Dynamic LeJEPA: Maximum Entropy Representations for Sequential Prediction and Latent Planning with Theoretical Guarantees

1. Introduction

In applied computer vision—particularly in data-starved domains like computational pathology, novel geometric mapping, and physical robotics—researchers face a fundamental crisis. When confronted with limited, new real-world sequential data, current deep learning frameworks act as “black boxes.” Practitioners are forced to rely on massive transfer-learning crutches (e.g., DINOv2) or spend endless epochs tuning heuristic stop-gradients, EMA schedules, and loss weights, never knowing if the internal representations have collapsed or if the true geometry of the data is actually understood.

The natural intuitive response to limited physical data is to inject domain knowledge—forcing physics constraints, geometric rules, or strong inductive biases into the latent space to help the model learn. However, empirically, these “Physics-Informed” approaches consistently fail or degrade representation quality. Why does helping the model actually hurt the model?

Joint-Embedding Predictive Architectures (JEPAs) offer a promising blueprint, predicting in latent space rather than pixel space. Recently, LeJEPA [1] provided the first theoretical foundation for JEPAs, proving that isotropic Gaussian embeddings minimize worst-case downstream risk (Theorem 1 of [1]). This result, combined with Sketched Isotropic Gaussian Regularization (SIGReg), enables training without heuristic stop-gradients or teacher-student architectures. However, LeJEPA was developed and validated on static images. Its application to sequential data—while empirically explored in V-JEPA  [5]—lacked theoretical grounding.

Dynamic LeJEPA as a Principled World Model. In the broader context of latent world models, this theoretical gap is a critical bottleneck. A world model comprises a state representation, a transition dynamics model, and an observation decoder. Without formal guarantees, practitioners are left guessing where to inject valuable domain priors (e.g., physics or geometric rules) without destroying the latent state—and, as we show, guessing wrong is the norm rather than the exception.

In this paper, we resolve the empirical paradoxes of sequential learning through mathematical revelation. Our theorems reveal that the intuition to constrain the model was correct, but the target was wrong. Theorem 1 and Theorem 5 prove that constraining the encoder violates Maximum Entropy, actively increasing downstream risk. However, we prove that physics and dynamics are not inherently harmful—they must simply be relocated: dynamics belong in the predictor (Theorem 6 shows any parametric model can be safely plugged in), and physics belongs strictly in the observation decoder (Theorem 5), never the encoder.

A representation-quality guarantee, however, is not by itself a guarantee that the representation is useful for what a world model is ultimately built to do: support decision-making. To ask whether the placement rule matters once the representation is actually used—not just measured—we build a latent planner directly on top of the trained Phase 3 models and evaluate it against the true future state rather than a proxy. Doing so surfaces a confound that, to our knowledge, is not commonly checked for in offline world-model evaluations: a planner optimizing against a learned decoder can find action sequences that score better under the model’s own cost than the real, recorded actions do, without those actions being any closer to the truth. We diagnose this directly, show it holds uniformly across all three ablation conditions, and report it as a necessary caution for anyone evaluating a JEPA-style world model’s control-readiness without a closed-loop, simulator-verified rollout.

This gives a complete theoretical answer through six main theorems, establishing a clean separation of concerns summarized in our contributions below.

1.1. Contributions

• Theoretical: We prove that prediction and representation learning are separable—prediction doesn’t constrain the embedding distribution (Theorem 4.1).
• World Model Architecture: We provide the first mathematically grounded design principles for latent world models, proving dynamics belong in the predictor and physics belong in the decoder (Theorems 2–6).
• Negative Result Explanation: We prove why prior physics-informed JEPA approaches fail: they constrain the encoder, not the predictor (Theorem 4.10).
• Sample Complexity Characterization: We prove that effective dimensionality metrics have a $1/K$ floor when $N<K$ (Proposition 6.1), establishing when distributional validation is feasible, and we empirically confirm this threshold across three orders of magnitude of $N/K$.
• Cross-Domain Validation (Phases 2–3): We validate the decoder-physics theorem across two domains—autonomous driving (nuScenes) and robotic manipulation (MimicGen)—with 60 seeded runs yielding statistically significant results in every comparison, and a refined scale-vs-structure decomposition isolating genuine maximum-entropy violation from ordinary variance shrinkage.
• Entropy–Utility Frontier (Phase 3): We characterize a frontier governing what deterministic encoders can achieve on low-intrinsic-dimension data: per-dimension variance matching alone is insufficient (correlated collapse), and optimization budget—not regularization weight—is the binding entropy constraint.
• Practical Guidance: We establish clear rules: maximum entropy in encoder, dynamics in predictor, physics in decoder—and characterize the data requirements for verifying each across domains.
• Toward Closed-Loop Validation and a Model-Exploitation Diagnostic (Phase 3): We probe the placement rule’s relevance for downstream control via a latent cross-entropy-method (CEM) planner built on the trained Phase 3 models, and introduce a model-exploitation diagnostic (action-space distance from the ground-truth trajectory) that reveals offline latent planning against a learned decoder inflates apparent planning accuracy uniformly across all three ablation conditions—a concrete, reproducible caution against evaluating a world model’s control-readiness without closed-loop, simulator-verified execution.

2. Related Work

2.1. World Models: Foundations and Positioning

The architectural decomposition we formalize—a state encoder, a learned transition (dynamics) model, and an observation decoder—was introduced by Ha and Schmidhuber’s World Models [2], which trained a variational encoder and an RNN-based dynamics model to let an agent plan entirely inside a learned latent simulator. Model-based reinforcement learning methods such as PlaNet and Dreamer [3] scaled this idea to pixel-based control by learning latent dynamics directly from images. More recently, LeCun’s position paper on autonomous machine intelligence [4] proposed the Joint-Embedding Predictive Architecture (JEPA) itself as the encoder/predictor component of exactly this world-model stack, arguing that a non-generative, latent-space predictor is necessary for the world model to ignore irrelevant pixel-level detail. Our contribution sits precisely at the seam these works leave open: Ha and Schmidhuber, and the Dreamer lineage, are empirically driven and offer no guarantee about what distribution the encoder should converge to; LeCun’s proposal names the JEPA encoder as the right component but does not prove that adding a dynamics predictor or a physics-constrained decoder preserves the encoder’s representational guarantees. Dynamic LeJEPA answers this open question directly: we prove that the world-model decomposition of [2] is compatible with LeJEPA’s maximum-entropy guarantee [1] only if physics and dynamics are placed in the predictor and decoder, never the encoder—turning an architectural convention from prior world-model work into a proven design constraint.

2.2. LeJEPA and Maximum Entropy

LeJEPA [1] proves that among all distributions $p\left(z\right)$ with fixed variance ${\sigma }^2$, the isotropic Gaussian $\mathcal{N}(0,{\sigma }^2{I}_K)$ minimizes worst-case downstream prediction risk for both linear and nonlinear probes. SIGReg enforces this distribution through characteristic function matching on random projections. Crucially, LeJEPA’s analysis is static: it considers a single embedding $z=f\left(x\right)$ without temporal structure. Our work extends this framework to the sequential setting while preserving the theoretical guarantees.

2.3. JEPA Variants for Sequential Data

V-JEPA [5] applies JEPA to video by predicting masked spatial regions. While empirically successful, it lacks theoretical analysis of how prediction affects the embedding distribution. I-JEPA [6] and others use prediction without formal guarantees.

2.4. Physics-Informed Learning

Physics-Informed Neural Networks (PINNs) [7] embed physical laws as soft constraints in supervised learning. Attempts to combine physics priors with self-supervised learning have shown mixed results, with performance often degrading when physical constraints are injected into the latent space. We provide the first theoretical explanation: physics constraints on embeddings violate maximum entropy, but constraints on decoders do not.

2.5. Maximum Entropy in Learning

The maximum entropy principle [8] states that, given constraints, one should select the distribution with maximum entropy. This principle underlies statistical mechanics and has connections to information theory [9]. Our work establishes LeJEPA as an implementation of this principle in representation learning.

3. Preliminaries

3.1. Notation for Sequential Setting

Table 1. Primary notation for sequential Dynamic LeJEPA.

Symbol	defin
$x_{1:T}$	Input sequence (images, observations)
$a_{1:T-1}$	Action sequence (robotics, driving)
$z_t=f_{\theta }\left(x_t\right)\in {\mathbb{R}}^K$	Latent embedding at time t
${\widehat{z}}_{t+1}=g_{\varphi }(z_t,a_t)\in {\mathbb{R}}^K$	Predicted next latent
$p\left(z\right)=\mathcal{N}(0,{\sigma }^2{I}_K)$	Target isotropic Gaussian
$\mathcal{P}=\{p:{\mathbb{E}}_p{[\parallel z\parallel }^2]=K{\sigma }^2\}$	Distributions with fixed variance
$\mathcal{H}\left(p\right)=-\int plogpdz$	Differential entropy
$p(z_{1:T}\parallel a_{1:T-1})$	Joint conditional distribution over time
$g^{*}=arg{min}_g{\mathcal{L}}_{\mathrm{pred}}(g,p)$	Optimal predictor for fixed p

Table 1. Primary notation for sequential Dynamic LeJEPA.

