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Low-Rank Dynamics-Effective Latent Carriers for Counterfactual Rollout in Learned World Models
Yang Liu, Yuming Chen
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
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Summary
This paper investigates the interpretability and controllability of learned world models by identifying low-rank latent carriers that enable counterfactual rollouts. Using a recurrent world model with a 192-dimensional hidden state in a two-object collision environment, the authors demonstrate that a rank-4 hidden-state patch can successfully redirect the model's autonomous future trajectory following a velocity edit. The study distinguishes 'dynamics-effective' interventions from mere output changes by requiring sustained, target-specific future computation without teacher forcing. A position-edit stress test serves as a negative control, showing that not all editable physical quantities constitute dynamics-effective carriers.
Entities (7)
Relation Signals (5)
World Model โ hashiddenstatedimension โ 192
confidence 98% ยท We study a recurrent world model with a 192-dimensional hidden state
Low-Rank Latent Carrier โ enables โ Counterfactual Rollout
confidence 95% ยท A single rank-4 patch at the anchor is sufficient to redirect a 12-step autonomous rollout
Rank 4 โ isminimumrankfor โ Dynamics-Effective
confidence 94% ยท On the registered rank grid, rank 4 is the smallest tested rank that satisfies the full development-panel criteria.
Velocity Edit โ issupportedby โ Low-Rank Latent Carrier
confidence 93% ยท For the tested velocity edits, we find a low-dimensional hidden carrier that can reliably redirect the modelโs future dynamics.
Position Edit โ failstoprove โ Dynamics-Effective
confidence 92% ยท A position-edit stress test provides a negative contrast... Thus, successful editing alone is not enough.
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Abstract
Abstract:World models may predict the future without making clear which parts of their hidden state actually drive those predictions. We ask whether a small, directly addressable hidden-state change can place a learned world model on the intended counterfactual trajectory and then let the model continue that future on its own. We study a recurrent world model with a 192-dimensional hidden state in a controlled two-object, two-dimensional collision environment. For a bounded family of local velocity edits, we first verify that the model can natively represent and roll out the edited future. We then construct candidate low-rank carriers from training-only factual-to-counterfactual hidden differences and learn a map from the factual state and requested edit to carrier coefficients. On the registered rank grid, rank 4 is the smallest tested rank that satisfies the full development-panel criteria. A single rank-4 patch at the anchor is sufficient to redirect a 12-step autonomous rollout, with no future observations, teacher forcing, or repeated correction. The frozen procedure satisfies the preregistered replication rule across independently trained checkpoints and remains usable across nearby intervention times. Random equal-norm, wrong-object, and wrong-time controls do not explain the effect. A position-edit stress test provides a negative contrast: the intended position patch can pass the raw rollout criteria, but no-patch and random controls can pass the same criteria, and wrong-object specificity is not established. Thus, successful editing alone is not enough. We use dynamics-effective to describe an intervention that changes the model's future computation in a sustained and target-specific way under autonomous rollout. The rank-4 result identifies a compact intervention interface for the tested velocity-edit family, not a closed four-dimensional state or an intrinsic state dimension.
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- Source: https://arxiv.org/abs/2608.15156v2
- Canonical: https://arxiv.org/abs/2608.15156v2
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Low-Rank Dynamics-Effective Latent Carriers for Counterfactual Rollout in Learned World Models Yang Liu * , Yuming Chen * College of Intelligent Robotics and Advanced Manufacturing, Fudan University Shanghai, China ly, yumingchen@fudan.edu.cn Abstract World models may predict the future without making clear which parts of their hidden state actually drive those predictions. We ask whether a small, directly addressable hidden-state change can place a learned world model on the intended counterfactual trajectory and then let the model continue that future on its own. We study a recurrent world model with a 192-dimensional hidden state in a controlled two-object, two-dimensional collision environment. For a bounded family of local velocity edits, we first verify that the model can natively represent and roll out the edited future. We then construct candidate low-rank carriers from training-only factual-to-counterfactual hidden differences and learn a map from the factual state and requested edit to carrier coefficients. On the registered rank grid, rank 4 is the smallest tested rank that satisfies the full development-panel criteria. A single rank-4 patch at the anchor is sufficient to redirect a 12-step autonomous rollout, with no future observations, teacher forcing, or repeated correction. The frozen procedure satisfies the preregistered replication rule across independently trained checkpoints and remains usable across nearby intervention times. Random equal-norm, wrong-object, and wrong-time controls do not explain the effect. A position-edit stress test provides a negative contrast: the intended position patch can pass the raw rollout criteria, but no-patch and random controls can pass the same criteria, and wrong-object specificity is not established. Thus, successful editing alone is not enough. We use dynamics-effective to describe an intervention that changes the modelโs future computation in a sustained and target-specific way under autonomous rollout. The rank-4 result identifies a compact intervention interface for the tested velocity-edit family, not a closed four-dimensional state or an intrinsic state dimension. 1 Introduction World models aim to construct an artificial agent with internal mechanisms for anticipating how its environment will evolve. The modern deep-learning formulation was popularized by Ha and Schmidhuber, who showed that an agent could learn a compact internal model of its environment and train its policy by simulating future trajectories within that model (Ha and Schmidhuber 2018). PlaNet subsequently showed that a model could learn latent dynamics directly from images and use them to plan future actions online (Hafner et al. 2019). Dreamer shifted the focus from planning with predicted observations to training the policy directly on imagined future trajectories inside the learned world model (Hafner et al. 2023). More recent systems have pushed the same idea toward much larger and more general interactive environments: Genie learns action-controllable generative environments from unlabeled video (Bruce et al. 2024), while V-JEPA 2 couples large-scale self-supervised video representation learning with an action-conditioned latent world model that supports zero-shot robot planning (Assran et al. 2025). Across these developments, the latent state has changed from an * Equal contribution Source codes and data can be found at https://github.com/lysea8282/dynamic-effective-latent-carriers engineering convenience into a central scientific object. The question is no longer just whether a model can predict the future, but what information its internal state keeps, what it leaves out, and whether the retained information can actually guide future predictions. This distinction has a long history in representation learning for control. DeepMDP formalized the idea that high-dimensional observations can be compressed into a continuous latent state while preserving reward and transition structure relevant to decision making (Gelada et al. 2019). Recent work has sharpened this functional viewpoint. Latent-state design has been framed in terms of distinct sufficiency requirementsโfor prediction, control, planning, memory, grounding, or counterfactual reasoningโrather than as a single architecture-independent notion of a good representation (K.W. Kim 2026). In parallel, work on physically viable world models argues that accurate prediction is not always enough for embodied reasoning. Two situations may look similar under passive observation but respond differently when something is changed. A useful world model should therefore preserve the information needed to tell such cases apart (Thorpe et al. 2026). These arguments place intervention and counterfactual behavior at the center of the state-representation problem. Physical dynamics provide a particularly demanding setting for this question. Benchmarks such as Physion have shown that learned models can acquire useful object-centered structure for predicting collisions, rolling, sliding, and other physical events, while still leaving a substantial gap to human physical prediction (Bear et al. 2021). Yet behavioral success on a benchmark does not determine how the necessary information is organized internally. A model may encode a physical quantity in a readily decodable form without using that readout as a state variable in its own transition computation. Conversely, the computationally relevant state may be distributed over many hidden coordinates and need not resemble the variables of a hand-designed physics engine. The broader scientific analogy is coarse-graining. In statistical physics, one seeks lower-dimensional macroscopic descriptions whose dynamics remain predictive even though the underlying microscopic system contains vastly more degrees of freedom. A recent rigorous example is the derivation of fluid equations from hard-sphere particle dynamics by Deng, Hani, and Ma (Deng, Hani, and Ma 2025). There, however, both the microscopic and macroscopic variables have explicit physical meaning. Learned neural world models pose a different identification problem. Their latent coordinates are learned, distributed, and generally non-identifiable up to changes of basis or other invertible reparameterizations. An effective latent state therefore need not consist of named variables such as position or velocity. It may instead appear as a low-dimensional subspace or local chart whose individual axes have no privileged physical interpretation, provided that manipulating this object changes the modelโs future dynamics in the intended way. Recent interpretability work has begun to expose this distinction empirically. Distributed Alignment Search shows that interpretable causal variables can align with distributed neural representations in non- standard bases rather than with single neurons or predefined axes (Geiger et al. 2024). Work on causal probing further demonstrates that an intervention should be judged not only by whether it changes the target property, but also by completeness and selectivityโwhether the desired representation has been manipulated sufficiently without indiscriminately corrupting other information (Canby et al. 2025). In world and video models, the pace of progress has accelerated. Joseph et al. find that physical information becomes accessible in characteristic regions of large video encoders and that motion direction is represented by a high-dimensional population code rather than a compact physics-engine-style coordinate (Joseph et al. 2026). Zhang reports approximately linear representations of game-state variables in two architecturally distinct learned environment simulators and shows that probe-derived hidden-state shifts can causally alter model predictions (Zhang 2026). Alam further demonstrates that Concept Activation Vectors can steer physical plausibility judgments in a video model when injected at appropriate layers (Alam 2026). Together, these studies establish that physical information can be readable, geometrically organized, distributed, and in some settings causally steerable. These studies still leave an important question open. Changing a hidden state may alter the modelโs next output, its plausibility score, or its predicted distribution, but this does not necessarily mean that the model has entered a new state from which it can generate the correct future on its own. For a world model, this distinction is important because an internal state is useful only if it can support the modelโs later predictions. In counterfactual physical reasoning, the stronger test is therefore not simply whether a hidden direction is related to a physical variable or can push an output in the desired direction. Instead, we ask whether a limited hidden-state intervention can start a new autonomous rollout that remains physically valid, while leaving unrelated parts of the state largely unchanged. In this sense, being able to read information from a representation is weaker than showing that the model actually uses it, and being able to edit a representation is weaker than showing that the edit can change the modelโs future dynamics. In this work, we study this question in a controlled world model with two moving objects. The model uses a 192-dimensional recurrent hidden state and is trained to predict both factual trajectories and locally edited counterfactual trajectories in a deterministic two-dimensional collision environment. Before studying hidden-state interventions, we first check that the model can produce the correct counterfactual future when it is given the fully edited current state. We then ask a more specific question: if we change one component of an objectโs velocity, does the required hidden-state change lie in a small part of the modelโs representation? More importantly, can we make this hidden-state edit directly from the current state and the requested velocity change, without first seeing the counterfactual hidden state? This makes the intervention addressable rather than simply reconstructing a hidden difference that is already known. For the tested velocity edits, we find a low-dimensional hidden carrier that can reliably redirect the modelโs future dynamics. On the preregistered rank grid, rank four is the smallest rank that passes the full set of tests. A single hidden-state patch at the intervention time is sufficient to launch a twelve-step autonomous counterfactual rollout, without future observations or repeated correction. The effect replicates across independently trained checkpoints and remains effective across several intervention times, while control interventions show that it is not simply caused by generic perturbation of the hidden state. A position-edit stress test provides an important negative contrast. Although some position patches improve raw outcome measures, similar effects can also appear under random or no-patch controls, and object specificity is not reliable. We therefore do not treat position editing as evidence for a dynamics- effective carrier. Together, these results show why successful editing alone is not enough: a stronger criterion is whether a targeted hidden-state change can place the model on the intended counterfactual trajectory and allow its own dynamics to carry that future forward. Our contributions are fourfold. First, we introduce an operational test for dynamics-effective hidden-state interventions: a one-shot intervention must be followed by an autonomous, physically valid counterfactual rollout, rather than merely changing an immediate readout. Second, for a bounded family of local velocity counterfactuals, we identify a low-dimensional and addressable hidden carrier that passes this test, including a preregistered rank sweep and replication in independently trained checkpoints. Third, we study how this carrier behaves across rollout time and use zero, random, wrong-object, full-hidden, and temporal reference conditions to test alternative explanations. Fourth, we use the position-edit stress test as a negative contrast, showing that a physically meaningful and editable quantity does not automatically provide evidence for a dynamics-effective latent carrier. 