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Kairos: A Regret-Aware Native World-Action Model Stack for Physical AI

Kairos Team, Fei Wang, Shan You, Qiming Zhang, Tao Huang, Zuoyi Fu, Zhisheng Zheng, Yunlong Xi, Feng Lv, Xiaoming Wu, Zeyu Liu, Cong Wan, Pu Li, Ruiqing Yang, Xiaoou Li, Wei Wang, Kangkang Zhu, Yuwei Zhang, Shi Fu, Zheng Zhang, Xiaoning Wu, Xuzeng Fan, Dacheng Tao, Xiaogang Wang

arXiv:2606.16533v3cs.AIcs.CV

TL;DR

Physical AI needs world models that preserve information for embodied decisions rather than simulate every future pixel, while current evaluations do not directly establish closed-loop regret reduction. Kairos addresses this with intervention-structured data, a unified understanding–generation–prediction stack, and deployment-aware inference, achieving strong performance across the evaluated capability and efficiency settings.

  • Problem

    Physical AI requires control-sufficient states for action consequences, failure risk, and deployment constraints, but existing evaluations do not directly validate real-robot closed-loop regret reduction.

  • Method

    Kairos organizes heterogeneous experience by intervention strength and unifies understanding, generation, prediction, temporal state maintenance, and deployment-aware inference around a shared world-action state.

  • Results

    Kairos achieves strong performance across embodied world-model, world-action, general world-model, long-horizon generation, and inference-efficiency evaluations.

  • Takeaways & Limitations

    Kairos establishes model-side, data-side, memory-side, and inference-side prerequisites for future regret-aware Physical AI.

  • Takeaways & Limitations

    Direct evidence of reduced real-robot closed-loop regret, including rollout correlation, failure prediction, safety filtering, recovery learning, and policy improvement, remains future work.

Abstract

from arXiv · show

We introduce \textbf{Kairos}, a regret-aware native world-action model stack for Physical AI. Kairos is motivated by the view that a physical world model should not aim to fully simulate all future pixels, but should learn and maintain the information most relevant to embodiment control: object state, spatial relations, contact conditions, task progress, action consequences, failure boundaries, and deployment uncertainty. Kairos establishes three model-side prerequisites toward this goal. First, it \textbf{learns} control-relevant information through a \textbf{Cross-Embodiment Data Curriculum}, which organizes open-world videos, human behavioral data, and robot interactions into an intervention-strength progression from passive physical observation to intentional behavior and embodied action grounding. Second, it \textbf{maintains} control-sufficient states through a unified \textbf{understanding, generation, and prediction architecture} equipped with \textbf{Hybrid Linear Temporal Attention}, where local, mid-range, and global temporal pathways support multi-timescale state maintenance under efficient inference. Third, it \textbf{deploys} these states through a \textbf{Deployment-Aware System Co-Design}, treating latency, memory footprint, and hardware compatibility as first-order constraints for future observation, action, and feedback loops. Experiments on embodied world-model benchmarks, world-action benchmarks, long-horizon generation, and inference-efficiency evaluation show that Kairos achieves superior performance while offering a favorable efficiency to capability trade-off.

1 Introduction

Kairos frames Physical AI world modeling around control-sufficient states rather than complete visual simulation, linking state learning, maintenance, prediction, and deployment. Its stack combines intervention-structured data, unified world-action modeling, multi-timescale temporal memory, and deployment-aware inference.