Symbol	defin
$x_{1:T}$	Input sequence (images, observations)
$a_{1:T-1}$	Action sequence (robotics, driving)
$z_t=f_{\theta }\left(x_t\right)\in {\mathbb{R}}^K$	Latent embedding at time t
${\widehat{z}}_{t+1}=g_{\varphi }(z_t,a_t)\in {\mathbb{R}}^K$	Predicted next latent
$p\left(z\right)=\mathcal{N}(0,{\sigma }^2{I}_K)$	Target isotropic Gaussian
$\mathcal{P}=\{p:{\mathbb{E}}_p{[\parallel z\parallel }^2]=K{\sigma }^2\}$	Distributions with fixed variance
$\mathcal{H}\left(p\right)=-\int plogpdz$	Differential entropy
$p(z_{1:T}\parallel a_{1:T-1})$	Joint conditional distribution over time
$g^{*}=arg{min}_g{\mathcal{L}}_{\mathrm{pred}}(g,p)$	Optimal predictor for fixed p

3.2. Assumptions

Assumption 3.1 

(Temporal Stationarity). The marginal embedding distribution is stationary: $p\left(z_t\right)=p\left(z\right)$ for all $t\in \{1,\dots ,T\}$.

Assumption 3.2 

(Markov Property). The sequence satisfies: $p\left(z_{t+1}\right|z_{\le t},a_{\le t})=p\left(z_{t+1}\right|z_t,a_t)$.

Assumption 3.3 

(Bounded Dynamics). The true transitions satisfy: $\mathbb{E}[\parallel z_{t+1}-z_t{\parallel }^2]\le L^2$ for some constant L.

Assumption 3.4 

(Loss Regularity). The prediction loss $\ell (y,y^{\prime })$ is $L_{\ell }$-Lipschitz in both arguments and bounded below by 0.

3.3. LeJEPA Background

Definition 3.5 

(LeJEPA Optimal Distribution [1]). The LeJEPA embedding distribution $p^{*}$ that minimizes worst-case downstream risk over linear probes satisfies:

$$p^{*}\left(z\right)=\mathcal{N}(0,{\sigma }^2{I}_K)$$

(1)

This is the maximum entropy distribution in $\mathcal{P}$.

Definition 3.6 

(SIGReg [1]). Sketched Isotropic Gaussian Regularization enforces Equation (1) by matching characteristic functions along random projections:

$${\mathcal{L}}_{\mathrm{SIGReg}}\left(z\right)={\displaystyle \frac{1}{\left|\mathcal{A}\right|}}\sum _{a\in \mathcal{A}}\mathcal{T}\left({\left\{a^{\top }{z}_n\right\}}_{n=1}^N\right)$$

(2)

where $\mathcal{T}$ is the Epps-Pulley statistic comparing projected samples to $\mathcal{N}(0,1)$.

Remark 3.7 

(Key Property). SIGReg operates on the marginal distribution $p\left(z_t\right)$ at each time step, not on the joint $p\left(z_{1:T}\right)$.

4. Main Results

4.1. Theorem 1: Prediction Doesn’t Change Optimal Embeddings

Theorem 4.1 

(Prediction-Entropy Separation). Let $p^{*}=arg{max}_{p\in \mathcal{P}}\mathcal{H}\left(p\right)$ be the maximum entropy embedding distribution. For any prediction loss of the form:

$${\mathcal{L}}_{\mathrm{pred}}(g,p)={\mathbb{E}}_{p(z_t,z_{t+1}|a_t)}\left[\ell (g(z_t,a_t),z_{t+1})\right]$$

(3)

where ℓ is an $L_{\ell }$-Lipschitz loss function and g is learnable, the optimal embedding distribution satisfies:

$$p_{\mathit{with}\mathit{prediction}}^{*}=p^{*}$$

(4)

Adding sequential prediction does not change the optimal embedding distribution.

Proof. 

Define the joint objective:

$$J(p,g)=-\lambda \mathcal{H}\left(p\right)+{\mathcal{L}}_{\mathrm{pred}}(g,p)$$

(5)

where $\lambda >0$ controls the entropy-prediction trade-off.

Step 1: Optimize predictor for fixed encoder. For any fixed p, define:

$$g^{*}\left(p\right)=arg\underset{g}{min}{\mathcal{L}}_{\mathrm{pred}}(g,p)$$

(6)

This is a standard regression problem (predict $z_{t+1}$ from $(z_t,a_t)$). By Assumption 3.4, the optimal $g^{*}\left(p\right)$ exists. Define the optimal prediction loss for given p:

$${\mathcal{L}}_{\mathrm{pred}}^{*}\left(p\right)={\mathcal{L}}_{\mathrm{pred}}(g^{*}\left(p\right),p)=\underset{g}{min}{\mathcal{L}}_{\mathrm{pred}}(g,p)$$

(7)

Step 2: Show ${\mathcal{L}}_{\mathrm{pred}}^{*}$ is invariant under entropy-preserving transformations. Let $\Phi :{\mathbb{R}}^K\to {\mathbb{R}}^K$ be an orthogonal transformation (rotation/reflection). Define the transformed distribution $p_{\Phi }\left(z\right)=p\left({\Phi }^{-1}z\right)$. Orthogonal transformations preserve both entropy and Gaussian structure:

$$\begin{array}{cc}\hfill \mathcal{H}\left(p_{\Phi }\right)& =\mathcal{H}\left(p\right)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill p_{\Phi }=\mathcal{N}(0,{\sigma }^{2}I)& \iff p=\mathcal{N}(0,{\sigma }^{2}I)\hfill \end{array}$$

(9)

The prediction loss transforms as:

$${\mathcal{L}}_{\mathrm{pred}}^{*}\left(p_{\Phi }\right)=\underset{g}{min}{\mathbb{E}}_{p_{\Phi }}[\ell (g(z,a),z^{\prime })]=\underset{\tilde{g}}{min}{\mathbb{E}}_p[\ell (\tilde{g}(\Phi z,a),\Phi z^{\prime })]$$

(10)

where $\tilde{g}=\Phi \circ g\circ {\Phi }^{-1}$. Since ℓ is Lipschitz and orthogonal transformations preserve distances:

$${\mathcal{L}}_{\mathrm{pred}}^{*}\left(p_{\Phi }\right)={\mathcal{L}}_{\mathrm{pred}}^{*}\left(p\right)$$

(11)

Step 3: Reduce to entropy maximization. The joint optimization becomes:

$$\underset{p}{min}J(p,g^{*}\left(p\right))=\underset{p}{min}\left[-\lambda \mathcal{H}\left(p\right)+{\mathcal{L}}_{\mathrm{pred}}^{*}\left(p\right)\right]$$

(12)

By the invariance shown in Step 2, the second term is constant over the set of distributions with equal entropy. By LeJEPA’s Theorem 1 [1], the maximizer of $\mathcal{H}\left(p\right)$ under this invariance is the isotropic Gaussian.

Step 4: Conclusion.

$$\begin{array}{cc}\hfill p_{\mathrm{with}\mathrm{prediction}}^{*}& =arg\underset{p}{min}\left[-\lambda \mathcal{H}\left(p\right)+{\mathcal{L}}_{\mathrm{pred}}^{*}\left(p\right)\right]\hfill \\ \hfill & =arg\underset{p}{max}\mathcal{H}\left(p\right)=p^{*}\hfill \end{array}$$

(13)

   □

Corollary 4.2 

(Extended Prediction Losses). Theorem 4.1 extends to:

(a) 

Multi-step prediction: ${\sum }_{h=1}^H\mathbb{E}[\parallel g{\left(z_t\right)}_h-z_{t+h}{\parallel }^2]$

(b) 

Masked prediction: $\mathbb{E}[\parallel g\left(z_{\mathrm{unmasked}}\right)-z_{\mathrm{masked}}{\parallel }^2]$

(c) 

Stochastic prediction: $\mathbb{E}[\parallel g(z_t,ϵ)-z_{t+1}{\parallel }^2]$ where $ϵ\sim \mathcal{N}(0,{\sigma }_{ϵ}^{2}I)$

(d) 

Distributional prediction: $D_{\mathrm{KL}}(p\left(z_{t+1}\right|g\left(z_t\right))\parallel p\left(z_{t+1}\right))$

All constrain the predictor g, not the embedding distribution p.

Remark 4.3 

(Interpretation). Theorem 4.1 establishes a clean separation resolving the black-box crisis:

• The encoder $f_{\theta }$ should produce maximum entropy embeddings (no constraints)
• The predictor $g_{\varphi }$ can learn any dynamics without affecting representation quality
• The loss only affects the predictor, not the embedding distribution

This is precisely the design philosophy of JEPA: predict in latent space, leave the representation unconstrained.

4.2. Theorem 2: Robotics — Action-Conditioned Prediction

Theorem 4.4 

(Robotics: Maximum Entropy Action-Conditioned Representations). Consider a robot with observations $x_t$, joint states $q_t$, and actions $a_t$. Let the encoder map to $z_t=f_{\theta }(x_t,q_t)$ and the predictor be ${\widehat{z}}_{t+1}=g_{\varphi }(z_t,a_t)$. For the objective:

$$J_{\mathrm{robot}}=-\lambda \mathcal{H}\left(f_{\theta }\left(z\right)\right)+\mathbb{E}\left[\parallel g_{\varphi }(z_t,a_t)-z_{t+1}{\parallel }^2\right]$$

(14)

the following hold:

(a) 

Maximum entropy encodings:  The optimal encoder produces $z_t\sim \mathcal{N}(0,{\sigma }^{2}I)$ regardless of the action space $\mathcal{A}$.

(b) 

Optimal predictor:  $g^{*}(z_t,a_t)=\mathbb{E}\left[z_{t+1}\right|Z_t=z_t,A=a_t]$ (conditional expectation)

(c) 

Task-relevant structure:  The representation encodes action-relevant information through predictability, not through embedding constraints.

