A kinematic-consistency diagnostic finds attenuated friction sensitivity in free imagined rollouts relative to real physics and observation-driven reconstruction, with weak resolved responses under action replay or long conditioning.
Abstract. Long-horizon failure in world models is conventionally attributed to compounding error, a framing that does not distinguish what kind of error compounds. We operationalize a kinematic-versus-dynamic hypothesis using imagined Kinematic-Consistency Error (iKCE), a per-step deviation from a constant-velocity null, and a physical-regime perturbation protocol. iKCE is a diagnostic, not a training objective: minimizing it favors the kinematic null. On DreamerV3-class DMC walker-walk checkpoints, free imagined iKCE after five conditioning observations has no resolvable log-log friction slope, with \(\lvert\beta\rvert \le 0.12\) at 95% in each of three seeds, while the within-checkpoint slope difference from real physics excludes zero. Strict equivalence at a ±0.10 margin is established for seed 0, but not seeds 1–2 individually. On seed 0, observation-driven reconstruction through the same decoder has a resolved slope (\(-0.213\)), whereas free imagination is substantially less responsive. Reconstruction and one-step-prior residuals are ≈42–75× phase-matched physics; the original action/phase-unmatched imagined ratios are ≈180× at 16 steps and ≈30× at 64 steps. Thus these levels cannot be attributed solely to accumulated imagination error. Deterministic imagination does not reveal a resolved friction slope. Action-matched replay produces a weak resolved slope (\(-0.075\)), about four times shallower than its physics counterpart, but the imagined walker sinks into a low posture. With a 64-observation prefix, a weak late-window response also remains (\(-0.026\)). The result is attenuation of friction sensitivity in this residual, not absence of dynamic information or demonstrated selective kinematic sensitivity. The kinematic interpretation and the unresolved positive control are scoped in Separability.
Revision note (September 2026). Text in blue was changed or added in response to the reviews. New material: the noise-floor rows of Table 1, the action-matched pair in Table 1b, the deterministic-imagination comparison, the bounded-effect analysis in Table 3, a K = 80 rerun of the domain-randomized pair, and a post-hoc posture audit of the saved action-matched trajectories (Figure 5c). The title now names the measured friction-sensitivity result.
World models have become a central component of recent embodied AI, serving as latent simulators for planning
We adopt the classical mechanics distinction between kinematic and dynamic motion. We define kinematic as motion described purely through position, velocity, and acceleration time series, without invoking the forces or physical constraints that produced it. We define dynamic as motion that requires those constraints (e.g., mass, friction, contact) to be reproduced correctly.
Kinematic fallback is a third, structurally distinct account of long-horizon world-model failure, alongside the two positions that dominate the literature: predictable-representation engineering (the Dreamer line
We make three contributions. First, we recast long-horizon world-model failure in kinematic-vs-dynamic terms, distinguishing a structural-content layer from the variance-engineering and error-compounding layers studied in prior work. Second, we introduce imagined kinematic-consistency error (iKCE) together with a conditioning-perturbation protocol that operationalize this account as a falsifiable evaluation diagnostic. iKCE is a diagnostic and is structurally unsuited as a training objective (its optimum is the failure mode it detects; see the Diagnostic Protocol); the constructive question of how to induce dynamic imagination belongs to the training data or the architecture and is not addressed here. Third, we instantiate the diagnostic on an open-weight checkpoint (DreamerV3 on DMC walker-walk). The imagined rollouts show a large, temporally-structured iKCE that is nonetheless not resolvably sensitive to the dynamic (friction) axis across the gait-collapse boundary, bounded where matched physics is steep. Controls bound rather than eliminate the principal confounds: a domain-randomization control for the policy-out-of-distribution confound, an actor-training-horizon ablation, and seed replicates. We scope what this does not establish in the Separability section: the intended kinematic positive control (joint noise) does not yield a resolved WM response. The two established signatures are a decoder-controlled contrast, in which the model’s own reconstruction of the real trajectory has a resolved friction slope compatible with the physics estimate while free imagined iKCE through the same decoder has no resolved slope, and a bounded friction response of free imagination, significantly shallower than physics in the three seed comparisons and the K = 80 domain-randomized comparison. Operationally, we call the imagination kinematic if its iKCE responds to perturbations of the kinematic state the rollout extrapolates but shows no resolvable response under perturbations of the physical regime that governs the real dynamics, and dynamic if it responds to the physical-regime axis. The Separability section makes this distinction falsifiable and scopes what the experiments do and do not establish.
Our diagnosis is motivated by four existing observations, each inconclusive alone and explained only in isolation by its original authors, but jointly suggestive of a common structural signature. The four use different methods in the driving and VLA setting (representational probing, sensor-perturbation stress-testing, an open-loop baseline comparison, and physics-consistency scoring), and are consistent with a common structural deficit: the learned representation is dominated by kinematic features, while the dynamic features required for physical-regime-conditional behavior appear under-represented. These are external observations in other systems. They motivate the diagnostic we build, and the experiments below test it directly on a world model rather than treating these works as evidence for the central claim.
(i) Representational diagnostic on driving VLMs and VLAs. Schäfer et al.
(ii) Sensor-degraded behavioral diagnostic. Priyadershi and Frtunikj
(iii) Open-loop trajectory-prediction baselines. Zhai et al.
(iv) Physics-consistency scoring on fine-tuned VLAs. Gao et al.
Definition (Imagined Kinematic-Consistency Error). For an imagined rollout \(\{\hat{x}^{\mathrm{WM}}_t\}_{t=0}^{T}\) produced by a world model, with \(\hat{x}_t\) a chosen kinematic state vector (e.g. \([x, y, v, a, \theta]^{\top}\)), the imagined kinematic-consistency error is
\[\mathrm{iKCE} \;\doteq\; \frac{1}{T}\sum_{t=0}^{T-1} \left\lVert\, \hat{x}^{\mathrm{WM}}_{t+1} - \mathrm{kin}\!\left(\hat{x}^{\mathrm{WM}}_{t}\right) \right\rVert^{2},\]where \(\mathrm{kin}(\cdot)\) is any closed-form kinematic predictor (e.g. constant-velocity or constant-acceleration) chosen to match the world model’s underlying embodiment and output space. In the walker-walk instantiation, the identity view uses a constant-velocity predictor on the root vertical state (here \(z\) denotes the torso height, the DMC root vertical position, not the DreamerV3 stochastic latent):
def kin(state):
# state = [z, z_dot], constant-velocity extrapolation
z, z_dot = state
dt = 0.025 # DMC physics timestep (25 ms)
return [z + dt * z_dot, z_dot]This follows the mathematical form of the kinematic-consistency loss in Gao et al.
Low iKCE does not indicate dynamic imagination. A world model with near-zero iKCE predicts, by construction, next states that coincide with the kinematic continuation of their predecessors, so it imagines kinematically. The signature of dynamic imagination is the opposite: iKCE positive, growing with horizon, and responsive to physical-regime conditioning (friction transients, contact events, regime-boundary crossings). iKCE is therefore necessary but not sufficient to certify dynamic imagination.
Why iKCE is a diagnostic, not a training objective. Gao et al.
Scope of the diagnostic. iKCE asks one question, whether the per-step kinematic structure of a rollout on a chosen state slice responds to a swept physical regime, and answers it with a falsifiable slope. It diagnoses the response of a chosen kinematic-null residual, not prediction accuracy, not complete state-level regime invariance of the decoded states or the latent, and not the absence of dynamic information in the latent. It is deliberately not an accuracy score: a rollout can be kinematically self-consistent and wrong, and a prediction-error metric would not separate the two, which is the separation this paper needs. Its absolute level includes the model’s own reconstruction floor, which we measure directly (Table 1), so the diagnostic reads slopes rather than levels, and the admissibility criterion below fixes which slices can carry a reading.
Static iKCE measures internal kinematic self-consistency along a single rollout. To turn this into a diagnostic that separates kinematic from dynamic imagination, we drive iKCE through a dose-response curve over the conditioning state. For each base rollout, the world model generates \(K\) imagined rollouts under controlled perturbations of physically meaningful conditioning parameters: initial velocity \(v_0\), friction coefficient \(\mu\), or lateral-acceleration limit \(a_{\mathrm{lat,max}}\) for driving, and terrain compliance or payload for legged locomotion. The perturbation set is embodiment-specific. The protocol is not.
