Temporal-Distance JEPA: Plan-Aware Representation Learning for Latent World Model Predictive Control

# Temporal-Distance JEPA: Plan-Aware Representation Learning for Latent World Model Predictive Control
Source: [https://arxiv.org/html/2607.25337](https://arxiv.org/html/2607.25337)
andJiaxuan XiongHong Kong Baptist UniversityHong KongChina

###### Abstract\.

Joint\-Embedding Predictive Architectures \(JEPAs\) learn world models by predicting in representation space rather than reconstructing pixels, making them a natural backbone for latent model predictive control from offline demonstration logs\. JEPA\-style training optimizes short\-horizon latent prediction, whereas planning requires a multi\-step ranking of imagined futures by goal progress\. Prior JEPA planners often inherit that ranking from embedding geometry, typically latent Euclidean distance, which arises as a byproduct of representation learning rather than as a progress cost mined from the logs\. We proposetemporal\-distance JEPA \(TD\-JEPA\), which retains the LeWM encoder–predictor backbone and mines a directed temporal cost from reward\-free trajectories: same\-trajectory step order supplies positive targets, cross\-trajectory pairs act as heuristic negatives, and a rollout\-consistency term matches the planner horizon\. The mined supervision serves two roles: as the deployed planning cost when progress is topological, and as a representation signal that improves Euclidean planning when contact geometry dominates\. Under locked evaluation, deploying the mined cost raises Two\-Room success to100\.0%100\.0\\%versus LeWM’s97\.4%97\.4\\%, while shared Euclidean planning on the same temporally trained checkpoint raises OGB\-Cube by14\.214\.2points over LeWM and improves Push\-T\. Against LeWM and the concurrent RC\-aux baseline under locked evaluation, TD\-JEPA matches or exceeds both methods on every environment\. Ablations show that the directed head, cross\-trajectory negatives, and rollout consistency each contribute\. TD\-JEPA narrows the train–plan gap for JEPA world\-model planners by discovering temporal progress structure in offline logs and co\-designing cost form with plan\-time deployment\. Code is available at[https://github\.com/HKBU\-KnowComp/TD\-JEPA](https://github.com/HKBU-KnowComp/TD-JEPA)\.

††copyright:none## 1\.Introduction

World models aim to capture enough structure about an environment that an agent can reason about futures without acting in the real world\(Li,[2026](https://arxiv.org/html/2607.25337#bib.bib3)\)\. In the planner\-facing role that motivates this paper, a world model encodes observations, imagines the consequences of candidate action sequences, and ranks those imagined futures by how close they bring the agent to a goal\. When the only available supervision is offline demonstration logs—observation–action trajectories without dense rewards—predictive dynamics alone do not yield a progress signal that the planner can trust\.

Joint\-Embedding Predictive Architectures \(JEPAs\) provide a natural foundation for this setting\(LeCun,[2022](https://arxiv.org/html/2607.25337#bib.bib2); Balestriero and LeCun,[2025](https://arxiv.org/html/2607.25337#bib.bib8)\)\. Rather than reconstructing pixels, a JEPA learns an encoder that maps observations into a latent space and a predictor that forecasts future embeddings from context and actions\. Useful world knowledge resides in the predictive structure of representations: if the model can anticipate what will happen next in embedding space, it has internalized aspects of the world’s dynamics without committing to a generative decoder\. LeWM instantiates this recipe for latent control\(Maeset al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib7)\)\. It trains an action\-conditioned encoder–predictor with next\-latent prediction, regularizes the representation with SIGReg to avoid collapse\(Balestriero and LeCun,[2025](https://arxiv.org/html/2607.25337#bib.bib8)\), and at test time searches over action sequences with CEM\(de Boeret al\.,[2005](https://arxiv.org/html/2607.25337#bib.bib29)\), scoring each imagined terminal latent by Euclidean distance to a goal embedding\.

This design yields a strong predictive backbone, yet it leaves a structural gap between what is learned and what planning needs\. JEPA\-style training optimizes short\-horizon prediction in latent space; latent model predictive control \(MPC\) instead needs a multi\-step ranking of futures by goal progress\. In LeWM and related JEPA planners, that ranking is typically inherited from embedding geometry: Euclidean distance between latents becomes the de facto planning cost\. Geometry can correlate with progress when local appearance and control advance together, but it is not trained to reflect steps\-to\-goal, directed reachability, or planner ranking\. Concurrent planning\-aware JEPA variants address pieces of this mismatch: value shaping with offline RL\(Destradeet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib21)\), temporal straightening of latent paths\(Wanget al\.,[2026b](https://arxiv.org/html/2607.25337#bib.bib22)\), horizon\-conditioned reachability auxiliaries\(Liet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib24)\), and progression subspaces\(Thilet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib23)\)\. These approaches either introduce value estimators beyond the demonstration logs, reshape geometry without an explicit plan cost, or treat temporal structure as an auxiliary rather than as a discoverable ranking signal for MPC\.

We therefore ask:*how can reward\-free demonstration trajectories yield temporal progress structure that narrows the train–plan gap for JEPA world\-model planners, and when should that structure be deployed as the plan cost versus used to shape representations for geometric planning?*Figure[1](https://arxiv.org/html/2607.25337#S1.F1)makes the ranking side of this gap concrete on Push\-T\. On held\-out demonstration pairs, LeWM’s latent Euclidean distance correlates with temporal separation at Spearmanρ=0\.65\\rho=0\.65, whereas a cost mined from the same logs tracks step order atρ=0\.95\\rho=0\.95\. Whether that mined signal also improves control—and under which plan\-time cost—is the empirical question addressed below\.

![Refer to caption](https://arxiv.org/html/2607.25337v1/x1.png)Figure 1\.Push\-T temporal\-gap alignment on held\-out demonstration pairs\.Two side\-by\-side scatter plots with x\-axis as true temporal distance and independent per\-panel y\-axes\. Left: LeWM latent L2 \(range 0–25\) is broadly scattered and weakly monotonic, with a shallow red linear\-fit line and Spearman rho 0\.65\. Right: TD\-JEPA learned d\-psi \(range 0–8\) is tightly increasing along a red linear\-fit line with Spearman rho 0\.95; no y=x reference is drawn because steps and learned energy do not share units\.We proposetemporal\-distance JEPA \(TD\-JEPA\)to address this challenge\. TD\-JEPA keeps the LeWM encoder–predictor and SIGReg backbone, and mines a goal\-conditioned energy that is low when a rolled\-out latent is compatible with reaching the goal\(LeCunet al\.,[2006](https://arxiv.org/html/2607.25337#bib.bib1)\)\. Because goal reaching under actions is generally asymmetric, the cost is directed rather than Euclidean\(Wanget al\.,[2023](https://arxiv.org/html/2607.25337#bib.bib9)\): it depends on the ordered pair of current and goal latents, so traveling fromAAtowardBBneed not cost the same as the reverse\. Supervision comes from trajectory mining rather than rewards\. Same\-trajectory step order supplies positive temporal targets; cross\-trajectory pairs act as heuristic negatives that discourage spuriously low off\-trajectory costs; and a rollout\-consistency term aligns the learned cost with the open\-loop horizon used by the planner \(Fig\.[2](https://arxiv.org/html/2607.25337#S3.F2)\)\.

The mined signal plays two roles, and plan\-time cost selection follows task structure\. When progress is mainly topological, as in navigation and reaching, the directed temporal cost is the natural planning objective\. When contact geometry dominates, the same temporally trained checkpoint is planned with latent Euclidean distance, so temporal supervision shapes the representation used by a geometric planner\. We deploydψd\_\{\\psi\}for topology\-dominated control and latentℓ2\\ell\_\{2\}for contact\-rich control; later sections refer to this design choice rather than re\-arguing it\.

Locked evaluations support both roles\. Deploying the mined cost improves Two\-Room and Reacher over LeWM; shared Euclidean planning on the same temporally trained checkpoint raises OGB\-Cube by14\.214\.2points and improves Push\-T\. Against LeWM and RC\-aux\(Liet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib24)\)under locked evaluation, TD\-JEPA matches or exceeds both methods on every environment\. Ablations show that the directed head, cross\-trajectory negatives, and rollout consistency each contribute, while contact\-phase diagnostics clarify when geometry must be selected at plan time\.

#### Contributions\.

- •A clear diagnosis of the JEPA train–plan gap\.We show that predictive JEPA world models learn dynamics from offline logs, while plan\-time scoring often relies on latent embedding geometry rather than an explicitly mined temporal\-progress cost, and that a log\-derived directed cost closes much of this ranking mismatch \(Fig\.[1](https://arxiv.org/html/2607.25337#S1.F1)\)\.
- •Temporal\-distance mining for JEPA planners\.We introduce TD\-JEPA, which keeps the LeWM backbone and adds directed temporal supervision, cross\-trajectory negatives, and horizon\-matched rollout consistency so that progress structure discovered in demonstrations serves as either a deployed planning cost or a representation signal \(Sec\.[3](https://arxiv.org/html/2607.25337#S3)\)\.
- •Evidence that cost form and deployment should be co\-designed\.Under matched evaluation, deploying the mined cost helps where topology governs progress, while shared Euclidean planning on the temporally trained checkpoint helps on contact\-rich tasks; ablations and contact\-phase analyses identify when each choice is appropriate \(Tables[2](https://arxiv.org/html/2607.25337#S4.T2),[5](https://arxiv.org/html/2607.25337#S4.T5),[4](https://arxiv.org/html/2607.25337#S4.T4), and[9](https://arxiv.org/html/2607.25337#S4.T9)\)\.