(d) 

Action invariance:  If $\pi \left(a\right|s)$ is the policy, the optimal z encodes state s without bias toward specific actions.

Proof. Proof of (a): Direct application of Theorem 4.1. The action $a_t$ enters only through the prediction loss, which doesn’t affect the optimal $p\left(z\right)$.

Proof of (b): For fixed encoder (hence fixed p), the prediction loss is a regression problem. By the properties of conditional expectation:

$$arg\underset{g}{min}\mathbb{E}\left[\parallel g(z_t,a_t)-z_{t+1}{\parallel }^2\right]=\mathbb{E}\left[z_{t+1}\right|Z_t=z_t,A=a_t]$$

(15)

This holds regardless of whether $p\left(z\right)$ is maximum entropy.

Proof of (c): By the data processing inequality, maximum entropy preserves all mutual information between z and any variable. If predicting $z_{t+1}$ from $(z_t,a_t)$ requires information about the task, then $z_t$ must encode it. Formally, let Y be the task-relevant information:

$$I(Z_t;Y)\ge I(Z_t;z_{t+1}|A_t)$$

(16)

By maximum entropy, $I(Z_t;Y)$ is maximized subject to the variance constraint.

Proof of (d): The optimal $p\left(z\right)$ is invariant under any transformation of the action representation. Whether actions are represented as joint angles, end-effector positions, or torques doesn’t affect the optimal embedding distribution.    □

Corollary 4.5 

(Linear Dynamics in Robotics). If the true dynamics are linear: $z_{t+1}=A{z}_t+B{a}_t+{ϵ}_t$ with ${ϵ}_t\sim \mathcal{N}(0,{\Sigma }_{ϵ})$, then:

1. 

The optimal predictor is: $g^{*}(z,a)=Az+Ba$

2. 

The maximum entropy encoder is unchanged

3. 

The learned matrices $(A,B)$ approximate the true dynamics

4.3. Theorem 3: Autonomous Driving — Ego-Motion Decomposition

Definition 4.6 

(Ego-Motion Decomposition). In autonomous driving, decompose the latent at time t as:

$$z_t=\left[\begin{array}{c}{z}_t^{\mathrm{ego}}\\ z_t^{\mathrm{static}}\end{array}\right]\in {\mathbb{R}}^{K_1+K_2}$$

(17)

where $z_t^{\mathrm{ego}}\in {\mathbb{R}}^{K_1}$ encodes ego-vehicle state (pose, velocity) and $z_t^{\mathrm{static}}\in {\mathbb{R}}^{K_2}$ encodes the static scene (geometry, semantics).

Theorem 4.7 

(Autonomous Driving: Ego-Motion Decomposition). Under the ego-motion decomposition, with dynamics:

$$\begin{array}{cc}\hfill {\widehat{z}}_{t+1}^{\mathrm{ego}}& =g_{\varphi }^{\mathrm{ego}}(z_t^{\mathrm{ego}},e_t)\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\widehat{z}}_{t+1}^{\mathrm{static}}& =z_t^{\mathrm{static}}\hfill \end{array}$$

(19)

where $e_t=(R_t,t_t)$ is the ego-motion (rotation, translation), the following hold:

(a) 

Independent maximum entropy:  Both $p\left(z^{\mathrm{ego}}\right)$ and $p\left(z^{\mathrm{static}}\right)$ individually satisfy maximum entropy

(b) 

Ego-motion predictor:  $g^{\mathrm{ego}*}$ learns the geometric transformation induced by $e_t$

(c) 

Static stability:  $z_t^{\mathrm{static}}$ provides stable scene representation invariant to ego-motion

(d) 

Dynamic object detection:  Deviations from ${\widehat{z}}_{t+1}^{\mathrm{static}}$ indicate dynamic objects without explicit supervision

Proof. Proof of (a): The decomposition $z=[z^{\mathrm{ego}},z^{\mathrm{static}}]$ is a linear projection onto orthogonal subspaces. The projection of a maximum entropy distribution is maximum entropy in the projected subspace. By Theorem 4.1 (which applies component-wise), both components are independently maximum entropy.

Proof of (b): The ego-motion prediction loss:

$${\mathcal{L}}_{\mathrm{ego}}=\mathbb{E}\left[\parallel g^{\mathrm{ego}}(z_t^{\mathrm{ego}},e_t)-{\widehat{z}}_{t+1}^{\mathrm{ego}}{\parallel }^2\right]$$

(20)

By Theorem 4.4(b), the optimal predictor learns the conditional expectation, which for rigid ego-motion with known calibration is a known geometric function.

Proof of (c): The static component satisfies $z_t^{\mathrm{static}}=z_{t+1}^{\mathrm{static}}$ for a static scene. Maximum entropy in $z^{\mathrm{static}}$ ensures all static scene information is preserved without bias toward any particular viewpoint.

Proof of (d): Let ${\Delta }_t=|{\widehat{z}}_{t+1}^{\mathrm{static}}-z_t^{\mathrm{static}}|$ measure the static scene change. Under maximum entropy, ${\Delta }_t$ is large only where the scene actually changes (dynamic objects). No explicit supervision is needed to detect dynamics.    □

Corollary 4.8 

(Depth Prediction from Static Features). Let $h_{\psi }$ be a depth decoder: ${\widehat{d}}_t=h_{\psi }\left(z_t^{\mathrm{static}}\right)$. Depth predictions are optimal when $z_t^{\mathrm{static}}$ is maximum entropy.

4.4. Theorem 4: Video Understanding — Multi-Step Prediction

Theorem 4.9 

(Video: Multi-Step Predictive Coding). For video sequences $x_{1:T}$, let the encoder produce $z_{1:T}$ and the predictor perform multi-step prediction:

$$[{\widehat{z}}_{t+1},\dots ,{\widehat{z}}_{t+H}]=g_{\varphi }\left(z_t\right)\in {\mathbb{R}}^{H\times K}$$

(21)

with loss:

$${\mathcal{L}}_{\mathrm{video}}={\displaystyle \sum _{h=1}^H}\mathbb{E}\left[\parallel g_{\varphi }{\left(z_t\right)}_h-z_{t+h}{\parallel }^2\right]$$

(22)

Then:

(a) 

Marginal maximum entropy:  $p\left(z_t\right)=\mathcal{N}(0,{\sigma }^{2}I)$ for all t

(b) 

Multi-step preservation:  Multi-step prediction doesn’t require multi-step entropy reduction

(c) 

Masked prediction safety:  V-JEPA style masked prediction doesn’t constrain embeddings

(d) 

Autoregressive danger:  Using predictions as inputs ($z_{t+1}=g\left(z_t\right)$ exactly) CAN create harmful constraints if applied as hard constraints

Proof. Proof of (a): Direct application of Theorem 4.1 to each time step independently.

Proof of (b): The multi-step loss constrains g, not $p\left(z_t\right)$. Each $z_{t+h}$ comes from the encoder distribution $p\left(z\right)$.

Proof of (c): V-JEPA masked prediction:

$${\mathcal{L}}_{\mathrm{mask}}=\mathbb{E}\left[\parallel g\left(z_{\mathrm{unmasked}}\right)-z_{\mathrm{masked}}{\parallel }^2\right]$$

(23)

This predicts masked spatial regions from unmasked regions. By Corollary 4.2(b), the optimal embedding is unchanged.

Proof of (d): Consider autoregressive unfolding with hard constraints:

$${\widehat{z}}_{t+1}=g\left(z_t\right),z_{t+1}={\widehat{z}}_{t+1}$$

(24)

This creates a constraint: $z_{t+1}$ must lie in the image of g. The support of $p\left(z_{t+1}\right)$ is constrained to $\mathrm{Im}\left(g\right)\cap \mathrm{supp}\left(p\right)$, which reduces entropy below maximum. However, if used as a soft target:

$${\mathcal{L}}_{\mathrm{video}}=\mathbb{E}\left[\parallel g{\left(z_t\right)}_1-z_{t+1}{\parallel }^2+{\lambda }_{\mathrm{ar}}{\parallel g{\left(z_t\right)}_1-g{\left(g\left(z_t\right)\right)}_1\parallel }^2\right]$$

(25)

the marginal $p\left(z_t\right)$ can still be maximum entropy.    □

4.5. Theorem 5: Decoder Physics Separation

Theorem 4.10 

(Physics-Informed Decoding is Valid). Let the encoder produce maximum entropy embeddings $z=f_{\theta }\left(x\right)$, and the decoder incorporate physics:

$$\widehat{y}=h_{\psi }\left(z\right),\mathrm{subject}\mathrm{to}C\left(\widehat{y}\right)=0$$

(26)

where $C:\mathcal{Y}\to {\mathbb{R}}^m$ is a physics constraint function. Then:

(a) 

Encoder optimal unchanged:   $p^{*}\left(z\right)=\mathcal{N}(0,{\sigma }^{2}I)$

(b) 

Any physics valid:  The decoder can use any physics constraint without affecting representation quality

(c) 

Information preservation:  Physics at the decoder level improves task performance without reducing representational capacity

Proof. Proof of (a): The decoder constraint $C\left(h_{\psi }\left(z\right)\right)=0$ operates on the output of the decoder, not on z itself. The encoder objective remains:

$$\underset{\theta }{min}\left[-\lambda \mathcal{H}\left(f_{\theta }\left(z\right)\right)+{\mathcal{L}}_{\mathrm{JEPA}}\right]$$

(27)

The decoder $h_{\psi }$ doesn’t appear in the encoder objective. Therefore $p^{*}\left(z\right)=\mathcal{N}(0,{\sigma }^{2}I)$ is still optimal.

