 1.0 72.0±1.072.0± 1.0 Accuracy (%) 60.8±0.860.8± 0.8 57.6±0.857.6± 0.8 Accuracy compares each question’s final designated answer with the withheld benchmark key, using all 125 questions as the denominator. AgentCue achieves mean accuracy of 60.8%60.8\%, compared with 57.6%57.6\% for History only (Table 1). The paired differences are 4.04.0, 4.04.0, and 1.61.6 percentage points, with mean 3.23.2 and sample standard deviation 1.391.39. Both methods designate answers for all 125 questions in every replicate. Supplementary .B.6 reports token use and elapsed time. Figure 4: Effectiveness, efficiency, and interpretability of AgentCue. (a) Final accuracy by domain; parentheses give question counts. (b) Accuracy over the common effective-session budget, using all 125 questions. Points and lines show means across three paired replicates; error bars and shading show one sample standard deviation. (c) One recorded choice in R1 links the fitted participation means of W15, the choice of inference, the use of version 1 of the published result of W26, and the correct designated answer, where Wii denotes worker i. Figure 4 connects the three evaluation dimensions. Final mean accuracy is higher for AgentCue in four domains and equal in legal documents, with the largest differences in financial and governmental documents. The accuracy curves in panel (b) coincide at 32 effective sessions, the end of the shared warm-up. At 64 effective sessions, mean accuracy is 26.1%26.1\% for AgentCue and 26.7%26.7\% for History only; AgentCue has higher mean accuracy at the remaining six checkpoints. Accuracies at intermediate checkpoints include published results of sessions in progress. Panel (c) traces one recorded choice, detailed below. Figure 5 links all 256 sessions with eight statistical summaries for the AgentCue trajectory in R1. Workers overlap in their task types and move among extract, compare, inference, and review, so the participation of no worker locks in to a single task type, the risk that Section 3.4 analyzes. The upper panels show jointly aligned estimates; the orthogonal alignment leaves every fitted mean unchanged. Supplementary .B.7 explains the construction. Figure 5: Worker sessions and statistical summaries for the AgentCue trajectory in R1. Upper panels show eight statistical summaries jointly aligned to the statistical summary at the final checkpoint by orthogonal Procrustes, with consecutive checkpoints paired by column. Linear axes show worker latent positions; insets enlarge the four task-type latent positions. The color scale links panels and arrows to the checkpoints t1,…,t8t_1,…,t_8. Lower bars show all 256 sessions of 32 workers, mapped from timestamps to the number of completed effective sessions. Equal-height stripes indicate task types of work published in a session; gray denotes no publication. The example in Figure 4(c) concerns changing security threats to courts and public officials. W26 publishes source-grounded evidence during warm-up. W15 reads version 1 of the published result of W26 and the statistical summary at the first checkpoint, then identifies inference as the task type with the largest fitted participation mean of W15, 1.7511.751. The recorded choice of W15 also cites adequate evidence published by other workers and the absence of an answer synthesis in progress. W15 uses W26’s result to designate the correct answer B while the first sessions of both branches still run, at 32 completed effective sessions. The Choice Record of W15 and the work records link the read of the statistical summary, the local comparison, the recorded choice, the result version, and the answer, and show how a worker combines fitted participation means with current needs and available evidence. 6 Discussion As an application of statistics to agent systems, our method replaces repeated searches of the shared board in large-scale decentralized MAS with a statistical summary of participation and collaboration records. In the experiment on LongBench v2 Multi-Doc QA, the statistical summary raised mean accuracy under a common effective-session budget and made the recorded choices of workers interpretable. More broadly, our method builds a stage on which statistics acts inside agent systems: the MAS generates network data, the statistical summary guides the MAS, and other statistical methods can join the loop. We plan two extensions of our method. First, owing to the cost of foundation-model calls, our experiments use moderate-scale MAS, and a direct next step is to scale the experiments up to 1,024 workers, as in Agensh (zhan2026agensh). Second, as workers accumulate private memories, early collaboration records may no longer reflect current collaboration. Drawing on continual learning (wang2024continual) and agent memory (park2023generative), the latent space model can down-weight the early collaboration records through discounted edge counts or time-varying latent positions (durante2014dynamic). Disclosure Statement. The authors report there are no competing interests to declare. Declaration of AI Use. OpenAI’s GPT-6 Astra was used to assist with literature searches and language editing, and GPT-5.6 Sol was used as the foundation model in the experiments. Data Availability Statement. LongBench v2: https://github.com/THUDM/LongBench. Supplementary Material. The supplementary file contains the theory and details of the LongBench v2 Multi-Doc QA experiment, including worker prompts, supplementary results, numerical estimation, computational resources, and figure construction. (PDF) References Supplementary File of “AgentCue: Statistical Summaries for Guiding Decentralized Multi-Agent Self-Organization” This supplementary file is organized as follows. Section .A presents the theoretical development, including model identifiability, supporting lemmas, and proofs of the main results. Section .B provides additional details of the LongBench v2 Multi-Doc QA experiment, covering the experimental design, worker prompts, network construction, supplementary results, numerical estimation, computational resources, and figure construction. Appendix .A Theoretical Development This section provides the theoretical results for the latent space model and its estimation. Section .A.1 establishes model identifiability up to a common orthogonal transformation of the latent positions. Section .A.2 develops supporting results on centering, concentration, parameter recovery, and local curvature. Section .A.3 proves the statistical error bounds in Theorem 1 and discusses their extension to larger cumulative activity. Section .A.4 proves the optimization bounds in Theorem 2 for a fixed number of updates at each checkpoint. Section .A.5 proves Corollary 1, which gives statistical error bounds and a sufficient condition for uniform consistency of computed estimates. Constants follow the convention in Section 4. .A.1 Identifiability As in fan2024generalized, identifiability follows from equal complete linear predictor matrices. Proposition 1 (Identifiability). Fix a checkpoint t. Consider two parameter tuples Θ(t)=((t),(t),(t),(t),(t)) ^(t)=( α^(t), β^(t), γ^(t), Z^(t), W^(t)) and Θ′(t)=(′(t),′(t),′(t),′(t),′(t)) (t)=( α (t), β (t), γ (t), Z (t), W (t)), where (t),(t)∈ℝn α^(t), β^(t) ^n, (t)∈ℝm γ^(t) ^m, (t)∈ℝn×d Z^(t) ^n× d, (t)∈ℝm×d W^(t) ^m× d, and the primed parameters have the same dimensions. Suppose rank((t))=drank( Z^(t))=d and both tuples satisfy: n⊤(t)=n⊤′(t)=⊤,m⊤(t)=m⊤′(t)=⊤,n⊤(t)=n⊤′(t)=0. 1_n Z^(t)= 1_n Z (t)= 0 , 1_m W^(t)= 1_m W (t)= 0 , 1_n β^(t)= 1_n β (t)=0. If their complete linear predictor matrices are equal, that is: (t)n⊤+n((t))⊤+(t)((t))⊤=′(t)n⊤+n(′(t))⊤+′(t)(′(t))⊤, α^(t) 1_n + 1_n( α^(t)) + Z^(t)( Z^(t)) = α (t) 1_n + 1_n( α (t)) + Z (t)( Z (t)) , ( .A.1) (t)m⊤+n((t))⊤+(t)((t))⊤=′(t)m⊤+n(′(t))⊤+′(t)(′(t))⊤, β^(t) 1_m + 1_n( γ^(t)) + Z^(t)( W^(t)) = β (t) 1_m + 1_n( γ (t)) + Z (t)( W (t)) , ( .A.2) then there is an orthogonal matrix (t)∈ℝd×d Q^(t) ^d× d such that: (′(t),′(t),′(t))=((t),(t),(t)),′(t)=(t)(t),′(t)=(t)(t).( α (t), β (t), γ (t))=( α^(t), β^(t), γ^(t)), Z (t)= Z^(t) Q^(t), W (t)= W^(t) Q^(t). Proof. Right-multiplying ( .A.1) by n 1_n and using centered worker latent positions gives: n((t)−′(t))+n⊤((t)−′(t))n=.n( α^(t)- α (t))+\ 1_n ( α^(t)- α (t))\ 1_n= 0. Summing its entries yields n⊤((t)−′(t))=0 1_n ( α^(t)- α (t))=0. Back-substitution gives (t)=′(t) α^(t)= α (t). Left-multiplying ( .A.2) by n⊤ 1_n and using the centering of (t),′(t),(t),′(t) β^(t), β (t), Z^(t), Z (t) gives n((t)−′(t))⊤=⊤n( γ^(t)- γ (t)) = 0 , so (t)=′(t) γ^(t)= γ (t). Right-multiplying the same equation by m 1_m, then using this equality and the centering of the task-type latent positions, gives m((t)−′(t))=m( β^(t)- β (t))= 0. Hence (t)=′(t) β^(t)= β (t). Removing the equal node effects from ( .A.1) and ( .A.2) leaves: (t)((t))⊤ Z^(t)( Z^(t)) =′(t)(′(t))⊤, = Z (t)( Z (t)) , (t)((t))⊤ Z^(t)( W^(t)) =′(t)(′(t))⊤. = Z (t)( W (t)) . The Gram-matrix equality implies that (t) Z^(t) and ′(t) Z (t) have the same column space. Since (t) Z^(t) has full column rank, there is a matrix (t)∈ℝd×d Q^(t) ^d× d such that ′(t)=(t)(t) Z (t)= Z^(t) Q^(t). Substituting this relation into the Gram-matrix equality and multiplying by a left inverse of (t) Z^(t) and its transpose gives (t)((t))⊤=d Q^(t)( Q^(t)) = I_d, where d I_d is the identity matrix. Thus (t) Q^(t) is orthogonal. Substituting the same relation into the cross-product equality and multiplying by a left inverse of (t) Z^(t) gives ((t))⊤=(t)(′(t))⊤( W^(t)) = Q^(t)( W (t)) . Therefore ′(t)=(t)(t) W (t)= W^(t) Q^(t). ∎ .A.2 Supporting Lemmas and Proofs We first verify invariance under centering. Let ¯=n−1∑i z=n^-1 _i z_i, ¯=m−1∑k w=m^-1 _k w_k, and b¯=n−1∑i(βi+i⊤¯) b=n^-1 _i( _i+ z_i w). Set: ic=i−¯,kc=k−¯,αic=αi+i⊤¯−12∥¯∥22, z_i^c= z_i- z, w_k^c= w_k- w, _i^c= _i+ z_i z- 12 z _2^2, βic=βi+i⊤¯−b¯,γkc=γk+¯⊤k−¯⊤¯+b¯. _i^c= _i+ z_i w- b, _k^c= _k+ z w_k- z w+ b. The positions and c β^c have zero sums. Expanding (ic)⊤ℓc( z_i^c) z_ ^c and (ic)⊤kc( z_i^c) w_k^c gives: αic+αℓc+(ic)⊤ℓc=αi+αℓ+i⊤ℓ,βic+γkc+(ic)⊤kc=βi+γk+i⊤k. _i^c+ _ ^c+( z_i^c) z_ ^c= _i+ _ + z_i z_ , _i^c+ _k^c+( z_i^c) w_k^c= _i+ _k+ z_i w_k. Thus all log-means and means are preserved. Subtracting the row average increases a uniform row-norm bound by at most a factor of two. Assumption 2 is imposed on the centered positions and their spectrum. Lemma S1. Fix u>0u>0. With latent row norms bounded by 2Rz,2Rw2R_z,2R_w, if every row sum of the mean matrices (θiℓA)( _i ^A) and (θikB)( _ik^B) lies between c0uc_0u and C0uC_0u, and every column sum of (θikB)( _ik^B) lies between c0nu/mc_0nu/m and C0nu/mC_0nu/m, then for all n≥n0n≥ n_0, where n0n_0 is a sufficiently large fixed constant, cu/n≤θiℓA≤Cu/ncu/n≤ _i ^A≤ Cu/n and cu/m≤θikB≤Cu/mcu/m≤ _ik^B≤ Cu/m. Constants depend only on the radii and marginal bounds. , Z, W may be rank deficient. Proof. In the A block, with xi=eαix_i=e _i, x+=∑ixix_+= _ix_i, si=∑ℓ≠iθiℓAs_i= _ ≠ i _i ^A, bounded positions give: e−4Rz2xi(x+−xi)≤si≤e4Rz2xi(x+−xi),c0u≤si≤C0u.e^-4R_z^2x_i(x_+-x_i)≤ s_i≤ e^4R_z^2x_i(x_+-x_i), c_0u≤ s_i≤ C_0u. ( .A.3) If the largest xix_i were at least x+/2x_+/2, then: (n−1)c0u≤∑j≠isj≤e4Rz2x+(x+−xi)≤2e4Rz2xi(x+−xi)≤2e8Rz2C0u.