## 2\.Related Work

Energy\-based learning and planning\.Energy\-based models associate low energy with compatible variable assignments and perform inference by energy minimization\(LeCunet al\.,[2006](https://arxiv.org/html/2607.25337#bib.bib1)\)\. Latent MPC fits this template by searching for action sequences that minimize terminal energy after imagined rollout\. Value\-guided and quasimetric methods learn energies or values from offline data, providing a foundation for goal\-conditioned planning costs\(Wanget al\.,[2023](https://arxiv.org/html/2607.25337#bib.bib9); Destradeet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib21)\)\. TD\-JEPA follows the same inference view but changes the knowledge source: instead of a value target, it mines a directed temporal cost from reward\-free step\-count labels and cross\-trajectory negative pairs in demonstration logs\(van den Oordet al\.,[2018](https://arxiv.org/html/2607.25337#bib.bib6); Kostrikovet al\.,[2021](https://arxiv.org/html/2607.25337#bib.bib28)\)\.

JEPA and LeWM\.Latent\-space planning from pixels predates JEPA control\. PlaNet learns stochastic latent dynamics and plans online with predicted rewards, while TD\-MPC jointly learns task\-oriented dynamics and a terminal value through temporal\-difference learning\(Hafneret al\.,[2019](https://arxiv.org/html/2607.25337#bib.bib31); Hansenet al\.,[2022](https://arxiv.org/html/2607.25337#bib.bib32)\)\. JEPAs instead learn predictive representations from data without pixel reconstruction; LeJEPA introduces SIGReg for stable self\-supervision, and LeWM combines this recipe with latent planning\(Balestriero and LeCun,[2025](https://arxiv.org/html/2607.25337#bib.bib8); Maeset al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib7)\)\. Recent extensions explore relational interventions, online adaptation, and temporal abstraction\(Namet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib16); Wanget al\.,[2026a](https://arxiv.org/html/2607.25337#bib.bib15); Zhanget al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib12); Masipet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib13)\)\. TD\-JEPA leaves the LeWM dynamics backbone unchanged and instead mines the goal\-progress cost consumed by the planner\.

The central distinction is where task structure is discovered\. PlaNet learns rewards with a reconstructive latent model, and TD\-MPC learns reward and value predictions through online interaction\. LeWM removes reward prediction and treats distance in a self\-supervised predictive representation as the goal cost\. TD\-JEPA stays reward\-free, but makes progress supervision explicit by mining temporal order from demonstration logs\. It therefore targets the missing objective between latent prediction and action search, rather than replacing the dynamics architecture or introducing a learned policy\.

Value\-shaped and temporal visual representations\.VIP learns a temporally smooth visual embedding whose distance acts as a goal\-conditioned reward, and LIV extends value\-shaped representation learning to language and image goals\(Maet al\.,[2023b](https://arxiv.org/html/2607.25337#bib.bib33),[a](https://arxiv.org/html/2607.25337#bib.bib34)\)\. TLDR learns temporal\-distance\-aware representations for goal selection, intrinsic reward, and goal\-conditioned policy learning\(Baeet al\.,[2024](https://arxiv.org/html/2607.25337#bib.bib35)\)\. These methods use values or temporal distances to shape rewards and policies\. TD\-JEPA instead mines a directed cost inside an action\-conditioned JEPA world model and uses that cost to rank imagined rollouts\. VIP and LIV provide reward signals to a downstream controller; TLDR trains goal\-conditioned policies and exploration objectives\. TD\-JEPA stays in the offline, reward\-free setting: no reward model and no policy learner, only a temporal cost discovered from logs and consumed by planning\.

Planning\-aware latent geometry\.Accurate short\-horizon prediction is insufficient when MPC searches over multi\-step rollouts in a latent space whose metric is not aligned with reachability\. Destrade et al\. shape JEPA embeddings so a quasimetric distance approximates a goal\-conditioned value function learned with IQL, then plan with MPC on that geometry\(Destradeet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib21)\)\. Unlike Value\-Guided JEPA, TD\-JEPA needs no IQL value estimator: it mines reward\-free step\-count labels on the LeWM\+SIGReg backbone and evaluates task\-dependent deployment ofdψd\_\{\\psi\}versus latentℓ2\\ell\_\{2\}\(Table[1](https://arxiv.org/html/2607.25337#S2.T1)\)\. Wang et al\. regularize latent*curvature*so Euclidean distance better tracks geodesic progress after temporal straightening\(Wanget al\.,[2026b](https://arxiv.org/html/2607.25337#bib.bib22)\)\. RC\-aux adds multi\-horizon open\-loop prediction and budget\-conditioned reachability supervision on top of LeWM, distinguishing states reachable within the planner’s horizon from merely eventually reachable ones\(Liet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib24)\)\. SD\-JEPA partitions the latent into orthogonal progression and content subspaces, training angular coordinates with a cosine\-margin triplet loss while retaining SIGReg on the remainder\(Thilet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib23)\)\. Hierarchical planners and latent\-planner JEPAs instead modify inference structure, using coarse latent models or action\-free subgoal predictors to reduce long\-horizon search difficulty\(Zhanget al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib12); Masipet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib13)\)\.

Table 1\.TD\-JEPA vs\. planning\-aware JEPA variants\.
Quasimetric and goal\-conditioned RL\.Quasimetric representations model directed reachability distances; optimal goal\-reaching cost\-to\-go is generally asymmetric\(Wanget al\.,[2023](https://arxiv.org/html/2607.25337#bib.bib9)\)\. Value\-guided JEPA approximates−V⋆\-V^\{\\star\}with latent distance via IQL\(Destradeet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib21); Kostrikovet al\.,[2021](https://arxiv.org/html/2607.25337#bib.bib28)\)\. Recent offline GCRL work further strengthens the link between temporal distances and goal reaching: TMD combines contrastive successor\-feature learning with quasimetric constraints to recover stitchable temporal distances\(Myerset al\.,[2025](https://arxiv.org/html/2607.25337#bib.bib17)\), while ProQ uses learned asymmetric distances as directional costs over latent keypoints for long\-horizon planning\(Kobandaet al\.,[2025](https://arxiv.org/html/2607.25337#bib.bib18)\)\. Temporal representation learning also studies continuous time structure outside JEPA control, including TD InfoNCE for future\-event prediction\(Zhenget al\.,[2024](https://arxiv.org/html/2607.25337#bib.bib19)\)and trajectory encoders that preserve multi\-scale spatiotemporal detail in mined mobility logs\(Zhouet al\.,[2025b](https://arxiv.org/html/2607.25337#bib.bib20)\)\. TD\-JEPA stays in this log\-mining view: it regresses a directed temporal cost on demonstration step counts withi<ji<jpairs and does not estimate a reward or value model\.

## 3\.Method

We keep the LeWM encoder–predictor backbone and mine adirected temporal costEψ≜dψE\_\{\\psi\}\\triangleq d\_\{\\psi\}from offline demonstration logs\. The method has three parts: the LeWM prediction backbone, a directed cost head calibrated by trajectory step order, and a horizon\-matched rollout loss on the same open\-loop operator used at plan time\. By default, latent MPC minimizesdψd\_\{\\psi\}after open\-loop rollout; on contact\-rich tasks the evaluation protocol instead plans with latentℓ2\\ell\_\{2\}on the same temporally trained checkpoint\.

#### Backbone prediction loss\.

Given observationoto\_\{t\}and actionata\_\{t\}, the LeWM backbone encodes observations and predicts the next latent:

\(1\)z^t\+1=Predϕ​\(Encθ​\(ot\),at\),zt\+1=sg​\(Encθ​\(ot\+1\)\)\.\\hat\{z\}\_\{t\+1\}=\\mathrm\{Pred\}\_\{\\phi\}\(\\mathrm\{Enc\}\_\{\\theta\}\(o\_\{t\}\),a\_\{t\}\),\\qquad z\_\{t\+1\}=\\mathrm\{sg\}\(\\mathrm\{Enc\}\_\{\\theta\}\(o\_\{t\+1\}\)\)\.The local dynamics term is the next\-latent prediction loss

\(2\)ℒpred=‖z^t\+1−zt\+1‖22\.\\mathcal\{L\}\_\{\\mathrm\{pred\}\}=\\\|\\hat\{z\}\_\{t\+1\}\-z\_\{t\+1\}\\\|\_\{2\}^\{2\}\.TD\-JEPA leaves this backbone and SIGReg regularization intact, then adds planning\-specific supervision for the cost optimized by MPC\.

![Refer to caption](https://arxiv.org/html/2607.25337v1/x2.png)Figure 2\.TD\-JEPA training and latent MPC pipeline\. Observations are encoded into latents; the LeWM predictor rolls out an open\-loop trajectory under candidate actions\. Training keeps the backbone prediction and SIGReg terms, and adds directed temporal\-distance supervision \(same\-trajectory step\-order positives and cross\-trajectory negatives\) plus a horizon\-matched rollout\-consistency loss on the planner’s terminal cost\. At plan time, CEM/iCEM search action sequences that minimize the learned directed costdψ​\(z^H,zg\)d\_\{\\psi\}\(\\hat\{z\}\_\{H\},z\_\{g\}\)after the same open\-loop rollout \(latentℓ2\\ell\_\{2\}may replacedψd\_\{\\psi\}on contact\-rich tasks\)\.Pipeline diagram showing observations encoded into latents, a predictor rolling out future latents, a directed cost head scoring the rollout against a goal, and CEM using that cost for planning\. Training losses include prediction, rollout consistency, temporal\-distance supervision, and SIGReg\.
### 3\.1\.Two roles: cost*class*vs\. cost*calibration*

TD\-JEPA uses two related objects: an asymmetric scalar cost and a forward temporal\-separation label\. By “directed temporal cost” we mean a non\-negative function of an ordered state–goal pair, not a vector displacement\. The directed head sets the form of the cost; temporal labels calibrate its values on demonstration data\.