  • Motivation: Physical AI requires states that preserve control-relevant information, action consequences, failure boundaries, and deployment uncertainty beyond visually plausible future simulation.The proposed state supports action selection, risk assessment, failure anticipation, and recovery planning while discarding irrelevant visual detail.
  • Motivation: Kairos defines regret as the excess physical cost induced when future planning uses compressed state Zt instead of the full observation–action history.Because history-based planners can emulate state-based planners, the resulting regret is nonnegative under the same admissible planner class.
  • System design: Kairos connects existing world-model capabilities to Physical AI requirements through a full-stack design rather than optimizing representation, generation, interaction, or deployment separately.The architecture is intended to move beyond static generative modeling toward a deployable world-action substrate.
  • Learning control-relevant information: The Cross-Embodiment Data Curriculum progresses from passive videos to intentional human behavior and embodied robot interaction, organizing experience by intervention strength.This structure addresses the mismatch between broad but ungrounded open-world data and scarce but actionable robot interaction.
  • Maintaining control-sufficient states: Kairos unifies understanding, generation, and prediction around a shared control-sufficient state, using hybrid temporal pathways to maintain relevant variables over multiple timescales.Understanding extracts abstractions, generation regularizes plausible future evolution, and prediction maps the shared state to future action and visual trajectories.
  • Deploying control-sufficient states: Deployment-Aware System Co-Design treats latency, memory, communication, and hardware constraints as modeling requirements for observation–action–feedback loops.The stack integrates runtime optimization, memory efficiency, hardware-aware execution, quantization, and scalable inference.

2 Model

Kairos centers its Physical AI architecture on a shared control-sufficient state rather than complete future-world simulation. Understanding constructs this state, Generation regularizes it through plausible futures, and Prediction jointly models future states and actions.

  • Kairos compresses heterogeneous observations and actions into a control-sufficient state retaining task-relevant physical, semantic, action, safety, and uncertainty variables.
  • Understanding, Generation, and Prediction are three interfaces to one shared world-action state, not independent modules.Understanding constructs Zt; Generation models plausible future observations; Prediction maps Zt to future state-action trajectories and executable actions.
  • World Generation regularizes the shared state toward physical plausibility, object permanence, task consistency, and long-horizon temporal coherence.
  • World Prediction couples future visual dynamics with future action tokens because the agent’s actions and future world states are interdependent.
  • Joint video-action modeling supports efficient action-only inference while remaining a model-side prerequisite for future regret-aware control.Direct improvement in real closed-loop behavior remains unvalidated.

2.2 Efficient Diffusion Transformer with Hybrid Linear Attention

Kairos replaces quadratic full-sequence temporal attention with hybrid local, mid-range, and global pathways designed to maintain control-relevant state over long horizons. The architecture combines sliding-window attention with gated linear memory for efficient recurrent generation and persistent causal context.

  • Standard DiT self-attention scales quadratically with sequence length, making long videos and high-resolution embodied observations computationally prohibitive.
  • Hybrid Linear Temporal Attention assigns local dynamics to SWA, mid-range dependencies to DSWA, and global causal memory to GLA.These pathways target motion and contact, interaction history and subtask transitions, and persistent scene-level information respectively.
  • GatedDeltaNet uses delta updates to remove outdated key–value associations and a decay gate to control forgetting of previous memory.The updated associative memory produces the output through retrieval with the current query.
  • GLA provides linear-complexity long-range propagation and preserves information such as object permanence, task progress, delayed effects, and failure history.
  • SWA and DSWA maintain constant per-step memory while GLA compresses historical context, enabling recurrent long-horizon generation with reduced memory bottlenecks.
  • The modular hybrid blocks support cross-modal integration by combining local spatial-temporal fusion with global cross-modal alignment.

2.3 Theoretical Scope and Analysis of Hybrid Multi-Scale Temporal Memory

The analysis formalizes why persistent temporal memory is necessary when long-horizon targets depend on events outside a recent window, and motivates a hybrid local, mid-range, and global architecture. Under stated assumptions, this hybrid design bounds long-horizon prediction error, but the theory does not establish real-robot closed-loop performance.

