--- title: "INTACT: Isomorphic Intent-to-Action Learning for Search-Free World Models" author: - Junhan Sun - Hao Zhao - Guofeng Zhang date: 2026-07-28 arxiv: 2607.26056v1 --- # Abstract Forward latent world models predict how actions change a scene, but recover actions for a desired change only through expensive test-time search. We introduce INTACT (**IN**tent-**T**o-**ACT**ion), an end-to-end JEPA that turns action-labeled, reward-free trajectories into a deployable intent-to-action interface. Each transition supplies physical intent $z_{t+1}-z_t$, while a future goal supplies deployment intent $\operatorname{sg}(z_g)-z_t$. The architecture is isomorphic between the local and goal motion-intent backbone-input graphs through an identical four-slot grammar and shared parameters, and between supported local and goal motion-intent families through action-law semantics induced by the same predictor rather than pointwise latent equality. INTACT also provides intact transfer from RGB evidence to action-effective latent intent coordinates and from intent families to their corresponding action-law families. Asymmetric endpoint gradients ground physical successors and fix future goals as anchors, joining representation learning and control without pointwise latent matching or globally linear dynamics. The resulting coordinates support a robust distributional action law: its conditional mean serves directly as a search-free policy, while sampling remains available for diversity or optional verification. On the four official LeWM tasks, one-epoch, zero-search models reach 85.78%, 100.00%, 97.67%, and 97.89% success. Optional local CEM centered on the Direct plan reaches 96.86% macro success using 384 instead of 9,000 candidate sequences, reducing sampling by $23.44\times$ while improving pure CEM by 16.00 points. One shared four-task encoder reaches 89.39% E5 Direct macro and improves every task over jointly trained LeWM, while predicted--expert action-family kNN tracks Direct success at $r=0.954$. Direct inference takes 2.9--5.5 ms. ![INTACT intent-to-action method and result overview](figures/teaser_geometry_v31.png) **INTACT turns action-aligned latent intent into search-free control.** A fixed E5 Cube audit compares INTACT Direct with LeWM CEM $300\times30$ from 300 matched starts; the remaining panels summarize multi-task latent geometry, the learned goal-conditioned operator, one-epoch control, local verification, latency, and action-family alignment. # Introduction {#sec:introduction} Latent world models learn a forward conditional: given a visual state and an action, predict what happens next. Goal-conditioned control then numerically inverts this map by proposing action sequences, rolling them through the model, and retaining those that approach a goal [@hafner2019planet; @hansen2024tdmpc2; @zhou2025dinowm; @maes2026lewm]. This division creates a representation--control asymmetry. Actions shape the latent during training, yet CEM or MPPI begins deployment from generic random or Gaussian actions [@deboer2005cem; @williams2017mppi]. Proposals become task-directed only after repeated model rollout, scoring, and refitting. The world model learns what an action will do, but not which action should realize a requested latent change. Action and latent are coupled in the forward predictor but semantically uncalibrated in the inverse direction needed for control. The missing supervision is already present in offline trajectories. Our core observation is that **even when demonstrations are unordered and training uses no reward, every action-labelled transition still reveals a state-conditioned motion intent and the action that realizes it**. Across many trajectories, these labels reveal families of state-conditional motion intents and their corresponding action families. Our insight is to represent a desired motion by a latent intent $m_t$, train $\pi_\eta(a_t\mid z_t,m_t)$ jointly with the world model, and let observed actions organize which latent intents require compatible control. The same intent construction is available at deployment, so the learned representation becomes an action proposal interface rather than only a scoring space. ```{=latex} \begin{figure*}[t] \includegraphics[width=0.99\textwidth]{figures/multitask_panel_c_v10_clear.pdf} \caption{\textbf{Shared-encoder INTACT training and recurrent control.