(n-1)c_0u≤ _j≠ is_j≤ e^4R_z^2x_+(x_+-x_i)≤ 2e^4R_z^2x_i(x_+-x_i)≤ 2e^8R_z^2C_0u. Taking n0>1+2e8Rz2C0/c0n_0>1+2e^8R_z^2C_0/c_0 rules out this possibility. Hence x+/2<x+−xi≤x+x_+/2<x_+-x_i≤ x_+ for every i. Summing ( .A.3) gives x+2≍nux_+^2 nu. The individual bounds then give xix+≍ux_ix_+ u, so xi≍u/nx_i u/n. Multiplying by the bounded inner-product exponential proves the A claim. For the B block, set xi=eβix_i=e _i, yk=eγky_k=e _k, x+=∑ixix_+= _ix_i, and y+=∑kyky_+= _ky_k. Write si,ck,s+s_i,c_k,s_+ for the row, column, and total sums, and a=e−4RzRwa=e^-4R_zR_w, b=e4RzRwb=e^4R_zR_w. Bounded inner products give: axiy+≤si≤bxiy+,aykx+≤ck≤bykx+,ax+y+≤s+≤bx+y+.ax_iy_+≤ s_i≤ bx_iy_+, ay_kx_+≤ c_k≤ by_kx_+, ax_+y_+≤ s_+≤ bx_+y_+. Using xiyk=(xiy+)(ykx+)/(x+y+)x_iy_k=(x_iy_+)(y_kx_+)/(x_+y_+) and si≍us_i u, ck≍nu/mc_k nu/m, s+≍nus_+ nu yields: a2b2sicks+≤θikB≤b2a2sicks+,θikB≍u/m. a^2b^2 s_ic_ks_+≤ _ik^B≤ b^2a^2 s_ic_ks_+, _ik^B u/m. ∎ The martingales below are indexed by work record r. Checkpoint t enters through the cumulative index StS_t. Assumption 3 gives: ∑ℓ≠iθ⋆,iℓA,(t)≍t,∑kθ⋆,ikB,(t)≍t,∑iθ⋆,ikB,(t)≍nt/m. _ ≠ i _ ,i ^A,(t) t, _k _ ,ik^B,(t) t, _i _ ,ik^B,(t) nt/m. ( .A.4) The entrywise bounds in (13) follow from Lemma S1 and are used to bound curvature. Define the weighted matrix increment and its centered cumulative sum: r=[n2Δr,mΔr],q=∑r=1qr−(r∣ℱr−1). H_r= [ n2 A_r,\ m\, B_r ], M_q= _r=1^q\ H_r-E( H_r _r-1)\. ( .A.5) Then q M_q is an n×(n+m)n×(n+m) matrix martingale. Write ∥⋅∥op · _op for the operator norm. The factor 1/21/2 compensates for symmetric duplication in A, while n,m n, m scale the two blocks. At checkpoint t, the first block of St M_S_t is n/2 n/2 times the observed (t) A^(t) minus the matrix of estimation targets (θ⋆,iℓA,(t))( _ ,i ^A,(t)). The second block is the corresponding B residual scaled by m m. Lemma S2. Under Assumptions 1 and 3, for every δ∈(0,1)δ∈(0,1), the following holds simultaneously at all checkpoints under study with probability at least 1−δ1-δ: ∥St∥op2≤C(n+m)[tlog2n+mδ+loglog(e+t)+log2n+mδ+loglog(e+t)2]. M_S_t _op^2≤ C(n+m) [t \ 2n+mδ+ (e^e+t) \+ \ 2n+mδ+ (e^e+t) \^\!2 ]. ( .A.6) Whenever the information threshold in Theorem 1 holds, the same event also gives: ct≤∑ℓ≠iAiℓ(t)≤Ct,ct≤∑kBik(t)≤Ct,cntm≤∑iBik(t)≤Cntm.ct≤ _ ≠ iA_i ^(t)≤ Ct, ct≤ _kB_ik^(t)≤ Ct, c ntm≤ _iB_ik^(t)≤ C ntm. ( .A.7) For fixed T≥1T≥ 1 and a∈(0,1)a∈(0,1), the following bound holds simultaneously for ⌈aT⌉≤t≤T aT ≤ t≤ T, with probability at least 1−δ1-δ: ∥St∥op2≤C(n+m)Tlog2n+mδ+log22n+mδ. M_S_t _op^2≤ C(n+m) \T 2n+mδ+ ^2 2n+mδ \. ( .A.8) If T≥Ca(1+m/n)log(2n+m)/δT≥ C_a(1+m/n) \(2n+m)/δ\, ( .A.7) also holds on this event. Proof. We first bound individual increments and conditional variances. Assumption 1 gives at most 2K2K nonzero entries in r H_r, with: ∥r∥F2=n2∑i<ℓ(Δr)iℓ+m∑i,k(Δr)ik≤K(n+m). H_r _F^2= n2 _i< ( A_r)_i +m _i,k( B_r)_ik≤ K(n+m). Conditional expectation preserves the corresponding operator-norm bound, so centered increments have norm at most 2K(n+m)2 K(n+m). For any vector x, applying Cauchy–Schwarz to the nonzero entries in each column gives: ⊤rr⊤≤2K∑ixi2n4∑ℓ≠i(Δr)iℓ+m∑k(Δr)ik. x H_r H_r x≤ 2K _ix_i^2 \ n4 _ ≠ i( A_r)_i +m _k( B_r)_ik \. Set r=r−(r∣ℱr−1) X_r= H_r-E( H_r _r-1). Conditional centering gives: (rr⊤∣ℱr−1) ( X_r X_r _r-1) =(rr⊤∣ℱr−1)−(r∣ℱr−1)(r∣ℱr−1)⊤ =E( H_r H_r _r-1)-E( H_r _r-1)E( H_r _r-1) ⪯(rr⊤∣ℱr−1). ( H_r H_r _r-1). The same identity holds with the factors transposed. Applying the preceding column bound and its row analogue, then ( .A.4), yields: ‖∑r≤St(rr⊤∣ℱr−1)‖op \| _r≤ S_tE( X_r X_r _r-1) \|_op ≤2Kmaxin4∑ℓ≠iθ⋆,iℓA,(t)+m∑kθ⋆,ikB,(t)≤C(n+m)t, ≤ 2K _i \ n4 _ ≠ i _ ,i ^A,(t)+m _k _ ,ik^B,(t) \≤ C(n+m)t, ‖∑r≤St(r⊤r∣ℱr−1)‖op \| _r≤ S_tE( X_r X_r _r-1) \|_op ≤2Kmaxn4max∑i≠ℓθ⋆,iℓA,(t),mmax∑ikθ⋆,ikB,(t)≤Cnt. ≤ 2K \ n4 _ _i≠ _ ,i ^A,(t),\ m _k _i _ ,ik^B,(t) \≤ Cnt. For a scalar row or column contribution 0≤Yr≤K0≤ Y_r≤ K: ∑r≤StVar(Yr∣ℱr−1) _r≤ S_tVar(Y_r _r-1) ≤∑r≤St(Yr2∣ℱr−1) ≤ _r≤ S_tE(Y_r^2 _r-1) ≤K∑r≤St(Yr∣ℱr−1)≤Ct,worker row,Cnt/m,task-type column. ≤ K _r≤ S_tE(Y_r _r-1)≤ casesCt,&worker row,\\ Cnt/m,&task-type column. cases Define the matrix variance process: Vq=max‖∑r≤q(rr⊤∣ℱr−1)‖op,‖∑r≤q(r⊤r∣ℱr−1)‖op.V_q= \ \| _r≤ qE( X_r X_r _r-1) \|_op,\ \| _r≤ qE( X_r X_r _r-1) \|_op \. The rectangular matrix Freedman inequality of tropp2011 states that, if ∥r∥op≤R X_r _op≤ R almost surely, then for any u,v>0u,v>0: ℙ∃q≥1:∥q∥op≥u,Vq≤v≤(2n+m)exp−u22(v+Ru/3).P \∃ q≥ 1:\ M_q _op≥ u,\ V_q≤ v \≤(2n+m) \- u^22(v+Ru/3) \. For x>0x>0 and variance budget v, the threshold 2vx+(2R/3)x 2vx+(2R/3)x therefore has failure probability at most (2n+m)e−x(2n+m)e^-x. This maximal event covers all work-record indices within the budget and hence applies to random StS_t. The scalar Freedman inequality for both signs controls the 2n+m2n+m marginal processes of the , A, B row sums and B column sums. For the time-uniform result, partition checkpoints into 2j≤t<2j+12^j≤ t<2^j+1, j≥0j≥ 0. On interval j, the variance budgets are C(n+m)2j+1C(n+m)2^j+1, C2j+1C2^j+1, and C(n/m)2j+1C(n/m)2^j+1. Allocate total failure probability at most Cδ/(j+1)2Cδ/(j+1)^2 to its matrix and marginal events, with the allocation constant chosen so that the sum is at most δ. Within this proof, take: xj=logC(2n+m)(j+1)2δ≤Clog2n+mδ+loglog(e+t).x_j= C(2n+m)(j+1)^2δ≤ C \ 2n+mδ+ (e^e+t) \. The matrix deviation is bounded by Cn+mtxj+xjC n+m\ tx_j+x_j\. Squaring proves ( .A.6). Worker row deviations are bounded by Ctxj+xjC\ tx_j+x_j\ and task-type column deviations by C(nt/m)xj+xjC\ (nt/m)x_j+x_j\. When mint,nt/m≥Cxj \t,nt/m\≥ Cx_j, each deviation is at most a fixed fraction of the corresponding sum of estimation targets, which gives ( .A.7). The information threshold in Theorem 1 suffices. For the bound over ⌈aT⌉≤t≤T aT ≤ t≤ T, apply the same maximal inequality directly over all work-record indices with variance budgets C(n+m)TC(n+m)T, CTCT, and CnT/mCnT/m. Taking x=logC(2n+m)/δx= \C(2n+m)/δ\ gives ( .A.8) and the marginal deviations. On t≥aTt≥ aT, the corresponding sums of estimation targets are at least caTcaT for worker rows and canT/mcanT/m for task-type columns, so the stated lower bound on T suffices for ( .A.7). ∎ Lemma S3. Suppose ( .A.7) holds, mint,nt/m \t,nt/m\ exceeds a fixed constant, and n≥n0n≥ n_0. Every estimate satisfying (10) then obeys: ctn≤θ^iℓA,(t)≤Ctn,ctm≤θ^ikB,(t)≤Ctm.c tn≤ θ_i ^A,(t)≤ C tn, c tm≤ θ_ik^B,(t)≤ C tm. ( .A.9) Under these conditions, the infimum in (10) is attained at a finite feasible parameter. Proof. Let D(t)D^(t) be one observed row sum of (t) A^(t) or (t) B^(t), or one column sum of (t) B^(t), and let D^(t) D^(t) be the corresponding fitted sum. Whenever D(t)>0D^(t)>0, we claim: D(t)D^(t)D(t)−1−logD^(t)D(t)≤εt.D^(t) \ D^(t)D^(t)-1- D^(t)D^(t) \≤ _t. ( .A.10) For row i of (t) A^(t), increasing only α^i(t) α_i^(t) by h multiplies its fitted means by ehe^h and changes the joint objective by D^(t)(eh−1)−D(t)h D^(t)(e^h-1)-D^(t)h. Taking h=log(D(t)/D^(t))h= (D^(t)/ D^(t)) decreases the joint objective by D^(t)−D(t)+D(t)log(D(t)/D^(t)) D^(t)-D^(t)+D^(t) (D^(t)/ D^(t)). The perturbation is feasible, so the decrease is at most εt _t. For row i of (t) B^(t), increase β^i(t) β_i^(t) by h(1−1/n)h(1-1/n), decrease every other β^j(t) β_j^(t) by h/nh/n, and increase every γ^k(t) γ_k^(t) by h/nh/n. This preserves n⊤^(t)=0 1_n β^(t)=0 and increases only that row’s log-means by h. For column k, change only γ^k(t) γ_k^(t). These three perturbations prove ( .A.10). Since infu>0,u∉[1/2,2](u−1−logu)>1/10 _u>0,\,u∉[1/2,2](u-1- u)>1/10, ( .A.10) and D(t)≥10ε0D^(t)≥ 10 _0 give D(t)/2≤D^(t)≤2D(t)D^(t)/2≤ D^(t)≤ 2D^(t). Lemma S1 then yields ( .A.9) and eα^i(t)≍t/ne α_i^(t) t/n, without requiring full-rank fitted positions or positive individual counts. For existence, the joint objective is bounded below and finite at the true parameter, so choose a feasible minimizing sequence. Its tail has optimization error at most max1,ε0 \1, _0\ and satisfies the preceding bounds. For each tail element Θ~(t) ^(t), centering gives: γ~k(t)=1n∑ilogθikB(Θ~(t)),β~i(t)=1m∑klogθikB(Θ~(t))−1nm∑j,klogθjkB(Θ~(t)), γ_k^(t)= 1n _i _ik^B( ^(t)), β_i^(t)= 1m _k _ik^B( ^(t))- 1nm _j,k _jk^B( ^(t)), maxi|α~i(t)−12log(t/n)|+maxi|β~i(t)|≤C,maxk|γ~k(t)−log(t/m)|≤C. _i | α_i^(t)- 12 (t/n) |+ _i| β_i^(t)|≤ C, _k | γ_k^(t)- (t/m) |≤ C. At fixed t, these bounds place the tail in a compact feasible set. Continuity gives a subsequence attaining the infimum. ∎ For parameter recovery, compare Θ=(,,,,) =( α, β, γ, Z, W) and Θ′=(′,′,′,′,′) =( α , β , γ , Z , W ). Both are centered and satisfy the radius constraints, and λmin(′)⊤′/n≥κ _ \( Z ) Z /n\≥κ. Define the differences of the complete linear predictor matrices: A D_A =(−′)n⊤+n(−′)⊤+⊤−′(′)⊤, =( α- α ) 1_n + 1_n( α- α ) + Z Z - Z ( Z ) , ( .A.11) B D_B =(−′)m⊤+n(−′)⊤+⊤−′(′)⊤. =( β- β ) 1_m + 1_n( γ- γ ) + Z W - Z ( W ) . The diagonal of A D_A is used for algebraic completion. The joint objective includes only the entries of A with i<ℓi< and all entries of B, which we call observed entries. For any ,∈ℝn×d U, X ^n× d with X of full column rank and smallest singular value σd() _d( X), tu2016 give: min⊤=d∥−∥F2≤∥⊤−⊤∥F22(2−1)σd2(). _ Q Q= I_d U- X Q _F^2≤ U U - X X _F^22( 2-1) _d^2( X). ( .A.12) Lemma S4. For A D_A in ( .A.11), we have: ∑i(A)ii2≤2n+8Rz2(2−1)nκ∥A∥F2. _i( D_A)_i^2≤ \ 2n+ 8R_z^2( 2-1)nκ \ D_A _F^2. ( .A.13) Consequently, for all n≥n0n≥ n_0 with a sufficiently large fixed n0n_0: ∥A∥F2≤4∑i<ℓ(A)iℓ2. D_A _F^2≤ 4 _i< ( D_A)_i ^2. ( .A.14) The two parameter tuples also satisfy distn,m2(Θ,Θ′)≤C∥A∥F2/n2+∥B∥F2/(nm)dist_n,m^2( , )≤ C\ D_A _F^2/n^2+ D_B _F^2/(nm)\, where C depends only on Rw,κR_w,κ. Together with ( .A.14), this implies that, for n≥n0n≥ n_0, equal means at observed entries identify the parameters up to a common orthogonal transformation. Proof. Set =−′ u= α- α and =⊤−′(′)⊤ G= Z Z - Z ( Z ) . Centering gives n= G 1_n= 0, so: ∥A∥F2=2n∥22+2(n⊤)2+∥F2. D_A _F^2=2n u _2^2+2( 1_n u)^2+ G _F^2. ( .A.15) By ( .A.12), choose Q such that ∥−′∥F≤∥F/2(2−1)nκ Z- Z Q _F≤ G _F/ 2( 2-1)nκ. Applying |∥i∥22−∥i′∥22|≤4Rz∥i−⊤i′∥2| z_i _2^2- z_i _2^2|≤ 4R_z z_i- Q z_i _2 rowwise gives: ∑i(A)ii21/2≤2∥2+4Rz∥F2(2−1)nκ≤2n+8Rz2(2−1)nκ1/2(2n∥22+∥F2)1/2. \ _i( D_A)_i^2 \^1/2≤ 2 u _2+ 4R_z G _F 2( 2-1)nκ≤ \ 2n+ 8R_z^2( 2-1)nκ \^1/2(2n u _2^2+ G _F^2)^1/2. Equation ( .A.15) proves ( .A.13). Choose a fixed n0n_0 such that the braced coefficient is at most 1/21/2. Subtracting the diagonal contribution from the complete norm proves ( .A.14). Equations ( .A.15) and ( .A.12) allow the same Q to satisfy: ∥−′∥22n≤∥A∥F22n2,∥−′∥F2n≤∥A∥F22(2−1)n2κ. α- α _2^2n≤ D_A _F^22n^2, Z- Z Q _F^2n≤ D_A _F^22( 2-1)n^2κ. The three B-block components are mutually orthogonal, yielding: ∥B∥F2=m∥−′∥22+n∥−′∥22+∥⊤−′(′)⊤∥F2. D_B _F^2=m β- β _2^2+n γ- γ _2^2+ Z W - Z ( W ) _F^2. The identity ′(−′)⊤=⊤−′(′)⊤−(−′)⊤ Z Q( W- W Q) = Z W - Z ( W ) -( Z- Z Q) W , together with σd2(′)≥nκ _d^2( Z Q)≥ nκ and ∥op2≤4mRw2 W _op^2≤ 4mR_w^2, gives: ∥−′∥F2m≤2∥⊤−′(′)⊤∥F2nmκ+8Rw2nκ∥−′∥F2. W- W Q _F^2m≤ 2 Z W - Z ( W ) _F^2nmκ+ 8R_w^2nκ Z- Z Q _F^2. Summing gives dimension-independent coefficients 1/2+1/2(2−1)κ+4Rw2/(2−1)κ21/2+1/\2( 2-1)κ\+4R_w^2/\( 2-1)κ^2\ for the normalized A D_A term and 1+2/κ1+2/κ for the normalized B D_B term. ∎ For small n, identifiability can fail. For example, take n=3,d=1,=n=3,d=1, W= 0, =(−1,0,1)⊤ Z=(-1,0,1) , = α= 0, and ′=/2 Z = Z/2, ′=(−3/8,3/8,−3/8)⊤ α =(-3/8,3/8,-3/8) . The two representations give identical off-diagonal A means. With the same , β, γ, they also give identical B means, but Z and ′ Z are not related by an orthogonal transformation. The next deterministic bound serves estimation and comparison of successive objectives. Lemma S5. Fix nonnegative observations , A, B, with A symmetric and zero-diagonal. For u>0u>0, suppose their row and column sums satisfy ( .A.7) with t replaced by u, and minu,nu/m \u,nu/m\ exceeds a fixed constant. Let the feasible reference Θ′ be centered, satisfy the radius constraints and λmin(′)⊤′/n≥κ _ \( Z ) Z /n\≥κ, and have means between cu/n,Cu/ncu/n,Cu/n for A and cu/m,Cu/mcu/m,Cu/m for B. Let Θ be feasible with global optimization error at most ε≤ε0 ≤ _0, where ε0 _0 is fixed. Define the mean matrices by (A(Θ′))iℓ=θiℓA(Θ′)( _A( ))_i = _i ^A( ) for i≠ℓi≠ , (A(Θ′))ii=0( _A( ))_i=0, and (B(Θ′))ik=θikB(Θ′)( _B( ))_ik= _ik^B( ), and set: =[n2−A(Θ′),m−B(Θ′)]. M= [ n2\ A- _A( )\,\ m\ B- _B( )\ ]. ( .A.16) For n≥n0n≥ n_0: distn,m2(Θ^,Θ′)≤C∥op2nu2+Cεnu.dist_n,m^2( , )≤ C M _op^2nu^2+C nu. ( .A.17) The normalized squared log-mean error obeys the same bound. Proof. The feasible marginal perturbations in Lemma S3 apply to these observations. Together with Lemma S1, they give θ^iℓA≍u/n θ_i ^A u/n and θ^ikB≍u/m θ_ik^B u/m. Objective difference. In ( .A.11), take Θ=Θ = and use the current reference Θ′ . At observed entries, (A)iℓ=logθ^iℓA/θiℓA(Θ′)( D_A)_i = \ θ_i ^A/ _i ^A( )\, with the analogous definition for B D_B. At a given entry, let Y be the observed count, θ^,θ′ θ,θ the fitted and reference means, and D=log(θ^/θ′)D= ( θ/θ ). Then: θ^−Ylogθ^−(θ′−Ylogθ′)=θ′(eD−1−D)−(Y−θ′)D. θ-Y θ-(θ -Y θ )=θ (e^D-1-D)-(Y-θ )D. Summing, using feasibility of the reference and the optimization error bound, yields: ∑i<ℓθiℓA(Θ′)e(A)iℓ−1−(A)iℓ+∑i,kθikB(Θ′)e(B)ik−1−(B)ik _i< _i ^A( )\e^( D_A)_i -1-( D_A)_i \+ _i,k _ik^B( )\e^( D_B)_ik-1-( D_B)_ik\ ( .A.18) ≤∑i<ℓAiℓ−θiℓA(Θ′)(A)iℓ+∑i,kBik−θikB(Θ′)(B)ik+ε. ≤ _i< \A_i - _i ^A( )\( D_A)_i + _i,k\B_ik- _ik^B( )\( D_B)_ik+ . Set =[A/n,B/m] F=[ D_A/ n,\ D_B/ m]. The noise sums in ( .A.18) equal ⟨,⟩=tr(⊤) M, F =tr( M F): the zero-diagonal A block of M has factor 1/21/2 against double counting. Taylor’s formula gives: θ′(eD−1−D)=D2∫01(1−v)(θ′)1−vθ^vv≥12minθ′,θ^D2.θ (e^D-1-D)=D^2 _0^1(1-v)(θ )^1-v θ^\,v\,dv≥ 12 \θ , θ\D^2. The reference and fitted mean bounds, followed by ( .A.14), therefore yield: cu∥F2≤cu1n∑i<ℓ(A)iℓ2+1m∥B∥F2≤⟨,⟩+ε,cu F _F^2≤ cu \ 1n _i< ( D_A)_i ^2+ 1m D_B _F^2 \≤ M, F + , where the first constant is decreased if needed. Low-rank noise bound. Equation ( .A.11) shows that the column space of F is contained in the span of ^,′,^−′,^−′,n Z, Z , α- α , β- β , 1_n. Thus rank()≤2d+3rank( F)≤ 2d+3. For a singular value decomposition =∑jσjjj⊤ F= _j _j a_j b_j : |⟨,⟩|≤∑jσj|j⊤j|≤∥op∑jσj≤2d+3∥op∥F.| M, F |≤ _j _j| a_j M b_j|≤ M _op _j _j≤ 2d+3 M _op F _F. The last step uses Cauchy–Schwarz and fixed d. Young’s inequality now gives: cu∥F2≤C∥op∥F+ε≤cu2∥F2+Cu∥op2+ε.cu F _F^2≤ C M _op F _F+ ≤ cu2 F _F^2+ Cu M _op^2+ . Thus ∥F2≤C∥op2/u2+Cε/u F _F^2≤ C M _op^2/u^2+C /u. Lemma S4 gives ( .A.17). The normalized log-mean error is at most ∥F2/n F _F^2/n. ∎ For local optimization, write a centered perturbation as =(,,,,) h=( a, b, c, U, V), with rows i⊤,k⊤ u_i , v_k in , U, V, and n⊤=0 1_n b=0, n⊤=⊤ 1_n U= 0 , m⊤=⊤ 1_m V= 0 . Use the norm: ∥n,m2=∥22+∥22+∥F2n+∥22+∥F2m. h _n,m^2= a _2^2+ b _2^2+ U _F^2n+ c _2^2+ V _F^2m. ( .A.19) The inner product ⟨⋅,⋅⟩n,m ·,· _n,m is the polarization of the norm ∥⋅∥n,m · _n,m on perturbations, and Equation (11) minimizes the discrepancy over common orthogonal transformations. Lemma S6. The minimum in (11) is attained, and distn,mdist_n,m satisfies the triangle inequality. For two tuples Θ,Θ′ , , let Q optimally align Θ′ to Θ and set Θ0=Θ′ _0= Q. For =Θ−Θ0 h= - _0, we have ∥n,m=distn,m(Θ,Θ0) h _n,m=dist_n,m( , _0) and: (0)⊤/n+(0)⊤/mis symmetric.