  • Necessity of Persistent Internal States: Persistent state is necessary whenever future targets depend on information outside the recent observation–action window.Theorem 1 gives positive excess risk when the full-history predictor cannot be recovered from the recent window.
  • Necessity of Persistent Internal States: Local temporal smoothness can remain visually plausible while long-horizon predictions fail after influential causes leave the context window.Examples include occluded-object identity, previous subtask outcomes, and prior contact states.
  • Approximate Sufficiency of Hybrid Multi-Scale Temporal Memory: Under branch-wise approximation error ε and contractive global-memory updates with factor ρ, the hybrid predictor satisfies an asymptotic long-horizon excess-risk bound.The bound attributes error to approximation quality and global-memory perturbation, with decoder and global-pathway Lipschitz constants L and LG.
  • Approximate Sufficiency of Hybrid Multi-Scale Temporal Memory: Contractive gated updates prevent global-memory perturbations from accumulating arbitrarily, with asymptotic error bounded by ¯ξ/(1 −ρ).Local and mid-range branches capture shorter temporal structures, while GLA propagates persistent causal state under bounded drift.
  • Interpretation for Kairos: The theory motivates persistent memory and separate SWA, DSWA, and GLA pathways for short-range, mid-range, and supra-window dependencies.Its scope is limited to bounding prediction error under stated assumptions; complementary real-robot validation remains future work.

3 Native Pretraining Paradigm for Physical AI

Kairos frames native pretraining as a curriculum over intervention strength, progressing from passive physical priors through intentional behavior to robot-grounded action–outcome modeling. The three-stage pipeline is designed to construct a control-sufficient state that supports counterfactual prediction, regret-aware alignment, and long-horizon embodied reasoning.

  • Cross-Embodiment Data Curriculum: Kairos organizes pretraining by intervention strength: open-world video, human-centric behavior, and robot trajectories progressively move learning toward action–outcome causation.CEDC is motivated by robotics’ interventional distribution shift, where agent actions change future data distributions.
  • Native Pretraining Paradigm: Together, the stages produce a world-action state that is increasingly control-sufficient rather than merely visually rich.The resulting state is intended to preserve variables relevant to physical cost, action consequences, task progress, safety, and recovery.
  • Stage I: Physical Pretraining: Stage I builds broad spatial–temporal and physical priors from large-scale open-world image and video data.The curriculum begins with passive physical observation before introducing intentional and embodied intervention.
  • Stage II: Embodied Pretraining: Stage II adds instruction-conditioned task structure through human-centric and robot-centric visual experience, increasing task-relevant information in Zt.Human behavior supplies intentional structure but does not directly provide final robot action grounding.
  • Stage III: Regret-Aware World-Action Training: Stage III aligns physical and intentional priors with executable robot actions, execution outcomes, safety risks, recovery costs, and regret-relevant preferences.Joint world-action training couples visual evolution and action prediction through a shared trajectory state, supporting counterfactual action candidates from the same Zt.
  • Native Pretraining Paradigm: Long-horizon temporal context is treated as necessary for maintaining object permanence, task progress, delayed effects, and failure history in extended sequences.The training infrastructure supports 720P and 15-second video settings.

4 Data

Kairos treats data as a control-sufficient data engine: value depends on information gained about control-relevant variables, not only video scale or visual quality. Its pipeline combines heterogeneous experience coverage with control-oriented curation, tagging, captioning, and efficient processing, while identifying alignment and closed-loop validation as future needs.