2 Related Work Our work connects three lines of research: how world models construct useful latent states, how neural representations can be probed and causally edited, and how physical structure is represented inside predictive video and world models. We organize the discussion around this progression from sufficiency, to intervention, to physical world-model interpretation. 2.1 Latent state, sufficiency, and action relevance A central goal of latent-state learning is to compress high-dimensional observations without discarding information needed for future behavior. DeepMDP formalized this idea by learning a continuous latent model that preserves reward and transition structure relevant to control (Gelada et al. 2019). In partially observable settings, Huang et al. (Huang et al. 2022) went further and defined action-sufficient state representations: compact states that preserve the information needed for downstream decision making. These works make sufficiency a functional property. A state is useful because it supports a task, not because its coordinates are easy to name. Recent work has made this distinction more explicit for world models. Kim (K.W. Kim 2026) argues that latent states should be judged by the function they must support, such as prediction, control, planning, memory, grounding, or counterfactual reasoning. Thorpe et al. (Thorpe et al. 2026) make a related argument for embodied AI: visually plausible prediction may be insufficient when two systems that look similar respond differently to intervention. Their physically viable world-model framework therefore emphasizes representations that preserve the distinctions needed to answer a particular intervention query. A parallel line of recent work asks which training signals make world-model latents useful for action. Yeom et al. (Yeom et al. 2026) compare several video representation families and find that temporal prediction is more strongly associated with action-relevant latent structure than reconstruction quality alone. Very recent work by Yan et al. (Yan et al. 2026) similarly argues that forward prediction in JEPA- style world models does not automatically make physical state easy to identify or use for planning, and improves downstream performance by adding explicit physical-state grounding during training. These studies strengthen the view that predictive success and task-relevant state structure should be evaluated separately. Our question is complementary. Rather than designing a new latent objective, we test whether an already trained recurrent state contains a compact intervention interface that can support autonomous counterfactual rollout. 2.2 From probing to causal intervention in neural representations Probing methods provide a simple way to ask what information can be recovered from hidden representations. Testing with Concept Activation Vectors (TCAV) uses linear directions in activation space to connect internal representations with human-defined concepts (B. Kim et al. 2018). However, high probe accuracy can be misleading. Hewitt and Liang (Hewitt and Liang 2019) showed that probes must be evaluated against control tasks because a powerful probe can learn structure that is not meaningfully present in the representation. This concern is especially important when a probe result is used to make a claim about how the underlying model computes. Several methods therefore move from readout to intervention. INLP repeatedly projects representations into null spaces that remove linearly decodable information (Ravfogel et al. 2020), while Amnesic Probing uses such removal to test whether an encoded property matters for model behavior (Elazar et al. 2021). LEACE later provided a closed-form linear erasure method designed to remove a concept while changing the representation as little as possible (Belrose et al. 2023). These approaches are mainly subtractive: they ask what happens when information is removed. Other methods study constructive edits. Concept Bottleneck Models expose human-specified intermediate concepts that can be corrected at test time, providing a clean example in which concept editing is built into the model architecture (Koh et al. 2020). Activation engineering shows that adding contrastive activation directions can steer model outputs without retraining (Turner et al. 2023). Interchange Intervention Training instead trains a network so that swapping selected internal representations produces the same effect as an intervention in a target causal model (Geiger et al. 2022). Distributed Alignment Search allows a causal variable to be spread across many hidden dimensions rather than tied to one neuron or coordinate, and searches for subspaces where internal swaps behave like interventions in a high-level causal model. (Geiger et al. 2024). Another literature also makes clear that causal editing needs controls. Canby et al. (Canby et al. 2025) evaluate causal probing methods in terms of completeness and selectivity: an intervention should change the targeted information strongly enough while avoiding broad, unrelated damage. Our work follows this general principle but applies a stricter downstream criterion suited to a world model. A hidden edit is not judged only by an immediate readout or a one-step output. It must start the intended counterfactual rollout, after which the modelโs own transition dynamics must continue that future. This is why random, zero, wrong-object, and full-hidden reference conditions are central to our evaluation rather than optional probe diagnostics. 2.3 Physical reasoning and object-centered dynamics Physical prediction has long motivated structured representations. Interaction Networks showed that learned object- and relation-centered computations can predict dynamics in systems ranging from n-body motion to rigid-body collisions (Battaglia et al. 2016). Later object-centric models continued this direction. SlotFormer models spatio-temporal interactions over learned object slots for long-horizon video prediction and planning (Wu et al. 2022), while OCVP separates temporal object dynamics from interactions between objects (Villar-Corrales, Wahdan, and Behnke 2023). OCK makes kinematic quantities such as position, velocity, and acceleration more explicit in an object-centric dynamics model (Song et al. 2025). These works show that structured state can improve physical prediction, while leaving open how physical information in a learned hidden representation is used by the modelโs own dynamics. A complementary benchmark literature tests whether models behave as if they understand physical regularities, i.e. IntPhys introduced controlled possible-versus-impossible videos for intuitive physics (Riochet et al. 2018). CLEVRER added descriptive, explanatory, predictive, and counterfactual questions around object motion and collision events (Yi et al. 2019). Physion broadened the range of phenomena and showed that object-centric models improve physical prediction while still remaining well below human performance (Bear et al. 2021). IntPhys 2 extends violation-of-expectation tests to permanence, immutability, spatio-temporal continuity, and solidity in more complex scenes (Bordes et al. 2025). These benchmarks are essential for measuring behavioral physical competence, but they generally do not identify the internal state variables used by the model. Recent representation-space work also suggests that intuitive physical regularities can emerge from self-supervised video prediction without an explicit physics engine: Garrido et al. (Garrido et al. 2025), for example, report improved violation-of-expectation performance for representation-predictive video models. This makes the internal question more important, not less. If physical behavior can emerge without explicit physical coordinates, then the relevant state may be distributed and may need to be identified by what it does rather than by whether its axes resemble a simulator state vector. 2.4 Physics representations, steering, and counterfactual consistency The closest recent work begins to directly inspect physical representations inside world and video models. Joseph et al. (Joseph et al. 2026) use layerwise probing, subspace geometry, patch-level decoding, and attention ablations to study physical variables in large video encoders. They find that physical information becomes accessible in characteristic intermediate layers and that motion direction can be represented by a high-dimensional population code rather than by a single compact coordinate. This is important for our setting because it argues against assuming that a physical variable should correspond to one neuron or one probe direction. Zhang (Zhang 2026) studies two architecturally different reinforcement-learning world models and finds approximately linear representations of game-state variables such as object position and score. Shifting hidden states along probe-derived directions changes model predictions, providing evidence that these representations are functionally used. Alam (Alam 2026) similarly shows that a probe-derived Concept Activation Vector can steer physical plausibility judgments in VideoMAE when injected into the appropriate layers. Together, these studies establish strong precedents for physical probing, distributed geometry, and causal steering in learned world models. Counterfactual evaluation is also becoming more explicit at the behavioral level. CRONOS tests whether video models respond consistently when scene factors such as viewpoint, appearance, or context are changed while the underlying physical event is held fixed (Begiristain, Dรผnkel, and Kortylewski 2026). Very recent work such as GAUGE moves further toward measurement-grounded physical evaluation by comparing generated or simulated trajectories against real-world observables and physical parameters (Wang et al. 2026). These benchmarks highlight a broader shift from visual plausibility toward intervention-sensitive and trajectory-level physical evaluation. Our work tests a stronger downstream criterion: whether a compact, addressable hidden-state edit can place a recurrent world model on the intended counterfactual trajectory and then let the model continue that future autonomously. The edit is applied once and evaluated with specificity controls. 3 Problem Formulation We now state the question studied in the rest of the paper. The goal is not to recover a physics state from the network. Instead, we ask whether a small, directly addressable change to the model hidden state can implement a local physical counterfactual and then allow the model to continue the new future on its own. This section introduces only the notation needed for that question. Formal definitions are collected in Appendix A, while the detailed carrier construction and coefficient fitting are deferred to the Method section and its corresponding appendices. 3.1 World-model setting We study a controlled two-object environment. Two circular objects, denoted A and B, move in a bounded two-dimensional arena. At each time step, each object has a two-dimensional position and a two- dimensional velocity. The simulator is deterministic, while the model receives numeric object-state observations with small observation noise. This restricted setting is useful because the physical intervention and the resulting counterfactual future can be defined exactly. For object j, the primitive object state consists of its horizontal and vertical positions and velocities. The joint physical state is the concatenation of the two object states: ํ ํ,ํก = [ ํฅ ํ,ํก , ํฆ ํ,ํก , ํฃ ํฅ,ํ,ํก , ํฃ ํฆ,ํ,ํก ] ํณ โโ 4 , ํโํด,ํต ํ ํก = [ ํ ํด,ํก ํณ , ํ ํต,ํก ํณ ] ํณ โโ 8 (1) Eq. (1) has two lines. The first line defines the primitive state of one object: t is the discrete time index and jโA,B identifies object A or B. For object j, ํ ํ,ํก โโ 4 contains x j,t and y j,t , the horizontal and vertical position coordinates, and v x,j,t and v y,j,t , the corresponding velocity components. The superscript T denotes transpose. The second line of Eq. (1) concatenates the two four-dimensional object states into the joint physical state ํฌ t โโ 8 . Thus ํฌ t contains the complete primitive simulator state of both objects at time t. The simulator also provides derived quantities such as relative position, relative velocity, pair distance, and contact state, but these are used as evaluation information rather than as additional independent state variables. The learned world model maintains a 192-dimensional recurrent hidden state. A factual observation history through an anchor time produces the factual anchor hidden state. Starting from this state, the model can be released to predict the future by repeatedly applying its own transition dynamics. No future observations are read after the anchor in the intervention experiments. The exact recurrent architecture, observation channels, and training procedure are reported in the experimental description and Appendix B. 3.2 Local physical counterfactual A counterfactual pair is constructed from the same physical history. The factual and counterfactual branches are identical before the anchor. At the anchor, we edit exactly one velocity component of exactly one object. For example, we may increase object Bโs horizontal velocity while leaving both object positions and all other primitive velocity components unchanged at that instant. Let a unit edit direction specify the edited object and velocity axis. A local velocity change of ํฟํฃ creates the counterfactual anchor state ํฌ ํก CF =ํฌ ํก ํน +ํฟํฃ ํ ํ,ํ (2) In Eq. (2), ฮดv is the signed magnitude of the requested velocity change. The index j specifies which object is edited, and ํโx,y specifies the velocity axis. The selector ํ ํ,ํ โโ 8 is zero in every component except the selected velocity coordinate, where it equals one. Eq. (2) therefore changes exactly one velocity component at the anchor while leaving both positions and all other primitive velocity components unchanged at that instant. The superscript F denotes the factual branch, superscript CF denotes the native counterfactual branch. This edit is local only at the intervention instant. Quantities that are algebraically determined by the edited state, such as relative velocity or center velocity, should change immediately. Other quantities, such as later positions, contact time, or the other objectโs post-collision velocity, may change later through the simulator dynamics. We therefore do not require every non-edited quantity to remain equal to its factual value. Only variables that are truly unaffected by the intervention should be preserved. For a native counterfactual reference, the edited anchor observation is passed through the modelโs normal observation/update path. This produces the native counterfactual hidden state: the state reached by the model when it is explicitly shown the edited current physical condition. Native counterfactual rollout therefore tests whether the model itself is capable of representing and propagating the requested counterfactual before we ask whether a smaller hidden-state intervention can reproduce the same future. The exact factual-counterfactual pair construction, intervention support, and native counterfactual route are detailed in Appendix C. 3.3 Dynamics-effective latent intervention A hidden-state edit is not considered successful merely because the decoder changes at the anchor. The stronger requirement is that a single edit places the model in a state from which its own transition dynamics produce the intended counterfactual future. Let a one-shot hidden perturbation be applied to the factual anchor state: ํก ํก โ =ํก ํก ํน +ํฟํก (3) In Eq. (3), ํ ํก ํน โโ 192 is the recurrent hidden state obtained from the factual history through the anchor time t. The perturbation ํฟํโโ 192 is applied once at that anchor, producing the patched hidden state ํ ํก โ . The superscript (*) labels the intervention route only; it does not mean that ํ ํก โ is itself the true physical state. We call this intervention dynamics-effective, within the registered evaluation domain, when the autonomous rollout from the patched anchor state matches the simulator-grounded counterfactual trajectory closely enough to pass the registered target, derived-state, downstream, preservation, and physical-validity checks. In compact notation, the central requirement is R ํป ( ํก ํก โ ) โํ ํก:ํก+ํป CF (4) where R H denotes the modelโs