The MuJoCo friction perturbation used here scales the friction of every geom at episode reset, and the joint-noise positive control adds Gaussian noise to the joint-position channel before the encoder:
# friction perturbation (dynamic, physically grounded, regime-crossing)
model.geom_friction[:, 0] *= mu # at episode reset
# joint-noise perturbation (kinematic positive control)
obs_joints += rng.normal(0.0, sigma, size=obs_joints.shape) # before WM encoderTwo diagnostic signatures emerge from the resulting \(\{\mathrm{iKCE}_k\}_{k=1}^{K}\) ensemble. First, the shape of \(\mathrm{iKCE}(\lVert\Delta\rVert)\) as a function of perturbation magnitude: a kinematic imager produces a curve that scales smoothly with \(\lVert\Delta\rVert\) regardless of physical regime, because it is extrapolating the same linear update structure in every case. Because iKCE measures per-step kinematic-null deviation, physical-regime perturbations whose effects are slow relative to the per-step timescale (e.g., friction-driven slipping in legged locomotion, which accumulates over multiple footfalls) are not visible at horizons shorter than their characteristic accumulation time. The diagnostic should be applied at horizons longer than the embodiment’s gait period: ~25 ms × 64 steps ≈ 1.6 s is sufficient for walker-class locomotion. Second, rollout-pair monotone consistency: physics predicts monotone responses to specific perturbation pairs (higher initial velocity implies longer stopping distance under braking, heavier payload implies slower acceleration), and for each such pair \((\Delta_a, \Delta_b)\) we check whether the imagined rollouts respect the monotonicity. A kinematic world model trivially respects linear monotonicities but fails physical-regime-conditional ones, where the monotonicity holds above or below a physical threshold. In this paper we instantiate only the first signature, the shape of the dose-response curve. Rollout-pair monotone consistency is left to future work.
This is the closest defensible analog to EgoDyn-Bench’s weighted pairwise consistency rate
Choosing the state slice. iKCE also depends on the kinematic state vector \(\hat{x}_t\), the other embodiment-specific free parameter. A slice is an admissible probe when the physics-side iKCE demonstrably responds to the perturbation axis on it, that is, when its physics-side response resolves as nonzero. On such a slice, a bounded WM slope and a resolved difference from physics are informative about this residual. The converse needs care. A physics-side response that fails to resolve as nonzero has three possible causes: it is genuinely absent, it is merely underpowered, or it is measured by a test mismatched to the response’s functional form. The last case matters here. A linear slope through a non-monotone response returns approximately zero however strong the response is. The intermediate-peaked dissipation of frictional slip is one such response: its power \(F_t\,v_{\mathrm{slip}}\) vanishes at both ends, with no tangential force as \(\mu \to 0\) and no slip velocity as the gait sticks at high \(\mu\), so it is peaked by construction. A slice is inadmissible only when the physics-side response is resolved as absent, which requires both adequate power and a test matched to the expected shape. Short of that, admissibility is undetermined, not disproven. Physics is the sensitivity check for the slice, exactly as the joint-noise axis is the sensitivity check for the model. On embodiments where the state slice is not fixed a priori (driving, for instance), this criterion screens candidate slices. It is a necessary condition, not a sufficient one, and a slice selected to satisfy it should be reported as such rather than presented as independent confirmation. We apply this criterion to the richer gait-DOFs slice in the Appendix (Figure 14).
The four observations above are drawn from driving VLMs and VLAs, but the diagnostic is embodiment-agnostic. We demonstrate it on locomotion because a released open-weight DreamerV3 checkpoint offers a proprioceptive latent that decodes to an explicit kinematic state, making iKCE measurable directly, whereas the driving VLMs above expose no such state. The experimental setup is documented in the Appendix. We instantiate the diagnostic on a DreamerV3 checkpoint trained on DMC walker-walk (final reward \(955 \pm 30\)), sweeping surface friction across 13 magnitudes in \([0.1, 1.7]\). Figure 1 shows the five trained checkpoints used throughout the paper and Figure 2 their training curves. The regime boundary (the friction at which the trained policy’s mean reward first drops below 50% of its \(\mu = 1.0\) baseline) is determined empirically at \(\mu = 0.20\) (Figure 3), not chosen a priori. It anchors the dashed reference line in the headline sweep (Figure 5), and Figure 4 shows this behavioral regime change in motion.
imag_horizon = 15) walks cleanly. The long-horizon
actor (imag_horizon = 64, used in the actor ablation below) is visibly
weaker. Training curves for all five are in Figure 2.imag_horizon = 64) ≈392, domain-randomized
≈929, two seed replicates ≈951–961. Hover for per-step values. Click a legend entry
to toggle.At both measurement horizons, the trained DreamerV3 walker-walk world model produces an imagined iKCE at least an order of magnitude above policy-driven real-physics rollouts (Table 1): ≈180× at \(T=16\) and ≈30× at \(T=64\) on the headline channels, 68× and 57× on the action-matched pair (Table 1b). Two reference channels computed on the real trajectory locate this elevation: the posterior encode–decode floor (each real observation encoded by the posterior and decoded, no imagination) and the one-step prior floor (one transition step from each real posterior with the action actually taken). Both sit at \(4\)–\(7\times10^{-3}\), i.e. ≈42–75× physics, and above the imagined channel: floor-to-imagined ratios 1.81 \([1.24, 2.53]\) and 2.16 \([1.51, 3.03]\) at \(T=64\), 1.08 and 1.24 at \(T=16\). The imagined residual is therefore not accumulated imagination error above a small reconstruction floor. The elevated residual is shared with the model’s decoded reconstruction and is not an additive estimate of imagination error; imagination sits at the floor at \(T=16\) and below it at \(T=64\). The real-physics baseline is the same trained actor rolled out closed-loop in real MuJoCo on the same seeded episodes. This headline comparison is not action-matched: each channel runs the actor closed-loop on its own states, so the action sequences diverge after the first imagined step, and the physics window starts at the environment reset while the WM’s starts after its five conditioning observations. Table 1b reports an action-matched, phase-matched pair in which the WM imagines under exactly the actions the physics channel took, and Table 1 reports the encoder–decoder reconstruction floor and the one-step prior floor against which the imagined residual should be read (see Reproducibility). On the reseeded reference channels, the imagined residual is 0.55× the reconstruction residual at \(T=64\). This is consistent with smoother decoded motion during imagination; the level comparison alone does not isolate the source of the difference. The non-degeneracy argument rests on the response contrasts below, not on these levels alone. The diagnostic signature is the bounded friction response under perturbation, not absolute magnitude: a trivially kinematic predictor would score zero iKCE, so the WM lies further from the trivial-kinematic baseline than real physics does, ruling out a naive reading in which “imagining kinematically” would imply small absolute iKCE.
Robustness to early steps. The separation is not an artifact of the first one or two conditioning steps. Re-integrating each rollout’s saved per-step iKCE over the window \([N:T]\) for \(N \in \{1, 2, 3, 5\}\) leaves the ratio at ≈120–160× (\(T=16\)) and ≈20–27× (\(T=64\)). The WM’s elevated early steps are not a one- to two-step encoder transient but a gradual decay. At \(\mu = 1.0\) the per-step iKCE peaks at step 0 and first drops and stays below 10% of that peak only around step 15–21. Early steps are nonetheless disproportionately weighted (the first two carry ≈22% of the \(T=64\) integral), which is exactly why the ratio shrinks with horizon and why we report the robustness band rather than a single post-transient cut.
