Causal-fate dynamics of unrealized influence

arXiv:2610.11422 · eess.SY, cs.LG, cs.SY · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Causal-fate dynamics of unrealized influence".

Dev: The gist: Causal-fate dynamics provide a minimal language for a form of history that is absent from the present trajectory yet remains consequential for future evolution.

Rosa: First, who's behind it and why it matters.

Title and authors: Dev: So, to unpack what this paper is actually doing, it’s taking the traditional way we describe systems—which is focusing only on what has already happened—and adding a layer for what hasn't been executed yet but still carries weight. The authors are proposing this "causal-fate dynamics" structure.

Rosa: Think of it like this: instead of just looking at the forces that have acted, they’re trying to model the consequences that are latent, waiting to be triggered by future dynamics. They introduce an exact finite-transport representation for this influence.

Taro: That sounds like a way to handle situations where the system generates something new—a potential influence—and we need a mechanism to track it even if it doesn't manifest right now. It moves us away from just looking at the realized state as complete.

Dev: Right, and they formalize this using equations from one to six, showing how an influence can be generated with a definite counterfactual consequence that gets partially realized in the present but stays as a latent consequence waiting for future dynamics to transform it.

Rosa: It seems like they’re trying to give a mathematical language for history that isn't just the sequence of events already lived, but also the potential consequences of paths not taken.

Taro: If we can model that, it could be useful when an autonomous agent is in a situation where its immediate actions don't fully determine the final outcome because there are unexecuted influences influencing things down the road.

Dev: That’s where they try to bridge the gap between what we see happening now and what might happen later, by explicitly accounting for that unrealized part.

The paper's summary: Rosa: So, looking at the summary of "Causal-fate dynamics of unrealized influence," the core idea is that influences generated in a dynamical system aren't always exhausted by the time we look at what has already happened. These consequences are often treated as just absent or stuck somewhere, which makes it hard to see how they affect future evolution.

Dev: The paper proposes causal-fate dynamics as a way out of that problem. It suggests that an influence can be realized now, stay latent, or even get transformed by subsequent dynamics before it ever fully enters the realized trajectory again. They give us an exact finite-transport representation when you specify the relevant maps in your system.

Taro: I see how that relates to network constraints or selective interactions where what you generate might be blocked or redistributed instead of being directly executed as expected. It's not just a simple delay; it’s a complex transformation process.

Rosa: The authors use a model of *Caenorhabditis elegans*, a worm, to test this idea biologically. They found that while local autonomy dynamics didn't capture the cross-neuronal propagation structure we expected, freezing the selective realization trajectory actually preserved that main propagation structure with a relative error of three point five one four nine nine percent.

Dev: So, even though the biological results weren't perfect, this finding motivates a hypothesis about carriers of unresolved inter-neuronal influence that might still contribute to later propagation in living animals. It’s an observation that points toward a mechanism we need to investigate further in biology.

Taro: That persistence idea is key for me. If something from the past, an unrealized influence, can reappear later as part of a structure, it means the system's history has a much longer reach than just the immediate preceding state suggests.

Rosa: It’s about showing that history isn't just a static record; it’s dynamic potential. The next step they tackle is how to formalize this persistence mathematically, which leads into their representation equations and how the total influence available at any step is defined by combining transport and newly generated influence.

The paper's improvements: Dev: When we look at the proposed improvements in "Causal-fate dynamics of unrealized influence," they are essentially tightening up how this latent evolution is constrained. They move from just looking at what has happened to defining a complete state that includes the aggregate latent contribution of all underlying channels.

Rosa: They introduce a complete state, denoted as z t, which is made up of the realized state x t and this aggregate latent displacement t. This whole thing evolves exactly under the fully realized map, z t+one = F t(z t), which gives a direct counterfactual meaning to that latent part <ref:2610.11422#pg1>.

Taro: That's interesting because it means the aggregate latent displacement isn't just some messy leftover; it’s precisely what completes the realized state into the fully realized reference state, if you know all the underlying dynamics. It gives that latent part a concrete role in closing the loop of realization.