autonomous rollout operator for H future transitions, starting from the supplied hidden state, and ํ ํก:ํก+ํป ํถํน denotes the simulator-grounded counterfactual trajectory over the same interval. In the main experiments, H = 12. The symbol โ means that the rollout satisfies the registered counterfactual-agreement and physical-validity criteria; it does not mean exact numerical equality at every state coordinate. In detail, it means that the direct target must follow the requested edit; derived quantities must be consistent with the edited state; dynamically coupled variables must follow the new counterfactual future rather than the factual future; truly unaffected quantities must remain stable; and the released rollout must remain physically valid. Future observations, teacher forcing, and repeated correction are not used after the one-shot edit. Exact metrics and thresholds are defined later in Appendix F. 3.4 Low-rank and addressable latent carrier The intervention becomes especially informative if the required hidden change can be restricted to a low- dimensional subspace. We represent a rank-r candidate carrier by a basis matrix and a coefficient vector: ํฟํก=ํ ํ ํ, ํ ํ โโ 192รํ , ํโโ ํ (5) In Eq. (5), r is the candidate carrier rank. ํผ ํ โโ 192รr is a matrix whose r columns span the candidate low-dimensional latent carrier, and ํโโ ํ is the coefficient vector for one particular intervention. Their product ํผ ํ ํis therefore a 192-dimensional hidden perturbation. In our implementation, ํผ ํ is obtained from an SVD-based construction described in Section 4 and Appendix D. Its individual columns are not assigned fixed physical meanings. The basis columns describe the shared latent directions available to the intervention family, while the coefficient vector specifies how far a particular edit moves along those directions. Individual basis columns are not assumed to correspond to named physical variables. A different basis can span the same subspace without changing the resulting 192-dimensional hidden perturbation. A low-rank carrier is useful only if a new intervention can be specified without first computing the native counterfactual hidden state. We therefore require an addressable map that predicts the coefficient vector from the factual physical state and the requested edit: ํ=ํ ( ํฌ ํก ํน ,ํ ) (6) In Eq. (6), ํ is the addressable coefficient map. Its first input, ํ ํก ํน , is the factual two-object primitive state at the intervention time. Its second input, e, is the requested local velocity edit: it specifies the edited object, the x or y velocity component, and the signed edit magnitude. The output c is the low-dimensional coefficient vector that determines the hidden perturbation. At evaluation time, g does not receive the native counterfactual hidden state. The corresponding test-time hidden intervention is then ํก ํก โ =ํก ํก ํน +ํ ํ ํ (7) Eq. (7) combines the shared carrier and the current edit. ํผ ํ provides the latent directions learned from training counterfactuals, while c specifies how far the present intervention moves along those directions. The resulting ํผ ํ ํ is added once to ํ ํก ํน to form ํ ํก โ . The model is then released without any further hidden- state correction. Eq. (6) and (7) define the operational object studied in this paper. The native counterfactual hidden state may be used on training data to learn the carrier and its coefficient targets, but it is not available to the addressable operator at evaluation time. The SVD construction of the basis and the ridge model used for the addressable map are method details and are therefore deferred to Section 4, Appendix D, and Appendix E. 3.5 Operational domain and claim boundary The statements above are intentionally local and bounded. A learned world model is only trained and tested over a finite range of physical states, intervention magnitudes, and rollout times. We therefore describe the domain in which an intervention claim is evaluated as ฮฉ=S tested รI tested รT tested (8) In Eq. (8), ฮฉ is the operational domain over which the paperโs intervention claim is evaluated. S tested is the tested region of factual physical states, I tested is the registered family and magnitude range of local interventions, and T tested is the set of evaluated intervention times. The symbol ร denotes a Cartesian product, not numerical multiplication. A result established on ฮฉ therefore applies only to combinations of state, intervention, and time that lie inside these tested boundaries; behavior outside ฮฉ is extrapolation unless separately evaluated. The three factors denote the supported physical-state region, the registered intervention family, and the evaluated anchor times, respectively. The experiments later exemplify the intervention family with bounded one-object, one-velocity-component edits. A numerical value outside the registered range can still be passed through the fitted linear operator, but this would be an extrapolation rather than an experimentally established result. The same caution applies to states or time scales outside the tested regime. Accordingly, finding a passing rank-r carrier does not mean that the model has discovered r named physical variables, that r is the intrinsic dimension of the physical system, or that the carrier is a complete effective state. The result is narrower: within the tested domain, a checkpoint-specific low-dimensional latent intervention can be addressed from a local physical edit and can be sufficient to redirect the modelโs autonomous dynamics. This bounded operational meaning is the claim tested by the methods below. 4 Method In this section we describe how the intervention is built and tested. The procedure has six stages: first verify that the trained world model can represent the requested counterfactual; then learn a low-rank carrier from training-only hidden differences; learn an addressable map from the physical edit to carrier coordinates; apply the resulting hidden patch once; evaluate the released rollout with positive and negative controls; and finally test whether the same carrier structure persists across nearby intervention times. Carrier fitting, rank selection, and confirmatory evaluation use separate data roles so that the tested rank is not chosen on the final evaluation outcomes. 4.1 Native counterfactual adequacy evaluations A low-rank intervention is interpretable only if the underlying model can first produce the intended counterfactual when the complete edited current state is provided. We therefore evaluate three reference routes before interpreting any low-rank result. G0 is the factual route: the model receives the factual history and is released from the factual anchor hidden state. G1 is the native counterfactual route: the pre- anchor history is kept unchanged, the edited anchor observation is passed through the modelโs normal update path, and the resulting native counterfactual hidden state is released autonomously. G2 checks whether the hidden-state patching interface works correctly. In Eq. (9), ํ indexes an intervention unit and ํก is the anchor time. We add the full native counterfactualโfactual hidden-state difference, i.e. ํฅํ ํ full to the factual state ํ ํก,ํ ํน , which should exactly reconstruct the native counterfactual hidden state โ ํก,ํ CF . ํฅํ ํ full =ํ ํก,ํ CF โํ ํก,ํ ํน , ํ ํก,ํ ํน +ํฅํ ํ full =ํ ํก,ํ CF . (9) Matching the native counterfactual rollout therefore confirms the intervention mechanism itself, rather than the sufficiency of any low-dimensional carrier. If G1 fails against the simulator counterfactual, the model itself is not adequate for that unit or stratum. If G1 succeeds but G2 fails to reproduce G1, the hidden intervention interface is invalid. Only after all G0-G2 tests are passed, can we validly evaluate low-rank carriers. The exact counterfactual construction and route definitions are given in Appendix C. 4.2 Constructing candidate low-rank carriers Carrier construction uses a dedicated carrier-fit split and is performed separately for each checkpoint. For every fit unit, we compute the difference between the native counterfactual hidden carrier and the factual hidden carrier at the same anchor. At the primary ํก = 7 interface, the registered fit matrix contains 1024 such 192-dimensional differences, as shown in Eq. (10). ํซ= [ ( ํฅํ 1 ) ํณ ( ํฅํ 2 ) ํณ โฎ ( ํฅํ ํ fit ) ํณ ] โโ ํ fit ร192 , ํ fit =1024 (10) Eq. (10) stacks one hidden difference per row. The fit-unit count is 1024. The rows come from a family of local velocity edits with different factual two-object states, edited objects, velocity axes, signs, and registered amplitudes. The matrix is not built from 1024 alternative hidden solutions to one fixed edit; it samples the hidden response to the registered velocity-edit family across different physical contexts. We use an uncentered singular-value decomposition of the raw difference matrix. No mean hidden difference is subtracted before the decomposition. ํซ=ํณํฎํฝ ํณ , ํผ ํ =ํฝ [ :,1:ํ ] โโ 192รํ . (11) In Eq. (11), L and V contain the left and right singular vectors and ฮฃ contains the singular values. Because each row of D lies in the 192-dimensional hidden coordinate system, the first r columns of V form the candidate carrier basis. The basis is checkpoint-specific. Its columns are latent directions, not named physical coordinates. Appendix D gives the complete construction, sign convention, rank grid, and second-moment diagnostics. For capacity analysis only, a fit or evaluation unit can be projected onto the candidate basis using its native hidden difference: ํ ํ oracle =ํผ ํ ํณ ํฅํ ํ . (12) The coordinate vector in Eq. (12) describes the true hidden difference inside the rank-r carrier. It uses the native counterfactual hidden state for the same unit and is therefore privileged. Oracle coefficients answer whether the subspace has enough capacity to carry the required hidden change, but they are not used as the primary test-time intervention mechanism. 4.3 Addressable coefficient operator The main intervention must be addressable without first computing the test unitโs native counterfactual hidden state. We therefore learn a small affine ridge map from the requested physical edit and the factual primitive state to the carrier coordinates. The edit is encoded by four sparse slots: horizontal and vertical velocity edits for object A, followed by horizontal and vertical velocity edits for object B. Exactly one slot is nonzero. The factual state contributes the current horizontal position, vertical position, horizontal velocity, and vertical velocity for each of the two objects. Eq. (13) provides details of the edit: ํ ํ = [ ํ ํ ํ ํก,ํ ํน ] โโ 12 , ํฅฬ ํํ = ํฅ ํํ โํ ํ ํํํฅ(ํ ํ ,10 โ6 ) ํ=1,2,3...12, ํ ฬ ํ =ํพ ํ ํณ ํ ฬ ํ +ํ ํ (13) In Eq. (13), ํ indexes fit examples, while ํ indexes the 12 non-intercept input features. The edit vector ํ ํ has four components and the factual physical state ํ ํก,ํ ํน has eight, so their concatenation ํ ํ lies in ํ ํํ . Each feature ํฅ ํํ is standardized using the mean ฮผ ํ and standard deviation ฯ ํ computed only on the carrier-fit split, with the standard deviation floored at 10 โ6 . Internally, the affine intercept is represented by a constant column of ones in the design matrix, giving 13 columns in total; in Eq. (13), the same intercept is written explicitly as ํ ํ . The rank-specific weights ํพ ํ and intercept ํ ํ are fitted by ridge regression with penalty ฮป=10 โ4 , while the intercept is left unpenalized. At evaluation time, the operator uses only the factual state, the requested edit, and the frozen standardization statistics from the carrier-fit split. It does not use the test unitโs native counterfactual hidden state, the true hidden difference, a future counterfactual trajectory, or oracle carrier coefficients. Appendix E gives the complete fitting objective and standardization rule. 4.4 One-shot autonomous rollout The predicted coefficient vector ํ ฬ ํ is converted back to a 192-dimensional hidden perturbation through the carrier basis and is added once to the factual anchor hidden state: ํ ํก,ํ โ =ํ ํก,ํ ํน +ํฝํผ ํ ํ ฬ ํ , ํฝ=1 for the primary evaluation. (14) Here ฮฒ is a scalar intervention strength. The primary evaluation fixes ฮฒ = 1, which corresponds to the full registered physical edit. After constructing the patched hidden state, the model is released for 12 future transitions. The anchor state and the 12 predicted future states give 13 decoded states in total. There are no future observation reads, no teacher forcing after the anchor, no per-step physical or hidden-state clamp, no future oracle state, and no decoder feedback outside the modelโs native transition path. The data used in this paper have passive zero actions, so the frozen action interface is present but receives zero actions throughout these rollouts. A low-rank edit is accepted only when the released rollout satisfies the registered direct-edit, immediate- derived, downstream-trajectory, unaffected-state, and physical-law criteria, while also passing engineering and autonomy eligibility. The metric roles, exact registered numerical thresholds, and aggregation rules are summarized in Appendix F. 4.5 Controls, rank selection, and fresh replication A decoded change alone is not enough to show that an intervention is dynamics-effective. We therefore compare each intervention with both reference conditions and specificity controls. References include the factual rollout, simulator counterfactual truth, native counterfactual rollout, and the full-hidden G2 route. Controls include sham and no-patch runs, random equal-norm perturbations, wrong-object and wrong- time edits, leakage checks, and object-identity/order checks. Since all actions are zero in this dataset, action shuffling is not informative and is marked as not applicable. Appendix G summarizes the purpose of each control. Rank is selected only on the development panel. Here, the development panel consists of three independently trained models (development models 1-3) with the same architecture and training procedure but different random seeds. The primary candidate grid is R = 1, 2, 4, 8, 12, 16, 20, 32; rank 64 is diagnostic, rank 0 is a no-patch reference, and rank 192 is the full-hidden reference. We evaluate the intervention separately in two trajectory strata. S1 contains cases in which neither the factual nor the counterfactual trajectory makes contact within the 12-step rollout horizon. S2 contains cases in which the factual trajectory includes a contact event and the velocity edit produces a nontrivial change in the subsequent trajectory. The two strata are evaluated separately and are not pooled. The selection rule chooses the smallest tested rank that passes both S1 and S2 for at least two of three development checkpoints, together with the mandatory controls: ํฬ grid =min ํโR:โ1 3 ํ=1 [ Pass ํ ( ํ,ํ1 ) โ Pass ํ ( ํ,ํ2 ) ] โฅ2 . (15) Eq. (15) defines a smallest passing rank on the registered grid only. In Eq. (15), ํฬ grid denotes the selected carrier rank, ํ is the preregistered candidate-rank grid, and ํ denotes one candidate rank being evaluated. The index ํ runs over the three independently trained development checkpoints. For a given rank ํ and checkpoint ํ, the S1 and S2 pass terms indicate whether that rank satisfies the registered evaluation criteria in the two trajectory strata, including the M1-M5 scientific metrics, the M6E and M7 eligibility gates, the coverage and median requirements, and the mandatory controls defined in Appendices F and G. The indicator function converts this joint pass condition into 1 or 0, so summing over q counts how many of the three development checkpoints pass both strata. Requiring this count to be at least two implements the preregistered 2-of-3 development-panel rule. The outer minimum then selects the smallest tested rank that satisfies the rule. Accordingly, the quantity defined by Eq. (15) is the smallest passing rank among the ranks that were actually tested; it is not an estimate of the modelโs intrinsic or complete state dimension. Once a rank is selected, that rank, ฮฒ, the feature definition, standardization procedure, ridge setting, decision thresholds, and panel rule are frozen before evaluation on separately trained fresh checkpoints. For each fresh checkpoint, we refit the carrier basis and ridge map using only that checkpointโs carrier- fit split, while keeping the method and rank fixed. The confirmatory split is used only to evaluate the result and is never used for fitting. This replication therefore asks whether the same preregistered procedure works across independently trained models. It does not copy a carrier learned from one checkpoint to another, and it does not retune the method after seeing the replication results. 