1. \(1\)Directed cost head \(geometry class\)\.The planner optimizes a directed functional form\. Goal\-reaching under actions is generally*asymmetric*: the cost of reachingzgz\_\{g\}fromzsz\_\{s\}can differ from the reverse cost\. Optimal goal\-reaching value functions and reachability costs are non\-negative and asymmetric, motivating quasimetric\-style parameterizations\(Wanget al\.,[2023](https://arxiv.org/html/2607.25337#bib.bib9)\)\. Raw latentℓ2\\ell\_\{2\}is symmetric \(‖zs−zg‖=‖zg−zs‖\\\|z\_\{s\}\-z\_\{g\}\\\|=\\\|z\_\{g\}\-z\_\{s\}\\\|\), so it cannot express this directionality by itself\. The MRN head in Eq\. \([5](https://arxiv.org/html/2607.25337#S3.E5)\) parameterizes a directed, non\-negative costdψ​\(zs→zg\)d\_\{\\psi\}\(z\_\{s\}\\\!\\to\\\!z\_\{g\}\)with a symmetric configuration term plus an asymmetric residual\. This parameterization encodes directionality by design\. A controlled Push\-T ablation keeps the same temporal labels but removes this MRN form: the ablated cost is‖gω​\(zs\)−gω​\(zg\)‖2\\\|g\_\{\\omega\}\(z\_\{s\}\)\-g\_\{\\omega\}\(z\_\{g\}\)\\\|\_\{2\}, a symmetric Euclidean distance between learned projection features\. We evaluate this symmetric temporal\-distance head under the same locked Push\-T protocol as the other component ablations \(Table[4](https://arxiv.org/html/2607.25337#S4.T4)\)\.
2. \(2\)Temporal\-distance supervision \(calibration signal\)\.Temporal labels assign numeric values to that cost\. The directed cost family gives TD\-JEPA the right shape, but it still needs labels that tie energy values to task progress\. Offline demonstrations supply a reward\-free supervision signal: along the same trajectory, the temporal gapτ​\(i,j\)=j−i\\tau\(i,j\)=j\-iis a proxy for steps\-to\-subgoal*on the data manifold*\. Eq\. \([6](https://arxiv.org/html/2607.25337#S3.E6)\) regressesdψ​\(zi,zj\)d\_\{\\psi\}\(z\_\{i\},z\_\{j\}\)toτ​\(i,j\)\\tau\(i,j\), while cross\-trajectory hinges penalize spuriously small costs for heuristic negative pairs\.

Composition\.Temporal labels set the magnitude ofdψd\_\{\\psi\}; the quasimetric\-style head makes that scalar depend on the order of its arguments\. Together they yield an asymmetric ranking function for goal progress\. TD\-JEPA keeps LeWM’s dynamics and anti\-collapse terms, and adds a planning term that supervises the*same*dψd\_\{\\psi\}used in Eq\. \([7](https://arxiv.org/html/2607.25337#S3.E7)\)\.

#### Relation to LeWM planning\.

LeWM’s cost‖z−zg‖22\\\|z\-z\_\{g\}\\\|\_\{2\}^\{2\}can be seen as a special case with identityϕsym\\phi\_\{\\mathrm\{sym\}\}and zero asymmetric residual\. TD\-JEPA adds a loss that encourages the plan\-time cost to rank same\-trajectory pairs bydψ​\(zi,zj\)≈j−id\_\{\\psi\}\(z\_\{i\},z\_\{j\}\)\\approx j\-i\. This focus on plan geometry is consistent with recent plan\-aware JEPA extensions\(Destradeet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib21); Wanget al\.,[2026b](https://arxiv.org/html/2607.25337#bib.bib22); Liet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib24)\)\.

### 3\.2\.Energy\-based and value\-function foundation

TD\-JEPA groundsdψd\_\{\\psi\}in two established frameworks for plan geometry: energy\-based learning\(LeCunet al\.,[2006](https://arxiv.org/html/2607.25337#bib.bib1)\)and quasimetric goal\-reaching RL\(Wanget al\.,[2023](https://arxiv.org/html/2607.25337#bib.bib9); Destradeet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib21)\)\. This section states the connection and the assumption used by our temporal supervision\.

#### Goal\-conditioned energy\.

In the energy\-based view\(LeCunet al\.,[2006](https://arxiv.org/html/2607.25337#bib.bib1)\), compatibility of variables\(x,y\)\(x,y\)is scored by an energyEθ​\(x,y\)E\_\{\\theta\}\(x,y\)with low values on desirable configurations\. For goal\-reaching from latent statezsz\_\{s\}to goalzgz\_\{g\}, define a*directed*energy

\(3\)Eψ​\(zs,zg\)≜dψ​\(zs,zg\)≥0\.E\_\{\\psi\}\(z\_\{s\},z\_\{g\}\)\\triangleq d\_\{\\psi\}\(z\_\{s\},z\_\{g\}\)\\geq 0\.Latent MPC selects actions that minimize terminal energy after rollout:a1:H⋆∈arg⁡mina1:H⁡Eψ​\(z^H​\(a1:H\),zg\)a^\{\\star\}\_\{1:H\}\\in\\arg\\min\_\{a\_\{1:H\}\}E\_\{\\psi\}\(\\hat\{z\}\_\{H\}\(a\_\{1:H\}\),z\_\{g\}\)\. TD\-JEPA*learns*EψE\_\{\\psi\}and uses the same functional form as the default train\-time and plan\-time energy\.

#### Link to optimal goal\-reaching costs\.

Consider deterministic dynamics with unit cost𝟏​\[st≠g\]\\mathbf\{1\}\[s\_\{t\}\\neq g\]and defineD⋆​\(s,g\)D^\{\\star\}\(s,g\)as the minimum number of steps needed to reachggfromss\. On mutually reachable states, this shortest\-path cost is non\-negative, vanishes on the diagonal, obeys the triangle inequality, and need not be symmetric; it is therefore a quasimetric \(or an extended quasimetric if unreachable pairs are assigned\+∞\+\\infty\)\(Wanget al\.,[2023](https://arxiv.org/html/2607.25337#bib.bib9)\)\. Value\-guided JEPA methods learnVθ​\(s,g\)≈V⋆​\(s,g\)V\_\{\\theta\}\(s,g\)\\approx V^\{\\star\}\(s,g\)with implicit Q\-learning and set plan cost to−Vθ\-V\_\{\\theta\}or a learned quasimetric surrogate\(Destradeet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib21)\)\. TD\-JEPA instead uses explicit temporal labels from offline demonstrations\. For a feasible trajectory segment fromoio\_\{i\}toojo\_\{j\}, its lengthτ​\(i,j\)=j−i\\tau\(i,j\)=j\-iupper\-bounds the shortest\-path distance:

\(4\)D⋆​\(oi,oj\)≤τ​\(i,j\)=j−i,dψ​\(Enc​\(oi\),Enc​\(oj\)\)≈τ​\(i,j\)\.D^\{\\star\}\(o\_\{i\},o\_\{j\}\)\\leq\\tau\(i,j\)=j\-i,\\qquad d\_\{\\psi\}\\big\(\\mathrm\{Enc\}\(o\_\{i\}\),\\mathrm\{Enc\}\(o\_\{j\}\)\\big\)\\approx\\tau\(i,j\)\.Equality holds only when the demonstrated segment is a shortest path, so we treat demonstration path length as an on\-manifold surrogate whose ordering reflects goal progress rather than asD⋆D^\{\\star\}itself\. Eq\. \([4](https://arxiv.org/html/2607.25337#S3.E4)\) is therefore an empirical*ranking\-calibration*step\. The MRN head supplies a directed parameterization inspired by the quasimetric structure ofD⋆D^\{\\star\}\(Wanget al\.,[2023](https://arxiv.org/html/2607.25337#bib.bib9)\); Eq\. \([6](https://arxiv.org/html/2607.25337#S3.E6)\) targetsτ​\(i,j\)=j−i\\tau\(i,j\)=j\-iin step units as a training signal, and Fig\.[1](https://arxiv.org/html/2607.25337#S1.F1)shows the resulting monotonic tracking\.