( Z_0) U/n+( W_0) V/m symmetric. ( .A.20) Equivalently, h is orthogonal to every tangent direction (,,,0,0)( 0, 0, 0, Z_0 , W_0 ) of common orthogonal transformations, where ⊤=− =- . Proof. Compactness of the orthogonal group and continuity give an optimal orthogonal matrix. For three tuples Θ1,Θ2,Θ3 _1, _2, _3, let 12 Q_12 align Θ2 _2 to Θ1 _1 and 23 Q_23 align Θ3 _3 to Θ2 _2. Invariance of ( .A.19) gives: distn,m(Θ1,Θ3) _n,m( _1, _3) ≤∥Θ1−Θ32312∥n,m ≤ _1- _3 Q_23 Q_12 _n,m ≤∥Θ1−Θ212∥n,m+∥(Θ2−Θ323)12∥n,m ≤ _1- _2 Q_12 _n,m+ ( _2- _3 Q_23) Q_12 _n,m =distn,m(Θ1,Θ2)+distn,m(Θ2,Θ3). =dist_n,m( _1, _2)+dist_n,m( _2, _3). For ⊤=− =- , optimal alignment implies: 0=ds∥Θ−Θ0exp(s)∥n,m2|s=0=−2tr(0⊤n+0⊤m)⊤.0= . dds - _0 (s ) _n,m^2 |_s=0=-2tr\! \ ( Z_0 Un+ W_0 Vm ) \. Orthogonality to every skew-symmetric matrix is equivalent to symmetry, proving ( .A.20). ∎ Lemma S7. Fix a centered reference Θ0 _0 with row norms bounded by 2Rz,2Rw2R_z,2R_w and λmin(0⊤0/n)≥κ1>0 _ ( Z_0 Z_0/n)≥ _1>0, where κ1 _1 is fixed. For a centered perturbation =(,,,,) h=( a, b, c, U, V), define the first-order log-mean changes: jA,iℓ()=ai+aℓ+i⊤0,ℓ+0,i⊤ℓ,jB,ik()=bi+ck+i⊤0,k+0,i⊤k.j_A,i ( h)=a_i+a_ + u_i z_0, + z_0,i u_ , j_B,ik( h)=b_i+c_k+ u_i w_0,k+ z_0,i v_k. ( .A.21) Every such h satisfies: 1n2∑i<ℓjA,iℓ()2+1nm∑i,kjB,ik()2≤C∥n,m2. 1n^2 _i< j_A,i ( h)^2+ 1nm _i,kj_B,ik( h)^2≤ C h _n,m^2. ( .A.22) If h also satisfies ( .A.20), then for n≥n0n≥ n_0 with a sufficiently large fixed n0n_0: 1n2∑i<ℓjA,iℓ()2+1nm∑i,kjB,ik()2≥c∥n,m2. 1n^2 _i< j_A,i ( h)^2+ 1nm _i,kj_B,ik( h)^2≥ c h _n,m^2. ( .A.23) Proof. Upper bound. Applying (x1+⋯+x4)2≤4∑jxj2(x_1+·s+x_4)^2≤ 4 _jx_j^2 and counting each worker at most n−1n-1 times gives: ∑i<ℓjA,iℓ()2 _i< j_A,i ( h)^2 ≤4(n−1)∥22+4Rz2∥F2, ≤ 4(n-1)\ a _2^2+4R_z^2 U _F^2\, ∑i,kjB,ik()2 _i,kj_B,ik( h)^2 ≤4m∥22+n∥22+4mRw2∥F2+4nRz2∥F2. ≤ 4\m b _2^2+n c _2^2+4mR_w^2 U _F^2+4nR_z^2 V _F^2\. Dividing by n2,nmn^2,nm and adding proves ( .A.22). First-order limit of the distance. Consider Θ0+ε _0+ h as ε→0 → 0. We first establish: limε→0distn,m2(Θ0+ε,Θ0)ε2=min⊤=−∥−(,,,0,0)∥n,m2. _ → 0 dist_n,m^2( _0+ h, _0) ^2= _ =- h-( 0, 0, 0, Z_0 , W_0 ) _n,m^2. ( .A.24) The choice =d Q= I_d bounds the minimum distance by |ε|∥n,m| | h _n,m. For optimal ε Q_ , the triangle inequality gives: ∥0(d−ε)∥Fn≤|ε|∥Fn+distn,m(Θ0+ε,Θ0)≤2|ε|∥n,m. Z_0( I_d- Q_ ) _F n≤ | | U _F n+dist_n,m( _0+ h, _0)≤ 2| | h _n,m. The spectral bound gives ∥ε−d∥F≤2|ε|∥n,m/κ1 Q_ - I_d _F≤ 2| | h _n,m/ _1. Hence ε=(ε−d)/ε _ =( Q_ - I_d)/ is bounded, and ε+ε⊤=−εε⊤ε _ + _ =- _ _ . Every subsequential limit is skew-symmetric, so substitution into the distance gives: lim infε→0distn,m2(Θ0+ε,Θ0)ε2≥min⊤=−∥−(,,,0,0)∥n,m2. _ → 0 dist_n,m^2( _0+ h, _0) ^2≥ _ =- h-( 0, 0, 0, Z_0 , W_0 ) _n,m^2. Conversely, each skew-symmetric gives the rotation exp(ε)=d+ε+O(ε2) ( )= I_d+ +O( ^2) and the matching upper bound. Full column rank of 0 Z_0 makes the quadratic coercive in , so the minimum exists. For h satisfying ( .A.20), = = 0 attains the minimum ∥n,m2 h _n,m^2. Lower bound. Apply Lemma S4 and ( .A.14) to Θ0+ε _0+ h and Θ0 _0. For any fixed h and sufficiently small |ε|| |, the row bounds can be enlarged to 3Rz,3Rw3R_z,3R_w, with constants independent of ,n,m h,n,m. The observed log-mean differences are: (A)iℓ=εjA,iℓ()+ε2i⊤ℓ,(B)ik=εjB,ik()+ε2i⊤k.( D_A)_i = j_A,i ( h)+ ^2 u_i u_ , ( D_B)_ik= j_B,ik( h)+ ^2 u_i v_k. For h satisfying ( .A.20), division by ε2 ^2 and ( .A.24) give: ∥n,m2 h _n,m^2 =limε→0distn,m2(Θ0+ε,Θ0)ε2 = _ → 0 dist_n,m^2( _0+ h, _0) ^2 ≤Climε→01n2∑i<ℓ((A)iℓε)2+1nm∑i,k((B)ikε)2 ≤ C _ → 0 \ 1n^2 _i< ( ( D_A)_i )^2+ 1nm _i,k ( ( D_B)_ik )^2 \ =C1n2∑i<ℓjA,iℓ()2+1nm∑i,kjB,ik()2. =C \ 1n^2 _i< j_A,i ( h)^2+ 1nm _i,kj_B,ik( h)^2 \. This proves ( .A.23). ∎ As in ( .A.16), the diagonal of A(Θ) _A( ) is set to zero. At checkpoint t, define residuals A(t)(Θ)=A(Θ)−(t) E_A^(t)( )= _A( )- A^(t) and B(t)(Θ)=B(Θ)−(t) E_B^(t)( )= _B( )- B^(t). Differentiating the joint objective block by block gives: (∇ℒ(t))i ( _ αL^(t))_i =∑ℓ≠i(A(t)(Θ))iℓ, = _ ≠ i( E_A^(t)( ))_i , ∇ℒ(t) _ βL^(t) =B(t)(Θ)m, = E_B^(t)( ) 1_m, ∇ℒ(t) _ γL^(t) =B(t)(Θ)⊤n, =\ E_B^(t)( )\ 1_n, ∇ℒ(t) _ ZL^(t) =A(t)(Θ)+B(t)(Θ), = E_A^(t)( ) Z+ E_B^(t)( ) W, ∇ℒ(t) _ WL^(t) =B(t)(Θ)⊤. =\ E_B^(t)( )\ Z. Each A edge appears once in the joint objective, and both endpoint derivatives enter the corresponding rows. On the centered parameter space with ( .A.19), ℒ(t)/(nt)L^(t)/(nt) has gradient: ∇n,mℒ(t)/(nt)=(∇ℒ(t)t,∇ℒ(t)t,m∇ℒ(t)nt,∇ℒ(t)t,m∇ℒ(t)nt). _n,m\L^(t)/(nt)\= P ( _ αL^(t)t, _ βL^(t)t, m _ γL^(t)nt, _ ZL^(t)t, m _ WL^(t)nt ). ( .A.25) The block weights make P orthogonal and give, for centered h: ⟨∇n,mℒ(t)/(nt),⟩n,m=Dℒ(t)(Θ)[]/(nt). _n,m\L^(t)/(nt)\, h _n,m=DL^(t)( )[ h]/(nt). Thus (17) is the gradient update in this weighted norm. Let η˙A,iℓ,η˙B,ik η_A,i , η_B,ik be the first-order changes in ( .A.21) at the current , Z, W. Along Θ+s +s h, the log-means have second derivatives 2i⊤ℓ2 u_i u_ and 2i⊤k2 u_i v_k. Differentiating once more yields: D2ℒ(t)/(nt)(Θ)[,]= D^2\L^(t)/(nt)\( )[ h, h]= 1nt∑i<ℓθiℓA(Θ)η˙A,iℓ2+∑i,kθikB(Θ)η˙B,ik2 1nt \ _i< _i ^A( ) η_A,i ^2+ _i,k _ik^B( ) η_B,ik^2 \ ( .A.26) +1nttr(⊤A(t)(Θ))+2nttr(⊤B(t)(Θ)). + 1nttr( U E_A^(t)( ) U)+ 2nttr( U E_B^(t)( ) V). The mean-weighted terms and residual terms jointly determine local curvature. Lemma S8. Suppose n≥n0n≥ n_0. Let Θ0 _0 be centered with fixed latent row bounds and λmin(0⊤0/n)≥κ1>0 _ ( Z_0 Z_0/n)≥ _1>0. Suppose its means satisfy ct/n≤θiℓA(Θ0)≤Ct/nct/n≤ _i ^A( _0)≤ Ct/n and ct/m≤θikB(Θ0)≤Ct/mct/m≤ _ik^B( _0)≤ Ct/m. There are fixed b0,r0,μ,L>0b_0,r_0,μ,L>0 such that the following residual condition suffices for the local bounds below: ∥A(Θ0)−(t)∥opt+mn∥B(Θ0)−(t)∥opt≤b0. _A( _0)- A^(t) _opt+ mn _B( _0)- B^(t) _opt≤ b_0. ( .A.27) For every centered Θ with ∥Θ−Θ0∥n,m≤r0/n+m - _0 _n,m≤ r_0/ n+m and every centered perturbation h: |D2ℒ(t)/(nt)(Θ)[,]|≤L∥n,m2. |D^2\L^(t)/(nt)\( )[ h, h] |≤ L h _n,m^2. ( .A.28) If h also satisfies ( .A.20) at Θ0 _0, then: D2ℒ(t)/(nt)(Θ)[,]≥μ∥n,m2.D^2\L^(t)/(nt)\( )[ h, h]≥μ h _n,m^2. ( .A.29) All constants are independent of n,m,t,Tn,m,t,T. Proof. Write 0=Θ−Θ0 h_0= - _0 within this proof. The norm definition bounds each worker scalar or latent row change by n∥0∥n,m n h_0 _n,m, and each task-type scalar or row change by m∥0∥n,m m h_0 _n,m. All these changes are therefore at most r0r_0. Expanding the inner products around the bounded reference positions gives: maxi<ℓ|logθiℓA(Θ)/θiℓA(Θ0)|∨maxi,k|logθikB(Θ)/θikB(Θ0)|≤Cn+m∥0∥n,m≤Cr0. _i< | \ _i ^A( )/ _i ^A( _0)\| _i,k| \ _ik^B( )/ _ik^B( _0)\|≤ C n+m h_0 _n,m≤ Cr_0. ( .A.30) For example, the A inner-product change is (i−0,i)⊤ℓ+0,i⊤(ℓ−0,ℓ)( z_i- z_0,i) z_ + z_0,i ( z_ - z_0, ), where the current row ℓ z_ is bounded by the reference radius plus r0r_0. Thus the local means remain within fixed multiples of t/n,t/mt/n,t/m. Mean-weighted terms. The first line of ( .A.26) is comparable to n−2∑i<ℓη˙A,iℓ2+(nm)−1∑i,kη˙B,ik2n^-2 _i< η_A,i ^2+(nm)^-1 _i,k η_B,ik^2. The derivative changes from the reference obey: ∥(−0)⊤+(−0)⊤∥Fn U( Z- Z_0) +( Z- Z_0) U _Fn ≤2∥Fn∥−0∥Fn, ≤ 2 U _F n Z- Z_0 _F n, ∥(−0)⊤+(−0)⊤∥Fnm U( W- W_0) +( Z- Z_0) V _F nm ≤∥Fn∥−0∥Fm+∥−0∥Fn∥Fm. ≤ U _F n W- W_0 _F m+ Z- Z_0 _F n V _F m. Their sum is at most C∥0∥n,m∥n,mC h_0 _n,m h _n,m. Lemma S7 and the reverse triangle inequality therefore bound the mean-weighted terms below by cF∥n,m2c_F h _n,m^2 for h satisfying ( .A.20) when r0r_0 is small. The upper bound ( .A.22), with enlarged fixed radii, gives CF∥n,m2C_F h _n,m^2 for every h. Residual terms. By |ex−1|≤e|x||x||e^x-1|≤ e^|x||x| and ( .A.30), entrywise mean changes are at most C(t/n)n+m∥0∥n,mC(t/n) n+m h_0 _n,m in A and C(t/m)n+m∥0∥n,mC(t/m) n+m h_0 _n,m in B. The operator norm of a symmetric matrix is at most its maximum absolute row sum, and the operator norm is always at most the Frobenius norm. Consequently: ∥A(Θ)−A(Θ0)∥opt+mn∥B(Θ)−B(Θ0)∥opt≤Cn+m∥0∥n,m≤Cr0. _A( )- _A( _0) _opt+ mn _B( )- _B( _0) _opt≤ C n+m h_0 _n,m≤ Cr_0. ( .A.31) Adding ( .A.27) bounds the same standardized residual sum at the current parameter by b0+Cr0b_0+Cr_0. The operator-norm definition gives: |tr(⊤A(t)(Θ))|nt |tr( U E_A^(t)( ) U)|nt ≤∥A(t)(Θ)∥opt∥F2n, ≤ E_A^(t)( ) _opt U _F^2n, 2|tr(⊤B(t)(Θ))|nt 2|tr( U E_B^(t)( ) V)|nt ≤2mn∥B(t)(Θ)∥opt∥Fn∥Fm. ≤ 2 mn E_B^(t)( ) _opt U _F n V _F m. Using 2ab≤a2+b22ab≤ a^2+b^2, the absolute residual contribution is at most C(b0+Cr0)∥n,m2C(b_0+Cr_0) h _n,m^2. Choose b0b_0, then r0r_0, sufficiently small that the residual contribution is at most cF∥n,m2/2c_F h _n,m^2/2. The lower bound under ( .A.20) holds with μ=cF/2μ=c_F/2, and the full-space absolute upper bound holds with a fixed L≥CF+cF/2L≥ C_F+c_F/2. The constants depend only on the fixed mean bounds, radii, and κ1 _1. ∎ Self-adjointness and ( .A.28) bound the Hessian operator norm by L. Integration along segments gives an L-Lipschitz gradient on the local ball. Equation ( .A.29) applies to directions satisfying ( .A.20) at the reference. .A.3 Proof of Theorem 1 Proof. We first derive the order of the cumulative number StS_t of work records. Each work record has total contribution at least one and, by Assumption 1, at most K. Taking conditional expectations given ℱr−1F_r-1 and summing along each path through r≤Str≤ S_t gives: St≤∑i<ℓθ⋆,iℓA,(t)+∑i,kθ⋆,ikB,(t)≤KSt.S_t≤ _i< _ ,i ^A,(t)+ _i,k _ ,ik^B,(t)≤ KS_t. The total A mean is half the sum of its row means, while the total B mean is the sum of its row means. By (12), the middle quantity lies between (3c0/2)nt(3c_0/2)nt and (3C0/2)nt(3C_0/2)nt. Hence St≍ntS_t nt, pathwise, including random StS_t. Work on the time-uniform event of Lemma S2. At any checkpoint satisfying the information threshold in Theorem 1, ( .A.7) holds and the smallest marginal sum is sufficiently large. Lemma S3 gives a finite exact minimizer. Apply Lemma S5 with u=tu=t and Θ′=Θ⋆(t) = _ ^(t). The matrix in ( .A.16) is then St M_S_t from ( .A.5). Write xt=log(2n+m)/δ+loglog(e+t)x_t= \(2n+m)/δ\+ (e^e+t) within this proof. Since t≥C(1+m/n)xtt≥ C(1+m/n)x_t, xt≥1x_t≥ 1, and ε0 _0 is fixed, ( .A.6) and ( .A.17) give: distn,m2(Θ^(t),Θ⋆(t)) _n,m^2( ^(t), _ ^(t)) ≤C∥St∥op2nt2+Cε0nt ≤ C M_S_t _op^2nt^2+C _0nt ≤C(n+m)xtnt1+xtt+ε0(n+m)xt≤C(n+m)xtnt. ≤ C (n+m)x_tnt \1+ x_tt+ _0(n+m)x_t \≤ C (n+m)x_tnt. The same deterministic bound controls (15). This proves (14) on the common concentration event, without further failure probability. The matrix bound is uniform over low-rank directions and therefore allows history-dependent true parameters. For the result over ⌈aT⌉≤t≤T aT ≤ t≤ T, use ( .A.8) and the corresponding marginal event. Write x=log(2n+m)/δx= \(2n+m)/δ\. For t≥⌈aT⌉t≥ aT , Lemma S5 gives: distn,m2(Θ^(t),Θ⋆(t))≤Can+mnTx+Can+mnT2x2+Caε0nT.dist_n,m^2( ^(t), _ ^(t))≤ C_a n+mnTx+C_a n+mnT^2x^2+C_a _0nT. The condition T≥Ca(1+m/n)xT≥ C_a(1+m/n)x ensures the marginal bounds and lets the first term absorb the other two. Taking maxima on the same event proves the claimed bounds over these checkpoints. The log-mean bound follows from the same deterministic inequality. ∎ Larger cumulative activity. Replace Assumption 3 by worker row sums of order (n+m)t(n+m)t and task-type column sums of order n(n+m)t/mn(n+m)t/m: c0(n+m)t≤∑ℓ≠iθ⋆,iℓA,(t)≤C0(n+m)t,c0(n+m)t≤∑kθ⋆,ikB,(t)≤C0(n+m)t, c_0(n+m)t≤ _ ≠ i _ ,i ^A,(t)≤ C_0(n+m)t, c_0(n+m)t≤ _k _ ,ik^B,(t)≤ C_0(n+m)t, ( .A.32) c0n(n+m)t/m≤∑iθ⋆,ikB,(t)≤C0n(n+m)t/m. c_0n(n+m)t/m≤ _i _ ,ik^B,(t)≤ C_0n(n+m)t/m. For n≥n0n≥ n_0, Lemma S1 gives: c(n+m)tn≤θ⋆,iℓA,(t)≤C(n+m)tn,c(n+m)tm≤θ⋆,ikB,(t)≤C(n+m)tm.c (n+m)tn≤ _ ,i ^A,(t)≤ C (n+m)tn, c (n+m)tm≤ _ ,ik^B,(t)≤ C (n+m)tm. Keep the feasible parameter set, model (1), Assumption 1, and the optimization-error bound ε0 _0 unchanged. Under Assumptions 1, 2, and these larger activity scales, St≍n(n+m)tS_t n(n+m)t. With probability at least 1−δ1-δ, every estimate satisfying (10), at every checkpoint with nt≥Clog((2n+m)/δ)+loglog(e+t)nt≥ C\ ((2n+m)/δ)+ (e^e+t)\, obeys: distn,m2(Θ^(t),Θ⋆(t))≤Cntlog2n+mδ+loglog(e+t).dist_n,m^2( ^(t), _ ^(t))≤ Cnt \ 2n+mδ+ (e^e+t) \. ( .A.33) For fixed a∈(0,1)a∈(0,1) and nT≥Calog(2n+m)/δnT≥ C_a \(2n+m)/δ\, the maximum normalized squared parameter error over ⌈aT⌉≤t≤T aT ≤ t≤ T is at most Calog(2n+m)/δ/(nT)C_a \(2n+m)/δ\/(nT) on one event of probability at least 1−δ1-δ. To verify these claims, summing ( .A.32) gives St≍n(n+m)tS_t n(n+m)t. Write ν=n+mν=n+m and ℓt=log(2n+m)/δ+loglog(e+t) _t= \(2n+m)/δ\+ (e^e+t) within this argument. The single-increment norm in ( .A.5) remains at most CνC ν, while the variance calculation in Lemma S2 replaces t by νtν t. The left and right variance bounds become Cν2tCν^2t and CnνtCnν t, hence the total bound is Cν2tCν^2t. Partitioning checkpoints into 2j≤t<2j+12^j≤ t<2^j+1 and applying Freedman gives ∥St∥op2≤Cν2tℓt+νℓt2 M_S_t _op^2≤ C\ν^2t _t+ν _t^2\, with failure probabilities allocated by checkpoint index. Worker margins now have scale νtν t, and task-type margins have scale nνt/mnν t/m. Their minimum scale νtmin1,n/mν t \1,n/m\ lies between ntnt and 2nt2nt: use n≤ν≤2n≤ν≤ 2n when m≤nm≤ n, and m≤ν≤2m≤ν≤ 