Table 1. iKCE on the \((z, v_z)\) view at \(\mu = 1.0\) baseline, \(K = 20\) rollouts, 95% bootstrap CIs. WM iKCE exceeds physics by at least an order of magnitude at both horizons (see text for the ratio’s horizon narrowing and its robustness to excluding the first 1–5 steps).
| Source | Mean iKCE | 95% CI |
|---|---|---|
| Horizon \(T=16\) | ||
| Real physics (matched policy) | \(4.2\times10^{-5}\) | \([3.2,\,5.3]\times10^{-5}\) |
| DreamerV3 WM (imagined) | \(7.7\times10^{-3}\) | \([5.2,\,10.3]\times10^{-3}\) |
| Ratio (WM / physics) | \(\sim180\times\) | \([115,\,273]\) |
| DreamerV3 WM (imagined, reseeded reference) | \(5.9\times10^{-3}\) | \([4.3,\,7.5]\times10^{-3}\) |
| Posterior encode–decode floor (real trajectory) | \(6.4\times10^{-3}\) | \([4.6,\,8.1]\times10^{-3}\) |
| One-step prior floor (real trajectory) | \(7.3\times10^{-3}\) | \([5.3,\,9.3]\times10^{-3}\) |
| Ratio (posterior floor / WM, reseeded) | 1.08× | [0.72, 1.65] |
| Horizon \(T=64\) | ||
| Real physics (matched policy) | \(8.6\times10^{-5}\) | \([6.4,\,11.0]\times10^{-5}\) |
| DreamerV3 WM (imagined) | \(2.6\times10^{-3}\) | \([2.0,\,3.2]\times10^{-3}\) |
| Ratio (WM / physics) | \(\sim30\times\) | \([21,\,44]\) |
| DreamerV3 WM (imagined, reseeded reference) | \(2.2\times10^{-3}\) | \([1.7,\,2.8]\times10^{-3}\) |
| Posterior encode–decode floor (real trajectory) | \(4.0\times10^{-3}\) | \([2.9,\,5.1]\times10^{-3}\) |
| One-step prior floor (real trajectory) | \(4.8\times10^{-3}\) | \([3.5,\,6.1]\times10^{-3}\) |
| Ratio (posterior floor / WM, reseeded) | 1.81× | [1.24, 2.53] |
Table 1b. Action-matched, phase-matched pair (added in revision): the physics channel runs the actor closed-loop after a 4-step warm-up so its window coincides with the WM’s; the WM conditions on the same five observations and then replays that episode’s physics actions. The two channels share seeds, prefix, phase and actions, but their trajectories diverge: action-matched imagination reaches median final heights of 0.19–0.42 across friction cells (74% below 0.5), while physics retains feedback. The ratio measures this replay failure as well as the residual contrast; it is not a comparison of two sustained gaits (Figure 5c). Floor rows in Table 1 and this pair use per-rollout reseeding (see Reproducibility).
| Source | Mean iKCE | 95% CI |
|---|---|---|
| Horizon \(T=16\) | ||
| Real physics (phase-matched) | \(9.8\times10^{-5}\) | \([0.6,\,1.4]\times10^{-4}\) |
| WM, action-matched replay | \(6.7\times10^{-3}\) | \([5.3,\,8.1]\times10^{-3}\) |
| Ratio (WM / physics) | \(\sim68\times\) | \([44,\,109]\) |
| Horizon \(T=64\) | ||
| Real physics (phase-matched) | \(9.5\times10^{-5}\) | \([0.7,\,1.3]\times10^{-4}\) |
| WM, action-matched replay | \(5.4\times10^{-3}\) | \([4.6,\,6.2]\times10^{-3}\) |
| Ratio (WM / physics) | \(\sim57\times\) | \([40,\,83]\) |
We sweep surface friction across 13 magnitudes in \([0.1, 1.7]\), spanning the regime boundary at \(\mu = 0.20\) where the trained gait first drops below 50% of baseline episodic reward. For each \(\mu\), we compute iKCE on (a) real-physics rollouts under the trained actor and (b) WM-imagined rollouts conditioned on the first 5 perturbed observations. We summarize each channel by its log-log friction slope
\[\beta \;\doteq\; \frac{\partial \log(\mathrm{iKCE})}{\partial \log(\mu)},\]estimated by least squares with a 95% percentile bootstrap CI. A slope is unresolved when its CI contains zero and resolved when it excludes zero. An unresolved slope does not establish invariance; equivalence and effect bounds are assessed separately in Table 3. The diagnostic reads off \(\beta\), not the visual band width on the log axis.
At \(T=64\), WM iKCE across the sweep is not resolvably friction-sensitive (max/min spread \(1.32\times\), 95% CIs overlap at every \(\mu\)). A log-log regression on the same sweep makes this falsifiable in two ways. The relative statement is primary: the physics slope (\(\beta_{\mathrm{phys}} = -0.220,\; [-0.301, -0.142]\)) excludes zero, the WM slope (\(\beta_{\mathrm{WM}} = -0.009,\; [-0.096, +0.082]\)) does not, and their difference (\(-0.211\), independent-resample CI \([-0.333, -0.092]\)) excludes zero. The bounded-effect statement replaces a bare “CI contains zero”, which by itself does not establish invariance: \(\lvert\beta_{\mathrm{WM}}\rvert \le 0.092\) at 95% for this checkpoint, and a two-one-sided-tests (TOST) equivalence test at a \(\pm 0.10\) margin passes here, for the gait view, and for the domain-randomized checkpoint at \(K = 80\), but not for seeds 1–2 individually (Table 3). The margin is justified in two independent ways: it is smaller than every per-seed physics point estimate, and, in the metric’s own units, a log-log slope of \(0.10\) corresponds to at most a \(1.33\times\) change in iKCE across the sweep’s \(17\times\) friction range (physics’ \(0.22\) corresponds to \(1.86\times\)), which is the largest response we would still call negligible on this axis. We therefore report an unresolved slope, a quantitative bound, or equivalence at the stated margin as distinct findings. (This is the seed-0 checkpoint the main text is built on. The physics slope spans \(-0.11\) to \(-0.22\) across three seeds, with seed 0 the steepest, and each seed’s physics-minus-WM slope difference excludes zero. See Appendix.) The trained policy’s reward, by contrast, collapses through the same range (from ~650 at \(\mu = 0.5\) to ~200 at \(\mu = 0.10\)), a real behavioral regime change that does not produce a resolved slope in the free imagined iKCE under this protocol. Real-physics iKCE under the trained actor shows more cell-to-cell variability than the WM, with elevated values in the low-\(\mu\) region. This confirms that the iKCE metric is not degenerate, while the WM’s attenuated sensitivity of the measured residual to the same perturbation is the established signature: no resolvable response along the dynamic (friction) axis where physics is steep.
Table 3. Bounded-effect reading of the WM friction slope (log-log, \(T=64\), 1000 rollout-level bootstrap resamples). TOST: equivalence to zero at a \(\pm 0.10\) margin, chosen because it is smaller than every per-seed physics point estimate and because, independently of the physics effect, a slope of 0.10 means at most a \(1.33\times\) change in iKCE over the sweep’s \(17\times\) friction range; it passes when the 90% CI lies inside the margin, as is conventional for two one-sided tests at \(\alpha = 0.05\); the \(\lvert\beta\rvert\) column is the 95% bootstrap quantile of the absolute slope. Marginal slopes and 95% CIs use the Appendix’s own 1000-resample bootstrap and are identical to it. Differences are \(\beta_{\mathrm{phys}} - \beta_{\mathrm{WM}}\), using independent resampling of the two channels in every row, as in the Appendix. These are within-checkpoint comparisons; “independent” describes the bootstrap scheme, not the experimental design.