### 3\.3\.Directed distance and temporal supervision

We instantiate the directed cost with the metric\-residual \(MRN\) parameterization\(Wanget al\.,[2023](https://arxiv.org/html/2607.25337#bib.bib9)\)\. Each latent is mapped to symmetric and asymmetric featuresϕsym​\(z\)∈ℝds\\phi\_\{\\mathrm\{sym\}\}\(z\)\\in\\mathbb\{R\}^\{d\_\{s\}\}andϕasym​\(z\)∈ℝda\\phi\_\{\\mathrm\{asym\}\}\(z\)\\in\\mathbb\{R\}^\{d\_\{a\}\}, and

\(5\)dψ​\(zs,zg\)=‖ϕsym​\(zs\)−ϕsym​\(zg\)‖2⏟symmetric component\+maxk⁡ReLU​\(ϕasym,k​\(zs\)−ϕasym,k​\(zg\)\)⏟directed residual\.\\begin\{split\}d\_\{\\psi\}\(z\_\{s\},z\_\{g\}\)&=\\underbrace\{\\\|\\phi\_\{\\mathrm\{sym\}\}\(z\_\{s\}\)\-\\phi\_\{\\mathrm\{sym\}\}\(z\_\{g\}\)\\\|\_\{2\}\}\_\{\\text\{symmetric component\}\}\\\\ &\\quad\+\\underbrace\{\\max\_\{k\}\\mathrm\{ReLU\}\\\!\\big\(\\phi\_\{\\mathrm\{asym\},k\}\(z\_\{s\}\)\-\\phi\_\{\\mathrm\{asym\},k\}\(z\_\{g\}\)\\big\)\}\_\{\\text\{directed residual\}\}\.\\end\{split\}The head is non\-negative by construction and can satisfydψ​\(zs,zg\)≠dψ​\(zg,zs\)d\_\{\\psi\}\(z\_\{s\},z\_\{g\}\)\\neq d\_\{\\psi\}\(z\_\{g\},z\_\{s\}\)\. The symmetric term captures shared configuration factors; the asymmetric residual encodes directed reachability\.

Expert demonstrations provide the calibration labels\. For indicesi<ji<jon the same trajectory, defineτ​\(i,j\)=j−i\\tau\(i,j\)=j\-ias forward steps along the observed segment \(not a shortest\-path metric\)\. We minimize

\(6\)ℒtd=𝔼i<j​\[ρ​\(dψ​\(zi,zj\)−τ​\(i,j\)\)\]\+𝔼i,g′∼𝒩​\[ReLU​\(m−dψ​\(zi,zg′\)\)\],\\begin\{split\}\\mathcal\{L\}\_\{\\mathrm\{td\}\}&=\\mathbb\{E\}\_\{i<j\}\\\!\\left\[\\rho\\\!\\big\(d\_\{\\psi\}\(z\_\{i\},z\_\{j\}\)\-\\tau\(i,j\)\\big\)\\right\]\\\\ &\\quad\+\\mathbb\{E\}\_\{i,\\,g^\{\\prime\}\\sim\\mathcal\{N\}\}\\\!\\left\[\\mathrm\{ReLU\}\\\!\\big\(m\-\\\!d\_\{\\psi\}\(z\_\{i\},z\_\{g^\{\\prime\}\}\)\\big\)\\right\],\\end\{split\}whereρ\\rhois Smooth L1,𝒩\\mathcal\{N\}permutes cross\-trajectory goals at the same time index, andm=η​\(T−1\)m=\\eta\(T\{\-\}1\)is a margin with window lengthTT\. The regression term fits on\-manifold progress; the hinge discourages spuriously low off\-trajectory costs, while admitting false negatives when trajectories share reachable states\. This is regressive EBM calibration without estimating a partition function: positives are shaped towardτ​\(i,j\)\\tau\(i,j\), and negatives are pushed above marginmm\(LeCunet al\.,[2006](https://arxiv.org/html/2607.25337#bib.bib1); van den Oordet al\.,[2018](https://arxiv.org/html/2607.25337#bib.bib6); Destradeet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib21)\)\. We treat demonstration step counts as an on\-manifold proxy for progress, not as shortest\-path distances\.

#### Train–plan coupling\.

TD\-JEPA trains the energy that serves as the default plan\-time cost:

\(7\)a1:H⋆=arg⁡mina1:H⁡dψ​\(z^H​\(a1:H\),zg\),a^\{\\star\}\_\{1:H\}=\\arg\\min\_\{a\_\{1:H\}\}d\_\{\\psi\}\\\!\\big\(\\hat\{z\}\_\{H\}\(a\_\{1:H\}\),z\_\{g\}\\big\),wherez^H\\hat\{z\}\_\{H\}is theHH\-step open\-loop rollout of the learned predictor\. CEM therefore ranks candidates with the energy shaped byℒtd\\mathcal\{L\}\_\{\\mathrm\{td\}\}by default\. For manipulation evaluation, the cost composes this trained energy with latent geometry as\(1−α\)​dψ\+α​ℓ2\(1\{\-\}\\alpha\)d\_\{\\psi\}\+\\alpha\\,\\ell\_\{2\}; Sec\.[4\.3](https://arxiv.org/html/2607.25337#S4.SS3)reports this blend as a planning\-time diagnostic\.

### 3\.4\.Latent MPC with CEM and iCEM

At plan time, MPC searches over action sequences that minimize the latent rollout cost in Eq\. \([7](https://arxiv.org/html/2607.25337#S3.E7)\), executes the first action block, and replans from the resulting observation\. Algorithms[1](https://arxiv.org/html/2607.25337#alg1)and[2](https://arxiv.org/html/2607.25337#alg2)summarize the two solvers used in our evaluations\. Standard CEM updates a diagonal Gaussian over action sequences from elite moments\(de Boeret al\.,[2005](https://arxiv.org/html/2607.25337#bib.bib29)\)\. iCEM adds temporally colored noise, elite reuse across refinements, and momentum\-smoothed mean/variance updates\(Pinneriet al\.,[2020](https://arxiv.org/html/2607.25337#bib.bib30)\)\. Both call the same latent\-rollout cost: open\-loop prediction for horizonHH, then scoring withdψd\_\{\\psi\}\(or latentℓ2\\ell\_\{2\}on contact\-rich manipulation\)\.

Algorithm 1Cross\-Entropy Method \(CEM\) for latent MPC1:Context

hth\_\{t\}, goal latent

zgz\_\{g\}, horizon

HH, candidates

NN, elites

KK, refinements

RR
2:Initialize action mean

μ\(0\)\\mu^\{\(0\)\}and standard deviation

σ\(0\)\\sigma^\{\(0\)\}
3:for

r=0,…,R−1r=0,\\ldots,R\-1do

4:for

n=1,…,Nn=1,\\ldots,Ndo

5:

ϵn∼𝒩​\(0,I\)\\epsilon\_\{n\}\\sim\\mathcal\{N\}\(0,I\);

An←μ\(r\)\+σ\(r\)⊙ϵnA\_\{n\}\\leftarrow\\mu^\{\(r\)\}\+\\sigma^\{\(r\)\}\\odot\\epsilon\_\{n\}
6:endfor

7:Set

A1←μ\(r\)A\_\{1\}\\leftarrow\\mu^\{\(r\)\}⊳\\trianglerightAlways evaluate the current mean

8:

Jn←LatentRolloutCost​\(ht,zg,An\)J\_\{n\}\\leftarrow\\textsc\{LatentRolloutCost\}\(h\_\{t\},z\_\{g\},A\_\{n\}\)for all

nn
9:

ℰ←\\mathcal\{E\}\\leftarrowindices of the

KKcandidates with lowest

JnJ\_\{n\}
10:

μ\(r\+1\)←Mean​\(\{An:n∈ℰ\}\)\\mu^\{\(r\+1\)\}\\leftarrow\\textsc\{Mean\}\(\\\{A\_\{n\}:n\\in\\mathcal\{E\}\\\}\);

σ\(r\+1\)←Std​\(\{An:n∈ℰ\}\)\\sigma^\{\(r\+1\)\}\\leftarrow\\textsc\{Std\}\(\\\{A\_\{n\}:n\\in\\mathcal\{E\}\\\}\)
11:endfor

12:return

μ\(R\)\\mu^\{\(R\)\}

Algorithm 2Improved Cross\-Entropy Method \(iCEM\)1:CEM inputs; color exponent

β\\beta, retained elites

MM, smoothing

γ\\gamma
2:Initialize

μ\(0\),σ\(0\)\\mu^\{\(0\)\},\\sigma^\{\(0\)\}and previous elites

ℰprev←∅\\mathcal\{E\}\_\{\\mathrm\{prev\}\}\\leftarrow\\varnothing
3:for

r=0,…,R−1r=0,\\ldots,R\-1do

4:Draw temporally colored noise

ϵ1:N∼𝒩β​\(0,I\)\\epsilon\_\{1:N\}\\sim\\mathcal\{N\}\_\{\\beta\}\(0,I\)
5:

An←Clip​\(μ\(r\)\+σ\(r\)⊙ϵn\)A\_\{n\}\\leftarrow\\textsc\{Clip\}\(\\mu^\{\(r\)\}\+\\sigma^\{\(r\)\}\\odot\\epsilon\_\{n\}\)for all

nn; set

A1←μ\(r\)A\_\{1\}\\leftarrow\\mu^\{\(r\)\}
6:Replace up to

MMcandidates with

ℰprev\\mathcal\{E\}\_\{\\mathrm\{prev\}\}
7:

Jn←LatentRolloutCost​\(ht,zg,An\)J\_\{n\}\\leftarrow\\textsc\{LatentRolloutCost\}\(h\_\{t\},z\_\{g\},A\_\{n\}\)for all

nn
8:

ℰ←\\mathcal\{E\}\\leftarrowthe

KKlowest\-cost candidates

9:

μ¯←Mean​\(ℰ\)\\bar\{\\mu\}\\leftarrow\\textsc\{Mean\}\(\\mathcal\{E\}\);