} Four visual domains use one encoder and task-specific predictor pairs. Within each task, graph-isomorphic local and goal calls enter the same INTACT Predictor through one matched input grammar, using attached local intent $z_{t+1}^k-z_t^k$ or detached goal intent $\operatorname{sg}(z_g^k)-z_t^k$. The Forward Predictor retains latent-dynamics and SIGReg supervision. At inference, the goal-conditioned call emits one action chunk for search-free Direct control; the world rollout is retained only for optional verification and replanning.} \label{fig:overview} \end{figure*} ``` Forward prediction already gives a latent predictive meaning; calling that latent wholly meaningless would be incorrect. What it does not specify is an intent-to-action semantics: which compressed distinctions should be immediately useful for converting a requested motion into control. In this sense, a forward-only bottleneck behaves like an uncalibrated funnel: it gathers the information needed to forecast observations while leaving its action geometry incidental. Intent-conditioned likelihood turns that funnel into a calibrated filter. Action-equivalent requests are organized for rapid readout, while the forward objective retains contacts, topology, and visual residuals needed for future prediction. Representation shaping is therefore not a cosmetic regularizer; it determines whether the learned state can be queried directly by a downstream controller. The name INTACTencodes three complementary notions of intactness. First, *representation intactness* asks whether RGB-to-latent compression keeps the controllable distinctions needed by the intent-to-action conditional. This is not synonymous with higher rank, more training epochs, or retaining every pixel detail. Second, *conversion intactness* asks whether relations among state-conditioned intent families survive their map into the corresponding action families. Third, ***isomorphic** intactness* has two levels. Structurally, the local and goal motion-intent calls use **isomorphic** backbone-input graphs: the same typed slots, parameter-labelled predictor, and action-law codomain. Semantically, their supported motion-intent families are **isomorphic** through the conditional action relation induced by that shared predictor: each conditional quotient class corresponds bijectively to its realizable action-law image, without requiring local and goal latents to be pointwise equal. Within either call, the first-order intent $m_t$ and its state-dependent interaction $z_t\odot m_t$ retain one matched feature grammar **intact**. The latter is a fixed state--intent bilinear feature---a low-cost second-order interaction in the inputs---not a learned second-order dynamics model. A frozen downstream head can read only what its encoder already exposes; it cannot reconstruct controllable distinctions discarded during pretraining. End-to-end action likelihood therefore shapes both what enters the latent filter and how it is converted into control. In the first sense, action-relevant intent must survive RGB compression **intact**; in the second, the relation between an intent family and its corresponding action family must survive conversion **intact**. This idea is not an additive recipe of two auxiliary losses. Taken alone, the local branch is physical inverse dynamics; taken alone, the deployment branch is GCSL-like goal-conditioned imitation. Neither isolated objective expresses the coupling that INTACT needs. The underlying asymmetry is $$\underbrace{z_{t+1}}_{\substack{\text{physically grounded}\\ \text{unavailable before acting}}} \qquad\text{vs.}\qquad \underbrace{z_g}_{\substack{\text{available before acting}\\ \text{not a one-step successor}}}. \label{eq:intro_endpoint_asymmetry}$$ INTACT resolves it by asking one conditional action operator to interpret both families, with $m\in\{m^{\rm local},m^{\rm goal}\}$, $$G_\eta:(z,m,a^-) \longmapsto p_\eta(a\mid z,m,a^-). \label{eq:intro_shared_operator}$$ The local successor remains attached and supplies physical reachability and representation shaping; the future goal is a stop-gradient deployment anchor. The two conditions are never forced together by latent $L_2$, while the Forward Predictor preserves world information richer than the immediate action. This shared, graph-isomorphic dual call is the coupling mechanism, not mere coexistence of two losses. We therefore treat the deployment condition as a *motion request*, not an imagined trajectory. Our waypoint attribution uses the horizon-normalized intent $m_{t,h}^{\rm waypoint}=(\operatorname{sg}(z_g)-z_t)/h$; the final model instead exposes the