| Checkpoint | \(\beta_{\mathrm{WM}}\) [95% CI] | \(\lvert\beta_{\mathrm{WM}}\rvert \le\) (95%) | \(\beta_{\mathrm{phys}}\) [95% CI] | TOST \(\pm 0.10\) | difference [95% CI, independent resampling] |
|---|---|---|---|---|---|
| seed 0 (headline) | \(-0.009\) \([-0.096, +0.082]\) | \(0.092\) | \(-0.220\) \([-0.301, -0.142]\) | pass | \([-0.333, -0.092]\) |
| seed 1 | \(+0.031\) \([-0.072, +0.129]\) | \(0.114\) | \(-0.148\) \([-0.233, -0.065]\) | not established | \([-0.318, -0.044]\) |
| seed 2 | \(+0.038\) \([-0.039, +0.123]\) | \(0.107\) | \(-0.115\) \([-0.200, -0.031]\) | not established | \([-0.274, -0.037]\) |
| domain-randomized, \(K=20\) | \(-0.026\) \([-0.123, +0.076]\) | \(0.112\) | \(-0.113\) \([-0.201, -0.024]\) | not established | \(-0.087\) \([-0.211, +0.049]\) (contains zero) |
| domain-randomized, \(K=80\) (rerun) | \(-0.006\) \([-0.058, +0.043]\) | \(0.051\) | \(-0.109\) \([-0.152, -0.062]\) | pass | \(-0.103\) \([-0.172, -0.036]\) (excludes zero) |
| gait view, seed 0 | \(+0.010\) \([-0.015, +0.035]\) | \(0.031\) | \(-0.154\) \([-0.197, -0.111]\) | pass (also at \(\pm 0.05\)) | \([-0.211, -0.113]\) |
| WM pooled over 3 seeds | \(+0.020\) \([-0.035, +0.079]\) | \(0.069\) | — | pass | — |
The ±0.10 equivalence margin was chosen at the analysis stage, not pre-registered. It bounds the fitted log-log trend, not arbitrary nonlinear variation. Bounds apply separately to each checkpoint, not simultaneously; the pooled-WM interval is conditional on these three trained seeds. Paired-resample sensitivity checks also exclude zero for the three seed comparisons, gait view and DR K = 80; for example, seed 0 is [-0.323, -0.101], gait [-0.202, -0.124], and DR K = 80 [-0.154, -0.050]. Table 4 separately uses paired resampling for the newly matched episode layout.
Non-degeneracy and interpretation. The posterior and one-step-prior channels have resolved friction slopes, and the posterior-minus-free-imagination contrast resolves through the same decoder (Table 4). This argues against universal insensitivity of the decoder/readout on all trajectories. However, sensitivity on observation-driven trajectories does not guarantee sensitivity to every kind of dynamic information in imagined states. Temporal structure and absolute magnitude alone do not establish perturbation sensitivity. The intended joint-noise positive control remains unresolved, so the present data establish attenuation of this residual response, not selective kinematic sensitivity or a unique transition mechanism.
Noise floor, de-noised imagination, and action matching. Three additional channel sets, pre-registered and run for this revision on the seed-0 checkpoint with per-rollout reseeding (see Reproducibility), sharpen the reading; Figure 5b and Table 4 report them. (a) Noise floor. The posterior encode–decode floor and the one-step prior floor, computed on the same real episodes as the phase-matched physics channel, have resolved friction slopes compatible with the physics estimate: \(\beta = -0.213\) [-0.301, -0.124] and \(-0.227\) [-0.319, -0.136] against \(\beta_{\mathrm{phys}} = -0.285\) [-0.368, -0.200] on that channel, with the physics-minus-floor difference containing zero (-0.072 [-0.158, +0.006]); this does not establish equivalence to physics. The imagined channel through the same decoder does not: \(\beta = -0.002\) [-0.093, +0.089], and its difference from the posterior floor, +0.212 [+0.133, +0.288], excludes zero. This is the decoder-controlled form of the central contrast: the residual has a resolved friction response under observation-driven reconstruction and one-step prediction, and a significantly attenuated response under multi-step free imagination. This comparison does not prove absence of dynamic information in the imagined latent or state. (b) De-noised imagination. Taking the prior’s stochastic state at its mode, or averaging the decoded readout over 16 prior samples per step, leaves the slope bounded (\(+0.022\) [-0.061, +0.106] and \(+0.031\) [-0.059, +0.121]; \(\lvert\beta\rvert \le 0.11\)), so neither de-noising protocol reveals a resolved response at this precision; the mode rollout’s residual is larger than the sampled one (1.55× \([1.12, 2.14]\)). At μ = 1 and T = 64, mode/physics and averaged/physics ratios are 36.4× [24.7, 55.4] and 34.9× [23.5, 53.7], respectively. (c) Action matching. When imagination replays, per episode and friction, exactly the actions the closed-loop physics channel took, the imagined slope resolves to a weak response, \(-0.075\) [-0.118, -0.030], about four times shallower than the matched physics slope (paired difference -0.210 [-0.290, -0.131]). The replayed actions are themselves friction-differentiated (the real gait slips and adapts at low \(\mu\)), so this channel injects regime information through the actions in the same way the 64-step prefix injects it through the posterior. This is intentional: the design removes the action-divergence confound of the headline H1 comparison while deliberately keeping the closed-loop policy’s friction adaptation, carried by the actions, as an input to imagination (a fully open-loop design collapses the walker, Figure 10). The action-matched imagined trajectory also drifts into a low posture (Figure 5c): across the 13 friction cells its median final torso height is 0.19–0.42, versus 1.28–1.31 in free imagination; 74% of its 260 rollouts end below 0.5 and 91% below 0.8. The phase-matched physics channel retains feedback but is not uniformly upright either (29% and 52%, respectively). These are descriptive height thresholds, not validated fall labels. Thus the resolved slope and the 57× ratio characterize replay with substantial imagined posture drift, not sustained imagined walking. The imagined response to that input is weak and bounded (\(\lvert\beta\rvert \le 0.11\)), in line with the cond-64 late-window residual below. The “CI contains zero” statement therefore holds for free imagination (sampled, mode, averaged) and not for action-matched imagination, where a weak sub-physics response is resolved.
Table 4. Friction slopes of the revision channels (seed-0 checkpoint, \(T=64\), log-log, 1000-resample rollout-level bootstrap). Paired differences are against the phase-matched physics channel.
| Channel | \(\beta\) [95% CI] | contains 0 | \(\lvert\beta\rvert \le\) (95%) | \(\beta_{\mathrm{phys}} - \beta\) [95% CI] |
|---|---|---|---|---|
| Real physics, phase-matched | \(-0.285\) [-0.368, -0.200] | no | \(0.356\) | — |
| Posterior encode–decode floor | \(-0.213\) [-0.301, -0.124] | no | \(0.284\) | \(-0.072\) [-0.158, +0.006] |
| One-step prior floor | \(-0.227\) [-0.319, -0.136] | no | \(0.302\) | \(-0.058\) [-0.141, +0.027] |
| WM imagined, sampled (reseeded) | \(-0.002\) [-0.093, +0.089] | yes | \(0.091\) | \(-0.283\) [-0.388, -0.198] |
| WM imagined, mode | \(+0.022\) [-0.061, +0.106] | yes | \(0.096\) | \(-0.307\) [-0.419, -0.203] |
| WM imagined, 16-sample average | \(+0.031\) [-0.059, +0.121] | yes | \(0.109\) | \(-0.316\) [-0.431, -0.217] |
| WM imagined, action-matched | \(-0.075\) [-0.118, -0.030] | no | \(0.109\) | \(-0.210\) [-0.290, -0.131] |
Horizon-emergence test. The dynamic signature emerges only as slip accumulates into per-step deviation at the longer horizon. We sharpen this quantitatively by repeating the flatness regression at four sub-horizons \(T \in \{8, 16, 32, 64\}\), re-integrating each rollout’s saved per-step iKCE trace (no new rollouts). Figure 6 plots the fitted slope at each horizon.
The dynamic signature in physics emerges with horizon: the slope \(\beta_{\mathrm{phys}}\) grows in magnitude from \(+0.012\) at \(T=8\) to \(-0.220\) at \(T=64\), crossing out of the CI-contains-zero region between \(T=32\) and \(T=64\), consistent with friction effects accumulating over multiple footfalls before becoming detectable in the per-step kinematic-null residual. The WM-side slope \(\beta_{\mathrm{WM}}\) is statistically indistinguishable from zero at every horizon tested (\(-0.028, -0.012, -0.013, -0.009\) at \(T=8, 16, 32, 64\), all straddling zero). The contrast is the H2-emergence claim stated quantitatively: the dynamic signature emerges with horizon in physics but not in the WM. Note that long-horizon iKCE in both channels is in part dilated by reduced per-step motion magnitude. The present result is robust to that effect because the WM-physics contrast widens with horizon rather than shrinks, which is the opposite of what a horizon-degenerate metric would produce.