  • Data Value and Control Information Density: Control information density measures information gain about control-relevant variables per unit data cost, including acquisition, annotation, compute, safety, and deployment costs.The concept is a data-side proxy for selecting experience that improves the control-sufficient state and regret-relevant decision making.
  • Data Value and Control Information Density: Kairos prioritizes near-boundary failure and recovery data, followed by near-boundary successes, contact-rich data, ordinary successful trajectories, and ordinary observation videos.These categories differ in diagnostic value for failure boundaries, recovery, contact dynamics, and action consequences.
  • Data Value and Control Information Density: Scale remains necessary for coverage and physical priors, but massive low-density observation data may leave models weak at failure anticipation, action–outcome reasoning, and safety filtering.Kairos therefore combines large-scale collection with filtering, tagging, captioning, and data-engineering infrastructure.
  • Data Collection: Kairos combines public and proprietary datasets to cover passive observation, human intervention, and robot intervention regimes rather than merely maximizing video count.The collection strategy is organized around the experience regimes required by CEDC.
  • Curation, Tagging, and Captioning: Control-oriented tagging and captioning aim to describe actions, state changes, risks, and causal relations, turning raw video into supervision for control-state learning.Tagging improves caption completeness and alignment with video content, while captioning is interpreted as control-state supervision.
  • Data Engineering Infrastructure: 30× end-to-end throughput improvement was achieved on a single machine with 8×4090 GPUs through parallelization, I/O optimization, and task scheduling.The reported improvements include distributed load balancing, concurrent decoding, zero-copy streaming, pipeline overlap, batching, and dynamic resource allocation.
  • Limitations and Future Directions: Future data development must address weak alignment among internet videos, human-centric data, and robot data, while directly validating imagined–real rollout correlation and closed-loop regret reduction.The paper also flags visual-quality metrics, caption hallucinations, and incomplete causal explanations as insufficient or unreliable without physical calibration.

5 Inference

Kairos treats inference as a deployment-critical mechanism for converting control-sufficient states into timely observation–action–feedback decisions. It combines rollout refinement, prompt self-alignment, low-step distillation, and hardware-aware optimization, while current self-evolution remains a proxy pending real-robot validation.

  • Deployment-aware inference: Inference must deliver control-relevant information within latency, memory, communication, and hardware constraints, rather than merely generate visually plausible futures.These constraints determine whether the model can support action selection, risk assessment, failure anticipation, and recovery planning before execution deadlines.
  • Self-evolution: Kairos’s rollout–evaluation–refinement loop generates candidate futures, evaluates them for physical plausibility, task progress, and risk, and refines subsequent decisions.Prompt Self-Alignment instantiates this paradigm by using the understanding module to score and rewrite prompts automatically.
  • Self-evolution: Prompt Self-Alignment is a prompt-level proxy that steers rollouts toward control-relevant variables such as object identity, contact relations, goals, actions, constraints, viewpoints, and state transitions.The paper explicitly distinguishes this mechanism from aesthetic prompt enhancement and notes that real-robot policy improvement remains unvalidated.
  • Efficient generation: A 4-step distilled generator preserves much of the teacher’s spatial structure, motion dynamics, physical interactions, visual quality, and temporal consistency while substantially reducing sampling cost.The qualitative results report stable realistic motion and largely preserved fine-grained details and coherent object trajectories.
  • Deployment-aware optimization: Kairos maintains strong generation performance on professional and consumer GPUs, with 480P generation reaching real-time throughput on NVIDIA A800 under the 4-GPU setting.Efficient memory utilization supports longer videos and higher-resolution content across hardware platforms.
  • Efficiency comparison: Kairos-4B achieves the lowest latency across tested 480P/720P and 5s/10s/15s scenarios, with 28×–85× lower latency than Cosmos-Predict2.5-14B.It also provides 6×–23× advantage over Cosmos-Predict2.5-2B, 2.5×–3.7× speedup over Wan2.2-5B, and linear scaling as workload increases.

6 Evaluation Results

Kairos shows strong performance across embodied world-model, world-action, physical reasoning, and long-horizon evaluations, including competitive results at 4B parameters. The evaluation also frames these results as proxy evidence rather than direct validation of real-robot closed-loop regret reduction.