4.6 Temporal tests The main carrier is first established at the reference anchor t = 7. We then ask two simple questions about how it behaves over time. B1 asks whether a similar low-rank carrier can be found independently at each anchor t โ 5, 6, 7, 8, 9. For each anchor, we fit a new checkpoint-specific carrier and addressable operator using only the carrier-fit data from that anchor, and then evaluate it at the same anchor. B2 asks a stronger question: can the carrier learned at t = 7 also be used at nearby times? To test this, we take the t = 7 basis, ridge map, feature standardization, selected rank, and ฮฒ exactly as fitted and apply them at t = 5, 6, 8, and 9, without refitting or recomputing the standardization. 4.7 Position-edit stress test Finally, we use position editing as a development-only stress test of the evaluation logic. At t = 7, one x or y coordinate is changed while all velocities and the other primitive fields are held fixed, and the simulator counterfactual future is regenerated from the edited state. The tested absolute position amplitudes are 0.05, 0.10, and 0.20. The same general question is then asked: does a low-rank edit show dynamics-specific behavior that is stronger than random, rank-zero, and wrong-object alternatives? This diagnostic is not allowed to redefine the primary velocity carrier or its confirmatory rank. Its construction boundary is summarized in Appendix C, while its outcome is reported separately in Results. 5 Results We report the evidence in the order of the experiment: native counterfactual adequacy, rank selection, fresh-checkpoint replication, temporal reuse, and the position-edit stress test. 5.1 The model natively supports the requested counterfactual rollouts Before evaluating any low-rank intervention, we first asked whether the trained world model itself could produce the requested counterfactual future. This is a necessary upper bound on the intervention experiment. If the model cannot propagate the counterfactual even when it is explicitly given the complete edited current state, then failure of a smaller hidden-state patch would say little about the organization of the latent representation. We evaluated the factual route G0, the native counterfactual route G1, and the full-hidden equivalence route G2 separately in the non-contact stratum S1 and the future-contact stratum S2. All three development checkpoints passed all six required route-by-stratum cells. Thus, the model could predict its ordinary factual future, assimilate the locally edited anchor state and continue the corresponding counterfactual future, and support the hidden-state intervention interface needed for the later low-rank experiments. We then repeated this model-adequacy test using three independently trained fresh checkpoints under the same frozen training and evaluation procedure. Again, all three checkpoints passed all six required cells, giving 18 of 18 passing cells across the fresh panel. The fresh evaluation contained 512 intervention units per checkpoint, split equally between S1 and S2. All G0 and G1 cell medians passed the registered scientific criteria, and the lowest joint unit-level coverage observed in any cell was 0.922, above the registered 0.80 requirement. Performance was therefore not perfect on every individual unit, but the result was stable under the preregistered aggregation rule rather than being driven by a small subset of successful trajectories. G2 provided a separate check of the hidden-state intervention interface. For every fresh checkpoint and evaluation unit, adding the complete native counterfactual-minus-factual hidden difference to the factual carrier reconstructed the native counterfactual hidden state and its subsequent rollout to the registered numerical tolerance. Across the fresh panel, all 1,536 seed-unit G2 evaluations passed this exact- equivalence check. This confirms that the later intervention assay is capable of placing the model at a specified hidden state without introducing an additional discrepancy through the patching mechanism itself. These results establish the prerequisite for the low-rank study: when given the complete edited current condition, the model can enter a counterfactual internal state and autonomously propagate the corresponding future in both non-contact and interaction-sensitive trajectories. They do not yet show that a low-dimensional carrier exists. Rather, they rule out two simpler explanations for a later low-rank failureโthe model being unable to represent the requested counterfactual, or the hidden-state patching interface itself being incorrect. 5.2 A rank-4 addressable carrier is the smallest passing tested rank After confirming that the model can produce the requested counterfactual future, we asked whether the full 192-dimensional hidden-state change was necessary. For each of the three development checkpoints at t = 7, we learned a low-rank carrier from the carrier-fit split and tested the preregistered ranks on a separate development split. At test time, the intervention coefficients were predicted only from the factual state and the requested velocity edit. The counterfactual hidden state of the evaluation example was never used. As summarized in Figure 1a, the rank sweep showed a clear transition on the tested grid. Rank 1 failed to pass both S1 and S2 for all three development checkpoints, while rank 2 passed both strata for only one of three checkpoints. At rank 4, all three development checkpoints passed both strata together with the mandatory controls, and all higher ranks on the primary grid also passed 3/3 checkpoints. Under the preregistered rule, which selects the smallest tested rank passing both strata for at least two of three development checkpoints, rank 4 was therefore selected. This is the smallest passing rank among the tested values 1, 2, 4, 8, 12, 16, 20, 32; rank 3 was not tested. The same selected rank was obtained with the oracle-capacity analysis and with the addressable operator. The oracle route uses the native counterfactual hidden difference to provide the best available coefficients inside a candidate subspace, whereas the addressable route must predict those coefficients from the factual state and edit request alone. Both selected rank 4 on the development panel. Thus, within this experiment, we observed no rank-selection gap between low-rank capacity and low-rank addressability. This does not imply that the two routes produce numerically identical trajectories, or that the fitted linear map is uniquely optimal. Figure 1b provides a complementary geometric view of the carrier spectrum. At rank 4, the cumulative uncentered second-moment ratios of the hidden-difference matrix were 0.824, 0.802, and 0.866 for the three development checkpoints. By rank 8, the corresponding cumulative capture rose to approximately 0.93-0.97. The intervention transition and the spectral profile are therefore related but not identical statements. The first four carrier directions capture a large fraction of the observed hidden-state change. However, we do not select the carrier rank based on explained variance or reconstruction quality alone. A rank is considered dynamics-effective only if the corresponding intervention produces the intended autonomous counterfactual rollout and passes the required controls. Figure 1. Development-stage rank transition and carrier spectrum. (a) Number of development checkpoints that pass both S1 and S2, including the mandatory controls, for the addressable-operator and oracle-capacity routes across the registered rank grid. The horizontal line marks the preregistered 2-of-3 panel requirement, and the vertical line marks the selected rank 4. (b) Cumulative uncentered second-moment ratio of the carrier-fit hidden-difference matrix for the three development models. The spectrum is a geometric diagnostic; rank selection is determined by autonomous-rollout success and controls rather than by a spectral threshold. Crucially, rank 4 passes as a rollout intervention, not merely as a hidden-space reconstruction: one anchor patch must support the full 12-step autonomous future and the registered controls. Thus, rank 4 is the smallest tested scale that is both addressable and dynamics-effective on the development panel; it is not an estimate of intrinsic state dimension. We next test the frozen procedure on independently trained checkpoints. 5.3 The frozen rank-4 procedure replicates under the fresh-checkpoint rule The development result in Section 5.2 could still have been specific to the checkpoints used for rank selection. We therefore froze the selected rank, intervention strength, feature definition, standardization rule, ridge setting, decision thresholds, and panel rule before evaluating separately trained fresh models. For each fresh model, the carrier basis and addressable coefficient map were fitted only on that modelโs carrier-fit split, using the same frozen rank-4 procedure. The confirmatory split was then used only for evaluation. No new rank sweep, operator comparison, beta adjustment, threshold change, checkpoint exclusion, or post-hoc retuning was allowed. The frozen procedure satisfied the preregistered fresh-checkpoint rule, with clear checkpoint-level heterogeneity (Figure 2a). Fresh model 1 achieved joint coverage of 0.867 in S1 and 0.914 in S2; fresh model 2 achieved 1.000 and 0.922; and fresh model 3 achieved 0.711 and 0.973. Thus fresh models 1 and 2 passed both strata, whereas fresh model 3 failed only S1. Because the confirmatory rule required at least two of three independently trained fresh models to pass both S1 and S2, the frozen rank-4 procedure replicated at the panel level. The Fresh model 3 S1 miss was specifically a joint-coverage failure. All six fresh model-by-stratum cells passed the registered M1-M5 median thresholds and the numerical and autonomy eligibility checks, and five of the six cells met the 0.80 joint-coverage requirement. The S1 miss therefore arose because too few individual units satisfied all registered conditions simultaneously, even though the cell-level medians and eligibility checks remained within threshold. Specificity controls further constrained the interpretation (Figure 2b). Sham and rank-zero routes satisfied the no-op criteria; wrong-object and wrong-time interventions behaved as required by the registered specificity tests; and leakage plus identity/order checks passed across the fresh panel. The independent random-orthogonal equal-norm perturbation produced 0/3 fresh-model target passes. Thus the confirmatory effect cannot be explained by patch magnitude alone or by a generic hidden-state perturbation. These controls establish specificity only relative to the registered alternatives. They do not show that the learned carrier is unique, that its coordinates align across checkpoints, or that all trained models organize the counterfactual in the same way. Figure 2. Fresh-checkpoint replication and specificity controls for the frozen rank-4 addressable carrier. (a) Joint unit coverage in S1 and S2 for each independently trained fresh model; the dashed line marks the registered 0.80 requirement. (b) Summary of registered reference and specificity-control outcomes. A satisfied control means that the control behaved according to its preregistered criterion, not that it reproduced the target counterfactual. Figure 2 therefore supports a bounded replication claim: the frozen procedure meets the 2-of-3 fresh- model rule, but the single subthreshold cell shows that reliability is not uniform across checkpoint-stratum combinations. The next question is whether this dynamics-effective carrier is tied to the reference anchor at t = 7 or persists across nearby intervention times. 5.4 The rank-4 carrier remains effective across the tested time window The results above establish the carrier at the reference anchor t = 7, but they do not show whether that result is tied to a single point in the rollout. We therefore separate temporal robustness into two tests. B1 asks whether a rank-4 addressable carrier can be established independently at several nearby anchors. B2 asks the stronger question of whether the exact carrier coordinates and addressable operator learned at t = 7 can be reused at nearby anchors without refitting. The distinction is important: B1 tests repeated existence, whereas B2 tests temporal reuse of the same latent intervention coordinates. Figure 3a summarizes B1. At each anchor t = 5, 6, 7, 8, and 9, each fresh model received an anchor- specific rank-4 carrier and addressable coefficient map fitted only on that anchorโs carrier-fit split; the rank, operator family, evaluation thresholds, and panel rule were unchanged. Every anchor met the preregistered panel criterion, with 2 of 3 fresh models passing both S1 and S2. Thus, a rank-4 addressable carrier could be re-established at every tested time. Because B1 allows a different basis and coefficient map at each anchor, however, it does not by itself show that the latent coordinate system is stable across time. Figure 3b reports the stronger B2 transport test. For each fresh model, we took the exact t = 7 basis, ridge operator, feature standardization, rank 4, and intervention strength and applied them unchanged at t = 5, 6, 8, and 9. No carrier refit, mapper refit, restandardization, rank change, or intervention-strength adjustment was allowed. All four transport targets again met the 2-of-3 fresh-model criterion. The t = 7 position in Figure 3b is the source anchor, not an additional transport result. These results show that the t = 7 intervention coordinates remain usable both earlier and later within the tested window. Figure 3. Temporal robustness of the rank-4 addressable carrier. Bars show the number of fresh models that pass both S1 and S2; the dashed line marks the registered 2-of-3 panel requirement. (a) B1 uses anchor-specific rank-4 fits at t = 5-9. (b) B2 applies the frozen t = 7 carrier and addressable operator directly at t = 5, 6, 8, and 9 with no refit; t = 7 is shown only as the source anchor. The contrast between the two panels is the key temporal result. B1 shows that low-rank dynamics- effective structure is recoverable throughout the tested window, while B2 shows that success does not require rotating or refitting the carrier separately at each time. We therefore describe the carrier as temporally stable within this bounded window. This is a local operational statement: it does not establish a globally time-invariant coordinate system, a unique physical gauge, a complete low-dimensional Markov state, or a universal rank-four representation outside the tested model, state, edit, and time domains. 