raw goal intent $m_t^{\rm goal}=\operatorname{sg}(z_g)-z_t$. The former imposes an equal-progress scale, while the latter retains the complete terminal residual and lets the nonlinear INTACT Predictor choose state-dependent progress. In both cases, the demonstrated action at the same state supervises the intent, and the same intent semantics are used at deployment. After their conditions are constructed, the two calls form **isomorphic typed predictor graphs**; their upstream gradient routes remain asymmetric. They do not pretend that a terminal goal is the next physical frame. This supervision is obtained *without reward labels or additional semantic annotations; multi-task training uses the known domain index only to select task-specific heads*. The resulting geometry is conditional. At fixed current state $z$, two endpoint conditions are equivalent when they induce the same expert action law: $$y\sim_z y' \quad\Longleftrightarrow\quad p_E^\star(a\mid z,y)=p_E^\star(a\mid z,y'). \label{eq:intro_quotient}$$ Distinct goals may share the same first action before a doorway and separate after a junction; a real successor and a goal-derived intent may be equivalent without being close in Euclidean latent space. This quotient applies only to the action condition. The complete JEPA latent must still preserve contacts, obstacles, and future visual information required by forward prediction. We call the resulting equivalence class a **conditional action quotient**. We instantiate this view in INTACT(**IN**tent-**T**o-**ACT**ion learning). INTACT comprises a *Forward Predictor* and one shared *INTACT Predictor*. The latter is called with $m_t^{\rm local}=z_{t+1}-z_t$ to preserve action-recoverable physical change and with $m_t^{\rm goal}=\operatorname{sg}(z_g)-z_t$ to map a pre-action goal intent to the demonstrated action block. These calls share feature grammar and parameters, but not necessarily their numerical distributions or encoder gradients. At inference, INTACT alternates the INTACT Predictor mean with the unchanged Forward Predictor and replans from real observations. Its [Direct]{.smallcaps} mode evaluates no candidates and calls no terminal latent cost; CEM remains available as an optional verifier. To our knowledge, INTACT is the first end-to-end latent-control architecture to combine all three ingredients: one shared proper action likelihood for observed physical-transition and deployable goal-intent condition families, attached physical-successor endpoints but stop-gradient future-goal anchors, and no pointwise loss that collapses the two latent condition families. The novelty is the complete statistical construction rather than inverse dynamics alone. It aligns the two families through the behavior they induce under the same conditional action operator, not by making their latent coordinates equal. The goal call itself is a goal-conditioned imitation objective. Our claim is not that INTACT lies outside behavior cloning; it is that goal-only, post-hoc frozen-representation, and independent-actor explanations are insufficient for the measured behavior of the complete system. Accordingly, goal-intent-only serves as the GCSL-like ablation in our factorial; the contribution of full INTACT is the end-to-end coupling of this deployment likelihood with JEPA dynamics and a shared local physical anchor. Equivalently, goal-intent-only identifies one familiar branch objective; Full INTACT begins when the attached physical and stop-gradient deployment conditions must be explained by the same parameterized action-law operator. This distinction also rules out a tempting shortcut. Making a physical successor close to a hand-designed latent waypoint can make a trajectory look straighter while discarding velocity, contact, or action-identifiable information. INTACT does not regress a goal intent onto a successor. It uses a probabilistic action likelihood to preserve the conditional family relation induced by the demonstrated action. Nor is this merely pointwise interpolation behavior cloning. The goal intent is constructed inside the jointly learned world representation, and its likelihood updates the current-state encoding while the future goal is stop-gradient. This goal-intent path has no physical-successor gradient; successor-based shaping is supplied by the separate physical inverse loss in the full objective. The Forward Predictor remains responsible for rolling the next request. Most importantly, actor-disabled CEM improves after