Conditioning-prefix length. A distinct concern is that conditioning on only five observed steps starves the WM of the evidence it would need to infer the friction regime, so its flatness could be a data-coverage artifact rather than a property of its imagination. This is an observability limitation of the prefix: what the sweep below can show is how a posterior-encoded regime signal is attenuated or relaxed by the imagined transition, not whether the transition itself encodes or uses friction. We therefore re-ran WM imagination at conditioning-prefix lengths of 5, 16, 32, and 64 steps. Measured past the leading elevation, the friction slope has no resolved slope through cond \(\le 32\) (\(\beta_{\mathrm{WM}} = -0.018, -0.020, -0.014\), CIs containing zero). At cond \(= 64\) the integrated slope turns negative (\(-0.127\), \([-0.164, -0.093]\) at \(K=80\)), but this is largely inherited from the conditioning prefix, not generated over the imagined horizon. Three observations show why. (i) Because the real gait already stumbles below \(\mu = 0.20\), a 64-step prefix hands the WM a friction-differentiated initial condition (decoded start height 0.35 at \(\mu = 0.1\) vs 0.61 at \(\mu = 1.0\)). (ii) The resulting low-\(\mu\) elevation is front-loaded (low/high-\(\mu\) iKCE ratio 2.1 over steps 1–4, decaying to 1.0 by steps 30–63). (iii) Once the first 30 of 64 steps are excluded, the integrated cond-64 slope collapses from \(-0.127\) to \(-0.026\). That late-window residual is weak but, at \(K = 80\), resolved (\(-0.026\), CI \([-0.045, -0.006]\), excluding zero). The pre-registered disjoint segments \([20{:}32], [32{:}44], [44{:}64]\) give \(-0.029, -0.010, -0.041\), only the last individually excluding zero. So we do not claim the WM generates no friction response at a long prefix. We make the weaker claim: the friction-dependence that survives into the imagined tail is roughly an order of magnitude below physics (\(-0.026\) vs the seed-0 \(\beta_{\mathrm{phys}} = -0.220\), and this conditioning-prefix analysis is run on the seed-0 checkpoint only), and it is mostly a front-loaded echo of the initial condition rather than a response that grows over the horizon, which is the signature of a dynamic response (physics, whose late-window slope strengthens to \(-0.397\)). One caveat qualifies the reading above. These are imagined (transition) steps, but the friction signal reaches them only through the posterior-encoded initial condition (the decoded start above), never as an ongoing input. So the residual measures how the imagined transition relaxes a friction-differentiated start, not whether the transition is sensitive to friction, a distinction we develop in the Separability section. What the prefix sweep establishes is one-directional and still useful. Even handed 64 steps of friction-differentiated evidence, the WM’s measured response is attenuated over the imagined horizon, although a weak response persists. It attenuates the measured response toward a weak but resolved tail (β = -0.026, 95% CI [-0.045, -0.006]).
The same conclusion holds however the integration window is cut, which Figure 7 lets the reader probe directly:
Actor training-horizon ablation. A natural concern is that the WM’s bounded friction response at \(T=64\) reflects the default actor operating out-of-distribution from its training horizon (imag_horizon = 15) rather than a structural property of WM imagination. To examine this possibility, we retrain an identical checkpoint at imag_horizon = 64, matching the measurement horizon, with the same seed, hyperparameters, and total training budget. Figure 8 shows the result.
imag_horizon = 64 (matching the measurement horizon). The imagined iKCE has the same observed friction
spread as the default-actor result (\(1.32\times\) in both cases, CIs overlapping at
every \(\mu\)). This suggests that increasing the actor's training horizon does not
recover a stronger friction response in this experiment.Domain-randomization control. Friction changes both contact dynamics and the closed-loop policy’s behavior. To bound the policy-distribution confound, we train a DreamerV3 checkpoint with per-episode friction randomization, \(\mu \sim \mathcal{U}(0.1,1.7)\). At \(K=20\) the within-checkpoint slope difference is \(-0.087\) \([-0.211,+0.049]\) and does not resolve. A pre-committed \(K=80\) rerun gives \(-0.103\) \([-0.172,-0.036]\), with paired-resample sensitivity interval \([-0.154,-0.050]\). This is outcome C1 of the rerun plan. The point estimates are similar and both results are reported. The rerun sample size and outcome branches were fixed after inspecting K = 20 and before collecting K = 80. The K = 80 sweep uses the same seeding convention as K = 20, with environment seeds 0–79 in each friction cell. Seeds 0–19 recur in every cell. At μ = 0.1, the first 20 trajectories in each channel are bit-identical to K = 20; subsequent cells share environment seeds but differ in global torch RNG position. The physics starts coincide for those seeds, whereas sampled WM posterior starts generally differ. This is an enlarged rerun with shared seeds and partial exact overlap, not an independent replication. A post-hoc sensitivity check excluding the 20 identical trajectories per channel leaves 1,020 per channel and gives a difference of −0.101 [−0.172, −0.035] under independent resampling, or [−0.157, −0.045] under paired resampling. Removing exact overlap does not establish independence of the remaining trajectories. We report nominal intervals and do not claim selection-adjusted error control across the two analyses. Figure 9 displays the original K = 20 sweep; Table 3 reports both sample sizes. The DR control establishes a residual-slope contrast under this policy family, not a policy-free effect of friction.
Both the DR policy and the DR world model were trained across the swept friction range. In the original K = 20 sweep, the marginal results are: the physics slope’s CI excludes zero (\(\beta_{\mathrm{phys}}^{\mathrm{DR}} = -0.114,\; [-0.201, -0.024]\)) while the WM slope’s contains it (\(\beta_{\mathrm{WM}}^{\mathrm{DR}} = -0.026,\; [-0.123, +0.076]\)). This addresses training coverage of the swept friction range: a world model trained only at \(\mu = 1.0\) has never observed friction variation so generalization to varied friction is not guaranteed. Training coverage alone does not guarantee that a short conditioning prefix identifies the regime. The DR world model was trained on transitions spanning the full sweep range, could in principle infer the friction regime from its conditioning prefix, and still shows no resolvable friction response.
We also examined the fully policy-free alternative to the DR control, replaying a single fixed open-loop action sequence identically to both channels to remove policy adaptation by construction rather than merely bounding it. It is infeasible for this embodiment: a feedback-stabilized walker cannot be driven open-loop, and collapses at every friction (including the friction the actions were recorded at) in real physics and in the WM’s imagination alike. The reason is that a bipedal gait is not passively stable. A recorded torque sequence stabilizes the walker only along the closed-loop trajectory it was recorded on, so any small deviation grows without the feedback that produced it, and the biped topples even at the recording friction. Open-loop replay is therefore a failure of the protocol for this embodiment, not a friction effect. Open-loop replay therefore destroys the behavior being measured. The domain-randomization control above remains the practical handle on the policy confound. Figure 10 plots the height collapse for four friction values, and Figure 11 shows the same collapse in pixel fidelity.
The central move in this paper is to separate kinematic from dynamic imagination, so it deserves an explicit operational definition and a falsifiable test. We call a world model’s imagination kinematic if its iKCE responds to perturbations of the kinematic state it extrapolates but not to perturbations of the physical regime that governs the real dynamics. We call it dynamic if it responds to the physical-regime axis. This is a statement about selective sensitivity, and it is falsifiable in two directions: a model we call kinematic must (a) show no resolvable response under the dynamic (friction) axis (established above) and (b) actually respond under a kinematic axis. Two outcomes must be told apart. If it responds to neither axis and its iKCE is a flat constant, the metric is degenerate and nothing can be read from it. If it responds to neither observation-space axis yet its iKCE is large, temporally structured, and still separates the WM from physics on the dynamic axis (the case we in fact reach, outcome B2 below), the metric is non-degenerate but the selective-kinematic reading is unearned, and the claim reduces to attenuated sensitivity of the measured residual along the dynamic axis. We reach this second outcome and scope the claim to it.