2m when m≥nm≥ n. Thus nt≥Cℓtnt≥ C _t ensures marginal concentration and finite minimizers. Applying Lemma S5 with u=νtu=ν t and Θ′=Θ⋆(t) = _ ^(t) yields: distn,m2(Θ^(t),Θ⋆(t))≤C∥St∥op2nν2t2+Cε0nνt≤Cℓtnt+Cℓt2nνt2+Cε0nνt.dist_n,m^2( ^(t), _ ^(t))≤ C M_S_t _op^2nν^2t^2+C _0nν t≤ C _tnt+C _t^2nν t^2+C _0nν t. The ratio of the second term in the last bound to the first is ℓt/(νt)≤n/(Cν) _t/(ν t)≤ n/(Cν). The fixed optimization error also lets the first term absorb the third. This proves ( .A.33). For fixed T, variance budget Cν2TCν^2T gives the matrix bound Cν2Tlog((2n+m)/δ)+νlog2((2n+m)/δ)C\ν^2T ((2n+m)/δ)+ν ^2((2n+m)/δ)\. The marginal variance budgets are CνT,CnνT/mCν T,Cnν T/m. Apply the same deterministic calculation over t≥aTt≥ aT. Finally, with x=log(2n+m)/δ≥1x= \(2n+m)/δ\≥ 1, the function loglog(e+t)/t (e^e+t)/t is bounded for t≥1t≥ 1, so supt≥1ℓt/t≤Cx _t≥ 1 _t/t≤ Cx. Thus n≥Cxn≥ Cx ensures the corresponding information threshold for every t≥1t≥ 1, and the supremum of the bound in ( .A.33) is at most Cx/nCx/n. The maximum error over ⌈aT⌉≤t≤T aT ≤ t≤ T has scale x/(nT)x/(nT), corresponding to cumulative activity of order n(n+m)Tn(n+m)T. An optimization analysis under these larger activity scales requires an analogue of Assumption 4 and parameter-block step sizes. .A.4 Proof of Theorem 2 Proof. Work on the time-uniform event of Lemma S2. We establish local geometry at exact minimizers, bound the displacement between successive minimizers, and then prove finite-step tracking. Write x=log(2n+m)/δx= \(2n+m)/δ\ and ℓt=x+loglog(e+t) _t=x+ (e^e+t). Step 1: Initial information and empirical local geometry. For any sufficiently small fixed ρ>0ρ>0, increasing the dimension-independent constant in the initial threshold, that is, the lower bound on t0t_0 in Theorem 2, ensures that, for every t0≤t≤Tt_0≤ t≤ T, all feasible estimates with global optimization error at most ε0 _0 satisfy: distn,m(Θ^(t),Θ⋆(t))≤ρn+m.dist_n,m( ^(t), _ ^(t))≤ ρ n+m. ( .A.34) If ρ≤minRz,Rw/4ρ≤ \R_z,R_w\/4, Assumption 2 then bounds their latent row norms by 5Rz/4,5Rw/45R_z/4,5R_w/4, leaving margins of at least 3Rz/4,3Rw/43R_z/4,3R_w/4 inside the radius constraints. Their normalized worker Gram matrices have minimum eigenvalue at least κ/4κ/4. For every exact minimizer, the left side of ( .A.27) can also be made at most b0/4b_0/4. The function ℓt/t _t/t decreases for t≥1t≥ 1, since tℓt′=t/(e+t)log(e+t)<1/e<ℓt _t =t/\(e^e+t) (e^e+t)\<1/e< _t. For b∗>0b_*>0, set ν=n+mν=n+m, ν~=ν2/n ν=ν^2/n, and t∗=b∗ν~xt_*=b_* νx. Here ν≤ν~≤ν2ν≤ ν≤ν^2 and x≥logν≥1x≥ ν≥ 1, so e+t∗≤(e+b∗)e3xe^e+t_*≤(e^e+b_*)e^3x. Consequently: supt≥t∗pℓt=ℓt∗b∗x≤2+loglog(e+b∗)+3b∗⟶0(b∗→∞). _t≥ t_* p _tt= _t_*b_*x≤ 2+ \ (e^e+b_*)+3\b_* 0 (b_*→∞). Choosing a fixed large b∗b_* controls ν~ℓt/t ν _t/t uniformly over t≥t0≥t∗t≥ t_0≥ t_*. This choice also ensures the information threshold of Theorem 1. Finite exact minimizers therefore exist, and all feasible estimates with the fixed optimization error bound satisfy distn,m2(Θ^(t),Θ⋆(t))≤Cνℓt/(nt)≤ρ2/νdist_n,m^2( ^(t), _ ^(t))≤ Cν _t/(nt)≤ρ^2/ν, proving ( .A.34). After optimal common alignment, each individual worker or task-type parameter row has error at most νdistn,m(Θ^(t),Θ⋆(t))≤ρ νdist_n,m( ^(t), _ ^(t))≤ρ. The true row bounds then prove the interior margins. Taking the infimum over unit vectors in ∥∥2≥∥∥2−∥(−)∥2 X v _2≥ Y v _2- ( X- Y) v _2 gives the singular-value perturbation bound: σd(^(t))n≥κ−∥^(t)−⋆(t)∥opn≥κ−distn,m(Θ^(t),Θ⋆(t)). _d( Z^(t)) n≥ κ- Z^(t)- Z_ ^(t) Q _op n≥ κ-dist_n,m( ^(t), _ ^(t)). If also ρ≤κ/2ρ≤ κ/2, the right side is at least κ/2 κ/2. Squaring proves the worker Gram bound. To bound empirical residuals, the first and second blocks of ( .A.5) are the true cumulative noise scaled by n/2 n/2 and m m. Taking a block does not increase the operator norm, so: ∥(t)−A(Θ⋆(t))∥opt+mn∥(t)−B(Θ⋆(t))∥opt≤3∥St∥opnt≤Cνℓtnt. A^(t)- _A( _ ^(t)) _opt+ mn B^(t)- _B( _ ^(t)) _opt≤ 3 M_S_t _op n\,t≤ C ν _tnt. ( .A.35) The last inequality uses ( .A.6) and t≥Cℓt≥ C _t to absorb the linear tail term. Common alignment preserves means. Expanding at each entry and using the radius constraints gives: maxi<ℓ|logθiℓA(Θ^⋆(t))θ⋆,iℓA,(t)|∨maxi,k|logθikB(Θ^⋆(t))θ⋆,ikB,(t)|≤Cνdistn,m(Θ^⋆(t),Θ⋆(t)). _i< | _i ^A( _ ^(t)) _ ,i ^A,(t) | _i,k | _ik^B( _ ^(t)) _ ,ik^B,(t) |≤ C νdist_n,m( _ ^(t), _ ^(t)). The entrywise, row-sum, and Frobenius calculations used in ( .A.31) now give: ∥A(Θ^⋆(t))−A(Θ⋆(t))∥opt+mn∥B(Θ^⋆(t))−B(Θ⋆(t))∥opt≤Cνdistn,m(Θ^⋆(t),Θ⋆(t)). _A( _ ^(t))- _A( _ ^(t)) _opt+ mn _B( _ ^(t))- _B( _ ^(t)) _opt≤ C νdist_n,m( _ ^(t), _ ^(t)). Adding ( .A.35) bounds the empirical residual by Cνℓt/(nt)Cν _t/(nt). Its square is Cpℓt/tCp _t/t, which the preceding threshold calculation makes arbitrarily small. In particular, the residual bound can be made at most b0/4b_0/4. Step 2: Stationarity, uniqueness, and initialization error. Increasing the fixed threshold constant if necessary, we show that every exact minimizer for t≥t0t≥ t_0 is unique up to a common orthogonal transformation and satisfies: ∇n,mℒ(t)/(nt)(Θ^⋆(t))=. _n,m\L^(t)/(nt)\( _ ^(t))= 0. ( .A.36) An initial estimate with optimization error at most ε0 _0 also satisfies: distn,m2(Θ^(t0),Θ^⋆(t0))≤Cε0/(nt0).dist_n,m^2( ^(t_0), _ ^(t_0))≤ C _0/(nt_0). ( .A.37) Step 1 gives positive margins inside the radius constraints at every exact minimizer. Thus, for any centered h, the perturbation Θ^⋆(t)+s _ ^(t)+s h is feasible for sufficiently small positive and negative s. The joint objective along this line is minimized at s=0s=0, so its derivative in s vanishes there. Since the derivative vanishes for every centered h, ( .A.36) follows. Two exact minimizers are within 2ρ/n+m2ρ/ n+m by ( .A.34). Choose ρ<r0/4ρ<r_0/4, optimally align them, and write =Θ^⋆,2(t)−Θ^⋆,1(t) h= _ ,2^(t)- _ ,1^(t). Lemma S6 gives ( .A.20) for h. Step 1 verifies Lemma S8 along the connecting segment. Stationarity and equal minimum values give: 0=ℒ(t)(Θ^⋆,2(t))−ℒ(t)(Θ^⋆,1(t))nt≥μ2∥n,m2.0= L^(t)( _ ,2^(t))-L^(t)( _ ,1^(t))nt≥ μ2 h _n,m^2. Thus = h= 0, proving uniqueness up to a common orthogonal transformation. The initial estimate and its exact minimizer both satisfy ( .A.34). Their optimally aligned difference lies in the same local ball and satisfies ( .A.20) at the exact minimizer. Repeating the integral lower bound proves ( .A.37): μ2distn,m2(Θ^(t0),Θ^⋆(t0))≤ℒ(t0)(Θ^(t0))−ℒ(t0)(Θ^⋆(t0))nt0≤ε0nt0. μ2dist_n,m^2( ^(t_0), _ ^(t_0))≤ L^(t_0)( ^(t_0))-L^(t_0)( _ ^(t_0))nt_0≤ _0nt_0. Step 3: Objective increments and a common local region. At the same parameter tuple Θ , subtracting successive objectives gives: ℒ(t)(Θ)−ℒ(t−1)(Θ)=−∑i<ℓΔAiℓ(t)logθiℓA(Θ)−∑i,kΔBik(t)logθikB(Θ).L^(t)( )-L^(t-1)( )=- _i< A_i ^(t) _i ^A( )- _i,k B_ik^(t) _ik^B( ). ( .A.38) For a parameter tuple with fixed latent row bounds, the normalized gradient change satisfies: ‖∇n,mℒ(t)/(nt)(Θ)−t−1t∇n,mℒ(t−1)/(n(t−1))(Θ)‖n,m2 \| _n,m\L^(t)/(nt)\( )- t-1t _n,m\L^(t-1)/(n(t-1))\( ) \|_n,m^2 ( .A.39) ≤Ct2[1n∑i∑ℓ≠iΔAiℓ(t)+∑kΔBik(t)2+mn2∑k∑iΔBik(t)2]. ≤ Ct^2 [ 1n _i \ _ ≠ i A_i ^(t)+ _k B_ik^(t) \^2+ mn^2 _k \ _i B_ik^(t) \^2 ]. The exponential terms cancel in ( .A.38). Up to the minus sign, its gradient has blocks: (Δ(t)n,Δ(t)m,(Δ(t))⊤n,Δ(t)+Δ(t),(Δ(t))⊤). ( A^(t) 1_n,\ B^(t) 1_m,\ ( B^(t)) 1_n,\ A^(t) Z+ B^(t) W,\ ( B^(t)) Z ). Write ri(t)=∑ℓ≠iΔAiℓ(t)+∑kΔBik(t)r_i^(t)= _ ≠ i A_i ^(t)+ _k B_ik^(t) and ck(t)=∑iΔBik(t)c_k^(t)= _i B_ik^(t). The scalar blocks are bounded by ri(t)r_i^(t) and ck(t)c_k^(t), and nonnegativity gives: ‖(Δ(t)+Δ(t))i⋅‖2 \| ( A^(t) Z+ B^(t) W )_i· \|_2 ≤∑ℓ≠iΔAiℓ(t)∥ℓ∥2+∑kΔBik(t)∥k∥2≤Cri(t), ≤ _ ≠ i A_i ^(t) z_ _2+ _k B_ik^(t) w_k _2≤ Cr_i^(t), ‖((Δ(t))⊤)k⋅‖2 \| (( B^(t)) Z )_k· \|_2 ≤∑iΔBik(t)∥i∥2≤Cck(t). ≤ _i B_ik^(t) z_i _2≤ Cc_k^(t). The weights in ( .A.25) give (1/n)t−2=1/(nt2)(1/n)t^-2=1/(nt^2) for worker blocks and (1/m)m/(nt)2=m/(n2t2)(1/m)\m/(nt)\^2=m/(n^2t^2) for task-type blocks. Projection cannot increase the norm, proving ( .A.39). The factor (t−1)/t(t-1)/t puts both objectives over ntnt. For t>t0t>t_0, we next use (16) and the initial threshold to show distn,m(Θ^⋆(t),Θ^⋆(t−1))≤r0/(4n+m)dist_n,m( _ ^(t), _ ^(t-1))≤ r_0/(4 n+m). We also verify Lemma S8 for the new objective ℒ(t)/(nt)L^(t)/(nt) with the old minimizer as reference. With ν=n+mν=n+m, (16) gives ∑i(ri(t))2≤K12ν _i(r_i^(t))^2≤ K_1^2ν and ∑k(ck(t))2≤K12nν/m _k(c_k^(t))^2≤ K_1^2nν/m. By nonnegativity: ∥Δ(t)∥F2=2∑i<ℓ(ΔAiℓ(t))2≤∑i(ri(t))2,∥Δ(t)∥F2≤∑k(ck(t))2. A^(t) _F^2=2 _i< ( A_i ^(t))^2≤ _i(r_i^(t))^2, B^(t) _F^2≤ _k(c_k^(t))^2. The operator norm is at most the Frobenius norm, so: ‖[n2Δ(t),mΔ(t)]‖op2≤n4∑i(ri(t))2+m∑k(ck(t))2≤Cn(n+m). \| [ n2 A^(t), m B^(t) ] \|_op^2≤ n4 _i(r_i^(t))^2+m _k(c_k^(t))^2≤ Cn(n+m). ( .A.40) Apply Lemma S5 to the new observations with u=t−1u=t-1 and reference Θ⋆(t−1) _ ^(t-1). The marginal scales are comparable because 1≤t/(t−1)≤21≤ t/(t-1)≤ 2, and the residual matrix is St−1+[n2Δ(t),mΔ(t)] M_S_t-1+[ n2 A^(t), m B^(t)]. The exact minimizer has zero optimization error. Equations ( .A.6) and ( .A.40) give: distn,m2(Θ^⋆(t),Θ⋆(t−1)) _n,m^2( _ ^(t), _ ^(t-1)) ≤C∥St−1∥op2+nνn(t−1)2 ≤ C M_S_t-1 _op^2+nνn(t-1)^2 ( .A.41) ≤Cνℓt−1n(t−1)+νℓt−12n(t−1)2+ν(t−1)2 ≤ C \ ν _t-1n(t-1)+ ν _t-1^2n(t-1)^2+ ν(t-1)^2 \ =Cνℓt−1n(t−1)1+ℓt−1t−1+n(t−1)ℓt−1≤Cνℓt−1n(t−1). =C ν _t-1n(t-1) \1+ _t-1t-1+ n(t-1) _t-1 \≤ C ν _t-1n(t-1). The last step uses t−1≥Cmaxn,ℓt−1t-1≥ C \n, _t-1\ and ℓt−1≥1 _t-1≥ 1. The old minimizer satisfies the same bound relative to Θ⋆(t−1) _ ^(t-1). The triangle inequality and Step 1 give: distn,m(Θ^⋆(t),Θ^⋆(t−1)) _n,m( _ ^(t), _ ^(t-1)) ≤distn,m(Θ^⋆(t),Θ⋆(t−1))+distn,m(Θ^⋆(t−1),Θ⋆(t−1)) _n,m( _ ^(t), _ ^(t-1))+dist_n,m( _ ^(t-1), _ ^(t-1)) ≤Cνℓt−1n(t−1)≤r04ν. ≤ C ν _t-1n(t-1)≤ r_04 ν. At the old minimizer, fitted means have scales (t−1)/n,(t−1)/m(t-1)/n,(t-1)/m, comparable to the new scales. Step 1 gives the radii, interior margins, and spectral bound. The new residual is the old residual minus the checkpoint increment, so the left side of ( .A.27) is at most: t−1tb04+∥Δ(t)∥Ft+mn∥Δ(t)∥Ft≤b04+Cn+mt. t-1t b_04+ A^(t) _Ft+ mn B^(t) _Ft≤ b_04+C n+mt. The initial threshold also implies t≥C(n+m)≥Cn+mt≥ C(n+m)≥ C n+m, making this bound at most b0b_0. Choose fixed b0,r0,μ,Lb_0,r_0,μ,L covering both sets of mean comparison constants if needed. Lemma S8 now applies to the new objective at the old reference. Step 4: Displacement of adjacent minimizers. Optimally align the old minimizer to the new one, retaining the notation, and set =Θ^⋆(t)−Θ^⋆(t−1) h= _ ^(t)- _ ^(t-1). By Lemma S6, h satisfies ( .A.20) at the old minimizer and its norm equals their distance. Step 3 puts the connecting segment in the new objective’s local ball at the old reference. Integrating the gradient difference gives: ⟨∇n,mℒ(t)/(nt)(Θ^⋆(t))−∇n,mℒ(t)/(nt)(Θ^⋆(t−1)),⟩n,m _n,m\L^(t)/(nt)\( _ ^(t))- _n,m\L^(t)/(nt)\( _ ^(t-1)), h _n,m =∫01D2ℒ(t)/(nt)(Θ^⋆(t−1)+s)[,]ds≥μ∥n,m2. = _0^1D^2\L^(t)/(nt)\( _ ^(t-1)+s h)[ h, h]\,ds≥μ h _n,m^2. The new and old minimizers have zero gradient for the new and old objectives, respectively. Combining the zero gradient of the old minimizer with ( .A.39), Cauchy–Schwarz yields: μ∥n,m2≤‖∇n,mℒ(t)/(nt)(Θ^⋆(t−1))‖n,m∥n,m.μ h _n,m^2≤ \| _n,m\L^(t)/(nt)\( _ ^(t-1)) \|_n,m h _n,m. The bound is immediate if ∥n,m=0 h _n,m=0. Otherwise, divide the preceding inequality by ∥n,m h _n,m and square. Applying ( .A.39) and then Assumption 4 gives, for every t0<t≤Tt_0<t≤ T: distn,m2(Θ^⋆(t),Θ^⋆(t−1))≤Ct2[1n∑i∑ℓ≠iΔAiℓ(t)+∑kΔBik(t)2+mn2∑k∑iΔBik(t)2]≤C(n+m)nt2.dist_n,m^2\!