5.5 Position edits do not satisfy the dynamics-effective evidence standard Having established a dynamics-effective carrier for local velocity edits, we next asked whether the same evidence standard would also support a different physical intervention family. At the reference anchor t = 7, we changed one x or y coordinate of one object by an absolute amount of 0.05, 0.10, or 0.20, while holding all velocities and the other primitive anchor fields fixed. The simulator counterfactual future was then regenerated from the edited state. The purpose was to test whether a directly specified position change could be implemented by a target-specific latent intervention under the same control logic used for velocity edits. The raw position results appeared positive, but the effect was not specific to the learned patch (as shown in Figure 4a). At every tested amplitude, the intended patch, the no-patch condition, and a random equal- norm perturbation all passed the same raw criterion in all three development models. The raw metric therefore could not tell the intended intervention apart from doing nothing or from applying an unrelated perturbation. The specificity tests tell a different story (Figure 4b). The intended position patch passed the raw criterion at all three amplitudes, but it failed the wrong-object test in every development model, and the full specificity criterion also failed throughout. The patch therefore did not reliably act on the intended object. Since the requested edit is object-specific, the raw improvement alone cannot be taken as evidence for a dynamics-effective position carrier. Figure 4. Position-edit stress test. (a) Number of development models satisfying the raw gate for the intended position patch, rank-zero/no-patch route, and independent random equal-norm route at each tested amplitude. All three routes pass in 3/3 models. (b) Registered outcomes for the raw and specificity criteria. The wrong-object and full-specificity conditions fail in 0/3 models at every amplitude. This analysis is development-only and does not alter the confirmatory velocity-carrier result. This negative contrast does not identify a single mechanism. Position-only edits may produce relatively weak downstream separation in some trajectories; a change that improves the decoded anchor state may still fail to place the recurrent model in the corresponding transition state; or a physically valid position counterfactual may require a more coupled latent update than changing one position coordinate alone. These possibilities are consistent with the observed lack of specificity, but the present diagnostic was not designed to distinguish among them. This result does not mean that position information is absent from the hidden state, or that position interventions can never be dynamics-effective. Our conclusion is narrower: the position-only intervention tested here did not meet the same evidence standard as the velocity intervention. An edit may change the model in a physically meaningful way and even improve the raw outcome, but this alone is not enough. To count as dynamics-effective, the edit must move the model toward the intended counterfactual future in a target-specific way, and the registered controls must not produce the same effect. 5.6 Simultaneous velocity edits reveal a boundary of compositional addressability We also examined a development-only extension in which both velocity components of the same object were changed simultaneously, while the total velocity-change magnitude remained within the support of the primary single-component intervention family. Across balanced S1 and S2 development units, the model natively supported these joint counterfactuals, and the frozen rank-4 carrier retained sufficient capacity to reproduce them. The original affine addressable route also satisfied the absolute joint-rollout criteria, and its direct joint coefficients matched the component-wise affine composition to numerical precision. Thus, the low-rank carrier and its affine coordinates extend beyond isolated single-component edits at the level of representational capacity and rollout efficacy. The stronger requirement of target-specific compositional control was not met. Compared with no-patch, random orthogonal equal-norm, wrong-object, and matched wrong-vector alternatives, the joint addressable interventions did not achieve the same specificity as the established single-component result, with the main loss occurring in the interaction-sensitive S2 stratum. Calibration confirmed that this difference was not an artifact of an overly strict specificity criterion. Layer-localization further showed that several of these distinctions were preserved by the native model and the rank-4 oracle projection but weakened at the addressable layer, although wrong-object specificity also showed an upstream native limitation in part of S2. Minimal joint-aware extensions of the addressable map did not remove this gap. We therefore treat simultaneous two-component velocity editing as a boundary of the present result. The rank-4 carrier can support bounded joint velocity changes, but carrier capacity does not by itself guarantee equally specific compositional addressability. In particular, the simple addressable maps tested here did not provide the same target-specific control for interaction-sensitive joint futures as they did for the primary single-component velocity family. All joint-edit analyses were development-only. This completes the empirical results. We next discuss what the positive velocity result does and does not establish about effective latent state, why the position contrast is informative, and which parts of the finding are likely to depend on the bounded model and intervention domain studied here. 6 Discussion The results support a compact dynamics-effective intervention interface for the tested velocity edits, while also showing clear limits on what that result means. This section focuses on those limits and on the broader role of effective-state analysis. 6.1 Dynamics-effective state is a functional, non-semantic, notion A central idea in this work is that information being present in a hidden representation does not mean that the model actually uses it as part of its working state. A probe may show that a physical quantity can be decoded, and a hidden-state edit may change the modelโs output, but neither result shows that the edited state can support the intended future. Our test therefore focuses on what happens after the intervention: we patch the hidden state once and then let the model generate the counterfactual trajectory on its own. The velocity result satisfies three requirements that make this stronger than a simple decoding or steering result. The intervention is addressable from the factual state and requested edit; it remains effective during twelve autonomous transitions; and it passes the registered specificity and integrity controls. Together, these requirements test whether the edited representation can actually take part in the modelโs future computation. This view is close to functional ideas of latent-state sufficiency. DeepMDP and action-sufficient state representation learning judge a representation by the function it can support rather than by whether its coordinates match named physical variables (Gelada et al. 2019; Huang et al. 2022). Recent work on world models has made the same distinction across prediction, control, memory, grounding, and counterfactual reasoning (K.W. Kim 2026). Our experiment shows one narrow form of counterfactual sufficiency: a compact hidden-state intervention is adequate to redirect the modelโs autonomous rollout for the registered velocity edits. We use dynamics-effective as an operational term, not a semantic one. The carrier need not contain axes that correspond to velocity, position, or other named physical variables. What matters here is that a compact, addressable edit changes the intended future in a sustained and target-specific way. 6.2 A low-rank carrier is not yet a closed low-dimensional state The rank-4 result solves a specific intervention problem. It identifies a small set of latent directions that can move the factual hidden state into a state from which the requested velocity counterfactual continues correctly. After that patch, however, the model still evolves its full 192-dimensional recurrent state. The result therefore gives a low-dimensional intervention interface, not a four-dimensional reduced world model. A closed low-dimensional state would require a stronger test. We would need to extract a compact state, update that compact state through time without relying on the remaining hidden coordinates, and still reproduce the future quantities that matter. The present experiments do not test this. Other hidden dimensions may still carry history, identity, interaction context, or other information needed by the recurrent dynamics. Rank four should also not be treated as a physical dimension. It is the smallest passing value on the tested grid, and rank three was not tested. The basis is not unique: different bases can span the same subspace and produce the same 192-dimensional patch. The rank is therefore a property of this tested intervention interface and evaluation procedure, not an estimate of the modelโs intrinsic state dimension. A natural next step is a closure test. If a compact representation derived from the carrier could update itself and reproduce the relevant future without returning to the full recurrent state, then the claim could move from a low-dimensional control interface toward a genuinely reduced effective state. 6.3 Temporal reuse and checkpoint variation show local rather than universal structure The temporal results strengthen the interpretation in a specific way. Re-fitting a rank-4 carrier at several anchors shows repeated existence, but the stronger result is that the carrier and addressable operator learned at t = 7 can be used directly at nearby anchors. Within each checkpoint, the same intervention coordinates remain useful across the tested window. This makes it less likely that the result depends on one accidental time point or on fitting a new basis at every anchor. Fresh-checkpoint replication shows that the same procedure can recover this structure in independently trained models, but not with identical strength in every checkpoint and stratum. This distinction matters because two learned models can behave similarly while using different internal coordinates. For this reason, the more useful replication target is functional: whether a compact, addressable carrier can be found by the same procedure and whether it remains effective during rollout. Exact axis-by-axis alignment across models is a separate and stronger question. 6.4 The position contrast shows why the controls are part of the claim The position stress test provides the clearest example of why raw improvement is not enough: the intended patch is not separated from no-patch, random, and wrong-object alternatives under the registered controls. The controls are therefore part of the evidence standard, not an optional extra. A dynamics-effective intervention must act on the intended target, at the intended time, in a way that unrelated perturbations do not reproduce, and the model must then continue the new future on its own. Without these comparisons, ordinary hidden-state sensitivity can be mistaken for a meaningful state-like intervention. The negative result should still be read narrowly. It does not show that position is absent from the hidden representation or that no position intervention could work. Instead, it shows that the tested position-only intervention family does not meet the same standard as the velocity intervention. More generally, a compact effective interface may depend on the intervention family: one class of counterfactuals may admit a small targeted edit while another may require a different or more coupled update. The simultaneous velocity-edit extension exposes a related boundary within the velocity family itself. The frozen rank-4 subspace retained joint-edit capacity, but this did not guarantee equally specific addressability when two velocity components were requested together. The fact that small joint-aware operator extensions did not close the interaction-sensitive specificity gap reinforces the distinction between what a latent subspace can carry and what a simple test-time map can reliably address. Effective intervention structure may therefore depend not only on the latent carrier, but also on the intervention family and on the mapping from an external request into that carrier. 6.5 Effective-state analysis as a diagnostic for world-model and World AI research One broader implication is that searching for an effective state may be useful as a model diagnostic even when interpretability is not the final goal. World models are usually judged by prediction, generation, planning, or control performance. A complementary question is whether a model can be moved into a well-defined counterfactual condition through a compact internal change and then continue that future with its own dynamics. This kind of test could distinguish models with similar task performance but different internal behavior. A model that needs large or non-specific hidden changes for a simple counterfactual may be more brittle or more entangled than one with a compact and stable intervention interface. These differences could help diagnose where a world model stores dynamics-relevant information and where its internal organization is fragile. Future work could test training objectives that encourage counterfactual consistency, target specificity, temporal stability, or compact intervention interfaces, and then ask whether these properties actually improve prediction, planning, or control. For broader World AI research, effective-state analysis may therefore provide one practical way to connect internal representation structure with the modelโs ability to change and continue imagined futures, without assuming in advance that the useful state must look like a hand-designed physics state. 6.6 Limitations and future directions The main limitation of this work is the controlled setting. We study one recurrent model family in a deterministic two-object environment with numeric object-state observations. The action interface is present, but all evaluated actions are zero. The results therefore do not establish the same structure for raw-video world models, action-conditioned agents, larger scenes, or real-world systems. The empirical claim is also narrow. The confirmatory result concerns bounded local velocity edits, a fixed rank grid, and a short-tested time window. The position experiment is development-only and gives a negative contrast rather than a second positive carrier. We also do not know whether the same effective rank or a linear carrier will appear in other architectures or at larger hidden widths. The development-only joint-edit analysis adds another boundary to this claim. Simultaneous same-object velocity-component edits remained representable by the model and by the frozen rank-4 carrier, but the tested addressable maps did not achieve the same full specificity as the single-component interface, especially in contact-sensitive trajectories. We therefore do not claim general compositional addressability for the velocity carrier. A further limitation is that the present study tests intervention sufficiency, not full dynamical closure. It also does not address partial observability, where a useful effective state may need to summarize history or behave more like a belief state. In larger models, a nonlinear manifold or a state-dependent local subspace may also be more appropriate than one fixed linear carrier. The most direct next steps are therefore to repeat the analysis across architectures and hidden dimensions, study partially observed settings, and move to action-conditioned world models. These extensions would test whether compact dynamics-effective interfaces remain useful as the model and task become more realistic, and whether the diagnostic ideas discussed above can eventually help design better world models rather than only describe existing ones. 7 Conclusion We show that, for a bounded family of local velocity edits, a 192-dimensional recurrent world model contains a compact addressable intervention interface: rank 4 is the smallest tested rank that passes the registered development criteria, and a single patch can launch a 12-step autonomous counterfactual rollout. Fresh-checkpoint replication, temporal reuse, and specificity controls support this result, while the position-edit stress test shows that apparent rollout improvement is not sufficient when controls can reproduce it. The result should therefore be read as evidence for a low-dimensional dynamics-effective intervention interface, not a closed four-dimensional state. Testing closure, history/belief-state compression, and transfer across architectures and hidden widths are natural next steps. AI Assistance Disclosure Generative AI tools, including ChatGPT and OpenAI Codex, were used as research-assistance tools for literature discovery and synthesis, refinement of research questions and experimental protocols, code implementation and debugging, analysis organization, and manuscript preparation. All experimental protocols, scientific decisions, formal runs, interpretation of results, and final claims were reviewed and approved by the authors. 