Using a per-step Kinematic-Consistency Error, we find a DreamerV3-class world model whose imagined rollouts stay invariant to a friction change that collapses the real gait, which we read as kinematic rather than dynamic imagination.
Abstract. Long-horizon failure in world models is conventionally attributed to compounding error, a generic framing that does not distinguish what kind of error compounds. We propose a kinematic-vs-dynamic reframing and operationalize it as the imagined Kinematic-Consistency Error (iKCE), a per-step diagnostic that measures how far a rollout departs from a closed-form kinematic null (the constant-velocity prediction of each next state), paired with a perturbation protocol that tests whether iKCE responds when physical conditions cross a regime boundary. We instantiate the diagnostic on a released DreamerV3 checkpoint
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. 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 invariant to the dynamic (friction) axis across the gait-collapse boundary, flat 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) resolves as a null. The two established signatures are the kinematic-null residual ≈180× above matched physics at a measurement horizon of \(T=16\) steps (narrowing to ≈30× at \(T=64\)) and this statistical invariance to the friction sweep. Operationally, we call the imagination kinematic if its iKCE responds to perturbations of the kinematic state the rollout extrapolates but stays flat 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.
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, WM flatness is informative. 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 matched policy-driven real-physics rollouts on the same conditioning (Table 1): ≈180× at \(T=16\), ≈30× at \(T=64\). The matched real-physics baseline is the same trained actor rolled out in real MuJoCo on the identical seeded episodes and conditioning prefixes the WM receives, so the two channels differ only in whether the rollout is imagined or simulated (see Reproducibility). The narrowing at the longer horizon is consistent with the per-step dilution noted in the limitations: the WM’s gradually-decaying per-step residual averages down the integrated metric. iKCE is therefore non-degenerate at both horizons. The diagnostic signature is friction-invariance under perturbation, not low 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) | \(\sim180\times\) | \([115,\,273]\) |
| 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) | \(\sim30\times\) | \([21,\,44]\) |
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 channel is flat when its \(\beta\) CI contains zero and steep when the CI excludes zero. The diagnostic reads off \(\beta\), not the visual band width on the log axis.
At \(T=64\), WM iKCE across the sweep is statistically flat (max/min spread \(1.32\times\), 95% CIs overlap at every \(\mu\)). A log-log regression on the same sweep makes this falsifiable: the WM slope’s 95% bootstrap CI (\(\beta_{\mathrm{WM}} = -0.009,\; [-0.096, +0.082]\)) contains zero, while the physics slope (\(\beta_{\mathrm{phys}} = -0.220,\; [-0.301, -0.142]\)) does not. (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 to which the WM’s imagined rollouts are blind. 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 invariance to the same perturbation is the established signature: invariance to the dynamic (friction) axis where physics is steep.
A natural alternative reading is that the WM’s iKCE is simply insensitive to all perturbations, a degenerate metric that would render the flatness vacuous. Two facts weigh against a fully degenerate reading. First, the WM’s iKCE is large and temporally structured (H1, and the per-step decomposition in the Appendix), not a flat constant. Second, the imagined rollouts do track the kinematic initial condition they are handed: when a long conditioning prefix carries a friction-differentiated state, the WM’s imagination inherits it (the conditioning-prefix analysis below). This is a posterior-sourced signal (the WM inherits the friction-differentiated start through the encoder, not through a transition that is itself sensitive to friction), so we use it only to rule out a constant readout, not as evidence about the transition. The Separability section develops the distinction. The cleanest test of the alternative would be an explicit kinematic positive control that the WM responds to while friction leaves it flat. The natural candidate, the joint-noise axis (right panel of Figure 5), does not provide it: a pre-registered powered rerun resolves the WM’s joint-noise response as a null, so the WM is insensitive to this observation-space kinematic axis as well. The non-degeneracy argument therefore rests on the two facts above (structure and initial-condition sensitivity) rather than on a positive control. With the metric shown non-degenerate, we read the flat-vs-steep result as kinematic-not-dynamic imagination, an interpretation we scope in the Separability section.
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. 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 stays flat 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 imagination does not amplify or sustain the regime signal. It relaxes it toward the friction-invariant tail.