local/goal-intent training, and conversion-fidelity metrics predict closed-loop SR across objective designs. The evidence therefore separates representation shaping, intent-to-action conversion, and optional search rather than assigning all gains to a terminal policy head. We deliberately separate the theory and evidence into single-task and multi-task regimes. In one task, physical inverse and goal-intent supervision are complementary: the inverse should improve the shared representation even when its actor is removed, while goal intent should primarily improve the deployed conditional. With a shared encoder across heterogeneous tasks, however, individually valid local- and goal-intent gradients can interact. We therefore test the minimal $2{\times}2$ factorial---neither action loss, physical inverse only, goal intent only, or both---with one shared encoder and task-specific small heads. This isolates whether local physical anchoring and deployable goal conditioning remain complementary under heterogeneous dynamics. Here \`\`unified" has an operational meaning: all four domains update one visual encoder in one training run, and the same deployment-supported action loss that produces control also shapes that encoder. This differs from training one world model per domain or freezing a pretrained LeWM and fitting a downstream controller or sampling prior. The published LeWM, C-JEPA, Fast-LeWM, GC-IDM, PRISM, and Qantara control results use task-specific checkpoints or cover only a subset of the suite. Among the interfaces surveyed in `\cref{tab:positioning}`{=latex}, none previously reports this four-domain joint setting; our novelty claim is deliberately restricted to that comparison set. The official four-task LeWM benchmark supports both parts of the account. With one full-data epoch and no candidate search, the final distance-preserving single-task model reaches 85.78%, 100.00%, 97.67%, and 97.89% on PushT, Cube, Reacher, and TwoRoom, with three training seeds per task. Earlier matched PushT attribution shows that actor-disabled CEM $30{\times}10$ improves from 42.22% for LeWM to 57.67% with inverse supervision, 61.44% with a matched goal-intent objective, and $69.44\!\pm\!1.64$% for the Waypoint INTACT model with SIGReg $0.03$. Because the actor is disabled, this is functional evidence beyond the INTACT Predictor proposal; frozen probes provide direct readout evidence. Direct uses no candidate search and only 2.9--5.5 ms in our audited planner-side measurements; matched PushT is 4.8 ms. Relative to the measured actor-initialized CEM $300{\times}30$ average of 1.48 s, this is about a $300\times$ planner-latency reduction. It removes a major model-based search bottleneck for high-frequency control, although it is not an end-to-end VLA latency benchmark. Over all 12 final checkpoints, the complete 576-job planner audit finds that Guarded A, a coherent Direct plan followed by reference-preserving $128{\times}3$ local raw-action CEM at $\sigma_0=0.25$, reaches 96.86% macro and 92.22% worst-task SR. Relative to matched pure CEM $300{\times}30$, it gains 16.00 macro points while reducing sampled candidate sequences from 9,000 to 384 (23.44$\times$). Ordinary $\sigma_0=0.25$ is only 0.08 macro point lower, making it a particularly simple strong default. For multi-task learning, we use the first complete controlled Math-SDPA matrix: six objective cells, three training seeds, and three evaluation seeds per checkpoint. At E1, Full INTACT reaches $42.69\!\pm\!7.19$% Direct macro SR, versus $35.75\!\pm\!7.58$% for matched waypoint-intent-only and $33.47\!\pm\!3.98$% for inverse-only. Relative to goal-intent-only, adding physical inversion raises PushT and TwoRoom by 12.89 and 15.33 points and reduces TwoRoom training-seed standard deviation from 31.13 to 10.58 points. At E5, one shared Goal-displacement INTACT encoder reaches $89.39\!\pm\!0.77$% macro Direct SR. Relative to the matched shared-encoder LeWM, it improves all four tasks by 5.66/32.23/12.56/42.44 points; it also exceeds the published task-specific LeWM macro and its Cube/Reacher scores. Against the matched three-seed goal-intent-only cell, the physical call adds $8.78\!\pm\!1.63$ macro points. With every action head disabled, pure CEM $300{\times}30$ rises from $66.17\!\pm\!2.67$% for LeWM to $70.08\!\pm\!1.13$% for Goal-displacement INTACT; adding Guarded A to the learned Direct plan further raises $89.39\!\pm\!0.77$% to $90.47\!