Proof of (b): For any physics constraint C, define the constrained decoder:

$$h^{*}=arg\underset{h:C\left(h\right(z\left)\right)=0}{min}{\mathcal{L}}_{\mathrm{task}}(h,z)$$

(28)

This is a constrained optimization over h, independent of the encoder.

Proof of (c): The encoder provides maximum entropy representations (maximum information by the data processing inequality). The decoder uses physics to interpret this information correctly. Formally:

$$\begin{array}{cc}\hfill I(Z;Y)& \le \mathcal{H}\left(Z\right)(\mathrm{data}\mathrm{processing}\mathrm{inequality})\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill I(Z^{*};Y)& =\mathcal{H}\left(Z^{*}\right)(\mathrm{maximum}\mathrm{entropy})\hfill \end{array}$$

(30)

The decoder with physics $h_{\psi }^{*}$ can extract Y from $Z^{*}$ as effectively as any other decoder, while potentially being more sample-efficient.    □

Corollary 4.11 

(Contrast with Encoder Physics). Adding the same physics constraint C to the ENCODER would violate maximum entropy and hurt representation quality. Therefore, physics constraints are valid at the decoder but invalid at the encoder.

Example 4.12 

(Physics-Informed Depth Decoding). For depth estimation, the decoder can enforce:

$${\nabla }^2\widehat{d}=0\mathit{where}\parallel {\nabla }_{\mathrm{rgb}}I\parallel <\tau$$

(31)

(edge-aware smoothness at non-edges). This is valid at the decoder level because it constrains $\widehat{d}$, not z.

4.6. Theorem 6: General Dynamics Learning

Theorem 4.13 

(Learning Parametric Dynamics Preserves Maximum Entropy). Let the dynamics be parameterized as ${\widehat{z}}_{t+1}=g_{\varphi }(z_t,a_t)$ where $g_{\varphi }$ belongs to a class $\mathcal{G}$ of parameterized functions. Then:

(a) 

Encoder optimal unchanged:  The optimal encoder is still maximum entropy regardless of $\mathcal{G}$

(b) 

Universal approximation:  Any dynamics compatible with the data can be learned

(c) 

Conditional expectation:  The learned dynamics approximate $\mathbb{E}\left[z_{t+1}\right|Z_t=z_t,A=a_t]$

Proof. Proof of (a) The dynamics model $g_{\varphi }$ is a parameterized predictor. By Theorem 4.1, any predictor doesn’t affect the optimal embedding distribution.

Proof of (b): By the universal approximation theorem, neural networks can approximate any continuous function to arbitrary precision given sufficient capacity.

Proof of (c): For any parametric family $\mathcal{G}$:

$$g^{*}=arg\underset{g\in \mathcal{G}}{min}\mathbb{E}\left[\parallel g(z_t,a_t)-z_{t+1}{\parallel }^2\right]$$

(32)

This is a constrained conditional expectation. As the capacity of $\mathcal{G}$ increases, $g^{*}$ converges to the true conditional expectation.    □

5. Unified Framework

Theorem 5.1 

(Unified Dynamic LeJEPA). For sequential prediction with encoder $f_{\theta }$, predictor $g_{\varphi }$, and decoder $h_{\psi }$, the optimal configuration satisfies:

$$\begin{array}{cc}\hfill \mathit{C}\mathbf{1}.& \mathit{Encoder}\text{:}{f}_{\theta }^{*}\left(x_t\right)\sim \mathcal{N}(0,{\sigma }^{2}I)\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \mathit{C}\mathbf{2}.& \mathit{Predictor}\text{:}{g}_{\varphi }^{*}(z_t,a_t)\approx \mathbb{E}\left[z_{t+1}\right|z_t,a_t]\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \mathit{C}\mathbf{3}.& \mathit{Decoder}\text{:}{h}_{\psi }^{*}\left(z\right)\mathit{subject}\mathit{to}C\left(h_{\psi }^{*}\left(z\right)\right)=0\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \mathit{C}\mathbf{4}.& \mathit{Separation}\text{:}\mathit{These}\mathit{optimizations}\mathit{are}\mathit{independent}\hfill \end{array}$$

(36)

Table 2. Dynamic LeJEPA design principles. Constraints on the encoder violate maximum entropy and hurt representation quality.

Component	Constraint Type	Max Entropy?	Valid?
Encoder $f_{\theta }$	None (only SIGReg)	Yes	Yes
Predictor $g_{\varphi }$	Prediction target	N/A	Yes
Decoder $h_{\psi }$	Physics $C\left(y\right)=0$	N/A	Yes

Table 2. Dynamic LeJEPA design principles. Constraints on the encoder violate maximum entropy and hurt representation quality.

Component	Constraint Type	Max Entropy?	Valid?
Encoder $f_{\theta }$	None (only SIGReg)	Yes	Yes
Predictor $g_{\varphi }$	Prediction target	N/A	Yes
Decoder $h_{\psi }$	Physics $C\left(y\right)=0$	N/A	Yes

6. Sample Complexity for Distributional Validation

Before presenting experimental results, we characterize when distributional properties can be reliably measured. This analysis is crucial for interpreting our experiments and guiding practitioners.

6.1. The Effective Dimensionality Floor

A standard metric for measuring representation quality is the effective dimensionality:

$$d_{\mathrm{eff}}={\displaystyle \frac{{\left({\sum }_{i=1}^K{\lambda }_i\right)}^2}{K{\sum }_{i=1}^K{\lambda }_i^2}}$$

(37)

where $\left\{{\lambda }_i\right\}$ are the eigenvalues of the covariance matrix. For an isotropic Gaussian $\mathcal{N}(0,{\sigma }^{2}I)$, $d_{\mathrm{eff}}=1$.

However, in practice we estimate $d_{\mathrm{eff}}$ from samples, which introduces a fundamental limitation:

Proposition 6.1 

(Effective Dimensionality Floor). Let $Z={\left\{z_i\right\}}_{i=1}^N\subset {\mathbb{R}}^K$ be i.i.d. samples with $N<K$, and let $\widehat{\Sigma }$ be the sample covariance with Ledoit-Wolf shrinkage [10] ${\widehat{\Sigma }}_{\gamma }=\gamma I+(1-\gamma )\widehat{\Sigma }$. Then:

$$d_{\mathrm{eff}}\left({\widehat{\Sigma }}_{\gamma }\right)\le {\displaystyle \frac{N-1}{K}}+\mathcal{O}\left({\displaystyle \frac{\gamma }{\overline{\lambda }}}\right)$$

(38)

where $\overline{\lambda }$ is the mean non-zero eigenvalue of $\widehat{\Sigma }$.

Proof. 

The sample covariance has rank $\le N-1$. Let ${\lambda }_1\ge \dots \ge {\lambda }_{N-1}>0$ be the non-zero eigenvalues and ${\lambda }_N=\dots ={\lambda }_K=0$. After shrinkage:

• Non-zero eigenvalues: ${\tilde{\lambda }}_i=(1-\gamma ){\lambda }_i+\gamma$
• Zero eigenvalues: ${\tilde{\lambda }}_i=\gamma$ for $i\ge N$

The effective dimensionality becomes:

$$\begin{array}{cc}\hfill d_{\mathrm{eff}}& ={\displaystyle \frac{{\left({\sum }_{i=1}^{N-1}{\tilde{\lambda }}_i+(K-N+1)\gamma \right)}^2}{K\left({\sum }_{i=1}^{N-1}{\tilde{\lambda }}_i^2+(K-N+1){\gamma }^2\right)}}\hfill \end{array}$$

(39)

When $\gamma \ll \overline{\lambda }$, the $(K-N+1)\gamma$ terms dominate, yielding:

$$d_{\mathrm{eff}}\approx {\displaystyle \frac{{\left[(K-N+1)\gamma \right]}^2}{K(K-N+1){\gamma }^2}}={\displaystyle \frac{K-N+1}{K}}\approx {\displaystyle \frac{N-1}{K}}$$

(40)

for $N\ll K$.    □

Corollary 6.2 

(Practical Validation Threshold). For reliable $d_{\mathrm{eff}}$ estimation, we require $N/K\ge 5$ (following Ledoit & Wolf [10]). For $K=256$, this means $N\ge 1280$.

6.2. Implications for Theorem Validation

Table 3. Sample complexity requirements for validating each theorem.

thm	Primary Metric	Min $\mathit{N}/\mathit{K}$
Th. 1 (Max Entropy)	$d_{\mathrm{eff}}$, scale ratio	$\ge 5$
Th. 3 (Ego Decomp.)	Component-wise $d_{\mathrm{eff}}$	$\ge 5$
Th. 5 (Decoder Physics)	Ablation comparison	$\ge 5$
Th. 2, 4, 6	Prediction loss only	$\ge 1$

Table 3. Sample complexity requirements for validating each theorem.

thm	Primary Metric	Min $\mathit{N}/\mathit{K}$
Th. 1 (Max Entropy)	$d_{\mathrm{eff}}$, scale ratio	$\ge 5$
Th. 3 (Ego Decomp.)	Component-wise $d_{\mathrm{eff}}$	$\ge 5$
Th. 5 (Decoder Physics)	Ablation comparison	$\ge 5$
Th. 2, 4, 6	Prediction loss only	$\ge 1$

This analysis motivates our phased experimental approach: we first demonstrate training stability on limited data (Phase 1), then validate distributional theorems with sufficient samples (Phases 2–3).