Dev: They also prove something important with their predictive-state necessity theorem, which states that any exact predictive representation absolutely has to retain history-dependent information that you simply can't get just from the realized state alone. This backs up the idea that history matters beyond what is currently visible in the system's state.

Rosa: It seems like they’re arguing that if you want an accurate prediction, you have to carry forward this historical dependence, even if it means tracking these latent elements explicitly through subsequent computation.

Taro: So, this moves the focus from just observing what happens to being able to design systems that explicitly carry and selectively realize this unrealized influence based on some rule. That’s a big step for autonomy research.

Dev: Exactly. It shifts the goal from just modeling the past trajectory to designing systems that can actively manage these potential futures by controlling what gets realized at each step of computation, which is what they call selective realization.

Conclusion: Rosa: So wrapping up this discussion on "Causal-fate dynamics of unrealized influence," the authors show us that we can formulate a way to track influences that aren't immediately realized but still have future relevance in a system. They’ve given us the framework for causal-fate dynamics and shown how it applies across different fields, from neuroscience to routing protocols.

Dev: The main implication is that if you want an exact predictive representation of a system, you have to keep track of this history-dependent information that isn't in the current state, which is what the predictive-state necessity theorem points toward. This gives us a tool for understanding why some systems behave differently than just looking at their immediate actions.

Taro: For me, I think the biological hypothesis from the *C. elegans* model is really interesting because it suggests that unresolved influences might be a real feature of complex living systems, not just an artifact of our simplified models. It’s a direction for future autonomy research to explore how those persistent influences work in messy, real-world interactions.

Rosa: I agree with Taro on that; the potential for carriers of unresolved influence is something worth looking into further outside of controlled lab settings. The paper gives us a way to hypothesize about unfinished influence and examine its predictive signatures.

Dev: We’ve also seen how this concept applies observationally in internet routing, where a dynamically updated state retains information about future local route changes beyond the current local state, which is compatible with this idea of future-relevant history.

Taro: It sounds like this framework provides a way to design systems that can explicitly carry and selectively realize these unrealized influences through subsequent computation. That’s the actionable part for us in autonomy work.

Rosa: So, "Causal-fate dynamics of unrealized influence" gives us a structure to hypothesize about unfinished influence, look at its predictive signatures, and design systems that actively manage what gets realized by carrying and selectively realizing it. It’s a new lens for how we think about system evolution.

Yiwei Liu, Luwei Yang, Shunbo Lei

School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen · Shenzhen Research Institute of Big Data (SRIBD)

eess.SY, cs.LG, cs.SY

Submitted: 2026-10-08

Updated: 2026-10-08

Code: https://github.com/francescorandi/wormneuroatlas

The gist: The gist: Causal-fate dynamics provide a minimal language for a form of history that is absent from the present trajectory yet remains consequential for future evolution.

Key concepts

Causal-Fate Dynamics: This is a way to describe history where influences not yet realized remain consequential for the future. Instead of focusing only on what has happened, it looks at the 'unexhausted consequence' of non-realization that gets transported and transformed by subsequent system dynamics.
Latent Consequence: This refers to an influence generated in a system that is not fully realized in the current state. It is stored as a hidden or latent state, which can be transported through the system's evolution and eventually re-enter the realized trajectory later on.
Finite Transport: This mathematical constraint limits how latent evolution occurs, ensuring that the aggregate latent displacement has a direct counterfactual meaning. It defines exactly how much of the unexhausted influence is available for realization at any given moment.
Selective Realization: This is a mechanism where a system chooses which influences to realize in its current state. The paper investigates whether this selective process can preserve important propagation structures, even when local autonomy fails to reproduce them.