\pm\!0.84$%. These matched controls separate representation/Forward usability, direct intent conversion, and optional verification. Separately, a 45-checkpoint E1--E5 simultaneous-dual-likelihood audit supplies evidence beyond objective rankings: predicted--expert action-family kNN and CKA have Pearson $r=0.954$ and $0.897$ with official SR, respectively, compared with $0.815$ for pointwise action $R^2$. After controlling epoch and cohort, kNN remains $r=0.902$. This gap indicates that preserving local and global action-family geometry is more informative than recovering every expert action by a single point estimate. Our contributions are: 1. We define a state-conditional action quotient whose classes are canonically isomorphic to the realizable action-law image, represent endpoint requests by latent displacement coordinates, and introduce an end-to-end *isomorphic implicit-training architecture*: one shared proper likelihood aligns observed-transition and deployment-intent condition families, with asymmetric gradient routing and without forcing their latent endpoints to coincide. A matched sharing ablation shows that two independent actors remain worse even at double capacity, directly supporting the shared-operator choice. 2. We identify the complementary matched intra-call grammar required by the INTACT head: the first-order intent $m_t$ must be paired with its matched state--intent bilinear interaction $z_t\odot m_t$. The new A--G control shows that centering the product alone gives only $+0.89$ points, whereas matching the centered main condition and interaction yields a $+3.89$-point residual--product interaction with the same direction in all three training seeds. This is a state-conditioned inverse-control factor, not a second-order dynamics model. 3. We develop a non-diffusion, autoregressive intent--forward controller that emits a state-conditioned action distribution rather than a single static answer. It supports zero-search Direct control and plan-centered local CEM: the best fixed $128{\times}3$ rule reaches 96.86% macro and 92.22% worst-task SR while using only 4.27% of the action-step proposals used by CEM $300{\times}30$. 4. We introduce native-coordinate action-family diagnostics---predicted--expert CKA and kNN overlap---and show across 45 eligible E1--E5 checkpoints that their correlations with official SR ($r=0.897/0.954$) exceed that of pointwise action $R^2$ ($r=0.815$); kNN remains $r=0.902$ after controlling epoch and cohort, exposing family-level intent-to-action preservation as the relevant mechanism. A 21,600-episode gauge intervention further shows that paired correspondence recovers 68.04% SR versus 9.46% after pair shuffling, while a Full backbone cannot give a LeWM actor an untrained deployment map (5.08%). 5. We show that joint intent supervision improves not only the deployed actor but also the shared representation: after removing the INTACT Predictor at evaluation, matched actor-disabled pure CEM still improves over LeWM, with frozen probes independently confirming more recoverable state and intervention information. 6. We give latent space an operational semantics by extending a Gaussian-regularized self-supervised predictor into a multi-task mapping between motion-intent families and action families. This principle is not specific to LeWM's encoder--encoder JEPA: it applies to general learned representations whenever action-labelled transitions are available. # Related Work {#sec:related} #### Latent world models and JEPAs. PlaNet and Dreamer learn recurrent latent dynamics through generative objectives [@hafner2019planet; @hafner2023dreamerv3]; TD-MPC2 learns compact dynamics jointly with control-relevant predictions [@hansen2024tdmpc2]. JEPA-style world models instead predict in representation space. DINO-WM plans with a frozen DINOv2 encoder [@zhou2025dinowm], PLDM trains end-to-end with a VICReg-style objective [@sobal2025pldm], and LeWM combines latent prediction with Sketched Isotropic Gaussian Regularization (SIGReg) for stable end-to-end learning [@balestriero2025lejepa; @maes2026lewm]. We retain LeWM's predictive backbone but ask how an action-conditioned deployment interface can shape, rather than collapse, that same representation. Two recent extensions improve the forward-model side without changing this search interface. C-JEPA uses object-level latent masking and reports efficient CEM on PushT [@nam2026cjepa]; Fast-LeWM predicts all action-prefix horizons in parallel and reduces the matched TwoRoom CEM solve from 54.4 to 28.3 seconds while