( _ ^(t), _ ^(t-1))≤ Ct^2 [ 1n _i\! \ _ ≠ i\! A_i ^(t)+ _k\! B_ik^(t) \^2+ mn^2 _k\! \ _i\! B_ik^(t) \^2 ]≤ C(n+m)nt^2. ( .A.42) Step 5: Gradient contraction and the inner iteration count. In the local region from Steps 1–2, use the current exact minimizer as reference. For a centered parameter Θ with distn,m(Θ,Θ^⋆(t))≤r0/n+mdist_n,m( , _ ^(t))≤ r_0/ n+m, we show that one update in (17) contracts distance by a fixed q0∈(0,1)q_0∈(0,1). Choose r0r_0 below the fixed boundary margin as well. Optimally align Θ to Θ^⋆(t) _ ^(t), and write h for the resulting difference and (t)=∇n,mℒ(t)/(nt)(Θ) g^(t)= _n,m\L^(t)/(nt)\( ). The difference h satisfies ( .A.20) at Θ^⋆(t) _ ^(t). A common orthogonal transformation preserves the joint objective and the local conditions. By stationarity, integration of ( .A.29) and ( .A.28) gives ⟨(t),⟩n,m≥μ∥n,m2 g^(t), h _n,m≥μ h _n,m^2 and ∥(t)∥n,m≤L∥n,m g^(t) _n,m≤ L h _n,m. Since P is included in the gradient, the update is Θ−η(t) -η g^(t), and: distn,m2(Θ−η(t),Θ^⋆(t)) _n,m^2( -η g^(t), _ ^(t)) ≤∥−η(t)∥n,m2 ≤ h-η g^(t) _n,m^2 ≤(1−2ημ+η2L2)∥n,m2=q02∥n,m2, ≤(1-2ημ+η^2L^2) h _n,m^2=q_0^2 h _n,m^2, where η=μ/(2L2)η=μ/(2L^2) and q0=1−3μ2/(4L2)1/2∈(0,1)q_0=\1-3μ^2/(4L^2)\^1/2∈(0,1). After contraction, distance remains below r0/n+mr_0/ n+m. Under some optimal orthogonal matrix, every latent row changes by at most r0r_0. The reference lies at least 3Rz/4,3Rw/43R_z/4,3R_w/4 inside the two boundaries. Choosing r0≤minRz,Rw/4r_0≤ \R_z,R_w\/4 leaves margins of at least Rz/2,Rw/2R_z/2,R_w/2. The update is therefore feasible. Choose a fixed integer J with q=q0J≤1/2q=q_0^J≤ 1/2, for example J≥log(1/2)/logq0J≥ (1/2)/ q_0. All of μ,L,q0,J,qμ,L,q_0,J,q depend only on fixed model constants. Step 6: Remaining in the local region. Write et=distn,m(Θ^(t),Θ^⋆(t))e_t=dist_n,m( ^(t), _ ^(t)) in this proof and subsequent references. Step 1 and a sufficiently large threshold constant ensure et0≤r0/(4n+m)e_t_0≤ r_0/(4 n+m). Bound ( .A.42) gives adjacent-minimizer distance at most C(n+m)/n/tC (n+m)/n/t. The threshold also implies t0≥C(n+m)/nt_0≥ C(n+m)/ n, so this distance can be made at most r0/(4n+m)r_0/(4 n+m) by increasing a dimension-independent constant. Suppose et−1≤r0/(4n+m)e_t-1≤ r_0/(4 n+m). The previous output initializes the new checkpoint, and the triangle inequality places it within r0/(2n+m)r_0/(2 n+m) of the new exact minimizer. Every inner step contracts by Step 5 and stays local. After J steps, the factor is at most q≤1/2q≤ 1/2, restoring et≤r0/(4n+m)e_t≤ r_0/(4 n+m). Induction proves that every checkpoint output and inner iterate stays in the local region and inside the radius constraints. Step 7: Tracking recursion. Iterating et≤qet−1+Cq(n+m)/n/te_t≤ qe_t-1+Cq (n+m)/n/t gives: et≤qt−t0et0+Cn+mn∑s=t0+1tqt−s≤qt−t0et0+C(n+m)/nt(1−q)2.e_t≤ q^t-t_0e_t_0+C n+mn _s=t_0+1^t q^t-ss≤ q^t-t_0e_t_0+ C (n+m)/nt(1-q)^2. The second inequality uses τ=t−sτ=t-s, t/s≤τ+1t/s≤τ+1, and ∑τ≥0(τ+1)qτ=(1−q)−2 _τ≥ 0(τ+1)q^τ=(1-q)^-2. Squaring with (a+b)2≤2a2+2b2(a+b)^2≤ 2a^2+2b^2 and applying ( .A.37) proves (18). Step 8: Subsequent global optimization errors. Align the computed estimate to the current exact minimizer. Their segment lies in the local region, where the reference gradient is zero. Integrating the Hessian upper bound ( .A.28) gives: 0≤ℒ(t)(Θ^(t))−ℒ(t)(Θ^⋆(t))≤Lnt2et2.0 ^(t)( ^(t))-L^(t)( _ ^(t))≤ Lnt2e_t^2. ( .A.43) Substitute Step 7 and et02≤Cε0/(nt0)e_t_0^2≤ C _0/(nt_0) to obtain the first upper bound in (19). For τ=t−t0τ=t-t_0: t0q2(t−t0)=(1+τ/t0)q2τ≤(1+τ)q2τ≤∑j=0∞(1+j)q2j=1(1−q2)2. tt_0q^2(t-t_0)=(1+τ/t_0)q^2τ≤(1+τ)q^2τ≤ _j=0^∞(1+j)q^2j= 1(1-q^2)^2. The initial threshold also gives (n+m)/t≤(n+m)/t0≤C(n+m)/t≤(n+m)/t_0≤ C, with a dimension-independent constant. Thus the global optimization error is at most C(1+ε0)C(1+ _0) uniformly. All steps are deterministic on the same statistical event, and (16) holds almost surely by Assumption 4. No additional failure probability is incurred. ∎ A sufficient condition for Assumption 4. Let a fixed K∗K_* bound all checkpoint increments: maxi∑ℓ≠iΔAiℓ(t)+∑kΔBik(t)≤K∗,max∑ikΔBik(t)≤K∗(1+n/m). _i \ _ ≠ i A_i ^(t)+ _k B_ik^(t) \≤ K_*, _k _i B_ik^(t)≤ K_*(1+n/m). Then (16) holds with K1=2K∗K_1= 2K_*. Using ri(t),ck(t)r_i^(t),c_k^(t) from Step 3, nonnegativity gives ∑kck(t)=∑i,kΔBik(t)≤∑iri(t)≤nK∗ _kc_k^(t)= _i,k B_ik^(t)≤ _ir_i^(t)≤ nK_*. Consequently: 1n∑i(ri(t))2≤K∗2,mn2∑k(ck(t))2≤mn2(maxkck(t))∑kck(t)≤K∗2(1+m/n). 1n _i(r_i^(t))^2≤ K_*^2, mn^2 _k(c_k^(t))^2≤ mn^2( _kc_k^(t)) _kc_k^(t)≤ K_*^2(1+m/n). The sum is at most 2K∗2(1+m/n)2K_*^2(1+m/n), proving (16). The converse fails: when m=nm=n, a single A edge of weight ⌊n⌋ n gives a weighted squared sum at most two, while its maximum row increment grows with dimension. Local stopping error. In the local region of Step 5, let (t)=∇n,mℒ(t)/(nt)(Θ^(t)) g^(t)= _n,m\L^(t)/(nt)\( ^(t)). Then: distn,m(Θ^(t),Θ^⋆(t))≤∥(t)∥n,m/μ,0≤ℒ(t)(Θ^(t))−ℒ(t)(Θ^⋆(t))≤Lnt2μ2∥(t)∥n,m2.dist_n,m( ^(t), _ ^(t))≤ g^(t) _n,m/μ, 0 ^(t)( ^(t))-L^(t)( _ ^(t))≤ Lnt2μ^2 g^(t) _n,m^2. For the aligned difference h, Step 5 and Cauchy–Schwarz give: μ∥n,m2≤⟨(t),⟩n,m≤∥(t)∥n,m∥n,m.μ h _n,m^2≤ g^(t), h _n,m≤ g^(t) _n,m h _n,m. For ≠ h≠ 0, divide by ∥n,m h _n,m and apply ( .A.43). For = h= 0, both bounds hold directly. For the numerical normalization, let npair=n(n−1)/2+nmn_pair=n(n-1)/2+nm. If ∥∇ℒ(t)/npair(Θ^(t))∥∞≤ε\| P∇\L^(t)/n_pair\( ^(t))\|_∞≤ , the n(d+2)n(d+2) worker and m(d+1)m(d+1) task-type coordinates give: ∥(t)∥n,m2≤npair2ε2t2d+2+(d+1)m2n2. g^(t) _n,m^2≤ n_pair^2 ^2t^2 \d+2+(d+1) m^2n^2 \. Together with the preceding bounds, this converts the stopping rule in Supplementary .B.5 into local distance and optimization errors. .A.5 Proof of Corollary 1 Proof. Allocate failure probability δ/2δ/2 to Theorem 2 and δ/2δ/2 to the event of Lemma S2 for ⌈aT⌉≤t≤T aT ≤ t≤ T, including ( .A.8). Their intersection has probability at least 1−δ1-δ, without an independence requirement. Replacing log(2n+m)/δ \(2n+m)/δ\ by log2(2n+m)/δ \2(2n+m)/δ\ adds only log2 2, absorbed into dimension-independent constants. On this intersection, all computed estimates are feasible and have global optimization error at most C(1+ε0)C(1+ _0). The optimization event also gives the marginal bounds ( .A.7) for ⌈aT⌉≤t≤T aT ≤ t≤ T. Write x=log(2n+m)/δx= \(2n+m)/δ\. Applying Lemma S5 with u=tu=t, Θ′=Θ⋆(t) = _ ^(t), and the fixed error bound gives, by ( .A.8) and t≥aTt≥ aT: max⌈aT⌉≤t≤Tdistn,m2(Θ^(t),Θ⋆(t)) _ aT ≤ t≤ Tdist_n,m^2( ^(t), _ ^(t)) ≤Ca(n+m)(Tx+x2)nT2+1+ε0nT ≤ C_a \ (n+m)(Tx+x^2)nT^2+ 1+ _0nT \ ≤Ca(n+m)xnT. ≤ C_a (n+m)xnT. The initial threshold gives T≥CxT≥ Cx, and x≥1x≥ 1 absorbs the fixed optimization error. The same argument controls (15), proving (20). For consistency, write ν=n+mν=n+m, s∗=2n+ms_*=2n+m, and let b∗b_* be the fixed threshold constant. For a fixed cutoff m∗>4m_*>4, each fixed δ∈(0,1)δ∈(0,1) eventually gives, along the stated sequence: νlogs∗nT≤logs∗b∗νlog(s∗/δ)≤(b∗m∗)−1,ν≥m∗,log(2m∗)/b∗log(1/δ),ν<m∗. ν s_*nT≤ s_*b_*ν (s_*/δ)≤ cases(b_*m_*)^-1,&ν≥ m_*,\\ (2m_*)/\b_* (1/δ)\,&ν<m_*. cases Given ε>0 >0, choose fixed m∗>4m_*>4 with (b∗m∗)−1<ε(b_*m_*)^-1< , then fixed δ making the second bound smaller than ε . Hence νlogs∗/(nT)→0ν s_*/(nT)→ 0 for every such sequence. The explicit sufficient condition (n+m)2log(2n+m)/(nt0)→0\(n+m)^2 (2n+m)\/(nt_0)→ 0 ensures the initial threshold for every fixed δ, since log(s∗/δ)≤Cδlogs∗ (s_*/δ)≤ C_δ s_* for a fixed CδC_δ. It also gives directly: νlogs∗nT≤1νν2logs∗nt0⟶0. ν s_*nT≤ 1ν ν^2 s_*nt_0 0. Together with t0≤⌈aT⌉t_0≤ aT , the explicit condition is the sufficient condition stated after Corollary 1. For every fixed δ, log(2n+m)/δ \(2n+m)/δ\ and log(2n+m) (2n+m) differ by at most a fixed factor, so (20) gives uniform consistency in probability over ⌈aT⌉≤t≤T aT ≤ t≤ T. If initialization succeeds with probability only 1−δinit1- _init, add its failure probability. When δinit→0 _init→ 0, allocate half of any target failure probability to concentration. The initialization failure is eventually below the other half, yielding unconditional consistency. ∎ Appendix .B Additional Experiment Details: LongBench v2 Multi-Doc QA This section supplements the LongBench v2 Multi-Doc QA experiment. Section .B.1 describes the paired design, shared warm-up, per-session limits, and checkpoint schedule. Section .B.2 reproduces the worker prompts and shared-board interface. Section .B.3 defines the task types assigned to published results and explains how these results contribute to the collaboration and participation networks. Section .B.4 reports outcomes and participation across replicates, together with workers’ recorded use of the statistical summaries. Section .B.5 specifies the penalized objective, initialization, optimization procedure, and convergence diagnostics. Section .B.6 reports token use, elapsed time, and estimation time. Section .B.7 explains the construction of the figures and documents the recorded choice illustrated in the main text. .B.1 Paired Design Each replicate begins with 32 warm-up sessions, which follow the History-only rules. Copies of the resulting shared board, private memories, and activation state initialize the AgentCue and History-only branches, each with 224 further sessions. A trajectory therefore contains 256 effective sessions, and a replicate uses 480 actual sessions. The AgentCue branch runs first; branches and replicates evolve independently. All sessions complete normally. Means and sample standard deviations summarize three replicates on the same 125 questions. The MAS has 32 persistent workers, with at most one concurrent session per worker and at most 32 concurrent sessions in total. Initial activations are spaced 20 seconds apart; later sessions replace completed ones asynchronously. Each session allows 32 actions on the shared board and 1,200 seconds, with private memory limited to 8,000 characters. The question-order seed is 20260930; activation seeds are 20261006–20261008 for R1–R3. Model sampling seeds are uncontrolled. All questions are accessible from the start through different initial question orders. Sources are divided into chunks of at most 1,400 words without removing content; answer keys remain withheld from workers. Both methods prioritize uncovered questions, answer-critical gaps, and useful existing works. Workers may perform any task type, revise their focus, and contribute to several questions per session. A first designated answer must use and explain a relevant published result of another worker; publishing an answer and designating the answer are separate actions. After all questions have answers, workers may review and correct the answers. Workers in the AgentCue branch also use the statistical summaries to narrow local choices, followed by checks of the current needs. Workers can choose otherwise with a concrete reason; no fixed number of alternatives is required. The first checkpoint at 32 completed effective sessions, which is common to both branches, and the final checkpoint at 256 follow completion of all active sessions. Intermediate checkpoints at 64–224 include published results of sessions in progress. Estimation at each intermediate checkpoint runs serially in the background while sessions continue. A session in the AgentCue branch reads the latest statistical summary with one action when the session first requests the statistical summary, keeps the same summary version, and computes subsequent local comparisons. The MAS estimates the parameters of the latent space model only in the AgentCue branch. The MAS copies the shared board into the two branches before the MAS publishes the statistical summary at the first checkpoint, so the shared board of the History-only branch never holds a statistical summary. .B.2 Worker Prompts The boxes reproduce the prompt components used in R1–R3. Each session prompt contains the identity and phase of the worker, the phase instruction, the common policy, the shared-board interface, the execution instruction, and the private memory of the worker. In these quotations, history_summary denotes AgentCue; common_warmup and history_only supply no statistical summary; “function” and “type” denote a task type, “artifact” denotes a published result, “submit” denotes designate, “complete estimate” and “packet” denote the statistical summary, and the statuses used and considered_not_used record whether the statistical summary informed a choice. Only line wrapping and pagination change. AgentCue: phase instruction. A latest complete fitted summary is published before this branch dispatches. Read the latest published complete packet once at your first statistical read, then reuse that immutable session version/token. New checkpoints fit and publish in the background without pausing collaboration; a later global version does not invalidate your already-read packet. Warm-up and History only: phase instruction. No statistical summary exists or is queried in this phase. Do not call summary or perform fitted-parameter calculations; use current state, experience and actual artifacts. Common worker policy. You and 31 peers jointly advance the fixed pool of 125 Multi-Doc QA questions. Every worker may answer any question and perform any of the four functions. Choose your own work, decomposition, collaborators and handoffs. Roles are not assigned, and questions do not require a fixed sequence of functions. Apply these priorities in order when choosing and continuing work: 1. **Complete the question pool within the fixed effective-session budget.** Aim to bring all 125 questions to source-grounded submitted answers. Give priority to work that moves unsubmitted questions toward actual submission: open a useful scope on a question without published results, fill an answer-critical gap, or carry existing results forward into an answer. Choose among these needs using current inputs, active claims and your ability to contribute; there is no required order of functions. When inputs are adequate, synthesize and submit promptly; peer PASS is not a prerequisite. When they are inadequate, contribute the needed evidence, comparison, reasoning or targeted check and leave a usable handoff. You need not perform every function or finish a whole question yourself. While suitable unsubmitted needs remain, defer routine polishing and repeated checks of submitted answers. A concrete error, evidence conflict or substantive quality benefit may justify submitted-question work; explain the tradeoff in the existing choice reason. After all questions are submitted, use remaining authorized sessions for useful verification and correction. 2. **Use familiar work and a revisable functional focus to serve that goal.