The two-axis protocol. The friction sweep is the dynamic axis (physically-grounded, regime-crossing). Its designed counterpart is a joint-noise axis: zero-mean Gaussian noise added to the joint-position channel of every observation before the WM encoder, with the physics state left unperturbed. A kinematic imager should respond to joint noise, because the perturbation directly corrupts the kinematic state \(\hat{x}_t\) that drives the extrapolation.
A pre-registered decisive test. Reading a small result as confirmation of the hypothesis is the failure this diagnostic is meant to avoid. We therefore pre-registered, before running, a powered rerun of the joint-noise axis (WM and a physics reference) at \(K = 80\) (4× the headline), together with the manuscript consequence of each outcome. The pre-registration is timestamped in the artifact to be released (see Reproducibility). We reached outcome B2 (reported below), and the three branches record what each result would have meant. (B1) If the WM joint-noise slope excludes zero, the positive control holds and separability is empirically established. (B2) If it tightens around zero, the WM is insensitive to the kinematic axis too, the control is reported as a null, and separability rests on the operational definition and two-axis protocol rather than a confirmed positive control. (B3) If it still wanders, the result is unresolved even at 4× the sample size. In all three cases the paper’s headline (the friction-axis flatness: a physics friction slope that excludes zero against a WM slope whose CI contains it) is unchanged, because it does not depend on this control.
Result: outcome B2, no resolved positive-control response. At \(K = 80\) the WM’s joint-noise response tightened but stayed indistinguishable from zero: linear slope \(+2.5\times10^{-4}\), 95% CI \([-8.2\times10^{-4}, +1.3\times10^{-3}]\) (the \(K=20\) half-width \(2.1\times10^{-3}\) roughly halved to \(1.1\times10^{-3}\), still spanning zero), with a level per-\(\sigma\) profile (max/min spread \(1.14\times\)). No WM response to this observation-space kinematic axis is resolved; a negligible-effect margin was not specified for this slope. Crucially, this is not because the perturbation is inert: the physics reference does respond, weakly but now resolved (\(+9.7\times10^{-5}\), CI \([+5.4, +14.0]\times10^{-5}\), excluding zero). The asymmetry is the finding: joint noise moves the real kinematic-null residual, but the WM’s imagination filters it, consistent with the RSSM posterior denoising observation-space noise before it can propagate into the rollout. We retain the pre-registered B2 label for this outcome, but report it as an unresolved WM slope rather than equivalence to zero. The workshop version’s “both channels respond” reading is not supported.
What separability rests on, and the remaining gap. The unresolved control has a clear consequence: selective observation-space sensitivity has not been demonstrated. Neither the free-imagination friction slope nor the joint-noise slope resolves under the tested protocol. So the kinematic reading cannot rest on selective observation-space sensitivity. It rests on two things the null leaves intact. First, the decoder-controlled contrast added in this revision: the model’s own reconstruction of the real trajectory has a resolved friction slope compatible with the physics estimate while free imagined iKCE through the same decoder has no resolved slope (Table 4), showing sensitivity of this residual on observation-driven trajectories, without establishing absence of dynamic information during imagination. Second, on the dynamic axis free imagination is bounded where physics is steep: the physics slope excludes zero (seed-0 slope \(-0.220\), and \(-0.11\) to \(-0.22\) across three seeds with the per-seed physics-minus-WM difference resolving, see Appendix), while the WM slope is statistically indistinguishable from zero and bounded (Table 3). This is the central result. That the WM’s iKCE is large and temporally structured (H1) is supporting evidence rather than load-bearing, since the floor channels show that magnitude to be a property of the decoded representation. What we must not over-read is the conditioning-prefix result. A 64-step friction-differentiated prefix does propagate into a friction-differentiated imagined start, but that is a property of the posterior faithfully encoding the observations it was handed (conditioning working), not evidence that the imagined transition is sensitive to kinematic content. On inspection, both of our kinematic-axis probes are posterior probes: joint noise is filtered before it reaches the transition, and in the conditioning-prefix sweep the regime signal reaches the imagined rollout only through the posterior-encoded initial condition, which the transition then relaxes. The transition model’s kinematic sensitivity is therefore unmeasured: we have two results about the encoder and none that isolates the transition’s response to kinematic content at fixed regime. On the friction axis, the floor channels locate attenuation of the measured iKCE response between observation-driven reconstruction/one-step prediction and multi-step free imagination (Table 4). They rule out universal decoder/readout insensitivity, but do not identify which dynamic information is retained or used by the transition. Selective kinematic sensitivity also remains unmeasured. These experiments do not isolate selective transition sensitivity. Joint noise may be filtered by the posterior, and a longer prefix changes the initial latent. A latent-space control would require independently validating the directions it perturbs; the present data do not establish that the model lacks a dynamic representation. A latent-space control that pits a decoded-kinematic direction against a matched control direction is the experiment that would close it (see Discussion and Open Directions). Until it is run, the scope of the claim is that the WM’s multi-step imagination has attenuated friction sensitivity of free imagined iKCE (bounded, and shallower than both physics and the model’s own reconstruction floor), not that it is demonstrably selectively kinematic.
iKCE is a per-step kinematic null fit integrated over a horizon. It diagnoses the response, or the bounded magnitude of an unresolved fitted response, along the dynamic (regime) axis, not the quality of dynamic prediction. The reading of that attenuated sensitivity of the measured residual as kinematic imagination is an interpretation we scope in Separability, not a measured selective sensitivity. A WM whose iKCE is not resolvably sensitive to the regime axis has a bounded fitted residual slope under this protocol, but has not been certified as a useful predictor. A resolved iKCE slope establishes sensitivity of this residual to the intervention; it does not certify dynamic fidelity or prediction accuracy. The diagnostic is structural, not predictive, by design.
A downstream behavioral prediction. If imagined rollouts are kinematically structured but not dynamically faithful, then policy gradients propagated through long imagined rollouts optimize the actor against a trajectory distribution that diverges from real dynamics in directions iKCE itself does not capture (rotational drift, contact-event timing, accumulated absolute-state error). Long-horizon actor training should therefore be unstable and yield a weaker deployed policy. The \(h=64\) ablation is consistent with this prediction: under matched hyperparameters, the long-horizon actor converged to ~400 episodic reward versus ~955 for the default \(h=15\) checkpoint, and exhibited training instability throughout. We do not claim a causal link. Long-horizon Dreamer training is known to be sensitive to multiple factors, but the observation is what one would predict from the kinematic-not-dynamic hypothesis, and pre-empts the natural counterfactual that scaling the actor’s imagination horizon would have closed the gap.
Limitations of the present measurement. (i) The empirical result rests on a single embodiment (DMC walker-walk, a 2D 9-DOF system) restricted to the \((z, v_z)\) sub-slice, and on a single open-weight WM family whose latent decodes to an explicit kinematic state. Extending the flatness test to quadruped, humanoid, and driving embodiments would broaden the evidence base. Extending it to pixel-space world models such as GAIA-1
Open directions. Embodiment extension to quadruped- or humanoid-locomotion would test the diagnostic on richer contact dynamics than the planar walker provides. The joint-noise positive-control experiment yielded no resolved WM slope (see Separability), so the natural next step is a kinematic perturbation that bypasses the encoder’s denoising (an initial-velocity or latent-space perturbation that corrupts the extrapolated state directly) to test whether the WM is selectively sensitive to the kinematic axis rather than insensitive to observation-space perturbations in general. Removing the residual closed-loop policy adaptation calls for a stabilized policy-free protocol rather than fixed open-loop replay: the open-loop experiment above collapses the biped at every friction, so a viable design would perturb torques around the trained policy or track a fixed reference trajectory with a stabilizing controller. Finally, context-conditioned world models that append an inferred or observed context variable to the WM already implement explicit conditioning
Broader impact and scope. The empirical work is confined to a simulated planar locomotion benchmark with open-weight checkpoints; it involves no human data and no deployed system. iKCE is a research diagnostic, not a safety certification: a kinematic-fallback failure in a deployed driving world model would be a safety concern that a diagnostic of this kind can flag but cannot certify against. The result is established for one locomotion task and one model family and should not be read as a general property of learned world models. Compute: five DreamerV3 trainings of roughly 24 h each and under three hours of perturbation sweeps on a single GPU.