retaining the same candidate budget [@gao2026fastlewm]. Both train a separate world model per reported control domain. INTACT instead asks whether a single shared encoder can expose a directly deployable action conditional across the four LeWM domains. The distinction is important: a forward-only encoder is not meaningless, but its compression criterion is prediction sufficiency rather than immediate action sufficiency. INTACT uses labeled transitions to calibrate that bottleneck without replacing its predictive role. #### Inverse dynamics and representation learning. Inverse models have long been used to isolate controllable visual structure [@pathak2017curiosity; @efroni2022exogenous] and support multitask imitation [@brandfonbrener2023inverse]. An early robotic precursor jointly learned forward and inverse models from unlabeled poking interaction, using inverse prediction to shape visual features and forward prediction to regularize their dynamics [@agrawal2016poking]. SMWM shows that one-step inverse supervision can regularize an end-to-end JEPA toward sensorimotor features [@ivashkov2026smwm]; broader probing work likewise links temporal pretraining to action recoverability [@yeom2026actionrelevant]. These results concern action-relevant representation, but a local inverse evaluated only on observed successors does not identify its behavior on a goal-derived intent. INTACTmakes this support mismatch explicit and keeps the learned conditional as a deployment interface. #### Hindsight goal-conditioned imitation. GCSL relabels a trajectory with a future state that it actually reaches and maximizes the corresponding goal-conditioned action likelihood, iterating between collection under the current policy and supervised updates [@ghosh2019gcsl]. Complementary formulations cast goal reaching as recursive future-state classification or learn a contrastive representation whose inner product is a goal-conditioned value function [@eysenbach2021clearning; @eysenbach2022contrastive]. INTACT's deployment branch has this GCSL-like statistical form, and goal-intent-only is the corresponding hindsight-imitation ablation in our factorial. It is not the original GCSL algorithm: it retains the JEPA forward objective and SIGReg and uses fixed offline expert trajectories rather than iteratively collecting its own. INTACT is not goal-only GCSL. Its deployment branch is a GCSL-like hindsight action likelihood inside a jointly trained JEPA, while the complete method additionally uses the same conditional operator on attached one-chunk physical successors to anchor local reachability. This distinction is structural rather than terminological. Goal-intent-only captures one branch call and is therefore the GCSL-like baseline reported in our tables; physical-inverse-only captures the other branch call. Only Full INTACT evaluates both on each transition through one weight-shared, graph-isomorphic predictor while routing endpoint gradients asymmetrically. Thus the shared operator is not cosmetic parameter tying: it is the mechanism that gives physically grounded and deployable condition families a common action-law semantics without matching their latent coordinates. #### Options, empowerment, and unsupervised skill discovery. The options framework treats closed-loop policies as temporally extended action units [@sutton1999options], while empowerment measures the information capacity from an agent's action channel to reachable outcomes [@klyubin2005empowerment]. Modern unsupervised RL often combines these ideas with a latent skill variable: DIAYN makes skills distinguishable from visited states [@eysenbach2019diayn]; DADS favors skills with predictable dynamics [@sharma2020dads]; CIC contrastively aligns skills with state transitions [@laskin2022cic]; and METRA learns a temporally grounded metric space in which a policy moves along latent directions [@park2024metra]. These methods share the useful interface $\pi(a\mid s,\xi)$, but solve a different statistical problem. Their code $\xi$ is generally sampled or discovered through interaction and optimized by intrinsic reward, mutual information, coverage, or temporal distance. Our $m$ is not a free behavior index: it is a state-conditional motion intent grounded in an offline demonstration, and the recorded action is its supervised target. Rather than discovering behaviors that make latent codes identifiable, we identify which latent requests are equivalent under the demonstrated action law. #### Metric learning