Remark 6.3 

(Two $d_{\mathrm{eff}}$ Estimators). Our implementation uses two related but distinct estimators of $d_{\mathrm{eff}}$. Thecovariance estimator(used for all final ablation comparisons, matching Equation (37) exactly) forms the full $K\times K$ sample covariance with Ledoit–Wolf shrinkage [10] and computes $d_{\mathrm{eff}}$ from its eigenvalues; this estimator is subject to the $N<K$ floor of Proposition 6.1. Thediagonal estimator(used only for the fast sample-complexity sweep, evaluated at up to ten sample sizes per run) instead applies Equation (37) directly to the K per-dimension marginal variances, which can each be estimated from $N\ge 1$ samples without requiring the full covariance matrix to have rank K. The diagonal estimator is thereforenotsubject to the same $N<K$ floor, and the two should not be numerically compared across sections—only the covariance estimator is used to test Theorem 4.10.

7. Experiments

We validate Dynamic LeJEPA through a phased experimental protocol across two domains. Rather than treating limited data as a barrier, we use each phase to rigorously characterize a different aspect of the theory: measurement reliability (Phase 1), statistical power in a high-dimensional domain (Phase 2), and domain transfer with a new representational regime (Phase 3).

7.1. Motivation: The Danger of Distributional Metrics in Low-Data Regimes

A fundamental challenge in developing theoretically grounded representations is evaluation during the initial R&D cycle. Before committing to large-scale compute, practitioners must debug architectures, loss weights, and gradient flows on small, lightweight datasets (e.g., nuScenes-mini, ∼4.4GB).

During this debugging phase, the standard metric for verifying Theorem 4.1 (maximum entropy) is effective dimensionality ($d_{\mathrm{eff}}$). However, we discovered a critical flaw in applying $d_{\mathrm{eff}}$ to data-limited settings. On nuScenes-mini, a perfectly stable encoder yielded $d_{\mathrm{eff}}\approx 0.004$. Standard literature would interpret this as catastrophic dimensional collapse.

As proven in Section 6 (Proposition 6.1), this interpretation is mathematically incorrect when $N<K$. This motivates our phased experimental design:

• Phase 1 (Debugging & Metric Characterization): Verify training stability and empirically validate the $1/K$ floor of $d_{\mathrm{eff}}$ on nuScenes-mini ($N/K=1.58$).
• Phase 2 (Full Theorem Validation): Scale to nuScenes Trainval ($N/K=55.5$) to definitively verify distributional theorems with statistical power.
• Phase 3 (Cross-Domain Transfer): Validate the placement rule on robotic manipulation (MimicGen, $N/K=855$) and characterize the entropy–utility frontier on low-dimensional data.

7.2. Phase 1: Sanity Checking and Metric Characterization on nuScenes-Mini

We explicitly utilize nuScenes-mini as a lightweight, rapid-iteration debugging protocol. The goal is not to claim state-of-the-art performance, but to verify: (i) the three-component objective (Theorem 5.1) optimizes without conflicts, and (ii) Proposition 6.1 accurately predicts the behavior of distributional metrics under sample starvation.

7.2.1. Experimental Setup

Data.

We use nuScenes-mini (∼4.4GB, 10 scenes, ∼4K LiDAR-camera sweeps). Temporal sequences are constructed from consecutive frames with $\Delta t<2$ seconds. Ego-motion $e_t=(R_t,t_t)\in {\mathbb{R}}^6$ is estimated via ICP between consecutive LiDAR point clouds. After filtering, we obtain $N=404$ in-distribution evaluation frames.

Architecture.

The encoder $f_{\theta }$ is a 6-layer Vision Transformer (patch size 16, embedding dim 192) producing $z_t\in {\mathbb{R}}^{256}$. Following Theorem 3, we decompose $K_1=64$ (ego) and $K_2=192$ (static). The predictor $g_{\varphi }$ is a 3-layer MLP conditioned on ego-motion. The decoder $h_{\psi }$ implements physics-informed depth prediction (Theorem 5, Example 4.12) with LiDAR supervision.

Training.

80 epochs, AdamW (weight decay ${10}^{-5}$), learning rates ${\eta }_{\mathrm{enc}}={10}^{-4}$, ${\eta }_{\mathrm{pred}}=3\times {10}^{-4}$, ${\eta }_{\mathrm{dec}}=3\times {10}^{-4}$, loss weights ${\lambda }_{\mathrm{SIGReg}}=10$, ${\lambda }_{\mathrm{pred}}=1$, ${\lambda }_{\mathrm{physics}}=0.5$.

Key Parameter.

The sample-to-dimension ratio is $N/K=404/256=1.58$, which is below the threshold of $N/K\ge 5$ established in Corollary 6.2. We proceed to explicitly characterize the consequences.

7.2.2. Training Stability and Gradient Separation

Figure 1 shows the evolution of all loss components:

The lack of a training-validation gap indicates no overfitting, serving as a successful sanity check for the architecture.

7.2.3. Variance Matching

Figure 2 shows the per-dimension variance distribution:

The mean per-dimension variance reaches ${\overline{\sigma }}^2\approx 0.9$, close to the target ${\sigma }^2=1.0$.

7.2.4. Empirical Validation of the $d_{\mathrm{eff}}$ Floor

Figure 3 reveals the exact pattern predicted by Proposition 6.1:

Control experiment. To empirically prove this is a metric artifact, we compute $d_{\mathrm{eff}}$ on a true isotropic Gaussian $\mathcal{N}(0,I_{256})$ using the exact same $N=404$ samples:

Table 4. Control experiment: $d_{\mathrm{eff}}$ with $N=404$, $K=256$. The near-identical values confirm that low $d_{\mathrm{eff}}$ reflects sample limitation, not embedding collapse.

Distribution	${\mathit{d}}_{\mathbf{eff}}$	Scale ratio
True $\mathcal{N}(0,I_{256})$	0.0041	1.00
Dynamic LeJEPA	0.0039	0.90

Table 4. Control experiment: $d_{\mathrm{eff}}$ with $N=404$, $K=256$. The near-identical values confirm that low $d_{\mathrm{eff}}$ reflects sample limitation, not embedding collapse.

Distribution	${\mathit{d}}_{\mathbf{eff}}$	Scale ratio
True $\mathcal{N}(0,I_{256})$	0.0041	1.00
Dynamic LeJEPA	0.0039	0.90

7.2.5. Task Performance Under Distributional Uncertainty

Figure 4 shows depth predictions from the physics-informed decoder:

7.2.6. Theorem 5 Ablation: Demonstrating Statistical Boundaries

Figure 5 shows the ablation attempt with 5 seeds:

Table 5. Phase 1 (nuScenes-mini) debugging outcomes. ✔: demonstrated; ≈: partial; —: bounded by $N/K$ limit.

Property	Status	Evidence
Training stability (C4)	✔	Figure 1
Variance matching	≈	scale ratio $=0.90$
$d_{\mathrm{eff}}$ floor validation	✔	Table 4
Isotropy proof	—	Requires $N/K\ge 5$
Depth decoding (C3)	✔	Figure 4
Theorem 5 ablation	—	Requires $N/K\ge 5$

Table 5. Phase 1 (nuScenes-mini) debugging outcomes. ✔: demonstrated; ≈: partial; —: bounded by $N/K$ limit.

Property	Status	Evidence
Training stability (C4)	✔	Figure 1
Variance matching	≈	scale ratio $=0.90$
$d_{\mathrm{eff}}$ floor validation	✔	Table 4
Isotropy proof	—	Requires $N/K\ge 5$
Depth decoding (C3)	✔	Figure 4
Theorem 5 ablation	—	Requires $N/K\ge 5$

7.3. Phase 2: Large-Sample Ablation and Cross-Scene Validation on nuScenes Trainval

Building directly on the sample-complexity boundary established in Phase 1 (Corollary 6.2: $N/K\ge 5$), Phase 2 scales sample count to a regime where distributional theorems can be tested with statistical power, rather than merely characterized as under-determined.

7.3.1. Experimental Setup

Data.

We use nuScenes Trainval Part 1 (41.24 GB) [12], an order of magnitude larger by archive size than nuScenes-mini. After scene-based filtering (11 scenes with complete camera–LiDAR–ego-pose data), we obtain a 70/30 scene-level split of 7 train scenes (14,197 image pairs) and 4 held-out validation scenes (5,263 image pairs). This yields $N/K_{\mathrm{train}}=55.5$ and $N/K_{\mathrm{val}}=20.6$—both far above the $N/K\ge 5$ threshold, and far beyond Phase 1’s $N/K=1.58$.

Architecture.

Identical encoder/predictor/decoder design to Phase 1 ($K=256$, decomposed as $K_1=64$ ego / $K_2=192$ static per Theorem 4.7), with two corrections identified during Phase 1 debugging: (i) the SIGReg loss weight was reduced from ${\lambda }_{\mathrm{SIGReg}}=100$ to ${\lambda }_{\mathrm{SIGReg}}=1$, since the original value produced an effective ${10}^5\times$ over-weighting of some projection terms; and (ii) the physics loss weight was increased from ${\lambda }_{\mathrm{physics}}=0.5$ to ${\lambda }_{\mathrm{physics}}=10$, since 0.5 was too weak to produce a measurable effect when the physics constraint was (deliberately, for the ablation) attached to the encoder rather than the decoder.

Training and Ablation Design.