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 friction-invariance 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 rule this out, 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). WM iKCE friction
spread is identical to the default-actor headline (\(1.32\times\) in both cases, CIs
overlapping at every \(\mu\)), confirming the friction-insensitivity of imagined
rollouts is not an artifact of the default actor's 15-step training horizon.Domain-randomization control. The most consequential confound of the physics-side signature is that the evaluation policy acted only at \(\mu = 1.0\) during training, so the elevated low-\(\mu\) physics iKCE may reflect in-distribution policy slipping under out-of-distribution friction rather than a genuine friction response of the contact dynamics. To quantify this confound, we train a fourth, otherwise-identical DreamerV3 checkpoint with per-episode domain randomization (DR) of friction, \(\mu \sim \mathcal{U}(0.1, 1.7)\), drawn at every episode reset (evaluation reward \(930 \pm 36\) across the full sweep range). Figure 9 shows the resulting friction and joint-noise sweeps.
Both the DR policy and the DR world model are therefore in-distribution at every friction value of the H2 sweep. Under matched DR conditions the H2 contrast retains its falsifiable form: 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 closes a data-coverage loophole: a world model trained only at \(\mu = 1.0\) has never observed friction variation and cannot have learned friction-conditional latent dynamics, so its flatness is partially guaranteed by construction. 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 remains friction-invariant.
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) stay flat 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 invariance 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 released artifact (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, the positive control is a null. 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 flat per-\(\sigma\) profile (max/min spread \(1.14\times\)). The WM does not respond to the kinematic axis we tested. 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 therefore report the joint-noise positive control as a null for the WM, and retract the workshop version’s “both channels respond” reading (the WM does not).
What separability rests on, and the remaining gap. The null has a sharp consequence: the WM is insensitive to both observation-space axes during imagination, the dynamic (friction) axis and the kinematic (joint-noise) axis alike. So the kinematic reading cannot rest on selective observation-space sensitivity. It rests on two things the null leaves intact. First, the WM’s iKCE is large and temporally structured, not a degenerate constant (H1, and the per-step decomposition in the Appendix). Second, on the dynamic axis the WM is flat 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. This is the central result. 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. This gap is structural, not an oversight. An observation-space kinematic perturbation is either filtered (joint noise) or enters only through the initial condition, while a latent-space perturbation cannot be cleanly labeled kinematic versus dynamic in a model that has no dynamic representation to contrast against. 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 imagined rollouts are invariant to the dynamic axis, not that they are demonstrably selectively kinematic.
iKCE is a per-step kinematic null fit integrated over a horizon. It diagnoses invariance to the dynamic (regime) axis, not the quality of dynamic prediction. The reading of that invariance as kinematic imagination is an interpretation we scope in Separability, not a measured selective sensitivity. A WM whose iKCE is invariant to the regime axis has been identified as dynamic-axis-invariant by our protocol, but has not been certified as a useful predictor. A WM with friction sensitivity has been certified as dynamic but not as accurate. 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 is now resolved (a null for the WM, 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
We recast long-horizon world-model failure as a question of kinematic versus dynamic imagination and made it measurable with the imagined Kinematic-Consistency Error and a conditioning-perturbation protocol. On a released DreamerV3-class walker-walk checkpoint, imagined rollouts stay invariant to a friction change that collapses the real gait: the physics-side iKCE slope excludes zero while the world model’s slope contains it, a contrast that holds across an actor-horizon ablation, domain randomization, three training seeds, and a second kinematic transform. What this establishes is invariance to the dynamic axis. Reading that invariance as kinematic imagination is an interpretation we scope rather than demonstrate, because the intended kinematic positive control resolved as a null and the imagined transition’s kinematic sensitivity remains unmeasured. The diagnostic and its pre-registered protocol carry over to any embodiment whose state admits a kinematic predictor, and the natural next step is a latent-space control that separates a decoded-kinematic direction from a matched control direction to test the kinematic reading directly.
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 and the cond-64 analysis) |
| 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 “statistically flat” 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 H2’s flatness claim survives a falsifiable statistical test across seeds. 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 within-checkpoint contrast is the paired quantity, 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 (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, \([-0.323, -0.102]\) 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 flat-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 ~0.13 (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 paired difference excludes zero for each.
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, friction-invariant tail. 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 gives the structure leg of the non-degeneracy argument (see Separability) the same interval-based footing as the flat-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 flat (\(\beta_{\mathrm{WM}} = +0.010\), 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 flat 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
flat (\(\beta_{\mathrm{WM}} = +0.010\), CI contains 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 (\(K = 80\)) resolves it as outcome B2: the WM’s response is a null 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 are released at [PLACEHOLDER FOR DOUBLE BLIND]. 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 the first 5 perturbed observations (encoder unroll), then roll out free imagination for the remaining \(T-5\) steps. 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).
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