and action-induced equivalence. Classical metric learning uses pair or triplet labels to make Euclidean embeddings invariant within a semantic class and separated across classes [@hadsell2006drlim; @schroff2015facenet]. In control, bisimulation metrics learn task-relevant invariances without pixel reconstruction, while goal-conditioned bisimulation extends this idea to functional equivariance between analogous goals [@zhang2021invariant; @hansenestruch2022bisimulation]. Action labels play an analogous organizational role here: across otherwise unordered trajectories, they reveal families of motion intents requiring similar control. The distinction is that INTACTdoes not impose a pairwise distance or collapse complete JEPA latents with the same action. Equivalence is defined at fixed state by equality of the conditional action law and learned through a proper action likelihood. Forward prediction and SIGReg preserve variations within an action fiber that remain important for future dynamics. In this sense our quotient is an output-distribution geometry, not a metric-learning loss on observations. #### Amortized and prior-guided world-model control. GLAMOR learns goal-conditioned inverse and prior models to guide random shooting [@paster2020glamor]. Biased-MPPI provides a general mechanism for fusing ancillary sampling distributions into MPPI [@trevisan2024biasedmppi]. More recently, PRISM predicts a current/goal-conditioned Gaussian trajectory prior from a frozen JEPA encoder and fuses it with MPPI by precision-weighted Product-of-Gaussians updates [@wang2026prism]. It addresses a train--deployment goal shift through the planner's deployment-time cost, but still evaluates sampled trajectories; its reported limitations include task-specific near-expert demonstrations and a unimodal local prior for potentially multimodal long-horizon behavior. GC-IDM first studies a pairwise inverse over consecutive frozen LeWM latents with plans produced by latent linear interpolation [@nguyen2026gcidm]. Its oracle action $R^2=0.993$ does not rescue invalid straight latent paths; on TwoRoom, its appendix reports 34% for interpolation, 48% after refinement and candidate selection, and 87% for CEM. Conversely, its final goal-conditioned controller reaches 100% TwoRoom SR with training $R^2$ around 0.20 by predicting useful directions and replanning. GC-IDM thus already shows that scalar coordinate recovery is not a sufficient theory of control. It avoids imagined latent paths in its final controller and conditions on encoded current/goal observations plus remaining horizon. INTACT takes another route: the waypoint intent is a request coordinate rather than a physical successor, and is placed on training support by pairing it with the demonstrated current action. INTACT also updates the JEPA encoder end to end and evaluates both search-free execution and the actor-disabled world model. Our claim is not the first inverse controller or learned sampling prior; it is that deployment-supported intent supervision can organize the representation and directly amortize action search. Statistically, the frozen-encoder stage used by GC-IDM and PRISM is closer to behavior cloning or prior fitting on top of a fixed visual representation: it can exploit information that the encoder already exposes, but its action loss cannot decide which distinctions that encoder should preserve. INTACT is unified in a stricter sense. The visual encoder, forward dynamics, and deployment-supported action conditional are optimized together, and the same encoder is shared across all four domains. Among LeWM and the LeWM-derived systems compared here, we are not aware of a prior report that jointly trains one four-domain encoder while allowing the deployed action conditional to shape that encoder. This is a scoped claim about the published interfaces in `\cref{tab:positioning}`{=latex}, not a claim that frozen-feature controllers are ineffective. #### Action quotients and visual fibers. QuoVLA formalizes a quotient of VLM prompt representations induced by the Bayes-optimal action-trajectory law and approximates it with quantization and a dual-branch VLA design [@wang2026quovla]. We share the general principle that representations distinguished by perception may be equivalent for action, and do not claim to originate action quotients. Our quotient acts on local/goal motion *intents at fixed JEPA state*, not on the complete VLM prompt representation; it predicts a transition-aligned action