  • Embodied World-Model Benchmarks: Kairos-4B achieves strong performance across embodied benchmarks despite its compact 4B parameter scale.The reported results support physical and instruction-grounded priors relevant to embodied world modeling.
  • Embodied World-Model Benchmarks: 9.30 total score on the WorldModelBench robot subset, with 2.36 Instruction Following and 4.96 overall Physics Adherence.Kairos-4B matches the 16B Cosmos3-Nano on Instruction Following and reaches perfect scores in Newtonian mechanics, fluid dynamics, and gravity.
  • Embodied World-Model Benchmarks: 0.618 AVG_Score on DreamGen Bench ranks first, while 0.538 AVG_PA also leads and 0.698 AVG_IF ranks second.These results are reported despite Kairos-4B having only 4B parameters and substantially larger competitors being included.
  • Embodied World-Model Benchmarks: 88.59 Domain Score and 82.57 Overall Score make Kairos-4B the best small-scale model on PAI-Bench TI2V.It remains essentially on par with the 16B Cosmos3-Nano, which scores 82.62 overall, while outperforming several larger models.
  • World-Action Benchmarks: 96.9 on RoboTwin 2.0 Clean and 96.1 average success rate demonstrate strong world-action benchmark performance.MotuBrain achieves the best Randomized score of 96.1, while Kairos leads on Clean and average success rate.
  • World-Action Benchmarks: Kairos achieves state-of-the-art performance on LIBERO-Plus, indicating robustness across perturbed evaluation settings and environmental variations.The benchmark emphasizes scene-level generalization, visual distribution shift, compositional manipulation reasoning, and long-horizon policy stability.
  • Ablation Studies: Jointly denoising future video and action tokens improves performance from 89.0 to 90.8.The ablation attributes this gain to coupling generation and prediction through joint attention.

7 Related Work

Related work progresses from pixel-level generation and latent dynamics toward interactive and unified world-action models. Efficient temporal architectures address the computational demands of long video sequences, while WAMs connect future-state prediction with action modeling.

  • Generative World Models: Generative world models synthesize temporally coherent visual futures, extending diffusion and autoregressive video-generation paradigms to large-scale world simulation.These approaches include pixel-space rendering and sequence modeling over discrete latent tokens.
  • Representational World Models: Latent-dynamics world models learn compact environment representations for predicting future states and training agents within imagined trajectories.Early systems such as World Models, PlaNet, and Dreamer established this simulated-interaction paradigm.
  • Interactive World Models: Interactive world models condition generation on actions or control signals to combine perception, dynamics modeling, and controllable environment generation.Examples include GAIA-1, Genie, and Cosmos.
  • World Action Models: World Action Models jointly model future states and actions, addressing the short-sightedness and weak physical grounding of reactive Vision-Language-Action policies.Cascaded WAMs use intermediate future representations before action extraction, whereas joint WAMs co-model decisions and dynamics in a shared space.
  • Efficient Temporal Modeling: Standard self-attention costs O(N^2) time and memory, motivating linear, sparse, recurrent, and state-space alternatives for long high-resolution sequences.Linear Transformers, Performer, RetNet, and Mamba reduce or restructure attention computation for scalability.

8 Conclusion and Future Works

Kairos presents a regret-aware world-action stack that learns, compresses, maintains, and deploys control-relevant information rather than simulating all future pixels. Its results are encouraging across several proxy capabilities, but direct closed-loop regret reduction remains future work.

  • Core Principle: Kairos defines a control-sufficient world-action state that preserves information needed for prediction, action selection, risk assessment, failure anticipation, and recovery planning.The state focuses on task-relevant variables rather than complete visual simulation.
  • Learning: The Cross-Embodiment Data Curriculum orders open-world videos, human-centric data, and robot interactions by increasing intervention strength.The progression moves from passive physical observation to intentional behavior and embodied action grounding.
  • Maintaining: The Native Understanding–Generation–Prediction Architecture exposes understanding, generation, and prediction as interfaces to a shared world-action state.Multimodal history, instruction, robot state, and physical context construct the state, while imagination and prediction use it.
  • Maintaining: Hybrid Linear Temporal Attention maintains control-relevant variables across local, mid-range, and global timescales.Sliding-window, dilated, and gated linear pathways target contact changes, subtask transitions, and persistent memory.
  • Deploying: Deployment-Aware System Co-Design treats latency, memory, communication, and hardware constraints as prerequisites for observation–action–feedback loops.Distillation, optimization, mixed parallelism, quantization, caching, and action-only pathways improve deployment readiness without proving closed-loop control.
  • Data Engine: The Data Engine for Control Information Density structures collection, curation, annotation, and processing around action consequences, contact dynamics, failures, recovery, and safety risks.Its intended value is control information density rather than raw data scale alone.
  • Evaluation: Kairos achieves strong results across embodied world-model, world-action, general world-model, long-horizon-generation, and inference-efficiency evaluations.The reported capabilities include physical plausibility, instruction grounding, joint prediction, long-horizon consistency, and deployment-oriented efficiency.
  • Future Work: Direct validation remains open for imagined–real rollout correlation, counterfactual action prediction, failure anticipation, safety filtering, recovery learning, and policy improvement in real robots.The report characterizes current evidence as component-level support rather than direct measurement of representation-induced regret.