** Among useful contributions to the first priority, prefer continuing work whose source familiarity, reusable results or current function reduces rereading, switching and repeated reasoning. Continue when relevant work is ready. Hand off or switch when an important uncovered need fits you better, the current input is unavailable, or a peer is better placed to continue. Familiarity and focus help choose how to contribute; they do not justify neglecting suitable unsubmitted needs. No worker has an exclusive or permanent function. 3. **In history_summary, use the current statistical summary first to narrow choices.** Within the first two priorities, use the fitted worker-function relations to identify suitable functions among actual pending tasks, and worker-worker relations to identify plausible collaborators. Once the summary and current need identify a suitable candidate, inspect that question/task and the relevant member's current workload, then proceed using the required artifacts and sources. Avoid browsing unrelated histories, questions or member profiles, or expanding comparisons merely to fill a quota. If support is weak, candidates are tied, the task is new, or relevant state/experience contradicts the suggestion, inspect only the additional history needed and choose with a concrete reason. The summary guides your choice; it does not assign work or rank question difficulty. This third priority applies only in history_summary. Use the supplied non-login shell; omit login or set login=false in exec_command. Do not request login=true, start a login-shell wrapper, or change shell settings. Run only the two quoted-heredoc interfaces below. Avoid preventable board refusals. Copy exact question, task, attempt and chunk IDs from actual responses; chunk numbering need not start at 0000. Before proposing first_answer, check current question_status and prefer an unsubmitted question without an active answer integration. Reuse an existing suitable open task. Refresh question_status immediately before CLAIM and pass that revision; concurrent changes may still require one targeted refresh. If another answer integration is active, choose useful complementary work or another question instead of repeatedly proposing/claiming competing syntheses. A concrete submitted-question review/correction remains allowed. Before finish, read every listed input artifact in this call, including your own earlier-session work; an overview, title or memory does not count. Copy short contiguous source quotations from the exact returned chunk and preserve punctuation and PDF hyphens. Whitespace and the seven Latin PDF ligatures (such as fi/fi) are equivalent; wording, punctuation, case, hyphens and source identity remain exact. If a result is supported by an actually read peer artifact, cite that A-... ID in evidence_refs and explain its contribution; do not copy its source quotes as if you had read those sources yourself. source_quotes may be empty when the actual artifact inputs supply the evidence. If you directly quote a source, read/search that source in this call first. In history_summary, prefer summary_use with just status, version and a concrete interpretation. Put genuine numerical comparisons and fallback reasoning in interpretation; optional alternatives are descriptive text or valid worker/function or worker/task objects, not a required comparison quota. After a documented helper rejects input or a local calculation fails, use its error/recovery to correct the specified issue in this same session and original budget. This includes malformed JSON, wrong fields/identities, self-comparison, missing cache tokens and rejected board actions. Do not repeat the same rejected payload unchanged or claim that rejected work was published/submitted. Empty, incomplete or missing-ok tool displays also permit continuing; inspect current state before retrying an uncertain write. Types describe independently usable delivered results: - **extract:** a source-located evidence package or useful search result, with objects, locations, dates, units and applicable conditions. - **compare:** an explicit alignment of objects, sources or periods on shared dimensions, with supported similarities, differences, conflicts, non-comparability or missing information. - **inference:** a new calculated result, deduction, candidate conclusion or final answer. State inputs or premises, the calculation or reasoning, the conclusion and its conditions. For a final answer, explain how the inputs support the option criteria. This function includes substantive reasoning and answer synthesis. - **review:** an independent check of another worker's identified artifact and version, with checks, findings, evidence and a PASS/FAIL verdict. Evidence, comparisons, calculations and candidate conclusions may be checked before a question has a submitted answer. A work may deliver several independently usable results with distinct acceptance criteria and therefore have multiple types. Finding quotations for a comparison, citing facts for a deduction, recalculating during verification and routine self-checks are supporting steps unless they produce additional independent results. Multiple calculations or answer deductions within one work count once toward inference. Copying or renaming the same content does not create a new contribution. Start with the compact overview of ready work, active claims and published results. Its coverage_opportunities shows separate factual examples of unsubmitted questions with and without published artifacts, including recent artifact IDs and active-work counts. Check both unmet-work and continuation needs; artifact presence alone does not establish sufficient evidence. In history_summary, read the current complete summary once next, before selecting work or collaborators. Then use targeted question/task/member reads for the candidate you choose; a broad snapshot is optional. Before you have relevant experience, explore a useful available scope and, where relevant, use the summary's activity support. You may choose randomly among similarly suitable alternatives. Register a specific scope and expected result before extensive source reading, and recheck the latest claims. Retain reusable search strategies, comparison frameworks, calculation methods, option criteria and verification checklists in private memory. Update your working focus from actual experience under the priorities above. Consider peers' published results and workload when choosing who should continue; do not wait for unavailable inputs or duplicate an active claim. Multiple genuine functional results may occur in one work or session. For an unsubmitted question with published artifacts, inspect the relevant exact versions and decide whether to fill a specific gap or synthesize the answer; do not repeat completed work. A targeted pre-submission check may resolve an answer-critical risk. Review the exact version and state which result it enables. Do not repeat checks just to obtain peer PASS or balance type frequencies. Publishing a synthesis with finish does not submit the answer. After publishing an eligible answer for an unsubmitted question, complete designate_final before moving on and check its designated=true response. That successful response confirms submission; a further state read is needed only if the result is unclear or a conflict requires it. You may also designate an eligible peer synthesis after actually reading its exact version. Do not wait for a peer review when the submission requirements are already met. If validation or a concurrent change prevents submission, inspect the relevant response/state and fix the specific issue within budget, or leave the exact answer/artifact IDs, blocking issue and next required action for continuation; do not claim it was submitted. Publish useful intermediate results with source locations, actual input artifact IDs and versions, boundaries and limitations. In the same result, identify the question, what the result enables, the specific remaining gap and next work with its required input IDs. This lets another worker continue without reconstructing your search. Keep a handoff concise; a separate invitation or duplicated task is not required for every result. One work may use results from several peers. An invitation, a read or a decorative citation is not completed cooperation. During common_warmup and history_only, do not query statistical summaries or perform fitted calculations. During history_summary, a current complete estimate is always published before your session starts; read its complete representation once for this session/version, then compare useful functions or possible collaborators locally. It contains all 32 worker positions Z, four function positions W, node effects alpha/beta/gamma, checkpoint metadata and activity support. Reuse it for changed candidates, new tasks and refreshed question state. After common c32, collaboration continues while statistical snapshots are fitted in the background. Your first full summary read receives the latest published complete version; cache that immutable version for this whole session. A newer global publication does not invalidate your cached version or its CLAIM/INVITE reasons. Refresh current task, question and artifact state as needed. Zero activity is missing record support, not lack of ability. The statistical service supplies numerical representation; you choose work and collaborators. Use the cached parameters for internal comparison or pass the returned local_cache token to the pure local_compute.py interface, which costs zero board actions. Its only cache read is this call's already obtained immutable parameters; no need to copy 140 numbers or fetch live state. For worker/function i,k compute z_i dot w_k and exp(beta_i+gamma_k+z_i dot w_k); for different workers i,j compute z_i dot z_j and exp(alpha_i+alpha_j+z_i dot z_j). These are fitted cumulative counts, not success probabilities or question difficulty. You do not need a candidate_summary request or returned task/worker alternatives to claim or invite. After statistical comparison, inspect current question/task state and member workload as needed. Before CLAIM refresh question_status; current revision and conflicts still apply. Record used/considered_not_used/single_option with the complete version and concrete reasoning. Weak support, ties, source experience or a new task can justify considered_not_used; unavailable means no fitted summary exists in this condition. Alternatives and adopted values are optional, not a comparison quota. Every response shows remaining session actions and the team session budget. Choose a scope you can publish within that budget and reserve actions for actual input reads, finish and, for an eligible answer, designate_final. As the team budget shrinks, favor feasible completion of pending answers and precise answer-critical gaps; avoid broad work whose useful result cannot be delivered in the remaining budget. If this session cannot finish an answer, publish actual reusable partial findings and an explicit handoff before ending. Do not spend the end of a session repeatedly comparing candidates. Submission priority does not permit fabricated evidence, unsupported guesses or relaxing publication validation. In common_warmup and history_only, choose using current work state, accumulated experience and relevant history. In history_summary, missing or invalid current parameters are a fault to report, not permission to continue without them. Workers who have not activated or delivered results lack statistical support; zero activity does not imply lack of ability. Retain autonomous exploration and targeted history fallback for weakly supported candidates. Expected cumulative participation and cooperation counts describe work history and relationships, not success rates. Statistical selection never replaces reading the chosen question, needed sources and actual input artifacts. A first final answer must use at least one relevant published result from another worker and explain its substantive contribution. Peer PASS is not required before submission. All 125 questions and source collections are accessible from the start. Ground answers in the provided collections. Do not use outside knowledge as a substitute for evidence. Only the frozen board interface and supplied-data pure calculation interface are permitted. Local calculation cannot fetch new state or replace source/artifact reads. Construct each request as a JavaScript object in the code-mode host and serialize it with `JSON.stringify(payload)` when building the documented quoted heredoc command. Source quotes, newlines and quotation marks must be serialized rather than manually escaped. A malformed JSON/object/action request at the fixed board entry is an ordinary rejection and consumes one of your 32 board actions. Correct it in this same session using the returned reason; it does not publish an artifact. At the action limit, finish without further board calls. At local_compute.py, rejected inputs cost zero board actions and return an ordinary error plus recovery instructions. A cooperation comparison must have different worker and peer IDs: explicitly remove your own worker ID from candidate peers. Use B for your own worker/function comparisons. Correct a rejected comparison locally and continue useful work in the original