σ¯←Std​\(ℰ\)\\bar\{\\sigma\}\\leftarrow\\textsc\{Std\}\(\\mathcal\{E\}\)
10:

μ\(r\+1\)←γ​μ\(r\)\+\(1−γ\)​μ¯\\mu^\{\(r\+1\)\}\\leftarrow\\gamma\\mu^\{\(r\)\}\+\(1\-\\gamma\)\\bar\{\\mu\};

σ\(r\+1\)←γ​σ\(r\)\+\(1−γ\)​σ¯\\sigma^\{\(r\+1\)\}\\leftarrow\\gamma\\sigma^\{\(r\)\}\+\(1\-\\gamma\)\\bar\{\\sigma\}
11:

ℰprev←ℰ\\mathcal\{E\}\_\{\\mathrm\{prev\}\}\\leftarrow\\mathcal\{E\}
12:endfor

13:return

μ\(R\)\\mu^\{\(R\)\}

### 3\.5\.Horizon\-aligned rollout consistency

CEM scoresHH\-step open\-loop rollouts\. We add a multi\-step teacher\-forcing loss on expert actions:

\(8\)ℒroll=1H​∑h=1H‖z^t\+h−sg​\(zt\+h\)‖22,\\mathcal\{L\}\_\{\\mathrm\{roll\}\}=\\frac\{1\}\{H\}\\sum\_\{h=1\}^\{H\}\\big\\\|\\hat\{z\}\_\{t\+h\}\-\\mathrm\{sg\}\(z\_\{t\+h\}\)\\big\\\|\_\{2\}^\{2\},withH=5H\{=\}5matched to the planner horizon\. Minimizingℒroll\\mathcal\{L\}\_\{\\mathrm\{roll\}\}reduces compounding error in the same rollout operator that appears inside Eq\. \([7](https://arxiv.org/html/2607.25337#S3.E7)\); together withℒtd\\mathcal\{L\}\_\{\\mathrm\{td\}\}, it couples*temporal*and*spatial*planning structure\.

### 3\.6\.Full objective

The complete TD\-JEPA objective is

\(9\)ℒ=ℒpred\+λroll​ℒroll\+λtd​ℒtd\+λsig​SIGReg​\(z\),\\mathcal\{L\}=\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\+\\lambda\_\{\\mathrm\{roll\}\}\\mathcal\{L\}\_\{\\mathrm\{roll\}\}\+\\lambda\_\{\\mathrm\{td\}\}\\mathcal\{L\}\_\{\\mathrm\{td\}\}\+\\lambda\_\{\\mathrm\{sig\}\}\\,\\mathrm\{SIGReg\}\(z\),whereℒpred\\mathcal\{L\}\_\{\\mathrm\{pred\}\}is the LeWM one\-step MSE\. The short\-window recipe usesλroll=0\.5\\lambda\_\{\\mathrm\{roll\}\}\{=\}0\.5,λtd=1\.0\\lambda\_\{\\mathrm\{td\}\}\{=\}1\.0,λsig=0\.09\\lambda\_\{\\mathrm\{sig\}\}\{=\}0\.09, prediction window 5, and rollout horizon 5\. SIGReg preserves the anti\-collapse prior from LeWM\(Balestriero and LeCun,[2025](https://arxiv.org/html/2607.25337#bib.bib8); Maeset al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib7)\)\. At plan time, CEM or iCEM \(Algs\.[1](https://arxiv.org/html/2607.25337#alg1)–[2](https://arxiv.org/html/2607.25337#alg2)\) optimizes Eq\. \([7](https://arxiv.org/html/2607.25337#S3.E7)\) withdψd\_\{\\psi\}by default; on contact\-rich tasks the main protocol uses latentℓ2\\ell\_\{2\}on the same checkpoint \(Sec\.[4\.3](https://arxiv.org/html/2607.25337#S4.SS3)\)\.

## 4\.Experiments

We address three questions: \(i\) Does temporal\-distance mining from reward\-free logs improve planning over LeWM and RC\-aux\(Liet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib24)\)under a fixed evaluation budget? \(ii\) Which mining ingredients drive the gains? \(iii\) When should the mined temporal cost be deployed, and when should latent geometry still be selected?

We study these questions on the four LeWM environments—Two\-Room, Reacher, Push\-T, and OGBench\-Cube—covering navigation, reaching, and contact\-rich manipulation\. TD\-JEPA retains the LeWM encoder–predictor backbone and is trained for ten epochs; full hyperparameters appear in Appendix[C](https://arxiv.org/html/2607.25337#A3)\. Unless noted otherwise, primary comparisons report mean±\\pmstd over ten independent seed runs with the checkpoint and episode manifest held fixed\. Diagnostic ablations and phase logs use three seeds; we note this whenever those tables appear\. Plan\-time defaults follow the two\-role design: TD\-JEPA deploys puredψd\_\{\\psi\}on Two\-Room and Reacher, and latentℓ2\\ell\_\{2\}on Push\-T and OGB\-Cube\. LeWM and RC\-aux use latentℓ2\\ell\_\{2\}throughout under the same locked manifests\. On Reacher, the planner aggregates terminal and trajectory\-mean costs with weightw=0\.3w\{=\}0\.3so that smooth progress is credited; terminal\-only scoring is a sensitivity check in Sec\.[4\.5](https://arxiv.org/html/2607.25337#S4.SS5)\.

### 4\.1\.Planning Performance

Table[2](https://arxiv.org/html/2607.25337#S4.T2)compares TD\-JEPA with LeWM and RC\-aux under matched episode manifests and planner budgets\. RC\-aux is evaluated with latentℓ2\\ell\_\{2\}planning on all four environments, matching its role in Table[1](https://arxiv.org/html/2607.25337#S2.T1)\.

Table 2\.Locked planning success \(%\) over ten independent seed runs\. TD\-JEPA deploysdψd\_\{\\psi\}on Two\-Room/Reacher and sharedℓ2\\ell\_\{2\}on Push\-T/OGB\-Cube \(Sec\.[4](https://arxiv.org/html/2607.25337#S4)\); LeWM and RC\-aux use latentℓ2\\ell\_\{2\}throughout\.
Under this locked protocol, TD\-JEPA matches or exceeds both LeWM and RC\-aux on every environment\. On Two\-Room and Reacher, deploying the mined costdψd\_\{\\psi\}yields the gains, showing the value of an explicit temporal plan cost over horizon\-budgeted reachability auxiliaries that still plan with geometry\. On Push\-T and OGB\-Cube, all three methods plan with latentℓ2\\ell\_\{2\}, so improvements reflect temporal representation learning rather than direct cost deployment\. The largest manipulation gain appears on OGB\-Cube, where TD\-JEPA improves over LeWM by14\.214\.2percentage points and slightly exceeds RC\-aux\. Thus temporal supervision helps in both roles: as a deployed plan cost on topology\-dominated tasks, and as a representation signal for geometric planning on contact\-rich tasks\.

For broader context, Table[3](https://arxiv.org/html/2607.25337#S4.T3)places the same locked TD\-JEPA numbers beside published offline RL, imitation learning, and latent world\-model baselines under their native protocols\. Those rows are not a matched head\-to\-head; all LeWM and RC\-aux claims in this paper use Table[2](https://arxiv.org/html/2607.25337#S4.T2)\.

Table 3\.Planning success \(%\) in literature context\. Prior methods are published point estimates under native protocols with different solvers, budgets, and episode draws; TD\-JEPA uses the locked protocol of Sec\.[4](https://arxiv.org/html/2607.25337#S4)\. Bold / underline: best / second best per column\. Rows are not strictly comparable\.
Relative to published baselines under non\-matched protocols, TD\-JEPA achieves the highest reported success on Two\-Room and Reacher and remains competitive on OGB\-Cube\. We next ablate training ingredients, then ask when puredψd\_\{\\psi\}should be deployed\.

### 4\.2\.Which Cost Components Matter?

We ablate the directed head, cross\-trajectory negatives, and rollout consistency on Push\-T \(Table[4](https://arxiv.org/html/2607.25337#S4.T4)\)\. Unlike the ten\-seed primary comparisons, these variants use three independent seed runs under a ten\-epoch training budget continued from four\-epoch checkpoints, with the same 50 validation episodes\. The columns report a diagnostic blend, puredψd\_\{\\psi\}, pure latentℓ2\\ell\_\{2\}, and iCEM; the full\-modelℓ2\\ell\_\{2\}entry of85\.3%85\.3\\%is therefore a three\-seed diagnostic and need not match the ten\-seed86\.0%86\.0\\%in Table[2](https://arxiv.org/html/2607.25337#S4.T2)\. The symmetric\-head row keeps the temporal regression and hinge but replaces the MRN residual with a Euclidean projection distance, separating cost form from supervision\.

Table 4\.Push\-T component ablations under the locked ten\-epoch protocol \(mean±\\pmstd over three independent seed runs\)\.
Each ablation hurts every planner setting relative to the full model\. Removing the directed residual causes the largest drop: temporal labels alone are not enough without the MRN form\. Removing rollout consistency hurts more than removing the hinge\. Cost form and supervision must therefore be co\-designed: a symmetric Euclidean head, missing negatives, or a missing horizon\-matchedℒroll\\mathcal\{L\}\_\{\\mathrm\{roll\}\}each weaken the planner\.