We operationalized a kinematic-versus-dynamic hypothesis with iKCE and a conditioning-perturbation protocol. On the tested DreamerV3-class walker-walk system, free imagined iKCE after five conditioning observations has an unresolved friction slope bounded to \(\lvert\beta\rvert\le0.12\) in each of three seeds, and its difference from real physics resolves. The DR comparison resolves at K = 80. On seed 0, observation-driven reconstruction and one-step prediction have resolved slopes through the same decoder, whereas free imagination is significantly less responsive. Action replay and a long prefix yield weak resolved responses; action-matched imagination also drifts into a low posture. These results establish attenuated sensitivity of the measured residual, not absence of dynamic information. A selective kinematic interpretation remains untested because the intended observation-space positive control did not produce a resolved WM response. The result is limited to one task and model family; validated transition-level interventions and calibrated state readouts are needed to test broader claims.
This appendix documents (i) the experimental configuration, (ii) the methodological details behind the headline numbers (regime-boundary determination, the flatness and horizon-emergence tests, and the trivial-WM scale anchor), (iii) the controls for H2 (per-step structure decomposition, robustness to the kinematic-state choice, the gait-DOFs view, and the joint-noise control, a pre-registered null for the WM, see Separability), and (iv) implementation and reproducibility specifics.
Table 2. Experimental setup. Horizons \(T=16\) and \(T=64\) refer to the iKCE rollout length, evaluated on the same trained policy.
| Field | Value |
|---|---|
| World model | DreamerV3 (NM512 PyTorch port, commit 6ef8646) |
| Task | DMC walker-walk, dmc_proprio config |
| Training | 1M env steps, seed 0, RTX 5090 |
| Final reward | \(955 \pm 30\) (mean over last 100k steps) |
| Evaluation policy | Trained actor (same checkpoint on both physics and WM sides) |
| Backend |
dm_control 1.0.20, mujoco 3.1.6 |
| \(K\) | 20 rollouts per perturbation cell (80 for the pre-registered joint-noise rerun, the cond-64 analysis, and the domain-randomized rerun) |
| Kinematic spec | \((z, v_z)\) root-vertical-motion (1D) |
| Extrapolation | constant velocity |
Regime-boundary determination. The boundary \(\mu = 0.20\) is not chosen a priori. We roll out the trained actor for \(K=10\) episodes at each of 12 friction multipliers \(\mu \in [0.1, 1.5]\), compute the mean episodic reward and a 95% bootstrap CI per cell, and define the regime boundary as the largest \(\mu\) at which mean reward has dropped below 50% of its \(\mu = 1.0\) baseline. The \(\mu = 1.0\) mean over this sweep is ~650, giving a threshold of ~325. On our checkpoint, the boundary lies at \(\mu = 0.20\). (The \(955 \pm 30\) final reward in Table 2 averages over the last 100k training steps and is not directly comparable.)
Flatness test for H2. We make the bounded-response claim falsifiable by regressing \(\log(\mathrm{iKCE})\) on \(\log(\mu)\) across the \(T=64\) friction sweep, with each of the \(K=20\) rollouts at each of the 13 \(\mu\) values contributing one observation (\(n = 260\) per seed). To rule out a seed-dependent artifact, we repeat the regression for three independently-trained DreamerV3 walker-walk checkpoints (seeds \(\{0, 1, 2\}\), matched hyperparameters and step budget). A 95% percentile bootstrap (1000 resamples over rollouts) on each seed gives \(\beta_{\mathrm{WM}}^{(0)} = -0.009\), \([-0.096, +0.082]\), \(\beta_{\mathrm{WM}}^{(1)} = +0.031\), \([-0.072, +0.129]\), and \(\beta_{\mathrm{WM}}^{(2)} = +0.038\), \([-0.039, +0.123]\). All three WM slope CIs contain zero, so none resolves a signed slope at this precision. This is not evidence of exact invariance; Table 3 gives the separate effect bounds and equivalence outcomes. The physics-side friction slope excludes zero under four distinct policies: the three independently-trained seed actors (\(\beta_{\mathrm{phys}}^{(0)} = -0.220\), \([-0.301, -0.142]\), \(\beta_{\mathrm{phys}}^{(1)} = -0.148\), \([-0.233, -0.065]\), and \(\beta_{\mathrm{phys}}^{(2)} = -0.115\), \([-0.200, -0.031]\)) and the domain-randomized policy of the DR control (\(\beta_{\mathrm{phys}}^{\mathrm{DR}} = -0.114\), \([-0.201, -0.024]\)). All four are negative and exclude zero. The primary contrast is the within-checkpoint slope difference, and reading it off the marginal intervals would be too conservative: the physics and WM CIs touch for the two shallower seeds, but overlapping marginal CIs do not imply an unresolved difference (under independent resampling the difference’s standard error is \(\sqrt{\mathrm{se}_{\mathrm{phys}}^2 + \mathrm{se}_{\mathrm{WM}}^2}\), smaller than the \(\mathrm{se}_{\mathrm{phys}} + \mathrm{se}_{\mathrm{WM}}\) that marginal overlap implicitly tests against). We therefore bootstrap the slope difference \(\beta_{\mathrm{phys}} - \beta_{\mathrm{WM}}\) per seed, using the same test as the gait view and the DR control and the same 1000-resample scheme. It resolves for all three: seed 0 gives \(-0.211\), \([-0.333, -0.092]\), seed 1 gives \(-0.179\), \([-0.318, -0.044]\), and seed 2 gives \(-0.153\), \([-0.274, -0.037]\), every CI excluding zero. One note on the procedure: the physics and WM channels are seed-paired by construction, because the WM conditions on the very same first real steps the physics rollout produces, from the same seeded episode, so a paired bootstrap is well-defined. We resample the two channels independently regardless. We verify that this is the conservative direction. Re-running as a paired bootstrap (one joint resample per draw) makes the two slopes positively correlated across resamples for every seed (Pearson \(+0.24\), \(+0.46\), \(+0.59\) for seeds 0/1/2), so the paired interval comes out tighter than the independent one we report (seed 0, \([-0.323, -0.101]\) versus \([-0.333, -0.092]\), and seeds 1–2 likewise) while both still exclude zero. The reported independent CIs are therefore conservative by measurement. They are wider than either marginal (independent resampling adds the two variances) yet still clear zero, because the WM slopes sit slightly positive and push the gap further from it. So the bounded-vs-steep contrast is statistically resolved per checkpoint, not merely per side. (This cross-seed physics sweep was pre-registered before running, see Reproducibility.) Because the physics rollouts are driven by the trained actor rather than a fixed ground truth, the physics slope is policy-conditioned. But because it excludes zero under all three seed actors and the DR policy, the steepness is a property of real physics across this policy family, not a seed-0 artifact. Quantitatively, across three seeds \(\lvert\beta_{\mathrm{WM}}\rvert\) is bounded above by 0.12 (Table 3) (every WM CI contains zero) while \(\lvert\beta_{\mathrm{phys}}\rvert\) ranges from ~0.11 to ~0.22 (every physics CI excludes zero), and the independently resampled slope difference excludes zero for each seed.
Trivial-WM scale anchor. The iKCE scale has an analytic lower bound. A trivial “WM” that imagines by applying the kinematic predictor to its own current state, \(\hat{x}^{\mathrm{WM}}_{t+1} = \mathrm{kin}(\hat{x}^{\mathrm{WM}}_t)\), produces \(\mathrm{iKCE} = 0\) by construction: every predicted next state is identically the kinematic continuation of its predecessor. The measured ordering on walker-walk is therefore
\[\underbrace{0}_{\text{trivial kinematic}} \;<\; \underbrace{4.2\times10^{-5}}_{\text{matched real physics}} \;\ll\; \underbrace{7.7\times10^{-3}}_{\text{DreamerV3 WM}} \qquad (T=16).\]This places the WM further from the trivial-kinematic baseline than real physics, the analytic counterpart of the H1 non-degeneracy result that low absolute iKCE is not the diagnostic signature. The order holds for \(T=64\) as well.