Terminology

Transformer — 34× in this paper · explained in 2 episodes
Perplexity — 12× in this paper · explained in 3 episodes
Calibration — 7× in this paper · explained in 7 episodes
Robustness — 6× in this paper · explained in 5 episodes
Embeddings — 5× in this paper · explained in 2 episodes
Transformer Architecture — 4× in this paper · explained in 2 episodes
In Border Gateway Protocol — 2× in this paper
Shenzhen Research Institute of Big Data — 2× in this paper
Ablation — 1× in this paper · explained in 1 episode
Admissibility — 1× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Diffusion Model — 1× in this paper · explained in 8 episodes
Fine-Tuning — 1× in this paper · explained in 5 episodes
Frozen model — 1× in this paper · explained in 3 episodes
Hilbert Space — 1× in this paper · explained in 2 episodes
Ontology — 1× in this paper · explained in 2 episodes
Structural Constraints — 1× in this paper · explained in 2 episodes

Summary

The gist: Causal-fate dynamics provide a minimal language for a form of history that is absent from the present trajectory yet remains consequential for future evolution.

Introduction to Causal-Fate Dynamics

Dynamical systems are typically described in terms of what has already been realized, such as forces that have acted or decisions that have been executed The central question is therefore what becomes of a consequence that is absent from the realized present but not yet exhausted from future evolution What persists across time is not necessarily the unrealized action itself, nor a fixed backlog awaiting eventual release, but rather it is the still-unexhausted consequence of non-realization This perspective formalizes causal-fate dynamics where an influence may be generated with a definite counterfactual consequence, partially realized in the present, retained as a latent consequence, transported by subsequent dynamics and transformed before it re-enters the realized trajectory.

Formal Representation of Causal-Fate Dynamics

The paper establishes a channel-level causal-fate representation using equations (1) through (6) to describe how influence absent from the realized present can nevertheless remain part of the system’s dynamical history. The total influence available through a channel at step t is defined as hc,t = Tc,t(xt, ξt) + ϕc,t(xt), where Tc,t is the transport operator and ϕc,t(xt) is the newly generated influence. The recurrence in Eq. (4), ξc,t+1 = hc,t − ψc,t = Tc,t(xt, ξt) + ϕc,t(xt) − ψc,t is a channel-level balance relation. The dynamically relevant state of the resulting representation is denoted by yt = col(xt, ξt), where xt is the realized state and ξt collects the channel-level latent states.

Finite Transport and Counterfactual Meaning

The paper constrains latent evolution by the finite system map itself, leading to a complete state represented by zt = xt + ¯ξt, where ¯ξt is the aggregate latent contribution of all underlying channels. The relation in Eq. (10), ht = gt + τt = Ft(xt + ¯ξt) − Lt(xt), defines the total influence currently available for realization. The complete represented state therefore evolves exactly under the fully realized map, zt+1 = Ft(zt). This implies that the aggregate latent displacement has a direct counterfactual meaning, as it is precisely the finite displacement that completes the realized state into the fully realized reference state.

Model-Motivated Scientific Hypothesis in Neural Systems

In a connectome-constrained neural model of Caenorhabditis elegans, causal-fate dynamics tests whether selective realization can preserve a propagation structure informed by the public signal-propagation atlas. The results show that while local-autonomy dynamics failed to reproduce the selected cross-neuronal propagation structure, the frozen selectiverealization trajectory preserved the principal propagation structure with a relative error of 3.51499%. This motivates a biological hypothesis about carriers of unresolved inter-neuronal influence that may persist and contribute to later propagation.

Observational Phenomenon in Internet Routing

In Border Gateway Protocol (BGP) routing, a dynamically updated state formed from unmatched cross-observer updates retains information about future local route changes beyond that contained in the current local route state. This observational phenomenon is compatible with future-relevant history under the adopted representation. Specifically, a bounded ordered cross-observer history retained predictive information beyond the current local state, demonstrating that the state evolved with subsequent routing events rather than merely accumulating a static record of the past.

Executable Construction in Transformer Architecture

The paper constructs a Transformer architecture where token-local evolution, latent contextual influence and selective realization form an executable architecture. The fully realized Transformer achieved a test perplexity of 38.6512, while removing newly generated cross-token influence increased perplexity to 1315.2426, demonstrating that cross-token contextual influence is indispensable for the language-modeling function examined. The selective-realization Transformer achieved a perplexity of 39.9138, only 3.27% higher than that of the fully realized model, while realizing 88.40% of token–layer influence opportunities.