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Physical State Grounding for JEPA World Models. arXiv preprint arXiv: 2608.06799 Yeom, Jewon, Hanseul Kim, Jeongjae Park, Sungmok Jung, Jaejin Lee, and Taesup Kim. 2026. "What Makes Video World Model Latents Action-Relevant: Prediction over Reconstruction." arXiv preprint arXiv:2606.07687. Yi, Kexin, Chuang Gan, Yunzhu Li, Pushmeet Kohli, Jiajun Wu, Antonio Torralba, and Joshua B Tenenbaum. 2019. "Clevrer: Collision events for video representation and reasoning." arXiv preprint arXiv:1910.01442. Zhang, Xinyu. 2026. "What Do World Models Learn in RL? Probing Latent Representations in Learned Environment Simulators." arXiv preprint arXiv:2603.21546. Appendix A. Formal Definitions and Notation This appendix collects the formal notation used in Section 3. It provides the meaning of the factual state, local counterfactual, native counterfactual hidden state, one-shot latent intervention, low-rank carrier, and bounded operational domain. Environment and model details are provided in Appendix B, and the exact counterfactual construction is given in Appendix C. The SVD carrier derivation, addressable ridge operator, and rollout evaluation details are given in Appendices D, E, and F, respectively. A.1 Core notation Table A1. Core notation and dimensions Symbol Meaning Dimension / domain ํโ Primitive two-object physical state at time t 8-dimensional real vector ํโ Numeric object-state observation at time t 2 ร 6 real array ํโ Model recurrent hidden carrier at time t 192-dimensional real vector ํก Intervention anchor time integer time index ํป Number of autonomous future transitions positive integer ํ Local edit specification: object, velocity axis, sign, and magnitude registered intervention family ํ ํก ํน Factual hidden state at the anchor 192-dimensional real vector ํ ํก ํถํน Native counterfactual hidden state after the edited anchor observation 192-dimensional real vector ํผแตฃ Rank-r candidate latent-carrier basis 192 ร r matrix c Coordinates of a specific edit in the carrier basis r-dimensional real vector ํ ํป Autonomous H-step rollout operator hidden state โ predicted trajectory ํบ Bounded operational evaluation domain tested state ร edit ร time domain A.2 Factual state and autonomous rollout For each object, the primitive state is ํฌ ํ,ํก = [ ํฅ ํ,ํก ,ํฆ ํ,ํก ,ํฃ ํฅ,ํ,ํก ,ํฃ ํฆ,ํ,ํก ] ํณ (A1) The joint two-object state is ํฌ ํก = [ ํฌ ํด,ํก ํณ ,ํฌ ํต,ํก ํณ ] ํณ โโ 8 (A2) Derived quantities such as relative velocity, center velocity, pair distance, and contact are deterministic functions of this state and are not treated as additional independent coordinates. Let the history/update map โฐ maps the observation history through time ํก to the recurrent hidden state ํก ํก ํน at the anchor and let the rollout operator R ํป generate the autonomous future ํ ฬ ํก:ํก+ํป ํน : ํก ํก ํน =โฐ ( ํ 0:ํก ํน ) (A3) ํ ฬ ํก:ํก+ํป ํน =R ํป ( ํก ํก ํน ) (A4) A.3 Same-history local counterfactual The factual and counterfactual branches share the same pre-anchor history: ํ 0:ํกโ1 CF =ํ 0:ํกโ1 ํน (A5) The only direct primitive change at the anchor is one registered velocity component: ํฌ ํก CF =ํฌ ํก ํน +ํฟํฃ ํ ํ,ํ (A6) Passing the edited anchor observation through the normal model update gives the native counterfactual hidden reference: ํก ํก CF =โฐ ( ํ 0:ํกโ1 ํน ,ํ ํก CF ) (A7) ํฅํก ํก native =ํก ํก CF โํก ํก ํน (A8) A.4 Low-rank and addressable intervention A rank-r candidate restricts the hidden perturbation to the column space of the carrier basis: ํฟํก=ํ ํ ํ (A9) For an addressable intervention, the coefficient vector is predicted from the factual physical state and the edit specification: ํ=ํ ( ํฌ ํก ํน ,ํ ) (A10) ํก ํก โ =ํก ํก ํน +ํ ํ ํ (A11) The basis itself is not unique. If Q is any orthogonal r-by-r matrix, rotating the basis and counter-rotating the coordinates gives the same 192-dimensional perturbation: ํ ํ ํ= ( ํ ํ ํ ) ( ํ ํณ ํ ) (A12) Equation (A12) is why individual carrier axes should not be identified with named physical coordinates. The behaviorally relevant object is the subspace together with the intervention it supports, not a unique orientation of the basis vectors. A.5 Dynamics-effective criterion and variable roles For evaluation, variables are separated by their role under the intervention. Direct targets are the explicitly edited velocity component. Contemporaneously derived variables, such as relative or center velocity, may change immediately. Dynamically downstream variables, such as later positions or post-contact velocities, should follow the simulator counterfactual. Only truly unaffected variables are required to preserve their factual value. Law and support constraints are evaluated after allowing the registered instantaneous intervention. ํ ฬ ํก:ํก+ํป โ =ํ ํป ( ํก ํก โ ) โํ ํก:ํก+ํป CF (A13) The approximation in (A13) is shorthand for the full multi-part decision rule; it is not a single Euclidean- distance test. Exact metrics, physical-validity checks, control comparisons, and pass thresholds are defined in Appendix F. A.6 Bounded operational domain All intervention claims are relative to the region over which the model and assay were actually evaluated: ํบ=S tested รํผ tested รT tested (A14) A result established on this domain does not automatically extend to larger intervention magnitudes, unseen physical regimes, longer effective time scales, different model architectures, or other environments. Such settings may be computationally reachable by the fitted operator, but they remain extrapolations until they are separately tested. Appendix B. Environment, Observation Interface, Model Architecture, and Training This appendix summarizes the environment and world model used in our experiments. It reports only what is actually implemented in the code and does not add assumptions that are not recorded there. For example, the simulator does not include explicit mass or restitution parameters, so we do not assign values to them here. B.1 Simulator and physical state Table B1. Simulator and physical-state specification Item Implemented value Scene Two circular objects moving in a two-dimensional square reflecting arena Primitive state per object x, y, horizontal velocity, vertical velocity Joint primitive state 2 ร 4 array; 8 scalar values Time step 0.25 Object radius 0.25 Valid object-center coordinate range [-4.75, 4.75] on each axis Process noise None; simulator is deterministic Stored episode 22 physical states (indices 0โ21) and 21 action slots Explicit mass / restitution Not represented as simulator parameters Collision detection uses the analytic earliest time of impact of two circles within one time step, obtained from the relative-motion quadratic. The simulator advances to the impact, computes the normal from the center displacement, applies the implemented normal impulse update, and then advances through the remaining portion of the step. Arena violations are handled by reflecting the out-of-range coordinate and flipping the matching velocity component until the center lies inside the allowed range. This code is consistent with equal-mass frictionless normal collision behavior, but the paper does not treat mass or restitution as explicit stored variables because they are absent from the scientific interface. B.2 Observation interface Table B2. Observation interface Property Value Per-time observation shape 2 ร 6 Per-object channels normalized noisy x, y, horizontal velocity, vertical velocity, visibility, identity Identity values -1 for one object slot and +1 for the other Directly observed when visible position, velocity, identity Not directly observed radius, contact Position-noise standard deviation 0.005 Velocity-noise standard deviation 0.01 Visibility schedule frames 0โ4 and the selected anchor are visible; physical channels at other frames are zeroed The factual and counterfactual members of a paired intervention unit share the same realized observation- noise array. Thus the hidden difference used for carrier fitting is not driven by an independent redraw of observation noise. Position and velocity are directly available to the model whenever the corresponding frame is visible; the study therefore does not claim that velocity must be inferred from images or from finite differences. B.3 Recurrent world model The model class is Deterministic Belief-like Model. It has 496200 trainable parameters and exposes a 192-dimensional carrier at the intervention interface. The carrier concatenates object-specific recurrent components and a global recurrent component as shown below. Table B3. Recurrent carrier composition and model-module dimensions Carrier block Width Object 0 deterministic recurrent state 48 Object 0 middle component 16 Object 1 deterministic recurrent state 48 Object 1 middle component 16 Global deterministic recurrent state 64 Total 192 Module Implemented dimensions Observation encoder 6 โ 128 โ 64 Action encoder 4 โ 128 โ 64 Message network 192 โ 256 โ 128 โ 64 Shared object GRU input 144, hidden 48 Global GRU input 192, hidden 64 Prior network 176 โ 256 โ 128 โ 16 Posterior network 176 โ 256 โ 128 โ 16 Decoder 192 โ 384 โ 192 โ 8 Normalization layers none Figure B1 summarizes the overall model architecture, the dimensions of the main modules, the 192- dimensional carrier composition, and the intervention point used in this study. Figure B1. Architecture of the Deterministic Belief-like Model and the 192-dimensional intervention carrier. The one-shot patch is applied to the full carrier after the recurrent update and anchor observation assimilation, before the autonomous future rollout. At time index t, the model performs its deterministic recurrent update, computes the prior and optional posterior quantities, selects the visible/update information, packs the full carrier, and stores the carrier for that time step. The intervention interface used in this paper is therefore post-update and post-observation- assimilation through t, but pre-transition for the future autonomous rollout. The one-shot patch is added to the complete 192-dimensional carrier at this interface. B.4 Training data and optimization Table B4. Training data and optimization settings Training item Value Dataset units 5,120 total units; 10,240 factual/counterfactual branches Training split 4,096 units Development-validation split 512 units Model-adequacy nonfinal split 512 units Training anchors t = 5, 6, 7, 8, 9 Batch construction 96 units / 192 branches; factual weight 0.5, counterfactual weight 0.5 Training duration 70 epoch-equivalents; 24 batches per epoch; 1,680 optimizer updates Optimizer AdamW; learning rate 0.0015; weight decay 0.00001; gradient clip norm 5.0 Scheduler Cosine annealing over 70 epoch-equivalents; minimum learning rate 0.00015 Checkpoint selection terminal model state after 1,680 updates; no best-validation selection loop The training objective combines rollout mean-squared error with auxiliary anchor-state, filtering-state, relative-state, and delta-velocity losses. Their frozen weights are 1.0, 0.2, 0.2, 0.25, and 0.25, respectively. The delta-velocity loss uses the recorded q90 scale 1.2762992187. Training does not use contact/event labels as auxiliary targets, does not supervise the later low-rank carrier or its rank, and does not perform a hyperparameter search over the intervention method. B.5 Checkpoint roles Table B5. Development and fresh-replication checkpoint roles Public role Internal seed Use in this paper Development model 1 291101 method development and rank selection Development model 2 291102 method development and rank selection Development model 3 291103 method development and rank selection Fresh replication model 1 291301 frozen-method replication Fresh replication model 2 291302 frozen-method replication Fresh replication model 3 291303 frozen-method replication All six checkpoints share the same architecture and 192-dimensional carrier interface. Carrier bases and addressable operators are checkpoint-specific; the study does not assume that the numerical basis coordinates are aligned across independently trained checkpoints. Appendix C. Counterfactual Pair Construction and Intervention Semantics This appendix specifies what is changed in a local counterfactual, what remains fixed at the intervention instant, and how simulator and model reference routes are constructed. The primary intervention family changes one velocity component of one object at a single anchor time. C.1 Same-history pairing Each factual/counterfactual pair begins from the same episode history. Before the anchor, the physical trajectory is identical. The paired observations also reuse the same realized observation noise rather than drawing independent noise for the counterfactual branch. ํ 0:ํกโ1 CF =ํ 0:ํกโ1 ํน , ํ 0:ํก CF =ํ 0:ํก ํน . (C1) In Eq. (C1), ํ denotes physical state and ํบ denotes the realized observation-noise sequence. The counterfactual branch differs only because the registered edit is applied at the anchor. C.2 Velocity-edit family The edit is encoded by four velocity slots. Exactly one slot is nonzero, identifying both the edited object and the edited axis. ํ= [ ํฅํฃ ํฅ,ํด ํฅํฃ ํฆ,ํด ํฅํฃ ํฅ,ํต ํฅํฃ ํฆ,ํต ] ํณ , โฅํโฅ 0 =1. (C2) The four entries correspond, in order, to the horizontal and vertical velocity edits of object A and the horizontal and vertical velocity edits of object B. The condition in Eq. (C2) requires exactly one of these four entries to be nonzero. The implemented family uses three absolute velocity increments with both positive and negative signs: | ํฅํฃ | โ0.066, 0.132, 0.264, ํฅํฃโยฑ | ํฅํฃ | . (C3) All positions and the other three primitive velocity components are left unchanged at the anchor. Derived quantities such as relative velocity, center velocity, summed velocity, and kinetic energy are recomputed from the edited primitive state and are allowed to change immediately. Later positions, distance, contact time, and later velocities are downstream consequences determined by simulator evolution from the edited state. The registered construction also enforces an edited-anchor speed support bound; the maximum recorded bound is approximately 0.6711. C.3 Simulator truth and model-native counterfactual After the primitive edit is applied, the deterministic simulator is restarted from the edited anchor state to generate the counterfactual future. The model-native G1 route uses the same factual pre-anchor history but supplies the edited anchor observation through the modelโs ordinary observation/update path. It then releases the model without future counterfactual observations. The G2 structural route injects the complete native hidden difference into the factual hidden state: ํ ํก G2 =ํ ํก ํน + ( ํ ํก CF โํ ํก ํน ) =ํ ํก CF . (C4) Equation (C4) is an algebraic identity. Its purpose is to verify that the hidden-state patching interface can reproduce the modelโs own native counterfactual state at the exact post-observation/pre-transition anchor interface. G2 is an interface control; G1 against simulator truth is the model-adequacy test. C.4 Evaluation strata Table C1. Evaluation strata and their operational roles Stratum Operational role S1: NO_CONTACT_WITHIN_HORIZON The edited counterfactual remains non-contacting over the 12-transition horizon; used to test local kinematics and autonomous propagation without later contact. S2: EDIT_AFFECTS_FUTURE_CONTACT The local velocity edit changes a future contact-related outcome or trajectory within the horizon; used to test whether downstream coupled consequences follow the edited physical future. S1 and S2 are always evaluated separately. A strong result in the easier non-contact regime is not allowed to compensate for a failure in the contact-sensitive regime. C.5 Anchor and rollout support The main low-rank experiment uses anchor t = 7 and 12 autonomous future transitions. The temporal extension evaluates anchors t = 5, 6, 7, 8, and 9. The anchor index is a discrete time-step index, not a time in seconds. With simulator time step 0.25, changing the physical speed too far outside the registered support would also change how much physical evolution can occur within one discrete step; such extrapolation is outside the current claim boundary. C.6 Position-edit diagnostic boundary The position intervention is a separate development-only diagnostic, not part of the confirmatory velocity family. At t = 7, one x or y coordinate is changed