We train for 30 epochs per run (AdamW, ${\eta }_{\mathrm{enc}}={10}^{-4}$, ${\eta }_{\mathrm{pred}}={\eta }_{\mathrm{dec}}=3\times {10}^{-4}$, weight decay ${10}^{-5}$, batch size 32) across three ablation conditions directly instantiating Theorem 4.10 and Corollary 4.11: no_physics (baseline, SIGReg only), decoder_physics (edge-aware depth smoothness, Equation (31), applied at the decoder), and encoder_physics (the same constraint applied directly to the encoder output, which Corollary 4.11 predicts should be harmful). Each condition is trained with 10 random seeds, for $3\times 10=30$ total runs, enabling a paired Wilcoxon signed-rank test across seeds.

7.3.2. Ablation Results: Testing Theorem 5

Table 6 reports the covariance-based $d_{\mathrm{eff}}$, H-ratio, scale ratio, and variance dispersion across all 30 seeded runs.

Table 7 reports the paired Wilcoxon signed-rank tests across the 10 shared seeds.

7.3.3. Scale-vs-Structure Decomposition

Define the scale gap and $d_{\mathrm{eff}}$ gap of an ablation condition c relative to baseline as ${\Delta }_{\mathrm{scale}}\left(c\right)=\mathrm{scale}\mathrm{ratio}(\mathtt{no}\_\mathtt{physics})-\mathrm{scale}\mathrm{ratio}\left(c\right)$ and ${\Delta }_{d_{\mathrm{eff}}}\left(c\right)=d_{\mathrm{eff}}(\mathtt{no}\_\mathtt{physics})-d_{\mathrm{eff}}\left(c\right)$. From Table 6:

$$\begin{array}{cc}\hfill {\Delta }_{\mathrm{scale}}\left(\mathrm{decoder}\right)& =0.304,{\Delta }_{\mathrm{scale}}\left(\mathrm{encoder}\right)=0.284\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\Delta }_{d_{\mathrm{eff}}}\left(\mathrm{decoder}\right)& =0.018,{\Delta }_{d_{\mathrm{eff}}}\left(\mathrm{encoder}\right)=0.046\hfill \end{array}$$

(42)

The two conditions lose a nearly identical amount of variance scale (0.020 difference), but the encoder condition’s $d_{\mathrm{eff}}$ gap is $2.6\times$ larger than the decoder condition’s. We define the residual as a structure violation:

$${\Delta }_{\mathrm{structure}}\triangleq {\Delta }_{d_{\mathrm{eff}}}\left(\mathrm{encoder}\right)-{\Delta }_{d_{\mathrm{eff}}}\left(\mathrm{decoder}\right)=0.028$$

(43)

Because $d_{\mathrm{eff}}$ measures the relative uniformity of the eigenvalue spectrum—not its absolute magnitude—a proportional shrinkage of all eigenvalues by the same factor leaves $d_{\mathrm{eff}}$ unchanged. What Theorem 4.10 protects is the encoder’s isotropy, not its absolute scale: the ${\Delta }_{\mathrm{structure}}=0.028$ gap isolates the genuine, non-proportional distortion of the eigenvalue spectrum, and this distortion is present only when the physics constraint acts on the encoder directly.

7.3.4. Sample Complexity Revalidation

Figure 6 repeats the Proposition 6.1 sanity check from Phase 1 at Phase 2 scale.

7.3.5. Cross-Scene Generalization

To test whether the trained encoder’s distributional properties are an artifact of aggregating the validation set, we compute $d_{\mathrm{eff}}$, H-ratio, and scale ratio separately for each of the 4 held-out validation scenes.

Table 8. Cross-scene statistics, 4 held-out validation scenes.

Metric	Mean ± Std
$d_{\mathrm{eff}}$	0.937 ± 0.012
H-ratio	0.844 ± 0.039
scale ratio	0.668 ± 0.079

Table 8. Cross-scene statistics, 4 held-out validation scenes.

Metric	Mean ± Std
$d_{\mathrm{eff}}$	0.937 ± 0.012
H-ratio	0.844 ± 0.039
scale ratio	0.668 ± 0.079

7.4. Phase 3: Robotic Manipulation on MimicGen

Building on the powered ablation protocol of Phase 2, Phase 3 transfers the validation to robotic manipulation—the domain Theorem 4.4 was designed for—and characterizes a new representational regime that the driving data of Phase 2 could not expose. Where the nuScenes imagery fills hundreds of independent dimensions, the threading task’s ∼10-dimensional manifold reveals an entropy–utility frontier that fundamentally changes what the encoder can achieve and how the physics-placement rule manifests.

Figure 7. Cross-scene generalization on 4 held-out validation scenes. Per-scene $d_{\mathrm{eff}}$ distributions are tightly clustered around the mean and far from the collapse threshold.

Figure 7. Cross-scene generalization on 4 held-out validation scenes. Per-scene $d_{\mathrm{eff}}$ distributions are tightly clustered around the mean and far from the collapse threshold.

7.4.1. Experimental Setup

Data.

We use the MimicGen two-arm-threading task [11]: 1,025 demonstrations of a bimanual threading task (14-DOF actions: 7 per arm, operational-space control), 218,858 frames at $84\times 84\times 3$ resolution from a third-person camera. Each demonstration contains 32-dimensional proprioception (joint positions, end-effector pose and quaternion, gripper state) for both arms. The task manifold is low-dimensional ($\sim 10$ effective DOF), which—as we show in Section 7.4.5—fundamentally changes the entropy landscape. We perform a demonstration-level 70/30 split (717 train / 308 validation demonstrations, fixed seed for paired-seed comparisons), yielding $N/K_{\mathrm{eval}}=78$ for distributional metrics.

Architecture.

The encoder $f_{\theta }$ is a 6-layer Vision Transformer (patch size 14, embedding dimension 192) processing the agentview image, concatenated with 32-dimensional proprioception, producing $z_t\in {\mathbb{R}}^{256}$. The predictor $g_{\varphi }$ is a 3-layer MLP conditioned on the 14-DOF action. The decoder $h_{\psi }$ is a 3-layer MLP reconstructing the 32-dimensional proprio signal. The physics constraint implements action-kinematics consistency: bounded-acceleration temporal smoothness and calibrated forward kinematics $\parallel ({\widehat{q}}_{t+1}-{\widehat{q}}_t)-W_{\mathrm{cal}}{a}_t{\parallel }^2$, where $W_{\mathrm{cal}}$ is a fixed least-squares map from actions to end-effector displacements ($R^2=0.788$) fit on training data only.

Training and Ablation.

Identical protocol to Phase 2: three conditions (no_physics, decoder_physics, encoder_physics), 10 seeds each, paired across conditions. 15 epochs per run. The SIGReg entropy weight was pre-registered at $\lambda =20$ from a sweep conducted before any ablation run. A budget replication at 30 epochs tests robustness. The full pipeline is deterministic: eight bit-identical seed-level reproductions were verified across sessions.

7.4.2. The Correlated-Collapse Discovery and SIGReg⁺

Our initial Phase 3 runs using the Phase 2 SIGReg exposed a failure mode invisible in the driving domain. The encoder converged to correlated collapse: per-dimension variances matched ${\sigma }^2$ exactly (scale $=1.0$, H-ratio $=1.0$) while the joint covariance had effective rank $\approx 6/256$ ($d_{\mathrm{eff}}=0.022$). On the high-dimensional nuScenes imagery, matching marginals implicitly produced a near-isotropic joint; on MimicGen’s $\sim 10$-dimensional task manifold, the encoder satisfied the marginal constraints while concentrating all variance in a rank-6 subspace.

We resolve this by reinforcing SIGReg’s joint-level terms—applied identically to all three ablation conditions—with (i) a covariance off-diagonal penalty, (ii) denser correlation sampling (500 → 2048 pairs), and (iii) projection weight normalization. This SIGReg⁺ restores the joint structure ($d_{\mathrm{eff}}$ from 0.022 to 0.18) while preserving marginal entropy. The discovery itself is a contribution: per-dimension variance matching is insufficient on low-intrinsic-dimension data; covariance-level structure must be enforced explicitly.

7.4.3. Ablation Results: Testing Theorem 5 in Robotics

Table 9. Phase 3 ablation on MimicGen (mean ± std over 10 seeds, covariance $d_{\mathrm{eff}}$, $N/K=78$).

Metric	no_phys.	dec._phys.	enc._phys.
$d_{\mathrm{eff}}$	0.1846	0.1740	0.1380
	±0.004	±0.003	±0.004
H-ratio	0.980	0.980	0.978
	±0.002	±0.001	±0.003
scale ratio	0.946	0.946	0.940
	±0.004	±0.003	±0.007
pred $R^2$	0.784	0.864	0.966
probe $R^2$ (action)	0.412	0.449	0.518
probe $R^2$ (proprio)	0.937	0.961	0.984

Table 9. Phase 3 ablation on MimicGen (mean ± std over 10 seeds, covariance $d_{\mathrm{eff}}$, $N/K=78$).

Metric	no_phys.	dec._phys.	enc._phys.
$d_{\mathrm{eff}}$	0.1846	0.1740	0.1380
	±0.004	±0.003	±0.004
H-ratio	0.980	0.980	0.978
	±0.002	±0.001	±0.003
scale ratio	0.946	0.946	0.940
	±0.004	±0.003	±0.007
pred $R^2$	0.784	0.864	0.966
probe $R^2$ (action)	0.412	0.449	0.518
probe $R^2$ (proprio)	0.937	0.961	0.984

Table 10. Paired Wilcoxon signed-rank tests on $d_{\mathrm{eff}}$, Phase 3 ($n=10$ paired seeds, Holm-corrected).