session. Shared-board interface. ## Board interface Shared-board requests must have exactly this quoted-heredoc form: ```sh python3 board.py <<'JSON' "action":"snapshot" JSON ``` The only other allowed command is the pure local_compute.py quoted-JSON interface described below. Do not inspect files, execute arbitrary code, use external sources, network tools or agents. Your session identity is bound by the controller. Each session permits 32 board actions including invalid actions, and 1200 seconds. The transport uses non-login shells. Omit login or set login=false; do not set login=true or wrap a command in a login shell. Failed requests still cost their actual board action. Ordinary rejections return failure_class="ordinary" and, where applicable, error_code/recovery with the exact input/chunk or next read to correct. They do not automatically execute that read, fix a payload or select work. Technical/protocol failures stop new dispatch. Every response includes the remaining team session budget. No new slot available means existing sessions may finish; do not assume another session will follow. shared_context.session_action_budget also gives this session's maximum, used and remaining actions. Keep actions for actual input reads, finish and designate_final. Start with overview; in history_summary read summary once next, then request the relevant question_status/task_status/member_profile and actual inputs for the selected work. A broad snapshot is optional. If the summary and current state identify a suitable candidate, proceed with targeted reads instead of browsing unrelated board history. In common_warmup and history_only, skip summary and fitted calculations. All 125 questions are open from the beginning. The initial question page is rotated by worker ID, not assigned. Use questions with offset and next_offset to browse all items; optional query/sub_domain/status filters. Choose any question. Each question has its own source collection. The common prompt prioritizes advancing unsubmitted questions to submission. Default question pages and functional ready-work examples sort unsubmitted questions first, without hiding submitted work. The overview also exposes submitted ready/review examples. You may select concrete submitted-question review/correction with a reason explaining its value relative to uncovered needs. No worker, function or question is assigned; artifact presence does not certify sufficient evidence. Read actions: - overview: compact ready-work counts/examples, active claims, unreviewed peer results and revisable working focuses. coverage_opportunities separately lists unsubmitted questions with and without published artifacts (counts and at most six rotated examples each), with recent artifact IDs/versions/types, published-synthesis counts and active-work/integration counts. Published results remain visible even after their task closes or is reviewed. This is included in the same overview action. Examples are factual leads, not judgments that an answer is ready or work assignments. Read question_status for complete exact versions and current claims; use targeted questions when the question text is needed. - snapshot: optional question/project/offset. - summary: one action returns the entire available 32-worker/four-function, two-dimensional representation: parameters version,dimension,worker_ids,type_keys,alpha,beta,gamma,Z,W, 140 numbers and 36 latent vectors, plus checkpoint/event cutoff, cumulative record interval, worker support and an own-session immutable local_cache token. No derived full matrix is sent. During common_warmup and history_only, do not call summary: these phases expose no fitted parameters or token and summary queries are rejected. In history_summary joint-c32 is published before the branch starts; later complete estimates publish asynchronously while collaboration continues. A missing, corrupt or inconsistent published packet is a stop condition, not unavailable fallback. A newer checkpoint still being fitted is normal; the latest published complete packet remains available. No automatic parameter delivery occurs in other responses. Cache the complete packet in this session. Same-version repeat requests still cost one board action and return only a cache reminder; changing candidate, task or question revision never requires another statistical read. The first actual full read pins the latest published version for this whole session. Background publication can advance latest_summary_version; session_summary_version identifies your already read packet. CLAIM/INVITE accept that exact actually-read version, and local_compute reuses its immutable token. A new session must perform its own first full read even if private memory contains an old packet. - question_status: question. Returns exact final, artifacts, reviews, active work, stage and revision; one state-read action, no statistical re-fetch. active_answer_integrations and first_answer_claim_possible_now report the current atomic conflict conditions, not evidence sufficiency. Overview ready-work examples also include claimable_now and blocking_answer_integration_ids, with blocked work still visible. These facts may change concurrently. Before proposing first_answer, inspect this question status; before CLAIM refresh its revision. If integration is active, use complementary useful work or another current need instead of duplicate synthesis claims. - task_status: task_id. Current task specification/state/owner; no statistics. - member_profile: worker. Current member workload/history and focus; no fitted parameters. - questions: optional offset/question/query/sub_domain/status; status is unsubmitted, awaiting_PASS or peer_PASS. - sources: question, optional offset. Returns real chunk IDs. - search: project="MultiDoc", question, pattern. Returns source excerpts. - read_source: project="MultiDoc", question, chunks=[one to three real IDs]. - read_artifact: attempt_id. Read each actual input's exact version before use. Pure local calculation (zero board actions): Use the exact summary.local_cache object returned by your complete summary read: "cache":"version":"the actual returned version","parameter_sha256":"the actual returned hash","comparisons":["kind":"B","worker":1,"function":"MultiDoc/extract","kind":"A","worker":1,"peer":2] Send that actual JSON (replace the two token placeholders) with exactly: ```sh python3 local_compute.py <<'JSON' ...the actual JSON request described above... JSON ``` The tool reads only the immutable parameter snapshot already obtained by this same bound call, then computes over it. You do not need to copy 140 numbers into each command. No caller-supplied path, other session, live board database/state, answer file, network, arbitrary code or board write is permitted. Alternatively, explicit summary.parameters plus comparisons may be supplied as numerical data; the trace checks the served version and hash. If local display or audit records are missing, continue your work in this original session; do not echo all parameters or repeat actions to supply audit evidence. Batch genuine comparisons (up to 4096 per request) and reuse the same token. Results give inner products and exp(beta_i+gamma_k+z_i dot w_k) or exp(alpha_i+alpha_j+z_i dot z_j), excluding self-cooperation. Numeric comparison/choice remains yours; the tool makes no work or role assignment. Pure calculations are recorded from command trace, consume session time/model usage and create no participation/cooperation. History-only has the identical tool but receives no fitted packet or token; do not fabricate them. Propose: "action":"propose","project":"MultiDoc","question":"exact ID", "expected_types":["extract","compare"],"description":"specific gap", "acceptance":"inspectable result","inputs":[],"parent":null,"tags":[],"purpose":"evidence", "target_attempt":null,"follow_up_reason":null purpose is evidence, first_answer, review or correction. Final-answer synthesis must use first_answer or correction. review is a focused task with review in expected_types and target_attempt set to its exact input artifact. correction requires follow_up_reason identifying the concrete issue. Expected types describe intentions only. Use a separate work/task per question when batching. Use existing work rather than reposting it as a new task. Claim: "action":"claim","task_id":"T-...","reason":"why this work now", "partner":null,"question_revision":"from question_status","follow_up_reason":null,"summary_use":"status":"unavailable","version":null, "alternatives":[],"interpretation":"concrete history or availability reason" Submitted-question work is permitted before full coverage; claiming it requires a concrete follow_up_reason for an error, conflict or quality benefit relative to uncovered work. Already-reviewed exact targets also need a new substantive reason. First-answer, version and duplicate-claim checks remain. summary_use for an available model is, for example: "status":"used","version":"joint-c32","interpretation":"actual numerical basis and current work need" Allowed statuses are used, considered_not_used, single_option and unavailable. Available-model choices require one complete read of this version in this session and legal selected task/worker identities. The actual CLAIM/INVITE records the selected object; no previously returned candidate pair or competitor list is required. Optional alternatives/adopted values are descriptive. unavailable requires no fitted model and null version. Always read actual input artifacts/exact versions and needed source text before reuse; statistics do not replace evidence. Prefer omitting alternatives and adopted values: the three-field status/version/interpretation object above is sufficient. If supplied, alternatives is a list of nonempty descriptive strings or objects worker:1,function:"MultiDoc/extract" / worker:1,task_id:"actual T-..."; structured identities are validated. Text is saved as an unverified declaration, not proof of a calculation or successful use. Put the actual comparison and reason in interpretation. Finish: "action":"finish","attempt_id":"A-...","lease":"returned lease", "artifact":"text":"actual result summary","inputs":[],"input_uses":, "source_quotes":["project":"MultiDoc","chunk":"real chunk ID","quote":"literal text"], "final_choice":null,"results":["id":"e1","kind":"evidence", "scope":"standalone","evidence_refs":["Q0"],"data":"facts":["source-located fact"]] There are four task types: extract, compare, inference, review. results[].kind uses the six delivery structures below; scope is standalone or supporting. Only standalone results add task types. All delivery data fields are required and must be nonempty. evidence_refs may be empty only for review; artifact.inputs and input_uses are empty when no published input is used. Use explicit "not applicable" or uncertainty where appropriate; do not fabricate findings to fill fields. Lists below contain strings unless otherwise stated. evidence_refs lists Q0/Q1/... for source_quotes or actual input A-... IDs. To ground an unsuccessful search, include artifact.search_records as ["question":"exact ID","pattern":"actual searched pattern"] and cite S0/S1/...; the board checks that you performed that search in this session. Describe the search scope and uncertainty; no hit from one query is not proof of absence. Every non-review result needs evidence_refs. Each artifact input must be used by a result or be its review target, and input_uses maps every input ID to an explanation of its actual contribution. Supporting results do not add types. - evidence -> extract: data.facts (list). source_quotes use short contiguous text from the exact cited chunk. Only whitespace and seven Latin PDF ligatures (f/f, fi/fi, fl/fl, ffi/ffi, ffl/ffl, st/st, st/st) are equivalent. No punctuation, case, hyphen, wording, cross-chunk or fuzzy substitution is accepted. The rule is recorded in each published source_quote_checks. Quote failures identify quote_index/chunk; unread-source/input failures name the precise read needed. If a read peer artifact is the evidence, cite its A-... ID with input_uses rather than copying its original-source quotes without a source read in this call. - comparison -> compare: data.objects (at least two distinct names), dimensions (list), findings (list). - calculation -> inference: data.inputs (list), procedure (string), result (string), units (string), time_basis (string). - inference -> inference: data.premises (list), reasoning (list), conclusion (string), conditions (string). - synthesis -> inference: data.choice (A/B/C/D), input_uses (list of actual input IDs), rationale (string), remaining_uncertainty (string). Set artifact.final_choice to the same choice. Standalone only. Use at least one relevant other-worker artifact and explain its contribution; it cannot be a decorative citation. - review -> review: data.target (input A-... ID), version (integer), checks (list), findings (string), verdict (PASS/FAIL). Standalone only; target another worker's exact version. This automatically publishes the peer review and records its participation. There is no separate review action. Do not create a duplicate