### 4\.3\.When Should the Mined Temporal Cost Be Deployed?

We next fix the TD\-JEPA checkpoint, episodes, solver, and budget, and vary only the plan\-time objective amongdψd\_\{\\psi\}, latentℓ2\\ell\_\{2\}, and their blend withα=0\.10\\alpha\{=\}0\.10\. Figure[3](https://arxiv.org/html/2607.25337#S4.F3)summarizes the comparison; exact values appear in Appendix Table[11](https://arxiv.org/html/2607.25337#A1.T11)\. With everything else held fixed, differences in success come from the cost surface seen by CEM\.

![Refer to caption](https://arxiv.org/html/2607.25337v1/x3.png)Figure 3\.Plan\-cost comparison with checkpoint, episodes, solver, and budget held fixed \(mean±\\pmstd over ten independent seed runs\)\.Grouped bar chart of planning success for directed d\-psi, latent l2, and their alpha 0\.10 blend on Two\-Room, Reacher, Push\-T, and OGB\-Cube\. D\-psi scores 100\.0, 97\.0, 69\.0, and 77\.0\. Latent l2 scores 99\.0, 95\.0, 86\.0, and 82\.2\. The blend scores 100\.0, 95\.6, 83\.4, and 81\.2\. Error bars show standard deviation over ten independent seed runs\.Plan\-cost comparison reveals two regimes\. In navigation\-dominated tasks, puredψd\_\{\\psi\}matches or exceedsℓ2\\ell\_\{2\}: Two\-Room reaches full success, and Reacher favors the temporal cost\. In contact\-rich manipulation,ℓ2\\ell\_\{2\}outperforms puredψd\_\{\\psi\}by a wide margin on Push\-T and by a smaller margin on OGB\-Cube; a small geometric blend partially closes but does not eliminate the gap\.

Table[5](https://arxiv.org/html/2607.25337#S4.T5)isolates the navigation regime by comparing LeWM’s nativeℓ2\\ell\_\{2\}planner with the same TD\-JEPA checkpoint underdψd\_\{\\psi\}and underℓ2\\ell\_\{2\}\. Deployingdψd\_\{\\psi\}exceeds LeWM in both environments, while planning the TD\-JEPA checkpoint withℓ2\\ell\_\{2\}falls slightly below LeWM on Reacher\. The navigation gain is therefore not explained by a better Euclidean landscape alone: where topology matters, the temporal cost itself helps\. Figure[4](https://arxiv.org/html/2607.25337#S4.F4)shows the Two\-Room training trajectory under the deployeddψd\_\{\\psi\}planner\. Early epochs are unstable under pure temporal scoring, then success climbs from74%74\\%at epoch 5 to full success by epoch 9, surpassing the locked LeWMℓ2\\ell\_\{2\}baseline\. This trajectory supports evaluating the final temporally trained checkpoint for navigation deployment under the two\-role protocol, rather than selecting an intermediate epoch by peak online score alone\.

Table 5\.Locked navigation costs: LeWMℓ2\\ell\_\{2\}versus TD\-JEPA underdψd\_\{\\psi\}andℓ2\\ell\_\{2\}\(mean±\\pmstd over ten independent seed runs\)\.
![Refer to caption](https://arxiv.org/html/2607.25337v1/x4.png)Figure 4\.Two\-Room planning success versus training epoch under puredψd\_\{\\psi\}\(iCEM\-30, seed 3072\)\. The dashed line marks locked LeWMℓ2\\ell\_\{2\}success \(97\.4%97\.4\\%\)\.On manipulation tasks, temporal training improvesℓ2\\ell\_\{2\}planning even thoughdψd\_\{\\psi\}ranks temporal gaps more accurately\. Table[6](https://arxiv.org/html/2607.25337#S4.T6)reports Spearman correlations on Push\-T over three independently trained checkpoints with 1,000 held\-out pairs each:dψd\_\{\\psi\}exceedsℓ2\\ell\_\{2\}on every split, yetℓ2\\ell\_\{2\}still plans better in Fig\.[3](https://arxiv.org/html/2607.25337#S4.F3)\. Better temporal ranking is therefore not the same as a better contact\-rich plan cost\. The two signals carry different information:dψd\_\{\\psi\}tracks progress, whileℓ2\\ell\_\{2\}preserves local geometry useful near contact\(Maeset al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib7)\)\. Offline CEM sweeps in Fig\.[5](https://arxiv.org/html/2607.25337#S4.F5)reinforce the same pattern: manipulation benefits from a geometric plan cost, while temporal training still helps when that geometric cost is used\.

Table 6\.Push\-T Spearman correlation with temporal separation \(three trained checkpoints\)\.
![Refer to caption](https://arxiv.org/html/2607.25337v1/x5.png)Figure 5\.Offline CEM diagnostics on manipulation\. \(a\) OGB\-Cube success across training epochs under offline\-CEM replanning\. \(b\) CEM plan\-cost sweep on Push\-T and OGB\-Cube versus blend weightα\\alphain\(1−α\)​dψ\+α​ℓ2\(1\{\-\}\\alpha\)\\,d\_\{\\psi\}\+\\alpha\\,\\ell\_\{2\}\.Two\-panel figure\. Left: OGB\-Cube success versus epoch for pure d\-psi and alpha 0\.10 blend, with LeWM and DINO\-WM published baselines\. Right: planning success versus blend weight for Push\-T and OGB\-Cube; both curves rise when a small geometric weight is added\.The mined temporal signal therefore helps navigation when deployed asdψd\_\{\\psi\}, and helps contact\-rich control when used as representation shaping for anℓ2\\ell\_\{2\}planner\. The next subsection analyzes*why*puredψd\_\{\\psi\}underperforms on Push\-T despite stronger temporal ranking\.

### 4\.4\.Why Pure Temporal Cost Is Not Enough on Push\-T

Unless stated otherwise, Push\-T evaluation in this paper uses the primary locked protocol withℓ2\\ell\_\{2\}planning in Table[2](https://arxiv.org/html/2607.25337#S4.T2)\. Here we explain why puredψd\_\{\\psi\}underperforms that geometric planner: under the locked cost matrix, puredψd\_\{\\psi\}reaches69\.0%69\.0\\%while latentℓ2\\ell\_\{2\}reaches86\.0%86\.0\\%\. We test whether failures occur before contact \(approach\) or during post\-contact alignment \(manipulation\)\.

MPC phase logging on the locked 50\-episode manifest answers this \(Table[7](https://arxiv.org/html/2607.25337#S4.T7)\)\. These logs use three seeds, so per\-cost success rates can differ slightly from the ten\-seed matrix in Fig\.[3](https://arxiv.org/html/2607.25337#S4.F3)\. Faileddψd\_\{\\psi\}episodes are almost entirely post\-contact, whereas latentℓ2\\ell\_\{2\}almost never fails before contact and roughly halves post\-contact failures\. Pure temporal cost is not enough because Push\-T success depends on fine pose and angle control*after*contact, where local geometry matters more than a progress ranking\.

Table 7\.Where Push\-T failures occur under locked CEM \(mean episode counts over three independent seed runs; 50 episodes each\)\.
Start geometry modulates the size of the gap but not its dominant mode \(Table[8](https://arxiv.org/html/2607.25337#S4.T8)\)\. Across locked ten\-seed outcomes,ℓ2\\ell\_\{2\}uniquely solves far more episodes thandψd\_\{\\psi\}\. Theℓ2−dψ\\ell\_\{2\}\{\-\}d\_\{\\psi\}gap is smaller on above\-median close starts and widens on far starts; splits on block–goal pose change the gap much less\. Far starts are harder overall, but Table[7](https://arxiv.org/html/2607.25337#S4.T7)shows that the dominant failure mode remains post\-contact control rather than failing to arrive\.

Table 8\.Push\-T success \(%\) by start agent–block distance \(locked ten\-seed protocol\)\. The gap column isℓ2−dψ\\ell\_\{2\}\-d\_\{\\psi\}\.
A proximity gate that uses temporal cost far from contact and geometry near contact does not recover most of theℓ2\\ell\_\{2\}score without retraining\. Table[9](https://arxiv.org/html/2607.25337#S4.T9)tests that idea with the same CEM\-30 checkpoint and ten independent seed runs\. A hard gate improves over puredψd\_\{\\psi\}but stays well below pureℓ2\\ell\_\{2\}; a soft gate matches theα=0\.10\\alpha\{=\}0\.10blend and still does not beat geometry alone\. The planner needs geometric scoring throughout CEM search, not only after contact is detected\.

Table 9\.Push\-T contact\-gated planning versus fixed costs \(locked CEM\-30; mean±\\pmstd over ten independent seed runs\)\.

### 4\.5\.Robustness and Calibration

How the cost is aggregated over a rollout matters as much as which cost is used\. Table[10](https://arxiv.org/html/2607.25337#S4.T10)varies planner aggregation on Reacher under the locked validation manifest\. The primary protocol mixes terminal and trajectory\-mean costs with weightw=0\.3w\{=\}0\.3, matching temporal\-distance supervision; the single\-run rates of96%96\\%for TD\-JEPA and94%94\\%for LeWM are consistent with, but not identical to, the ten\-seed means in Table[2](https://arxiv.org/html/2607.25337#S4.T2)\. Terminal\-only scoring withw=1w\{=\}1is a sensitivity check that favors geometry and understates the temporal cost\. Under the progressive aggregation used in the primary protocol, CEM and iCEM agree and TD\-JEPA matches or exceeds LeWM\. This choice affects only planner aggregation, not the checkpoint or training objective\.