A Contributors

The report lists advisors, project leads, core contributors, and contributors for Kairos.

  • Advisors: Dacheng Tao and Xiaogang Wang are listed as advisors.
  • Project Leads: Fei Wang, Shan You, and Qiming Zhang are listed as project leads.
  • Core Contributors: Tao Huang and Zuoyi Fu are listed as core contributors.
  • Contributors: Zhisheng Zheng, Yunlong Xi, Feng Lv, Xiaoming Wu, Zeyu Liu, Cong Wan, Pu Li, Ruiqing Yang, Xiaoou Li, Wei Wang, Kangkang Zhu, Yuwei Zhang, Shi Fu, Zheng Zhang, Xiaoning Wu, Xuzeng Fan are listed as contributors.

B Theoretical Analysis

The theoretical analysis establishes why bounded recent windows can fail for long-horizon targets and gives conditions under which hybrid multi-scale memory achieves controlled excess risk.

  • Necessity Result: A finite recent window incurs strictly positive irreducible excess risk whenever the Bayes-optimal predictor depends on history outside that window.This establishes the necessity of a persistent internal state for such long-horizon targets.
  • Sufficiency Result: A hybrid multi-scale temporal memory has an explicit excess-risk bound when the Bayes predictor factorizes into shared, short-range, mid-range, and contractive global-memory branches.The bound depends on branch-wise approximation errors and a geometrically discounted global-memory perturbation term.
  • Architectural Interpretation: The analysis links short- and mid-range pathways to localized and intermediate motion while persistent global memory propagates supra-window context with controlled drift.This formal correspondence supports the architectural rationale for Hybrid Linear Temporal Attention.

B.1 Problem Setup and Theoretical Scope

The setup compares prediction from complete observation–action history with prediction from a bounded recent window, then defines persistent and exact sufficient internal states. The stated scope is architecturally general, while motivating a four-component temporal factorization.

  • Theoretical scope: The subsequent necessity theorem uses only process-history filtrations, while the sufficiency theorem introduces shared, short-range, mid-range, and global memory components.This factorization is the theoretical counterpart to the unified understanding–generation–prediction substrate and its temporal pathways.
  • History and prediction information: The complete history includes observations through time t and prior actions, while the recent window retains only the latest w steps.The associated sigma-fields represent information available from these observation–action histories.
  • History and prediction information: Full-history and recent-window predictors are defined through their respective information sets and optimal risks.The framework uses conditional prediction under the history-generated and window-generated sigma-fields.
  • Internal states: A persistent internal state recursively propagates compressed historical information instead of recomputing predictions from a bounded local context.Its update is represented by measurable maps Φ_t, with an initial state M_1.
  • Internal states: An exact sufficient state recursively updates and decodes precisely the historical information relevant to predicting the target.The definition requires measurable state transitions and a measurable decoder.

B.2 Necessity of Persistent Internal States

The section formalizes why bounded recent windows can miss earlier causes that determine long-horizon outcomes. When the full-history Bayes predictor depends on supra-window information, persistent state is necessary and local predictors incur irreducible excess risk.