review summary task. Supporting recalculations do not add inference. calculation, inference and synthesis are three delivery formats of the single inference task type. A work containing both contributes only one inference participation. Use inference in expected_types for any of these formats. After publication, results are immutable. An identical finish retry is idempotent; a changed retry is rejected. Reopen is for genuine new corrective work, not for reposting a result. Publish actual partial findings and state remaining gaps; completion or correctness is not needed for a useful participation record. In artifact.text, make unfinished work easy to continue: identify what this result enables, the remaining answer-critical gap and next work with actual required input IDs. Do not create a redundant task or invitation just to record a handoff. Other actions: - focus: function (extract/compare/inference/review or null), reason. Revisable nonexclusive working focus; it creates no participation count. - designate_final: question, attempt_id of a published synthesis; peer review is not required first. Only relevant other-worker inputs satisfy collaboration. finish alone does not submit an answer. After publishing an eligible answer for an unsubmitted question, reserve and perform this action before moving on; result.designated=true confirms submission without an additional polling action. An eligible peer synthesis may also be designated after actually reading its exact version. If a rejection or concurrent change blocks submission, address the specific issue within budget or leave the artifact ID, blocking reason and next required action in the handoff. Report submission only after confirmation. - reopen: task_id, reason (specific supported correction). - invite: worker, task_id, reason, summary_use in the same format as claim. Invitations are nonexclusive; other workers may accept the work. Return final JSON with exactly summary (string), published_attempts (list of your own actual A-... IDs), memory (string), useful_work_remaining (boolean). Preserve useful context in memory. Invalid/no-progress completions still consume a session. Request construction: in the code-mode host, use `const payload = ...; const cmd = "python3 board.py <<'JSON' " + JSON.stringify(payload) + " "; text(await tools.exec_command(cmd, shell:"/bin/zsh", login:false));`. The command must still contain exactly one documented helper and one quoted heredoc. If a value contains a line exactly `JSON`, use its JSON-escaped string representation; never place a literal delimiter line in the body. A malformed board request returns `failure_class:ordinary`, `error_code:INVALID_BOARD_REQUEST`, a raw-input SHA-256, recovery instruction and action-budget header; it consumes one action without publishing or executing the requested action. Correct the payload in the same session. After 32 actions, `BOARD_ACTION_LIMIT` rejects without further execution. A malformed local calculation returns `failure_class:ordinary`, `error_code:INVALID_LOCAL_REQUEST`, the raw-input SHA-256, zero board actions and a correction instruction before any cache read or arithmetic. Local cache/calculation errors and uncertain displays are warnings; correct them within this same session. Exclude the acting worker from A comparison peers; use B for worker/function comparisons. A rejected query in a condition without a summary exposes no parameters and permits continuing with state/history. Missing delivery records are warnings and do not require replay or extra evidence from the worker. Actual forbidden commands, confirmed conflicting parameter versions/hashes, damaged execution provenance and a missing published summary remain integrity failures. Execution instruction. EXECUTION PARAMETERS: Use tools.exec_command with shell="/bin/zsh" and login=false. Omit workdir so it inherits this session workspace; do not invent a directory or shell path. Keep cmd exactly one documented quoted-JSON board.py or local_compute.py invocation. Use the returned recovery instruction to correct ordinary board rejections; do not repeat an unchanged rejected request. If tool output is empty, incomplete, or lacks ok, continue in this same session. For an uncertain write, read current state to check whether it already succeeded before any correction; do not blindly repeat writes. Correct local input errors within the existing time/action budget. .B.3 Published Results and Count Construction Each work record stores the worker who produced the work, the source references, and the exact versions of the published results that the work uses or reviews. A fixed program checks required fields, quotations, and input references, and classifies independently reusable results as in Table .B.1. The classification ignores benchmark correctness. Table .B.1: Task types defined by independently reusable results. Task type Required independently reusable result Extract Source-located facts, values, conditions, or a documented search result. Compare Named objects, shared dimensions, and evidence-supported similarities, differences, or conflicts. Inference Inputs or premises, reasoning, conclusion, and applicable conditions. Answer synthesis also gives an option and explains its use of published results of other workers. Review The exact version of another worker’s published result, checks, findings, evidence, and a PASS or FAIL verdict. Each work contributes at most once to each task type in (t) B^(t). Supporting quotations, review calculations, and self-checks add no separate task types. Several deductions or answer syntheses in one work count once as inference. Identical publication retries add no counts. Using or reviewing published results of other workers contributes once per work and worker–worker pair to (t) A^(t); invitations and reads alone add nothing. A work may contribute to multiple task types and pairs. .B.4 Outcomes and Summary Use Table .B.2 reports replicate outcomes, and Tables .B.3– .B.4 summarize participation. All six trajectories designate answers for all 125 questions. All 32 workers perform extract and inference; compare involves 31–32 workers and review 25–31. PASS concerns a review of the designated answer version and differs from benchmark correctness. Table .B.2: Final results by replicate. Difference is in percentage points; PASS columns count designated answers with a PASS verdict (AgentCue/History only). Replicate AgentCue History only Difference PASS Correct + PASS Correct Accuracy (%) Correct Accuracy (%) R1 76 60.8 71 56.8 4.0 36/36 22/20 R2 77 61.6 72 57.6 4.0 33/41 18/23 R3 75 60.0 73 58.4 1.6 92/47 55/24 Table .B.3: Participation and collaboration (mean ± sample standard deviation). Task-type counts are column sums of (T) B^(T); collaboration is ∑i<ℓAiℓ(T) _i< A_i ^(T). Measure AgentCue History only Published works 359.7±26.8359.7± 26.8 348.0±2.6348.0± 2.6 Extract 160.7±3.2160.7± 3.2 162.3±11.2162.3± 11.2 Compare 140.0±18.4140.0± 18.4 153.3±6.4153.3± 6.4 Inference 246.0±56.0246.0± 56.0 250.7±51.4250.7± 51.4 Review 71.3±31.071.3± 31.0 57.0±5.357.0± 5.3 Collaboration count 284.7±46.9284.7± 46.9 248.7±3.2248.7± 3.2 Distinct collaborating pairs 209.3±26.0209.3± 26.0 190.3±1.5190.3± 1.5 Table .B.4: Participation by replicate. E/C/I/R denote extract/compare/inference/review; the worker column counts contributors to each task type. Replicate Method Works E C I R Workers E/C/I/R Collaboration R1 AgentCue 350 163 119 290 56 32/32/32/28 274 R1 History only 349 175 156 288 51 32/32/32/25 250 R2 AgentCue 339 162 153 183 51 32/32/32/27 244 R2 History only 345 154 146 192 59 32/32/32/29 251 R3 AgentCue 390 157 148 265 107 32/31/32/31 336 R3 History only 350 158 158 272 61 32/32/32/30 245 Each AgentCue branch provides the statistical summary to all 224 new sessions. R1–R3 record 285, 288, and 299 requests for local comparisons, which contain 1,687, 1,559, and 1,784 local comparisons. Recorded choices use the statistical summary 237, 197, and 214 times and consider but decline the statistical summary 58, 84, and 114 times; a session can make several choices. Workers read the first seven summary versions. The final statistical summary follows all worker activity. .B.5 Numerical Estimation Each estimation uses the latent space model (1) with d=2d=2 on all 32 workers and four task types, including zero-count pairs. Let x stack the entries of Θ(t) ^(t). The penalized objective is f(t)()=ℒ(t)()624+0.0132∥F2+0.014∥F2,624=(322)+32×4.f^(t)( x)= L^(t)( x)624+ 0.0132 Z _F^2+ 0.014 W _F^2, 624= 322+32× 4. ( .B.1) Penalty coefficients are fixed throughout. Let P subtract column means from Z and W and the mean from β, leaving α and γ unchanged. Projected updates are s+1=s−ηs∇f(t)(s). x_s+1= P\! \ x_s- _s P∇ f^(t)( x_s) \. ( .B.2) The step size ηs _s in ( .B.2) results from Armijo backtracking, which halves a trial step size until the sufficient-decrease condition with coefficient 10−410^-4 holds. The first trial step size is 11. Write Δs=s+1−s x_s= x_s+1- x_s and Δs=∇f(t)(s+1)−∇f(t)(s) g_s= P∇ f^(t)( x_s+1)- P∇ f^(t)( x_s). Subsequent trial step sizes alternate across iterations between the two Barzilai–Borwein step sizes Δs⊤Δs/(Δs⊤Δs) x_s x_s/( x_s g_s) and Δs⊤Δs/(Δs⊤Δs) x_s g_s/( g_s g_s), restricted to [10−6,104][10^-6,10^4]; when Δs⊤Δs≤10−20 x_s g_s≤ 10^-20, the trial step size is 1.5ηs1.5 _s, capped at 10410^4. The first checkpoint uses five random starts, with seeds 42–46, centered N(,2)N( 0, I_2) latent positions, and node effects initialized from the observed network counts. The smallest finite penalized objective selects the estimate. Later checkpoints start from all parameter blocks of the previous estimate. Each start allows 20,000 iterations, stopping at ∥∇f(t)(s)∥∞≤10−5\| P∇ f^(t)( x_s)\|_∞≤ 10^-5. All 24 estimates, eight checkpoints in each of the three AgentCue trajectories, satisfy this stopping rule (Table .B.5). The local stopping error in Supplementary .A.4 converts this stopping rule into bounds on the distance to the exact minimizer and on optimization error. Table .B.5: Estimation diagnostics over eight checkpoints. Centering columns give maximum absolute sums of latent-position columns or node effects. Method Estimates Iterations Gradient norm Z W β AgentCue 24 111–469 2.33×10−62.33× 10^-6–9.95×10−69.95× 10^-6 1.33×10−151.33× 10^-15 1.11×10−161.11× 10^-16 6.22×10−156.22× 10^-15 .B.6 Computational Resources Table .B.6 reports the final token use and elapsed time of each 256-session trajectory. Cached input and reasoning output are subsets of input and output; cache-write input is zero. Worker spans, that is, the elapsed times of worker activity, cover the 224 new branch sessions and exclude the warm-up, the intervals between branches, and the estimation after completion. Warm-up spans are 11.57, 11.56, and 9.60 minutes. The seven additional estimations in the AgentCue branch take 0.307, 0.297, and 0.337 seconds per replicate, and estimation at intermediate checkpoints overlaps worker activity. AgentCue uses more tokens and elapsed time and improves accuracy under the common effective-session budget. Table .B.6: Computational resources (mean ± sample standard deviation). Tokens include warm-up; worker spans exclude it. Additional estimation time covers checkpoints 64–256; History only involves no estimation. Measure AgentCue History only Input tokens (millions) 415.97±2.04415.97± 2.04 390.06±2.93390.06± 2.93 Cached input (millions) 392.85±1.91392.85± 1.91 367.76±2.92367.76± 2.92 Output tokens (millions) 2.05±0.042.05± 0.04 1.94±0.041.94± 0.04 Reasoning output (millions) 0.60±0.010.60± 0.01 0.56±0.020.56± 0.02 Post-warm-up worker span (minutes) 29.81±0.7029.81± 0.70 24.33±3.3524.33± 3.35 Additional estimation time (seconds) 0.314±0.0210.314± 0.021 – .B.7 Figure Construction and Choice Record Figure 4(a,b) reports means and sample standard deviations across R1–R3. Domain accuracies use their question counts; checkpoint accuracies use all 125 questions. Figure 5 jointly aligns R1’s worker and task-type latent positions to the statistical summary at the final checkpoint by orthogonal Procrustes, preserving inner products. Linear axes and insets retain the original scale. Session opening and closing times map to the cumulative number of completed effective sessions. Equal-height stripes mark task types of published work; gray marks no publication. The eight statistical summaries are published at their checkpoints. Figure 4(c) follows one recorded choice in R1. W26 publishes evidence version 1 at 22 completed effective sessions. W15 reads the evidence and the statistical summary at the first checkpoint, computes the four fitted participation means shown, and identifies inference as the task type with the largest fitted participation mean of W15. The recorded reason of W15 also cites adequate evidence published by other workers and the absence of an answer synthesis in progress. W15 uses version 1 of the result of W26 to designate the correct answer B while new sessions remain active, at 32 completed effective sessions. The linked read of the statistical summary, local comparison, recorded choice, and answer illustrate how the statistical summary enters a choice.