Comparison	p	${\mathit{p}}_{\mathbf{Holm}}$	$\mathbf{\Delta }{\mathit{d}}_{\mathbf{eff}}$
dec. vs. none	0.0020	0.0059	+0.0105
enc. vs. none	0.0020	0.0059	+0.0466
enc. vs. dec.	0.0020	0.0059	−0.0361

Table 10. Paired Wilcoxon signed-rank tests on $d_{\mathrm{eff}}$, Phase 3 ($n=10$ paired seeds, Holm-corrected).

Comparison	p	${\mathit{p}}_{\mathbf{Holm}}$	$\mathbf{\Delta }{\mathit{d}}_{\mathbf{eff}}$
dec. vs. none	0.0020	0.0059	+0.0105
enc. vs. none	0.0020	0.0059	+0.0466
enc. vs. dec.	0.0020	0.0059	−0.0361

The two-sided signature is unambiguous: encoder-level physics simultaneously destroys entropy ($d_{\mathrm{eff}}$: $0.185\to 0.138$) and inflates predictability (pred $R^2$: $0.78\to 0.97$)—the encoder is dragged toward the ≤14-dimensional image of the action space. Decoder-level physics is exactly scale-neutral: ${\Delta }_{\mathrm{scale}}=-0.0002$ at $p=1.0$. The gap ratio is $\mathbf{4}.\mathbf{43}\times$—sharper than Phase 2’s $2.6\times$.

7.4.4. Spectral Evidence

The spectral decomposition reveals a signal peak ($\sim 20$ directions at 5–$9{\sigma }^2$) atop a noise plateau; the encoder-physics constraint compresses the plateau by $-30\%$ while concentrating the top-10 share from 30.8% to 38.7%—the structure violation visible directly in eigenvalues.

Figure 8. Embedding covariance eigenspectra on MimicGen (15 epochs, seed 0, $N=20,000$; log-scale). The encoder-physics model (red) exhibits a lower noise plateau (0.320 vs. 0.460) and higher top-direction concentration (38.7% vs. 30.8%)— the spectral signature of Corollary 4.11.

Figure 8. Embedding covariance eigenspectra on MimicGen (15 epochs, seed 0, $N=20,000$; log-scale). The encoder-physics model (red) exhibits a lower noise plateau (0.320 vs. 0.460) and higher top-direction concentration (38.7% vs. 30.8%)— the spectral signature of Corollary 4.11.

Figure 9. Phase 3 ablation on MimicGen ($N/K=78$, 10 seeds per condition, covariance $d_{\mathrm{eff}}$). Left:$d_{\mathrm{eff}}$ boxplots per condition with complete seed separation. Center: prediction $R^2$ and probe $R^2$ per condition—the two-sided Corollary 4.11 signature. Right: multi-step rollout error vs. horizon.

Figure 9. Phase 3 ablation on MimicGen ($N/K=78$, 10 seeds per condition, covariance $d_{\mathrm{eff}}$). Left:$d_{\mathrm{eff}}$ boxplots per condition with complete seed separation. Center: prediction $R^2$ and probe $R^2$ per condition—the two-sided Corollary 4.11 signature. Right: multi-step rollout error vs. horizon.

7.4.5. The Entropy–Utility Frontier

Phase 3 exposes a representational regime that Phases 1–2 could not: the entropy ceiling is data-limited. On nuScenes’ high-dimensional imagery, a deterministic encoder fills $K=256$ independent directions with real signal variation ($d_{\mathrm{eff}}=0.99$). On MimicGen’s $\sim 10$-dimensional task manifold, the extra directions must come from amplified input noise—and the prediction and reconstruction losses actively suppress that noise. A $\lambda$-sweep revealed logarithmic saturation of $d_{\mathrm{eff}}$ in $\lambda$, and a budget replication showed that doubling the training budget raised $d_{\mathrm{eff}}$ by 3.7× more than doubling λ: optimization budget, not regularization weight, is the binding entropy constraint.

Figure 10. Sample complexity on MimicGen (decoder-physics model). The covariance $d_{\mathrm{eff}}$ stabilizes at $N/K\ge 5$, validating Corollary 6.2 at Phase 3 scale.

Figure 10. Sample complexity on MimicGen (decoder-physics model). The covariance $d_{\mathrm{eff}}$ stabilizes at $N/K\ge 5$, validating Corollary 6.2 at Phase 3 scale.

7.4.6. Budget Replication: The Violation Deepens

At 30 epochs, the encoder-physics gap grows from 0.047 to 0.070 ($+\mathbf{49}\%$), and the scale/H-ratio differences vanish entirely. The placement rule’s effect is budget-monotone and purely structural: a transient artifact would shrink with budget, not grow.

Figure 11. Budget replication: Phase 3 ablation at 15 vs. 30 epochs (10 seeds per arm per budget). The encoder-physics violation deepens with budget (0.047 → 0.070, $+49\%$), ruling out a transient training artifact. All scale and marginal-entropy differences vanish at 30 epochs (Holm n.s.).

Figure 11. Budget replication: Phase 3 ablation at 15 vs. 30 epochs (10 seeds per arm per budget). The encoder-physics violation deepens with budget (0.047 → 0.070, $+49\%$), ruling out a transient training artifact. All scale and marginal-entropy differences vanish at 30 epochs (Holm n.s.).

7.4.7. Cross-Task Transfer

Applying the trained encoders to three foreign MimicGen tasks reveals embeddings collapse on all foreign tasks ($d_{\mathrm{eff}}$ 0.035–0.088 vs. 0.185 in-task). A proprio-ablation control confirms the collapse is driven by visual domain shift. Notably, the physics-placement ordering persists cross-task: encoder-physics degrades 2.1–3.0× more than the baseline on every foreign task. The max-entropy property is distribution-specific; the placement rule is not.

7.4.8. Toward Closed-Loop Validation: Latent Planning and the Model-Exploitation Confound

Every result so far evaluates the placement rule through representation-quality metrics ($d_{\mathrm{eff}}$, H-ratio, probe $R^2$). None of them test whether the resulting representation is actually useful for the thing a world model is built to do: support goal-directed action selection. We probe this directly with a latent cross-entropy-method (CEM) planner [13] built on the frozen, trained Phase 3 encoder–predictor–decoder triples, in the style of model-predictive control with learned dynamics [14].

Setup.

At a randomly sampled anchor frame $t_0$ in a held-out demonstration, we encode $z_{t_0}=f_{\theta }(x_{t_0},q_{t_0})$ and define a horizon-H goal as the ground-truth recorded end-effector (EEF) pose at $t_0+H$ (not a model-generated proxy; see below). CEM searches over H-step action sequences $a_{t_0:t_0+H-1}\in {\mathbb{R}}^{H\times 14}$, rolling each candidate through the frozen predictor $g_{\varphi }$ and scoring it by decoded-EEF distance to the goal at every step, iterating the sampling distribution toward the elite candidates. We use $H=8$, a population of 256 candidates, 4 CEM iterations, an elite set of 32, and 200 anchors per model, evaluated on the three seeds regenerated in Section 7.4.1’s deterministic checkpoint pipeline. Two baselines calibrate the result: real-action replay (roll the predictor forward using the true recorded actions instead of CEM’s search—the ground-truth solution, and hence an expected lower bound on achievable error) and a random-action rollout (an upper bound).

An earlier version of this evaluation defined the goal as the model’s own decoded reconstruction of the future frame, $h_{\psi }\left(f_{\theta }\left(x_{t_0+H}\right)\right)$, rather than the true recorded pose. That design is self-referential—it lets CEM optimize against a target generated by the very model being scored—and produced CEM errors below real-action replay for every ablation condition, which is not possible for a genuine planning task, since replaying the true actions is itself the ground-truth solution. We report the corrected, ground-truth-goal version throughout; the self-referential version is documented here only to flag the failure mode for others building similar offline evaluations.

The confound.

Table 11 and Figure 12 show CEM error below the real-action replay floor for all three modes even after the goal-leakage fix above. Since replay executes the literal ground-truth action sequence, no planner can legitimately do better against the true future state; a lower CEM error therefore indicates that CEM is finding action sequences that reduce the learned decoder’s own cost without being genuinely closer to the truth than the recorded trajectory—a well-documented failure mode in model-based control known as objective mismatch [15]: an optimizer with unrestricted access to a learned model’s cost surface will exploit that surface’s imperfections rather than solve the underlying task, and nothing in this offline evaluation can distinguish the two without executing the plan in the real environment.

Exploitation diagnostic.

To test this directly, we measure the $L_2$ distance between CEM’s chosen action sequence and the true recorded actions at the same anchor, normalized by the per-step action-space range ($\parallel a_{max}-a_{min}{\parallel }_2=6.298$ across the 14 action dimensions).

Table 12. CEM action distance from the true recorded trajectory (mean ± std over 3 seeds), as a fraction of the full per-step action range ($\parallel a_{max}-a_{min}{\parallel }_2=6.298$).

Mode	Action distance	% of range
no_phys.	1.615 ± 0.020	25.6%
dec._phys.	1.689 ± 0.003	26.8%
enc._phys.	1.622 ± 0.016	25.8%

Table 12. CEM action distance from the true recorded trajectory (mean ± std over 3 seeds), as a fraction of the full per-step action range ($\parallel a_{max}-a_{min}{\parallel }_2=6.298$).