by an absolute magnitude 0.05, 0.10, or 0.20. The selected position field and its visible normalized anchor observation are changed, all velocities and other primitive anchor fields are held fixed, and simulator truth is regenerated from the edited state. This diagnostic asks whether the evaluation can distinguish a physically meaningful but potentially non- specific edit from the velocity carrier. It cannot redefine the primary rank or velocity claim. C.7 Development-only simultaneous velocity-vector extension A separate development-only extension tests simultaneous changes to the horizontal and vertical velocity components of the same object at t = 7. Both requested components are nonzero, while the other object and all position coordinates are unchanged directly at the anchor. The Euclidean magnitude of the two- component velocity change is restricted to the largest magnitude supported by the primary single- component velocity family, and the edited-anchor speed remains inside the same registered support. The joint-edit units use the ordinary S1 and S2 semantics of the main experiment; no contact-to-no- contact event flip or grazing-contact construction is required. The analysis is kept separate from the confirmatory single-component velocity family and cannot redefine the selected rank or the primary claim. Appendix D. Low-Rank Carrier Construction This appendix gives the linear-algebra construction behind the candidate carrier. The construction is deliberately simple: it uses the hidden change that the trained model itself produces for registered local counterfactuals, then asks whether those changes share a low-dimensional linear geometry. D.1 Hidden-difference matrix For each carrier-fit intervention unit ํ, the native counterfactual and factual carriers are taken at the same anchor interface and differ only because of the registered anchor edit: ํฅํ ํ =ํ ํก,ํ CF โํ ํก,ํ ํน โโ 192 . (D1) For each checkpoint, the fit split provides 1,024 such vectors. They are stacked as rows: ํซ=[ ( ํฅํ 1 ) ํณ โฎ ( ํฅํ ํ ) ํณ ]โโ ํร192 , ํ=1024. (D2) ํซ is built from raw hidden differences. No test unit enters this matrix, and no result from method- development or confirmatory evaluation is used to choose the basis. D.2 Uncentered singular-value decomposition The implemented carrier uses an uncentered singular-value decomposition of the raw local-edit hidden- difference matrix. In particular, the mean hidden difference is not subtracted before factorization: ํซ=ํณํฎํฝ ํณ . (D3) The columns of V live in the original 192-dimensional hidden coordinate system. The candidate rank-r basis is therefore ํผ ํ =ํฝ [ :,1:ํ ] , ํผ ํ ํณ ํผ ํ =ํ ํ . (D4) The columns of the basis in Eq. (D4) are orthonormal. To make saved artifacts deterministic, each basis vector is assigned a sign so that its largest-magnitude loading is positive. This sign convention has no scientific meaning; reversing a basis vector and its coefficient leaves the perturbation unchanged. D.3 Projection and oracle capacity The orthogonal projector onto the candidate carrier and the corresponding oracle coordinates are ํท ํ =ํผ ํ ํผ ํ ํณ , ํ ํ oracle =ํผ ํ ํณ ํฅํ ํ , ํฅํ ฬ ํ oracle =ํผ ํ ํ ํ oracle . (D5) The oracle reconstruction is the closest rank-r projection of the native hidden difference in Euclidean norm. Because the oracle coordinate vector is computed from the same unitโs native counterfactual hidden state, it is a capacity diagnostic only. It cannot be used as the primary addressable test-time operator. D.4 Descriptive second-moment capture For geometry diagnostics, we report the cumulative fraction of raw-difference second-moment energy captured by the first r singular directions: ํ ํ = โ ํ ํ 2ํ ํ=1 โ ํ ํ 2 192 ํ=1 . (D6) In Eq. (D6), the numerator sums the squared singular values retained by the first r directions, and the denominator sums them over all stored directions. Because the decomposition is uncentered, this ratio describes the second moment of the raw local-edit differences rather than the variance of mean-centered data. The ratio is descriptive only and is not interpreted as a physical dimension or used as the rank- selection gate. D.5 Registered rank sets R primary =1,2,4,8,12,16,20,32, R diag =64, R ref =0,192. (D7) The primary grid is fixed before reading development rank outcomes. Rank 64 is a geometry diagnostic. Rank 0 is a no-patch reference, and rank 192 is the full-hidden reference. The method reports the smallest passing rank on the primary grid; it does not claim that no untested rank, nonlinear coordinate system, or different carrier family could provide another solution. D.6 Checkpoint and split boundaries A separate basis is fitted for each checkpoint. No cross-checkpoint basis alignment is assumed. At the reference anchor, each checkpoint uses its own 1024-unit carrier-fit matrix. Method-development data may evaluate a fitted basis but do not refit it. Confirmatory data are not used for rank sweeps or method changes. For the fresh replication checkpoints, the already frozen construction is applied once under the preregistered confirmatory protocol. Appendix E. Addressable Coefficient Operator The carrier basis answers where a hidden edit may live. The addressable operator answers a different question: given a new factual physical state and a requested velocity edit, where should that edit be placed inside the carrier without reading the unitโs native counterfactual hidden state? E.1 Input encoding ํ ํ = [ ํ ํ ํ ํก,ํ ํน ] โโ 12 , ํ ํ โโ 4 , ํ ํก,ํ ํน โโ 8 . (E1) The edit vector has four slots: horizontal and vertical velocity edits for object A, followed by horizontal and vertical velocity edits for object B. Exactly one is nonzero. The factual primitive state contributes eight values: horizontal position, vertical position, horizontal velocity, and vertical velocity for each object. Thus the non-intercept input has 12 dimensions. The implementation stores a leading intercept column explicitly, so the fitted design matrix has 13 columns. E.2 Fit-only standardization Each non-intercept feature is standardized using mean and scale values computed on the fit split only: ํฅฬ ํ,ํ = ํฅ ํ,ํ โํ ํ max ( ํ ํ ,10 โ6 ) . (E2) In Eq. (E2), the two fit statistics are the mean and standard deviation of feature k on the carrier-fit split. A scale floor of 10 โ6 prevents division by a numerically tiny standard deviation. The same frozen mean and scale are reused at evaluation time; target-anchor transport in the temporal B2 test does not restandardize the features. E.3 Coefficient targets On the carrier-fit split, the native hidden differences are available, so the rank-r regression targets are the oracle coordinates: ํ ํ oracle =ํผ ํ ํณ ํฅํ ํ , ํช ํ =ํซํผ ํ โโ ํรํ . (E3) The target matrix ํช ํ in Eq. (E3) contains one r-dimensional coefficient vector for each carrier-fit unit. It is obtained by projecting each 192-dimensional hidden difference in ํซ onto the rank-r carrier basis ํผ ํ . Each row of ํช ํ is therefore the coefficient target used to train the addressable map. These targets are available only during fitting and are not available for a new evaluation unit. E.4 Ridge-affine fit ( ํพ ํ ,ํ ํ ) =argmin ํพ,ํ โ โฅ โฅ ํ ํ oracle โ ( ํพ ํณ ํ ฬ ํ +ํ ) โฅ โฅ 2 2 ํ ํ=1 +ํโฅํพโฅ ํน 2 , ํ=10 โ4 . (E4) The weight matrix maps standardized physical/edit features to carrier coordinates, and the affine intercept provides the offset. The frozen ridge coefficient is 10 โ4 . The intercept is not penalized. No nonlinear mapper, hyperparameter search, or outcome-based tuning is used in the primary operator. E.5 Test-time inference and patch ํ ฬ =ํ ํ ( ํ ํก ํน ,ํ ) =ํพ ํ ํณ ํ ฬ +ํ ํ . (E5) At test time, the fitted addressable map receives only the factual state and requested edit. It cannot access the native counterfactual hidden state, the true hidden difference, the simulator future, or future observations. The predicted coefficient is then converted to a hidden perturbation: ํ ํก โ =ํ ํก ํน +ํฝํผ ํ ํ ฬ , ํฝ=1. (E6) The primary intervention uses ฮฒ = 1. Once the patched hidden state is formed, the addressable operator is no longer consulted during the rollout. This separation is what makes the carrier addressable: the physical edit is translated into a low-dimensional hidden displacement without reconstructing the answer from a test-time counterfactual hidden state. E.6 Oracle capacity versus addressability Table E1. Oracle-capacity and addressable coefficient routes Route Coefficient source Allowed interpretation Oracle capacity Same-unit native hidden difference projected onto the rank-r carrier basis. Tests whether the rank-r subspace can carry the required hidden difference; privileged upper bound only. Addressable operator Predicted from the factual primitive state and the registered edit. Primary intervention route; tests whether the carrier can be reached without a test- time native counterfactual hidden state. E.7 Joint-edit composition and operator extensions For the simultaneous velocity extension, the original four-slot edit vector can contain the two nonzero velocity entries belonging to the same object. With the original frozen affine operator, the coefficient predicted from the joint request was also compared with the coefficient obtained by composing the two corresponding single-component requests and subtracting the zero-edit offset: ํ ํฅํฆ = ํ ํฅ + ํ ํฆ โ ํ 0 The direct and component-wise composed coefficients agreed to numerical precision, and the resulting direct and composed rollouts were numerically identical. This verifies the algebraic composition implied by the affine map, but it does not by itself establish target-specific dynamics-effective composition. A later operator-only development test kept the world model and the checkpoint-specific rank-4 carrier fixed. The first candidate refitted the same affine feature family using equal numbers of the original single-component carrier-fit units and new joint-edit fit units. The second candidate added only two same- object joint-product features, one for each object, to allow a minimal non-additive correction. Both candidates used the same frozen rank, intervention strength, ridge setting, and original feature standardization for the pre-existing features. Both also preserved the original single-component development result. Neither candidate closed the joint-edit specificity gap, so no extended operator was selected. Appendix F. Rollout Metrics, Eligibility Gates, and Registered Thresholds This appendix gives the exact evaluation kernel used for the low-rank velocity-carrier and temporal- stability experiments. The five scientific metrics are shared across both evaluations. They score the decoded anchor, the released twelve-step future, and physical consistency after the intervention. Numerical eligibility, autonomy metadata, stratum definitions, and higher-level aggregation rules are reported separately so that engineering validity is not conflated with scientific performance. F.1 Evaluation notation and frozen normalization scales For evaluation unit ํ, ํก ํ denotes the intervention anchor and ํป = 12 the number of autonomous future transitions. The denormalized decoded state at relative index ํ is denoted by ํฟ ํ,ํ ฬ and has shape 2 ร 4, with object rows 0 and 1 (corresponding to objects A and B) and primitive columns ordered as ํฅ,ํฆ,ํฃ ํฅ ,ํฃ ํฆ . Relative index ํ = 0 is the decoded anchor, and ํ = 1,...,12 are the twelve released future states. ํฟ ํ,ํ ํํํ denotes the reference state used by the metric kernel to evaluate a given route. For the primary patched, native-counterfactual, wrong-object, and full-hidden G2 routes, ํฟ ํ,ํ ํํํ =ํฟ ํ,ํ ํถํน ; for the factual/no-edit control, ํฟ ํ,ํ ํํํ =ํฟ ํ,ํ ํน . The edited object index is ํ ํ โ0,1. The velocity-axis index is ํ ํ โ0,1, so the edited primitive column is ํ ํ =2+ํ ํ . The frozen primitive scale matrix ํฌ ํ is applied coordinatewise: Table F1. Frozen primitive-coordinate normalization scales ํ ํ ํ ํ ํ ํ Object 0 (A) 0.64721 0.62436 0.40208 0.39198 Object 1 (B) 0.68680 0.55883 0.40071 0.35781 M2 uses a 13-dimensional derived vector ํ ( ํ ) . Its entries, in order, are the two coordinates of relative position ํฉ 1 โํฉ 0 , the two coordinates of relative velocity ํฏ 1 โํฏ 0 , the two center-position coordinates ( ํฉ 0 +ํฉ 1 ) /2, the two center-velocity coordinates ( ํฏ 0 +ํฏ 1 ) /2, the two summed-velocity coordinates ํฏ 0 +ํฏ 1 , one kinetic-energy-like scalar 1 2 โ โฅ ํ ํฏ ํ โฅ 2 2 , inter-object distance, and a distance-based contact indicator. The corresponding frozen scales ํฌ ํท are: Table F2. Frozen scales for the 13-dimensional derived representation d Derived feature ํ ํ Frozen scale ํ ํ ํท 1 relative_position_x 1.04125 2 relative_position_y 0.87639 3 relative_velocity_x 0.74275 4 relative_velocity_y 0.79797 5 center_position_x 0.32511 6 center_position_y 0.32665 7 center_velocity_x 0.06288 8 center_velocity_y 0.07177 9 summed_velocity_x 0.12576 10 summed_velocity_y 0.14354 11 0.5 x sum of squared speeds 0.06497 12 inter-object distance 0.35839 13 ํdistanceโค0.50000001 1.00000 The scale registry was calibrated once and then read as frozen constants by the evaluators. For continuous coordinates, the stored scale estimator is max ( IQR/1.349, 1.4826 MAD, 0.01 ) ; the contact-indicator scale is fixed to 1. No M1-M4 formula adds an epsilon, clipping operation, intervention-amplitude normalization, or metric-specific weight. F.2 Anchor fidelity and future rollout metrics (M1-M4) M1 measures direct fidelity of the edited velocity coordinate at the decoded anchor. ํ 1,ํ = | ํฟ ฬ ํ,0,ํ ํ ,ํ ํ โํฟ ํ,0,ํ ํ ,ํ ํ ref | ํ ํ ํ ,ํ ํ ํ , ํ ํ =2+ํ ํ . (F1) Here ํ ํ is the edited object, ํ ํ is the x/y velocity-axis selector, and ํ ํ is therefore the ํฃ ํฅ or ํฃ ํฆ primitive column. M1 is a single normalized absolute error. It does not average over other coordinates or over time. M2 measures fidelity of the complete 13-dimensional derived representation at the anchor. ํ 2,ํ = โ 1 13 โ ( ํ ํ ( ํ ฬ ํ,0 ) โํ ํ ( ํ ํ,0 ref ) ํ ํ ํท ) 2 13 ํ=1 . (F2) The index ํ runs over the derived features listed in F.1. Each coordinate error is divided by its own frozen derived scale, and the metric is the unweighted RMS across the 13 normalized errors. The hard contact feature changes at distance 0.50000001 and has denominator exactly 1. M3 evaluates the autonomous future after the anchor. ํ 3,ํ = โ 1 8ํป โ ( ํฟ ฬ ํ,ํ,ํ,ํ โํฟ ํ,ํ,ํ,ํ ref ํ ํ,ํ ํ ) 2 3 ํ=0 1 ํ=0 ํป ํ=1 , ํป=12. (F3) The anchor ํ=0 is excluded. M3 is one RMSE over all 12 future states, both objects, and all four primitive fields, giving 96 normalized scalar errors per unit. No contact-conditioned weighting, pre/post- contact split, or missing-frame censoring is used. M4 checks preservation at the anchor for primitive coordinates recorded as unaffected by the intervention. ํ 4,ํ =max ( ํ,ํ ) : ํผ ํ,ํ,ํ =1 | ํฟ ฬ ํ,0,ํ,ํ โํฟ ํ,0,ํ,ํ ref | ํ ํ,ํ ํ . (F4) ํ ํ is the record-provided 2ร4 Boolean unaffected-primitive mask. For the registered one-component velocity edits, the frozen generators mark seven primitive slots as unaffected and the edited velocity slot as affected. M4 takes the maximum, not the mean, of the selected normalized anchor errors. There is no future preservation term in M4. F.3 Post-edit law and geometry consistency (M5) M5 is not a single physical residual. It is the maximum of five terms: one-step simulator replay, circle overlap, arena-boundary violation, excessive per-step displacement, and consistency between the recorded factual-counterfactual anchor difference and the declared one-slot intervention. Let S denote the exact one-step simulator used by the evaluator, including the recorded action, analytic two-circle collision handling, remaining-time motion, and reflecting walls. ํ ํ,ํ dyn = โ 1 8 โ( ํ ฬ ํ,ํ+1,ํ,ํ โ [ S ( ํ ฬ ํ,ํ ,ํ ํ,ํก ํ +ํ )] 1,ํ,ํ ํ ํ,ํ ํ ) 2 3 ํ=0 1 ํ=0 , 0โคํ<ํป. (F5) ํ dyn compares each decoded next state with the exact simulator result obtained by initializing the simulator from the decoded current state. The subscript 1 selects the resulting next state of the one-step simulator. All eight primitive coordinates are normalized by ํฌ ํ before the RMSE is taken. ํ ํ overlap = 1 0.01 max 0โคํโคํป max(0.5โ โฅ โฅ ํ ฬ ํ,ํ,1 โํ ฬ ํ,ํ,0 โฅ โฅ 2 , 0). (F6) ํ overlap penalizes positive overlap between the two radius-0.25 circles. The penalty is evaluated on all 