  • Problem: Long-horizon futures can differ despite identical recent observations and actions because earlier occlusions, delayed effects, or task stages remain relevant.These examples motivate dependence on historical information outside the recent window.
  • Necessity theorem: Theorem 3 establishes that supra-window dependence necessitates a persistent internal state.The result is framed as an information requirement rather than a restriction on a particular neural architecture.
  • Risk gap: The excess risk of restricting prediction to the recent window is given by an exact identity involving the discrepancy between full-history and window-conditioned predictors.The identity makes the penalty quantitatively measurable rather than merely qualitative.
  • Risk gap: Whenever the full-history predictor depends on information outside the recent window, every bounded-window predictor incurs irreducible excess risk.In world modeling, this corresponds to future consistency requiring persistent state beyond the visible short-range past.
  • Practical interpretation: If the historical cause of a future target cannot be recovered from the recent window, purely local temporal models have a nonzero prediction gap regardless of optimization quality.The target may be a future coordinate, object-identity variable, collision outcome, or task-progress indicator.

B.3 Approximate Sufficiency of a Hybrid Multi-Scale Temporal Memory

The sufficiency analysis models a hybrid multi-scale temporal memory whose shared, short-range, mid-range, and global components approximate the Bayes predictor. Stable gated global updates complement local and mid-range pathways for long-horizon state maintenance.

  • Sufficiency principle: The hybrid factorization is approximately sufficient when the Bayes predictor admits the corresponding decomposition and the global recurrent memory evolves stably.The result formalizes complementary temporal roles rather than relying only on local context.
  • Multi-scale decomposition: The exact decomposition contains a shared predictive representation, short-range state, mid-range state, and global recurrent causal memory.These components are combined through a measurable decoder.
  • Architecture correspondence: The implemented architecture maps short- and mid-range components to SWA and DSWA, and the global component to the GLA pathway.The shared representation is reused across understanding, generation, and prediction tasks.
  • Global memory update: The global-memory branch uses a gated delta update with decay gate α, writing strength β, current value v, and key k.The update combines decay with a key-conditioned correction to the state.

Then the exact and learned global memories satisfy

This subsection introduces approximation-error quantities for the non-global branches and separately defines one-step gate approximation errors. These quantities assess how closely learned components match their corresponding ideal temporal components.

  • Approximation errors: Non-global branch errors quantify approximation quality for the shared predictive substrate and the mid-range branch.The definitions distinguish component-level approximation errors within the hybrid predictor.
  • Approximation errors: One-step gate approximation errors are defined separately for the gated global-memory update.This isolates errors introduced during individual memory-update steps.

Also define the initial global-memory discrepancy

The analysis establishes contraction and approximate sufficiency for Kairos’s hybrid multi-scale temporal memory under stated decomposition, Lipschitz, and contraction assumptions. It bounds long-horizon prediction degradation through branchwise approximation and controlled global-memory perturbations.

  • Contraction: The gated delta update is contractive when α + β∥k∥2 < 1, yielding a contraction factor ρ < 1.The resulting Frobenius-norm discrepancy contracts between any two memory states.
  • Theorem 4: Theorem 4 assumes a Bayes-predictor decomposition, coordinate-wise Lipschitz decoding, and the contraction conditions of Lemma 3.Under these assumptions, the theorem derives global-memory and long-horizon excess-risk bounds.
  • Global-memory error bound: The global-memory branch has a bounded asymptotic error governed by the worst-case one-step perturbation and geometric damping.The asymptotic contribution is bounded by LG¯ξ/(1 −ρ), while prior memory discrepancies are discounted over time.
  • Long-horizon excess-risk bound: The hybrid predictor’s long-horizon excess risk is controlled by non-global approximation error together with the contractively accumulated global-memory term.This gives approximate sufficiency when the Bayes predictor factorizes across the four architectural roles.
  • Exact sufficiency: Exact sufficiency holds in the realizable case when the learned hybrid state recovers the Bayes decomposition at every step and the initial global-memory state is aligned.Under these conditions, εU_t = 0 for all t and e0 = 0.
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