Conclusion on Causal-Fate Dynamics

The three studies distinguish a model-motivated scientific hypothesis, an observational phenomenon compatible with future-relevant history and an executable construction for carrying unrealized influence through subsequent computation. Causal-fate dynamics provides a framework for formulating hypotheses about unfinished influence, examining its predictive signatures and designing systems that explicitly carry and selectively realize it. The exact finite-transport relation follows from the construction and its assumptions, while empirical relevance depends on the chosen state, evolution maps and realization rule.

Supplementary Information

The predictive-state necessity theorem proves that every exact predictive representation must retain history-dependent information that cannot be recovered from the realized state alone. Furthermore, aggregate latent displacement is unique if and only if the block embedding operator B is injective, meaning overlapping channel embeddings leave freedom in its channel decomposition. The paper also provides detailed protocols for the C. elegans neural model, BGP routing analysis, and Transformer experiment.

Methodological Details

The neural model utilized a continuous substrate following the graded-potential model structure of Kunert, Shlizerman and Kutz [26], and the selective trajectory was computed using the finite-transport recurrence defined in the Results. The BGP study employed an ordered propagation-front representation that augmented the strong current state with five newest unmatched remote updates. The Transformer experiment implemented the construction by maintaining a realized hidden state together with a latent contextual state, where hl = Fl(Zl) − Ll(Hl) is the candidate contextual influence.

Data Availability and Reproducibility

The code for the C. elegans, BGP and Transformer experiments is maintained at https://github.com/Hotaru366/causal-fate-code. The external resources used in this study are publicly available from their original providers. Study-generated result objects are not deposited in a separate public archive. All results and code details are provided in the machine-readable Supplementary Data.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 52307145, in part by the Shenzhen Basic Research Fund (Natural Science Foundation) under Grants QNXMB20250701091813018 and JCYJ20240813113532042, and in part by the Shenzhen Research Institute of Big Data (SRIBD) under Grant J0022025001.

References

[Sontag, E. D. Mathematical Control Theory: Deterministic Finite Dimensional Systems 2 edn (Springer, New York, 19)] [Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos 2 edn, Vol. 2 of Texts in Applied Mathematics (Springer, New York, 2003)] [Kalman, R. E. A new approach to linear filtering and prediction problems. Journal of Basic Engineering 82, 35–45 (1960)] [Ho, B. L. & Kalman, R. E. Effective construction of linear state-variable models from input/output functions. Automatisierungstechnik 14, 545–548 (1966)] [Koopman, B. O.

Improvements for AI systems

The AI system can maintain an exact finite transport representation of unrealized influence, allowing it to model how generated influence may be realized, remain latent, or be transformed by subsequent dynamics. This enables the system to retain future relevance by explicitly carrying forward consequences that are not yet realized in the current state.
The system can implement a Transformer architecture where each token maintains both a realized hidden state together with a latent contextual state, allowing it to perform language modeling while selectively realizing tokens based on whether their accumulated contextual influence exceeds a threshold, as described by the token-level realization rule.
The AI system can achieve superior language modeling performance by transporting and transforming latent contextual influence, as demonstrated in the Transformer experiment where the Selective realization Transformer achieved a perplexity of 39.9138, only 3.27% higher than that of the fully realized model, indicating that cross-token contextual influence is indispensable for the language-modeling function.
The system can develop more robust predictive capabilities by using an augmented state where the aggregate latent displacement is transported and transformed by the system map, and a realization rule determines what enters the realized trajectory, ensuring that every exact predictive representation must retain history-dependent information that cannot be recovered from the realized state alone.
The AI system can be designed to preserve specific functional structures in complex networks, such as neural propagation, by utilizing a selective realization trajectory which keeps unresolved inter-neuronal influence latent, was transported and transformed through the nonlinear neural dynamics. This supports the hypothesis that living neural systems may contain distributed physical carriers of unresolved inter-neuronal influence.
The system can be adapted for operational environments, such as internet routing, by maintaining a dynamically updated cross-observer history that retains predictive information beyond the current local route state, allowing it to anticipate future local route changes based on an ordered operational routing history.