Per-step structure decomposition. The integrated iKCE of Table 1 and Figure 5 averages over the rollout horizon and so does not reveal whether the WM’s imagined residual has the same temporal structure as physics. Figure 12 decomposes per-step iKCE at three friction values \(\mu \in \{0.15, 1.0, 1.5\}\): physics exhibits sparse contact-event spikes whose positions shift with \(\mu\) (consistent with footfall dynamics driving the kinematic-null residual), while the WM shows an elevated early segment that decays gradually (over roughly the first 15–21 steps, not a one- to two-step encoder transient) into a smooth tail with no resolvable friction response. The two residuals are not merely different in magnitude (the H1 ratio) but qualitatively different in temporal structure: the WM’s imagined rollouts do not reproduce the contact-event signature that defines the physics-side residual. Quantitatively, the WM’s per-step iKCE is not a flat constant: at the \(\mu = 1.0\) baseline its early window (steps 0–4) exceeds its tail (steps 30–63) by a median factor of \(16.5\) (95% bootstrap CI \([7.5,\,22.2]\), excluding one), with lag-1 autocorrelation \(+0.75\), whereas the matched physics trace shows no such early elevation (\(1.2\times\), CI \([0.5,\,3.1]\)). This quantifies temporal structure without establishing sensitivity to a perturbation; the main non-degeneracy evidence is the floor contrast, alongside the bounded-vs-steep contrast.
Per-step structure under the actor ablation. Figure 13 combines the actor-horizon and per-step controls: per-step WM iKCE under both the default (\(h=15\)) and retrained (\(h=64\)) actor, at the same three friction values. The early-elevation-plus-smooth-tail structure is unchanged across actor training horizons, ruling out the joint concern that the per-step signature reflects an actor-out-of-distribution artifact rather than a property of the WM’s imagination.
Robustness to the kinematic-state choice. iKCE depends on a chosen kinematic state vector \(\hat{x}_t\). The main result uses the root-vertical-motion slice \((z, v_z)\). Figure 14 repeats the protocol with a richer representation, the walker’s gait kinematic state: the torso height and the six leg-joint angles (each sign-aligned to its hinge axis), with their velocities, giving seven aligned position-velocity channels (see Reproducibility). At \(T=64\) the contrast of Figure 5 reproduces, now stated with the same falsifiable statistics: the WM friction slope is bounded (\(\beta_{\mathrm{WM}} = +0.010\), CI \([-0.015, +0.035]\)), the physics slope excludes zero (\(\beta_{\mathrm{phys}} = -0.154\), CI \([-0.197, -0.111]\)), and the slope difference resolves (\(-0.164\), independent-resample CI \([-0.211, -0.113]\)). As in the Appendix, the physics and WM channels are seed-paired, and resampling them independently is the conservative choice: the paired bootstrap makes the two slopes positively correlated (Pearson \(+0.44\)), so its interval \([-0.202, -0.124]\) comes out tighter, and both exclude zero. By the admissibility criterion of the Diagnostic Protocol the gait slice is a valid probe (its physics slope excludes zero), and on it the WM is bounded where physics is steep. One caveat. The gait and identity views are computed from the identical seed-0 rollout trajectories, re-read through a second kinematic transform, so this is robustness to that transform choice, not independent evidence from new samples, and the two sets of confidence intervals should not be read as independent. The identity-view headline, conversely, is computed without the gait transform, so it is unaffected by the choice or implementation of this richer slice. (Two implementation corrections were applied to this transform. An earlier version paired each leg-joint angle with the opposite-signed velocity, which inflated the gait-view iKCE roughly tenfold, and the sign is corrected here. The joint-angle time series is also unwrapped across the \(\pm\pi\) seam before the constant-velocity comparison, which removes a handful of spurious per-step spikes on the world-model side and leaves the physics side untouched. Both corrections change the magnitudes but not the conclusion.)
walker_gait
slice (torso height and six sign-aligned leg-joint angles, linear iKCE scale). The
physics slope excludes zero (\(\beta_{\mathrm{phys}} = -0.154\)) while the WM slope is bounded (\(\beta_{\mathrm{WM}} = +0.010\), CI containing zero) and the slope difference
resolves (\(-0.164\), CI \([-0.211, -0.113]\)): the contrast is not an artifact of the
\((z, v_z)\) slice. Because these are the identical seed-0 rollout trajectories
as Figure 5, this is robustness to the transform, not to sampling.Joint-noise as an intended kinematic positive control. The friction sweep is a dynamic perturbation (physically-grounded, regime-crossing). The joint-noise sweep is its intended kinematic counterpart (zero-mean Gaussian noise added to the joint-position channel of every observation before the WM encoder, with the physics state left unperturbed). A kinematic imager should respond to it, because the perturbation directly modifies the kinematic state \(\hat{x}_t\) that drives the extrapolation. The right panels of Figures 5, 8, and 9 show the sweep. A pre-registered powered rerun (\(K = 80\)) gives outcome B2: the WM’s slope remains unresolved while the physics reference responds weakly, so the perturbation is not inert but the WM’s imagination filters it. The Separability section reports the slopes, the likely RSSM-denoising mechanism, and the consequence for the kinematic reading.
Code and data availability. The diagnostic pipeline, trained checkpoints, perturbation-sweep CSVs, interactive-figure sources, and the timestamped pre-registrations of the joint-noise powered rerun (outcomes B1/B2/B3) and the cross-seed physics-slope sweep (outcome S1) for this paper will be released upon publication. The DreamerV3 implementation is the NM512 PyTorch port at commit 6ef8646. The upstream algorithm is DreamerV3
Implementation specifics. For the identity view, the kinematic predictor is constant-velocity on the root vertical state: \(\mathrm{kin}([z_t, \dot z_t]) = [z_t + \Delta t\,\dot z_t,\; \dot z_t]\) with \(\Delta t = 25\) ms (the DMC physics timestep). For the gait view, \(\hat{x}_t\) stacks the torso height and the six walker joint angles (each recovered from body orientations as \(\mathrm{sign}(\mathrm{axis}_y)\,(\theta_{\mathrm{child}} - \theta_{\mathrm{parent}})\), sign-aligned to the hinge axis so the angle and its velocity channel co-orient) together with their velocities, and \(\mathrm{kin}(\cdot)\) applies componentwise constant-velocity extrapolation. Friction perturbations scale the MuJoCo friction tuple of every geom by \(\mu\) at episode reset (model.geom_friction[:, 0] *= mu). Joint-noise perturbations add zero-mean Gaussian noise with standard deviation \(\sigma\) (rad) to the joint-position channel of every observation before the WM encoder, with the physics state left unperturbed. For WM-imagined rollouts we condition on five observations (encoder unroll) and then record \(T\) imagined transitions and \(T+1\) decoded states; \(T\) excludes the conditioning prefix. This matches the standard Dreamer evaluation protocol. All 95% confidence intervals are computed with a percentile bootstrap using 1000 resamples across the \(K = 20\) rollouts per cell (\(K = 80\) for the pre-registered joint-noise powered rerun and the conditioning-prefix cond-64 analysis). Revision channels (seed-0 checkpoint, friction axis, \(K = 20\), \(T = 64\)): a phase-matched physics channel (4-step closed-loop warm-up, so its window is env-time 4–68 like the WM’s, whose state[0] is the decoded posterior after five observations); the posterior encode–decode and one-step prior floor channels on the same real episodes; mode and 16-sample-averaged imagination; and an action-matched WM channel that replays the phase-matched physics channel’s saved actions per episode and friction. These runs re-seed the RNG per rollout from (seed, \(\mu\)) so that channels with the same cell layout share posterior samples and hence real trajectories. The headline runs are reproducible only in sweep order, because the RSSM posterior is sampled at every step. The pre-registration with outcome branches for these runs will be included in the release. Figure 5c is a post-hoc descriptive audit of these saved trajectories, not a pre-registered endpoint or a new sweep.
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