Table 10\.Reacher solver and rollout\-cost aggregation on the locked validation manifest\. Primary protocol:w=0\.3w\{=\}0\.3;w=1w\{=\}1is a terminal\-only sensitivity check\.
Held\-outdψd\_\{\\psi\}rises monotonically with demonstrated step separation \(Appendix Fig\.[6](https://arxiv.org/html/2607.25337#A2.F6)\), so the head learns the intended ordering even if its absolute scale need not match the true step count\. Together with Table[6](https://arxiv.org/html/2607.25337#S4.T6), this separates two claims: the cost ranks time well, but good ranking need not suffice for contact\-rich control\.

## 5\.Conclusion

TD\-JEPA narrows the train–plan gap in JEPA world models by mining a directed temporal cost from reward\-free demonstration logs on the LeWM encoder–predictor backbone\. The mined signal has two roles: as the deployed plan cost when progress is topological, and as a representation signal when contact\-rich control prefers latent geometry\. Locked evaluation supports both choices: deployingdψd\_\{\\psi\}improves Two\-Room and Reacher over LeWM, while shared\-ℓ2\\ell\_\{2\}planning on the same temporally trained checkpoint raises OGB\-Cube by14\.214\.2points and improves Push\-T\. Contact\-rich control still prefers geometry at plan time even whendψd\_\{\\psi\}ranks temporal gaps better, and proximity\-gated hybrids do not recover pure geometric scoring\. Component ablations show that the directed residual, cross\-trajectory hinge, and rollout consistency each contribute under locked planning\.

## References

- J\. Bae, K\. Park, and Y\. Lee \(2024\)TLDR: unsupervised goal\-conditioned reinforcement learning via temporal distance\-aware representations\.InProceedings of the Conference on Robot Learning,Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p4.1)\.
- R\. Balestriero and Y\. LeCun \(2025\)LeJEPA: provable and scalable self\-supervised learning without the heuristics\.External Links:2511\.08544Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p2.1),[§2](https://arxiv.org/html/2607.25337#S2.p2.1),[§3\.6](https://arxiv.org/html/2607.25337#S3.SS6.p1.6)\.
- P\. de Boer, D\. P\. Kroese, S\. Mannor, and R\. Y\. Rubinstein \(2005\)A tutorial on the cross\-entropy method\.Annals of Operations Research134,pp\. 19–67\.Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p2.1),[§3\.4](https://arxiv.org/html/2607.25337#S3.SS4.p1.3)\.
- M\. Destrade, O\. Bounou, Q\. Le Lidec, J\. Ponce, and Y\. LeCun \(2026\)Value\-guided action planning with JEPA world models\.External Links:2601\.00844Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p3.1),[§2](https://arxiv.org/html/2607.25337#S2.3.3.5.1.1),[§2](https://arxiv.org/html/2607.25337#S2.p1.1),[§2](https://arxiv.org/html/2607.25337#S2.p5.2),[§2](https://arxiv.org/html/2607.25337#S2.p6.2),[§3\.1](https://arxiv.org/html/2607.25337#S3.SS1.SSS0.Px1.p1.3),[§3\.2](https://arxiv.org/html/2607.25337#S3.SS2.SSS0.Px2.p1.10),[§3\.2](https://arxiv.org/html/2607.25337#S3.SS2.p1.1),[§3\.3](https://arxiv.org/html/2607.25337#S3.SS3.p2.8)\.
- D\. Ghosh, A\. Gupta, A\. Reddy, J\. Fu, C\. Devin, B\. Eysenbach, and S\. Levine \(2020\)Learning to reach goals via iterated supervised learning\.External Links:1912\.06088,[Link](https://arxiv.org/abs/1912.06088)Cited by:[§4\.1](https://arxiv.org/html/2607.25337#S4.SS1.24.7.7.4)\.
- D\. Hafner, T\. Lillicrap, I\. Fischer, R\. Villegas, D\. Ha, H\. Lee, and J\. Davidson \(2019\)Learning latent dynamics for planning from pixels\.InProceedings of the 36th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.97,pp\. 2555–2565\.Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p2.1)\.
- N\. A\. Hansen, H\. Su, and X\. Wang \(2022\)Temporal difference learning for model predictive control\.InProceedings of the 39th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.162,pp\. 8387–8406\.Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p2.1)\.
- A\. Kobanda, W\. Radji, M\. Petitbois, O\. Maillard, and R\. Portelas \(2025\)Offline goal\-conditioned reinforcement learning with projective quasimetric planning\.External Links:2506\.18847Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p6.2)\.
- I\. Kostrikov, A\. Nair, and S\. Levine \(2021\)Offline reinforcement learning with implicit Q\-learning\.External Links:2110\.06169Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p1.1),[§2](https://arxiv.org/html/2607.25337#S2.p6.2),[§4\.1](https://arxiv.org/html/2607.25337#S4.SS1.27.10.10.4),[§4\.1](https://arxiv.org/html/2607.25337#S4.SS1.30.13.13.4)\.
- Y\. LeCun, S\. Chopra, R\. Hadsell, M\. Ranzato, and F\. J\. Huang \(2006\)A tutorial on energy\-based learning\.InPredicting Structured Data,Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p5.2),[§2](https://arxiv.org/html/2607.25337#S2.p1.1),[§3\.2](https://arxiv.org/html/2607.25337#S3.SS2.SSS0.Px1.p1.4),[§3\.2](https://arxiv.org/html/2607.25337#S3.SS2.p1.1),[§3\.3](https://arxiv.org/html/2607.25337#S3.SS3.p2.8)\.
- Y\. LeCun \(2022\)A path towards autonomous machine intelligence\.Note:OpenReview preprintVersion 0\.9\.2External Links:[Link](https://openreview.net/forum?id=BZ5a1r-kVsf)Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p2.1)\.
- F\. Li \(2026\)A functional taxonomy of world models: renderers, simulators, planners, and the loop that connects them\.Note:SubstackPublished June 3, 2026External Links:[Link](https://drfeifei.substack.com/p/a-functional-taxonomy-of-world-models)Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p1.1)\.
- W\. Li, G\. Li, K\. Maeda, T\. Ogawa, and M\. Haseyama \(2026\)Predictive but not plannable: RC\-aux for latent world models\.External Links:2605\.07278Cited by:[Appendix C](https://arxiv.org/html/2607.25337#A3.SS0.SSS0.Px2.p1.14),[§1](https://arxiv.org/html/2607.25337#S1.p3.1),[§1](https://arxiv.org/html/2607.25337#S1.p7.1),[§2](https://arxiv.org/html/2607.25337#S2.3.3.6.2.1),[§2](https://arxiv.org/html/2607.25337#S2.p5.2),[§3\.1](https://arxiv.org/html/2607.25337#S3.SS1.SSS0.Px1.p1.3),[§4\.1](https://arxiv.org/html/2607.25337#S4.SS1.12.12.12.12.5),[§4](https://arxiv.org/html/2607.25337#S4.p1.1)\.
- Y\. J\. Ma, V\. Kumar, A\. Zhang, O\. Bastani, and D\. Jayaraman \(2023a\)LIV: language\-image representations and rewards for robotic control\.InProceedings of the 40th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.202,pp\. 23301–23320\.Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p4.1)\.
- Y\. J\. Ma, S\. Sodhani, D\. Jayaraman, O\. Bastani, V\. Kumar, and A\. Zhang \(2023b\)VIP: towards universal visual reward and representation via value\-implicit pre\-training\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p4.1)\.
- L\. Maes, Q\. Le Lidec, D\. Scieur, Y\. LeCun, and R\. Balestriero \(2026\)LeWorldModel: stable end\-to\-end joint\-embedding predictive architecture from pixels\.External Links:2603\.19312Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p2.1),[§2](https://arxiv.org/html/2607.25337#S2.1.1.1.2),[§2](https://arxiv.org/html/2607.25337#S2.p2.1),[§3\.6](https://arxiv.org/html/2607.25337#S3.SS6.p1.6),[§4\.1](https://arxiv.org/html/2607.25337#S4.SS1.42.25.25.5),[§4\.3](https://arxiv.org/html/2607.25337#S4.SS3.p4.7)\.
- S\. Masip, J\. Swinnen, Y\. Hu, R\. Detry, and T\. Tuytelaars \(2026\)FF\-JEPA: long\-horizon planning in world models with latent planners\.External Links:2606\.09311Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p2.1),[§2](https://arxiv.org/html/2607.25337#S2.p5.2)\.
- V\. Myers, B\. C\. Zheng, B\. Eysenbach, and S\. Levine \(2025\)Offline goal\-conditioned reinforcement learning with quasimetric representations\.Note:Advances in Neural Information Processing SystemsCited by:[§2](https://arxiv.org/html/2607.25337#S2.p6.2)\.
- H\. Nam, Q\. L\. Lidec, L\. Maes, Y\. LeCun, and R\. Balestriero \(2026\)Causal\-jepa: learning world models through object\-level latent masking\.External Links:2602\.11389,[Link](https://arxiv.org/abs/2602.11389)Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p2.1)\.
- C\. Pinneri, S\. Sawant, S\. Blaes, J\. Achterhold, J\. Stueckler, M\. Rolinek, and G\. Martius \(2020\)Sample\-efficient cross\-entropy method for real\-time planning\.InProceedings of the Conference on Robot Learning,Cited by:[§3\.4](https://arxiv.org/html/2607.25337#S3.SS4.p1.3)\.
- V\. Sobal, W\. Zhang, K\. Cho, R\. Balestriero, T\. G\. J\. Rudner, and Y\. LeCun \(2025\)Stress\-testing offline reward\-free reinforcement learning: a case for planning with latent dynamics models\.In7th Robot Learning Workshop: Towards Robots with Human\-Level Abilities,External Links:[Link](https://openreview.net/forum?id=jON7H6A9UU)Cited by:[§4\.1](https://arxiv.org/html/2607.25337#S4.SS1.34.17.17.5)\.
- L\. Thil, J\. Read, R\. Kaddah, and G\. Doquet \(2026\)Subspace\-decomposed JEPAs: disentangling progression and content in latent world models\.External Links:2605\.31111Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p3.1),[§2](https://arxiv.org/html/2607.25337#S2.3.3.7.3.1),[§2](https://arxiv.org/html/2607.25337#S2.p5.2)\.
- A\. van den Oord, Y\. Li, and O\. Vinyals \(2018\)Representation learning with contrastive predictive coding\.External Links:1807\.03748Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p1.1),[§3\.3](https://arxiv.org/html/2607.25337#S3.SS3.p2.8)\.
- T\. Wang, A\. Torralba, P\. Isola, and A\. Zhang \(2023\)Optimal goal\-reaching reinforcement learning via quasimetric learning\.InProceedings of the 40th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.202,Honolulu, HI, USA,pp\. 36411–36430\.Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p5.2),[§2](https://arxiv.org/html/2607.25337#S2.p1.1),[§2](https://arxiv.org/html/2607.25337#S2.p6.2),[item 1](https://arxiv.org/html/2607.25337#S3.I1.i1.p1.6),[§3\.2](https://arxiv.org/html/2607.25337#S3.SS2.SSS0.Px2.p1.10),[§3\.2](https://arxiv.org/html/2607.25337#S3.SS2.SSS0.Px2.p1.13),[§3\.2](https://arxiv.org/html/2607.25337#S3.SS2.p1.1),[§3\.3](https://arxiv.org/html/2607.25337#S3.SS3.p1.2)\.
- Y\. Wang, O\. Bounou, Y\. LeCun, and M\. Ren \(2026a\)AdaJEPA: an adaptive latent world model\.External Links:2606\.32026Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p2.1)\.
- Y\. Wang, O\. Bounou, G\. Zhou, R\. Balestriero, T\. G\. J\. Rudner, Y\. LeCun, and M\. Ren \(2026b\)Temporal straightening for latent planning\.External Links:2603\.12231Cited by:[§1](https://arxiv.org/html/2607.25337#S1.p3.1),[§2](https://arxiv.org/html/2607.25337#S2.p5.2),[§3\.1](https://arxiv.org/html/2607.25337#S3.SS1.SSS0.Px1.p1.3)\.
- W\. Zhang, B\. Terver, A\. Zholus, S\. Chitnis, H\. Sutaria, M\. Assran, R\. Balestriero, A\. Bar, A\. Bardes, Y\. LeCun, and N\. Ballas \(2026\)Hierarchical planning with latent world models\.External Links:2604\.03208Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p2.1),[§2](https://arxiv.org/html/2607.25337#S2.p5.2)\.
- C\. Zheng, R\. Salakhutdinov, and B\. Eysenbach \(2024\)Contrastive difference predictive coding\.Note:International Conference on Learning RepresentationsCited by:[§2](https://arxiv.org/html/2607.25337#S2.p6.2)\.
- G\. Zhou, H\. Pan, Y\. LeCun, and L\. Pinto \(2025a\)DINO\-wm: world models on pre\-trained visual features enable zero\-shot planning\.External Links:2411\.04983,[Link](https://arxiv.org/abs/2411.04983)Cited by:[§4\.1](https://arxiv.org/html/2607.25337#S4.SS1.38.21.21.5)\.
- S\. Zhou, Y\. Chen, S\. Shang, L\. Chen, B\. He, and R\. Shibasaki \(2025b\)Blurred encoding for trajectory representation learning\.InProceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining,New York, NY, USA,pp\. 4132–4143\.External Links:[Document](https://dx.doi.org/10.1145/3711896.3736861)Cited by:[§2](https://arxiv.org/html/2607.25337#S2.p6.2)\.