Mode	Action distance	% of range
no_phys.	1.615 ± 0.020	25.6%
dec._phys.	1.689 ± 0.003	26.8%
enc._phys.	1.622 ± 0.016	25.8%

CEM’s chosen actions sit roughly a quarter of the full action range away from the ground truth—not a rounding error—while still scoring lower on-model cost than replaying the true actions. Critically, this gap is uniform across all three ablation conditions (25.6–26.8%, well within seed noise), which rules out the hypothesis that entropy collapse specifically makes encoder-physics easier to exploit: all three learned decoders are exploitable to essentially the same degree. This is itself informative. It means the ranking in Table 11 (encoder_physics < decoder_physics < no_physics on CEM error) most plausibly reflects the same effective-dimensionality geometry already documented in Table 9—a lower-$d_{\mathrm{eff}}$ decoder cost surface is easier to drive down per unit of action-space deviation, independent of whether the resulting trajectory is any more physically valid—rather than a genuine control-quality advantage for any one condition.

What this does and does not establish.

We therefore do not interpret Table 11 as evidence that decoder-physics improves downstream control: the exploitation confound is uniform enough across conditions that it cannot be ruled out as the dominant driver of the ranking, and nothing here has been checked against a live environment. What we do establish is (i) a reproducible demonstration that offline latent planning against a learned decoder can produce misleadingly favorable numbers for any ablation condition, and (ii) a simple diagnostic—action-space distance from ground truth, relative to replay—that practitioners can and should run before treating an offline planning result as evidence of control competence. Closing this gap rigorously requires executing the CEM-selected action sequences in a live simulator (e.g., robosuite/MimicGen) and measuring true task success; we identify this as the concrete next step toward a fully closed-loop validation of the placement rule and leave it to future work.

7.4.9. Phase 3 Summary

Table 13. Phase 3 (MimicGen robotics) validation outcomes.

Property	Status	Evidence
Thm 2(a) max entropy	≈	frontier regime
Thm 2(b) $g^{*}\approx \mathbb{E}[·]$	✔	$R^2=0.80$–$0.99$
Thm 2(c) task info	✔	probes 0.41 / 0.94
Thm 2(d) action inv.	≈	mode-independent
Thm 5 / Cor. IV.11	✔	Table 9 and Table 10
Budget robustness	✔	Figure 11
Cross-task ordering	✔	Section 7.4.7
Closed-loop planning	× confounded	Section 7.4.8

Table 13. Phase 3 (MimicGen robotics) validation outcomes.

Property	Status	Evidence
Thm 2(a) max entropy	≈	frontier regime
Thm 2(b) $g^{*}\approx \mathbb{E}[·]$	✔	$R^2=0.80$–$0.99$
Thm 2(c) task info	✔	probes 0.41 / 0.94
Thm 2(d) action inv.	≈	mode-independent
Thm 5 / Cor. IV.11	✔	Table 9 and Table 10
Budget robustness	✔	Figure 11
Cross-task ordering	✔	Section 7.4.7
Closed-loop planning	× confounded	Section 7.4.8

8. Discussion

8.1. Why Prior Physics-Informed JEPAs Failed

• Pixel-space constraints violate Theorem 4.10 because they implicitly constrain the encoder through decoder gradients.
• Latent-space constraints directly violate Theorem 4.1 by constraining the embedding distribution.
• Both approaches reduce entropy below optimal, increasing worst-case downstream risk.

8.2. Cross-Domain Synthesis: What Three Phases Established

The three-phase experimental arc mirrors how theoretical claims should be validated: Phase 1 exposed a measurement artifact that would have invalidated all downstream conclusions; Phase 2 provided the first powered test in the high-dimensional driving domain; Phase 3 transferred the test to robotics and discovered two phenomena invisible in driving.

Table 14. Cross-domain synthesis of the physics-placement rule.

Property	Phase 2	Phase 3
$d_{\mathrm{eff}}$ (unconstrained)	0.986	0.185
$d_{\mathrm{eff}}$ (encoder-phys)	0.939	0.138
Gap ratio	$2.6\times$	$4.4\times$
Wilcoxon p	≤0.002	0.002
Scale effect	$-0.28$	$-0.006$ (n.s.)
Data intrinsic dim.	high	$\sim 10$
$d_{\mathrm{eff}}$ achievable	0.99	frontier-limited
Failure signature	scale + structure	purely structural
Budget robustness	—	$+49\%$ (30ep)
Cross-task ordering	—	holds (2–3×)

Table 14. Cross-domain synthesis of the physics-placement rule.

Property	Phase 2	Phase 3
$d_{\mathrm{eff}}$ (unconstrained)	0.986	0.185
$d_{\mathrm{eff}}$ (encoder-phys)	0.939	0.138
Gap ratio	$2.6\times$	$4.4\times$
Wilcoxon p	≤0.002	0.002
Scale effect	$-0.28$	$-0.006$ (n.s.)
Data intrinsic dim.	high	$\sim 10$
$d_{\mathrm{eff}}$ achievable	0.99	frontier-limited
Failure signature	scale + structure	purely structural
Budget robustness	—	$+49\%$ (30ep)
Cross-task ordering	—	holds (2–3×)

The driving domain, with its high-dimensional imagery, allows the encoder to satisfy the maximum-entropy target almost exactly—and the placement rule manifests as a scale-vs-structure decomposition. The robotics domain, with its $\sim 10$-dimensional task manifold, sits at an entropy frontier where the placement rule’s effect is purely structural and sharper ($4.4\times$ vs. $2.6\times$). The rule holds in both regimes; its signature differs.

8.3. Practical Design Recommendations

1.

Never apply physics constraints to the encoder output z.

2.

Always place physics at the decoder.

3.

The predictor can use any architecture without affecting the encoder’s guarantees.

4.

On low-dimensional data, monitor the covariance eigenvalue spectrum, not just per-dimension variance.

5.

Optimization budget, not regularization weight, is the primary lever for entropy on low-dimensional data.

6.

Never treat an offline planning result as evidence of control competence without checking it against a real-action-replay floor and an action-space exploitation diagnostic (Sec. 7.4.8); an optimizer with unrestricted access to a learned model’s cost surface will exploit it.

8.4. Limitations and Toward Closed-Loop Validation

The representation-quality claims in this paper—Theorem 5, Corollary IV.11, and their empirical validation across Phases 1–3—rest on statistically powered, ground-truth-referenced metrics ($d_{\mathrm{eff}}$, H-ratio, Wilcoxon tests on 10 paired seeds) and we consider them solidly established. The closed-loop planning probe in Sec. 7.4.8 is different in kind: it was designed to test whether the placement rule matters for downstream control, and it returned a negative methodological result rather than a positive control-quality one. We were careful not to round this off into a headline claim. Two things are true simultaneously: (i) the CEM-based planning error ordering across ablation conditions is directionally consistent with the representation-quality ordering already established via $d_{\mathrm{eff}}$, and (ii) the same ordering can be fully explained by uniform model exploitation rather than genuine differences in control competence, and our data cannot distinguish the two possibilities without a live environment. We report both facts rather than the more publishable-sounding half of them.

We see this as valuable precisely because it is not the result we set out to find. Offline evaluation of learned world models is common practice, and the specific failure mode demonstrated here—a planner exploiting a learned decoder’s cost surface to score better than the ground-truth trajectory itself—is easy to miss without an explicit replay-floor check of the kind we introduce in Sec. 7.4.8. We consider the diagnostic itself (real-action replay as a floor, plus normalized action-space distance from ground truth as an exploitation signal) to be of independent, general use for anyone evaluating a JEPA-style or other learned world model’s control-readiness offline. The concrete next step for Dynamic LeJEPA specifically is executing the CEM-selected action sequences in a live simulator (robosuite/MimicGen) and measuring true task success under each ablation condition—at that point, and only at that point, would a claim that the placement rule improves downstream control be justified.

9. Conclusions

We proved through six theorems that prediction and representation learning are strictly decoupled in Joint-Embedding Predictive Architectures, yielding a world-model design principle with provable guarantees: encoder maximum-entropy, predictor dynamics, decoder physics. We validated this principle across two domains through 60 seeded runs, demonstrating that the placement rule is statistically significant in every comparison, deepens with training budget, and extends to out-of-distribution tasks. Phase 3 additionally discovered the correlated collapse failure mode— invisible on high-dimensional data—and characterized the entropy–utility frontier governing deterministic encoders on low-dimensional task manifolds.

We additionally probed whether the placement rule matters beyond representation quality, building a latent CEM planner on the trained Phase 3 models and evaluating it against ground truth rather than a model-generated proxy. This probe did not yield a clean downstream-control win: a model-exploitation diagnostic showed that offline planning against a learned decoder inflates apparent accuracy uniformly across all three ablation conditions, meaning the observed planning-error ordering cannot presently be attributed to genuine control competence rather than exploit of the decoder’s cost surface. We report this as a methodological contribution in its own right—a diagnostic that any offline evaluation of a learned world model’s control-readiness should run before trusting its numbers—and as the concrete open problem separating Dynamic LeJEPA’s current, representation-level guarantees from a fully validated, closed-loop world model.

The broader significance: Dynamic LeJEPA establishes that the intuition to inject domain knowledge into world models was correct—the target was simply wrong. Physics does not belong in the encoder, where it destroys the representational guarantee; it belongs in the decoder, where it is provably free. Whether that guarantee translates into better closed-loop control remains open, and we have shown precisely why answering it requires more than an offline planner and a learned cost function.

All code, trained checkpoints, ablation results, and an interactive demo accompanying this paper are publicly released (see Code, Data, and Demo Availability above) so that both the placement-rule results and the planning confound can be independently reproduced and, we hope, extended toward the closed-loop validation this work identifies as still missing.