13 decoded states and scaled by 0.01. The inner maximum with zero implements the positive-part operation. ํ ํ arena = 1 0.01 max ํ,ํ,ํโํฅ,ํฆ max (| ํฬ ํ,ํ,ํ,ํ | โ4.75, 0 ) . (F7) ํ arena is the maximum positive excess of any decoded object-center coordinate beyond the valid arena bound | ํ ํ | โค4.75, again scaled by 0.01. ํ ํ step = 1 0.01 max 0โคํ<ํป ํโ0,1 max( โฅ โฅ ํ ฬ ํ,ํ+1,ํ โํ ฬ ํ,ํ,ํ โฅ โฅ 2 โ0.25max( โฅ โฅ ํ ฬ ํ,ํ,ํ โฅ โฅ 2 , โฅ โฅ ํ ฬ ํ,ํ+1,ํ โฅ โฅ 2 ) โ10 โ8 , 0). (F8) ํ step checks the position displacement over each of the 12 transitions against the larger endpoint speed multiplied by the simulator time step 0.25, with a numerical allowance of 10 โ8 . It uses both objects and is scaled by 0.01. ํ ํ =ํ ํ,0 CF โํ ํ,0 F , ํ ํ decl = โฅ โฅ ํ ํ โํ ํ โฅ โฅ โ 10 โ8 . (F9) ํ ํ is the complete recorded simulator counterfactual-minus-factual difference at the anchor. ํ ํ is a 2ร4 zero matrix except at the declared edited slot ( ํ ํ ,ํ ํ ) , where it equals the same counterfactual-minus- factual anchor difference for that slot. ํ ํ decl therefore checks the intervention record itself at tolerance scale 10 โ8 ; it does not compare the modelโs decoded anchor with the declaration. ํ 5,ํ =max max 0โคํ<ํป (ํ ํ,ํ dyn ,ํ ํ overlap ,ํ ํ arena ,ํ ํ step ,ํ ํ decl ) . (F10) M5 is the maximum across the twelve replay residuals and the four remaining penalties. The terms are not summed or weighted. The declared anchor edit is allowed because replay starts from the already edited decoded anchor. Collision consistency is enforced through the exact simulator replay rather than by a separate contact-time penalty, and boundary reflection is included in that replay. There is no additional speed-support, explicit energy-conservation, or out-of-distribution penalty in M5. If a decoded output is nonfinite, the law-residual path returns infinity. Frozen experiments always use ํป=12 transitions. F.4 M6E numerical eligibility M6E is a per-unit engineering eligibility gate rather than a scientific metric. In the main low-rank evaluation it requires finite final hidden state and decoded trajectory, together with loose explosion bounds: ํธ ํ main โfinite ( ํก ํ final ) โงfinite ( ํ ฬ ํ,0:ํป ) โง( โฅ โฅ ํก ํ final โฅ โฅ 2 โค10 6 )โง( โฅ โฅ ํ ฬ ํ,0:ํป โฅ โฅ โ โค10 6 ). (F11) The hidden vector in Eq. (F11) is only the final 192-dimensional hidden state after the twelve transitions. Intermediate hidden states and the anchor hidden state are not separately checked by M6E. The temporal evaluation uses the same gate and adds a per-decoded-state 8-dimensional ํฟ 2 bound: ํธ ํ temp โํธ ํ main โง(max 0โคํโคํป โฅ โฅ vec ( ํ ฬ ํ,ํ ) โฅ โฅ 2 โค10 6 ). (F12) M6E does not itself test serialization completion, generic execution completion, output shape, dataset support, or intermediate-hidden finiteness. Such failures occur elsewhere in the execution path rather than being encoded as additional M6E conditions. An M6E failure is never removed from the denominator: it causes the unit to fail joint coverage and also prevents the cell from satisfying the all-eligible requirement. F.5 M7 autonomous-rollout structure M7 records the structural conditions of the released rollout. For every scored route in the frozen 12- transition assay, the following values are attached: Table F3. Required structural metadata for the autonomous rollout (M7) M7 field Required value transition_count 12 decoder_count 13 observation_reads_after_anchor 0 teacher_forcing_steps 0 state_clamp_steps 0 future_oracle_reads 0 pass True The decoder count includes one anchor decode plus one decode after each of the twelve transitions. These values are literal structural metadata asserted by the frozen rollout construction rather than runtime- instrumented counters. The rollout reads the recorded action once per transition, but M7 has no separate action-use counter or action-path Boolean. F.6 Full-hidden structural equivalence (G2) The full-hidden reference route tests the intervention wiring by starting a second rollout from a copy of the same native counterfactual hidden state used by the native route. ํ ํ =max ํ,ํ,ํ,ํ | ํ ํ,ํ,ํ,ํ ํบ2 โํ ํ,ํ,ํ,ํ ํบ1 | , G2Passโํ ํ โค10 โ6 . (F13) G1 denotes the native-counterfactual rollout and G2 the copied-full-hidden rollout. The rollout discrepancy ํ ํ is the maximum absolute difference over every unit in the stratum, all 13 decoded states, both objects, and all four primitive fields. Hidden discrepancy is not independently computed: ํ โ is exactly zero by construction because G2 receives a copy of the same ํ ํถํน . Only ํ ํ is numerically gated, with the inclusive tolerance 10 โ6 . F.7 Programmatic S1/S2 strata Thresholds are applied separately to two pre-registered strata. Define the contact-step detector ฮ over an anchor-plus-future state sequence as follows, where ํ ํ is the 2ร2 array of object velocities and the Frobenius norm covers both objects and both velocity components: ฮ ( ํ ) = ํโ1,...,ํป:โฅํ ํ,1 โํ ํ,0 โฅ 2 โค0.500001 โจ โฅํ ํ โํ ํโ1 โฅ ํน >10 โ5 . (F14) The detector returns only relative future indices ํ=1,...,ํป; the anchor ํ=0 is not itself returned as a contact step. A step is flagged either by center distance โค0.500001 or by a velocity-array jump larger than 10 โ5 . ํ1โฮ ( ํ 0:ํป ํน ) =โ โง ฮ ( ํ 0:ํป ํถํน ) =โ. (F15) S1 therefore requires both the factual and counterfactual branches to have empty contact-step lists across the 12-transition window. ํ2โฮ ( ํ 0:ํป ํน ) โ โ โง max ํ>ํก ํ ,ํ,ํ | ํฟ ํ,ํ,ํ ํถํน โํฟ ํ,ํ,ํ ํน | โฅ10 โ4 . (F16) S2 requires factual contact in the 12-transition window and a sufficiently large post-anchor factual- counterfactual trajectory difference. The second condition is evaluated over all stored post-anchor states in the frozen generator rather than being explicitly sliced to ํป. Importantly, S2 does not require counterfactual contact, a changed contact/noncontact status, or a changed contact time. The evaluator consumes the frozen stratum label rather than recomputing it. F.8 Registered thresholds and unit/cell decision rule The same frozen threshold registry is used by the main low-rank and temporal evaluations. Comparisons are inclusive: ties at the threshold pass. A nonfinite unit metric fails before the numerical comparison. Table F4. Registered M1-M5 thresholds for S1 and S2 Metric S1 threshold S2 threshold M1 0.38876 0.46698 M2 0.65309 0.65681 M3 0.28745 0.55140 M4 0.33938 0.25616 M5 20.26019 10.47116 For unit ํ in stratum ํ , ํ ํ is the joint scientific threshold condition, ํธ ํ is the joint numerical/structural eligibility condition, and ํฝ ํ is the final joint unit-pass condition: ํ ํ โโ 5 ํ=1 [ finite ( ํ ํ,ํ ) โงํ ํ,ํ โคํ ํ ,ํ ] , ํธ ํ โํ6ํธ ํ โงํ7 ํ , ํฝ ํ โํ ํ โงํธ ํ . (F17) M6E and M7 are therefore part of the unit-level joint pass condition; they are not merely annotations attached after the scientific metrics. Coverage= ํ joint-pass ํ expected . (F18) ํ joint-pass is the number of materialized units for which ํฝ ํ is true. ํ expected is fixed by the registered cell design: 256 units for each checkpoint ร stratum cell in the main low-rank confirmatory evaluation, and 128 units for each checkpoint ร anchor ร stratum cell in the temporal evaluation. A failed or ineligible unit remains in this denominator. CellPassโํถํํขํํกํํพโงํธํํํํํพโงํถํํฃํํํํํํํพโงํํํํํํํํพโงControlsPass. (F19) In Eq. (F19), Cํํขํํกํํพ means ํ materialized =ํ expected ; Eํํํํํพ means every unit satisfies ํธ ํ ; Cํํฃํํํํํํํพ means Coverageโฅ0.8; and Mํํํํํํํพ means that, for every ํ=1,...,5, the raw cell median of ํ ํ is finite and no larger than the corresponding stratum threshold. ControlsPass denotes the mandatory control conditions defined in Appendix G. S1 and S2 are evaluated separately and are never pooled before the cell decision. F.9 Checkpoint and panel aggregation A checkpoint-level primary result requires both strata to pass for the same checkpoint: SeedPass ํ =CellPass ํ,ํ1 โงCellPass ํ,ํ2 . (F20) For a three-checkpoint confirmatory panel, the pre-registered primary rule is two passing checkpoints out of three: PanelPassโ | ํโ1,2,3:SeedPass ํ | โฅ2. (F21) The temporal tests apply the same two-of-three checkpoint rule independently at each registered anchor or transport target. The overall temporal existence result requires every anchor panel at 5, 6, 7, 8, and 9 to pass, while frozen-anchor transport requires every target panel at 5, 6, 8, and 9 to pass: ExistenceOverallโโ ํกโ5,6,7,8,9 PanelPass ํก , TransportOverallโโ ํกโ5,6,8,9 PanelPass ํก . (F22) Individual trajectory units are not treated as independent model-level replications. Negative-control panel requirements, including the random-orthogonal carrier, remain separate control conditions and are specified in Appendix G. F.10 Joint-edit scoring and paired-specificity calibration For a same-object two-component velocity edit, the two direct velocity errors are evaluated separately using the original primitive-coordinate normalization and the original stratum-specific M1 threshold. A joint direct-target summary uses the larger of the two normalized velocity errors; the future summary remains M3. The unaffected-primitive mask contains six slots because both velocity components of the edited object are direct targets, and the declaration-consistency term in M5 permits exactly those two direct velocity changes. Because the absolute main-experiment thresholds can be satisfied by more than one nearby joint-edit route, the development-only composition analysis also uses a paired specificity comparison on matched units. For each intended-versus-control pair, the control must have a larger median direct-target error and a larger median future error, and at least 0.80 of paired units must be worse than the intended route for both summaries. This 0.80 comparator was calibrated retrospectively against the already established single-component positive result and was found to be compatible with that baseline. It is used only to diagnose the joint-edit extension and does not alter the registered thresholds or decisions of the primary experiment. Appendix G. Control Routes and Temporal Generalization Protocols This appendix collects the controls that make a hidden-state intervention interpretable and the temporal protocols used after the main t = 7 experiment. The controls are designed to distinguish a targeted dynamics-facing edit from generic hidden-state perturbation, interface errors, timing errors, or data leakage. G.1 Positive and structural reference routes Table G1. Reference and control routes used in the intervention assay Route / control Purpose FACTUAL_NO_EDIT Checks ordinary factual rollout from the same model and interface. SIMULATOR_COUNTERFACTUAL_TRUTH External physical reference generated from the edited simulator state. NATIVE_COUNTERFACTUAL_FULL_HIDDEN (G1) Checks whether the model can natively represent and propagate the complete edited current state. G2_FULL_HIDDEN_EQUIVALENCE Checks that hidden injection reproduces the native counterfactual route to the registered numerical tolerance. SHAM_EDIT Exercises the edit/operator path with zero physical change; should behave as a no-op. RANK_ZERO Explicit no-patch reference for the low-rank intervention family. RANDOM_ORTHOGONAL_EQUAL_NORM Tests whether an equal-size but unrelated hidden perturbation can mimic the targeted result. WRONG_OBJECT_EDIT Tests object specificity by routing the edit to the other object while keeping the edit family comparable. WRONG_TIME Tests timing specificity by applying the registered mechanism at an adjacent or otherwise registered incorrect anchor. LEAKAGE_NEGATIVE_TEST Verifies that building the intervention does not require counterfactual future payload. IDENTITY_ORDER_CONTROL Verifies object-slot identity and decoder/output ordering. The dataset actions are all zero. For that reason, an action-shuffle route contains no scientific contrast and is marked not applicable rather than counted as a successful negative control. G.2 B1: per-anchor carrier existence B1 asks whether an addressable rank-4 carrier can be established independently at several absolute anchors. The tested anchors are t = 5, 6, 7, 8, and 9. For each checkpoint and anchor, the basis, feature standardization, and ridge operator are fitted only on that anchorโs carrier-fit split. The confirmatory evaluation then uses the frozen rank-4 setting once on the corresponding sealed cells. This assay supports an existence question at each time; it does not require the numerical basis at one anchor to equal the basis at another. Table G2. Temporal data allocation for the B1 per-anchor analysis Temporal data role Per anchor ร stratum Total across five anchors and two strata Carrier fit 128 units 1280 units Method development 128 units 1280 units Confirmatory nonfinal 128 units 1280 units G.3 B2: exact transport of the t = 7 coordinate system B2 asks a stronger question. For each checkpoint, the exact t = 7 artifact is transported to target anchors t = 5, 6, 8, and 9. The transported artifact includes the t = 7 carrier basis, ridge weights and intercept, feature means and scales, the frozen rank-four setting, and full intervention strength. No target-anchor basis fit, mapper fit, feature restandardization, rank change, or intervention-strength change is allowed. B1 at the target anchor serves as the refitted reference; B2 tests whether the same frozen coordinate system can be reused. For both B1 and B2, each anchor/stratum cell uses the same M1โM5, M6E, M7, control, coverage, and 2-of-3 checkpoint rules defined in Appendix F. The temporal pairing analysis is descriptive and does not change the cell or panel gate. G.4 Temporal interpretation boundary Repeated success at several anchors does not by itself prove that the latent coordinates are globally time- invariant. Exact t = 7 transport over the tested nearby anchors is stronger evidence, but it is still bounded to the registered model family, physical state support, edit family, and time range. Neither B1 nor B2 establishes a unique physical gauge, a globally closed low-dimensional Markov state, belief-state sufficiency, or transfer to new architectures and environments. G.5 Joint-edit controls and development boundary The simultaneous velocity analysis uses four matched scientific controls. RANK_ZERO / NO_PATCH tests whether the factual hidden state is already close enough to the intended joint future. RANDOM_ORTHOGONAL_EQUAL_NORM tests an unrelated hidden perturbation with the same patch norm. WRONG_OBJECT_JOINT applies the same signed two-component request to the other object. MATCHED_WRONG_VECTOR keeps the object and edit-vector norm fixed but changes the vector direction, using a component swap when it changes the request and a one-component sign flip otherwise. The latter control tests whether the intervention distinguishes the velocity-vector direction rather than only the total edit magnitude. Sham, leakage, and identity/order checks remain engineering or integrity controls; wrong-time is retained only as a secondary diagnostic for this extension. On the final development analysis, factual, native joint-counterfactual, full-hidden, and frozen rank-4 oracle prerequisite routes passed both strata for all three development checkpoints. The original frozen affine operator, a joint-aware affine refit, and a minimal joint-correction ridge map each passed the intended joint-rollout criterion in 3/3 checkpoints. However, none of the three operators satisfied the complete paired-specificity panel for the no-patch, random, wrong-object, or matched wrong-vector controls; the failures were concentrated in S2. The layer-localization analysis showed that no-patch, random, and matched wrong-vector distinctions retained at the rank-4 oracle layer were not recovered with the same strength by the addressable maps. Wrong-object specificity in S2 also showed a native- model ceiling on the development panel. Accordingly, the joint-edit extension is not treated as a second positive intervention family. It shows that the frozen rank-4 carrier has capacity for bounded simultaneous velocity-component changes, while simple addressable maps trained or minimally extended for joint requests do not achieve the same target- specific control as the primary single-component interface. No joint-aware operator was nominated for confirmatory testing, and no fresh-checkpoint joint-edit claim is made.