## Appendix ADetailed Plan\-Cost Results

Table[11](https://arxiv.org/html/2607.25337#A1.T11)lists the exact values behind Fig\.[3](https://arxiv.org/html/2607.25337#S4.F3)\. The epoch\-10 TD\-JEPA checkpoint, locked 50\-episode manifests, environment\-specific solvers, and ten independent seed runs are held fixed; only the plan cost changes\. The Push\-T and OGB\-Cubeℓ2\\ell\_\{2\}cells match the TD\-JEPA row of Table[2](https://arxiv.org/html/2607.25337#S4.T2)\.

Table 11\.Matched plan\-cost matrix \(mean±\\pmstd success over ten independent seed runs\)\.

## Appendix BTemporal\-Cost Calibration

![Refer to caption](https://arxiv.org/html/2607.25337v1/x6.png)Figure 6\.Learneddψd\_\{\\psi\}versus held\-out temporal separation\.Line plot of mean learned d\-psi versus binned temporal separation\. The mean cost rises monotonically across bins, with error bars showing within\-bin variation\.On held\-out same\-trajectory pairs, meandψd\_\{\\psi\}rises monotonically with demonstrated step separation\. Absolute scale need not match the true step count; the planner uses the induced ranking\.

## Appendix CTraining and Evaluation Protocol

#### Training\.

The short\-window recipe usesNpred=5N\_\{\\mathrm\{pred\}\}\{=\}5\(windowT=8T\{=\}8with history size 3\), rollout horizon 5,λroll=0\.5\\lambda\_\{\\mathrm\{roll\}\}\{=\}0\.5,λtd=1\.0\\lambda\_\{\\mathrm\{td\}\}\{=\}1\.0, andλsig=0\.09\\lambda\_\{\\mathrm\{sig\}\}\{=\}0\.09\. Directed head: hidden dimension 512;ϕsym,ϕasym∈ℝ128\\phi\_\{\\mathrm\{sym\}\},\\phi\_\{\\mathrm\{asym\}\}\\in\\mathbb\{R\}^\{128\}\. All runs share a ViT\-tiny encoder \(∼\{\\sim\}15M parameters, patch 14, image 224\), 192\-dimensional latent, history size 3, AdamW with learning rate5×10−55\{\\times\}10^\{\-5\}and weight decay10−310^\{\-3\}, bf16 mixed precision, and 10 epochs\. Push\-T uses effective batch 128 \(32×432\{\\times\}4accumulation\)\.

#### Planning defaults\.

MPC uses horizonH=5H\{=\}5, goal offsetG=25G\{=\}25, 50 episodes, and 300 candidate action sequences per CEM/iCEM iteration unless noted\. Two\-Room: iCEM\-30 with puredψd\_\{\\psi\}and terminal weightw=1w\{=\}1\. Reacher: iCEM\-30 with puredψd\_\{\\psi\}andw=0\.3w\{=\}0\.3\(terminal mixed with trajectory\-mean cost\)\. Push\-T and OGB\-Cube locked LeWM/RC\-aux comparisons \(Table[2](https://arxiv.org/html/2607.25337#S4.T2)\): latentℓ2\\ell\_\{2\}on the temporally trained checkpoint \(CEM\-30 and CEM\-10;w=1w\{=\}1\), matching LeWM’s geometric plan cost\. RC\-aux uses the same locked manifests and latentℓ2\\ell\_\{2\}planning on all four environments \(CEM/iCEM settings as LeWM\), with multi\-horizon rollout and budgeted\-reachability training losses as in\(Liet al\.,[2026](https://arxiv.org/html/2607.25337#bib.bib24)\)\. Table[3](https://arxiv.org/html/2607.25337#S4.T3)places the same locked TD\-JEPA numbers in literature context beside published point estimates; it is not a matched head\-to\-head\. Blendα=0\.10\\alpha\{=\}0\.10and puredψd\_\{\\psi\}appear in the matched cost matrix and contact diagnostics \(Fig\.[3](https://arxiv.org/html/2607.25337#S4.F3), Tables[11](https://arxiv.org/html/2607.25337#A1.T11)and[9](https://arxiv.org/html/2607.25337#S4.T9)\)\. Push\-T LeWM uses the released checkpoint; other environments use our LeWM checkpoints\. iCEM usesnoise\_beta=2\.0\{=\}2\.0, smoothing coefficient0\.10\.1, andn\_elite\_keep=5\{=\}5\.

#### Statistical scope\.

Unless noted otherwise,±\\pmvalues are sample standard deviations \(n−1n\{\-\}1\) over independent seed runs with the checkpoint and episode manifest held fixed\. Push\-T ablations \(Table[4](https://arxiv.org/html/2607.25337#S4.T4)\) use three training\-seed runs; paired LeWM/RC\-aux comparisons, the cost matrix, and contact\-gated Push\-T results use ten plan seeds \(Tables[2](https://arxiv.org/html/2607.25337#S4.T2),[11](https://arxiv.org/html/2607.25337#A1.T11), and[9](https://arxiv.org/html/2607.25337#S4.T9)\)\. Push\-T rank correlations \(Table[6](https://arxiv.org/html/2607.25337#S4.T6)\) use independently trained checkpoints\.