block and coexists with an explicit forward world model. FiberTune identifies the complementary risk that action supervision leaves visually meaningful directions within an action fiber unconstrained and therefore vulnerable to collapse [@lin2026fibertune]. This supports our architectural separation: the actor may quotient action-equivalent conditions, while forward prediction and SIGReg preserve the richer world latent. #### Unified video--action models. UWM couples video and action diffusion to expose forward, inverse, policy, and generation queries [@zhu2025uwm]; DreamGen converts imagined futures to actions through a separate inverse or latent-action model [@jang2025dreamgen]. Qantara trains a JEPA for several dispatch-dependent control paradigms using bridge flow [@rakhimov2026qantara]. These systems motivate a unified conditional view. Our scope is deliberately smaller and mechanistic: a compact reward-free JEPA, a proper endpoint action likelihood, and controlled evidence linking the statistical unit of supervision to latent geometry and direct control. ```{=latex} \begin{table*}[t] \scriptsize \caption{Positioning among closely related methods. ``Action$\rightarrow E$'' means that action supervision updates the visual/world encoder, and ``Shared-4T $E$'' means one encoder is jointly trained over all four official LeWM domains. The final column states each method's actual training-to-control path; a cross denotes an unreported interface, not an impossible extension.} \label{tab:positioning} \setlength{\tabcolsep}{2.4pt} \renewcommand{\arraystretch}{1.08} \begin{tabularx}{\textwidth}{@{}lcc >{\arraybackslash}p{1.48cm} >{\arraybackslash}p{1.62cm} cc>{\arraybackslash}X@{}} \toprule Method & Action$\to E$ & Shared-4T $E$ & Condition & Action obj. & Direct & Search & Training-to-control path \\ \midrule DINO-WM \citep{zhou2025dinowm} & ✗ & ✗ & action & none & ✗ & ✓ & Frozen image encoder; CEM ranks latent rollouts \\ C-JEPA \citep{nam2026cjepa} & ✗ & ✗ & action/proprio. & none & ✗ & ✓ & Frozen object features; CEM searches masked rollouts \\ LeWM \citep{maes2026lewm} & ✗ & ✗ & action & none & ✗ & ✓ & Joint JEPA prediction; CEM inverts the forward model \\ Fast-LeWM \citep{gao2026fastlewm} & ✗ & ✗ & action prefix & none & ✗ & ✓ & LeWM training; parallel action prefixes are searched \\ SMWM \citep{ivashkov2026smwm} & ✓ & ✗ & successor & local action & ✗ & ✓ & Local inverse shapes the encoder; deployment still uses CEM \\ GC-IDM \citep{nguyen2026gcidm} & ✗ & ✗ & goal + horizon & next action & ✓ & ✗ & Frozen LeWM; a separate goal/horizon actor acts directly \\ PRISM \citep{wang2026prism} & ✗ & ✗ & state/goal & Gaussian prior & ✗ & ✓ & Frozen LeWM; a separately trained prior initializes MPPI \\ Qantara \citep{rakhimov2026qantara} & ✓ & ✗ & dispatch-dependent & action flow & ✓ & ✓ & Joint JEPA; bridge-flow heads serve dispatch modes \\ QuoVLA \citep{wang2026quovla} & ✓ & ✗ & visual/language & action chunk & ✓ & ✗ & VLM action quotient; quantized intent drives a VLA actor \\ INTACT{} & ✓ & ✓ & local/goal intent & action block & ✓ & opt. & One shared JEPA/actor law; Direct by default, search verifies \\ \bottomrule \end{tabularx} \end{table*} ``` # INTACT: Theory and Method {#sec:method} INTACT augments a forward JEPA with one conditional action operator used twice: once on a realized physical transition and once on a condition available before acting. The two calls share parameters and feature grammar but deliberately have different endpoint gradients. We first give the single-task construction and then extend it to a shared multi-task encoder. ```{=latex} \begin{figure*}[t] \includegraphics[width=0.98\textwidth]{figures/planning_pipeline.pdf} \caption{\textbf{INTACT converts mandatory search into optional verification.} LeWM samples raw actions from an uninformed Gaussian and uses forward rollouts to become goal-directed. INTACT alternates the raw goal displacement, the shared conditional action mean, and the unchanged Forward Predictor to produce a Direct plan. It can execute immediately or locally verify that plan with $K=128$, $I=3$, and $\sigma_0=0.25$.} \label{fig:planning_pipeline} \end{figure*} ``` ## Problem Setting and the Forward-Action Gap `\Cref{fig:overview,fig:planning_pipeline}`{=latex} show the same system from its training and deployment views. In `\cref{fig:overview}`{=latex}, superscript $k$ indexes a visual domain and its task-specific heads; below we rename that index $\tau$ and reserve $k$ for temporal position inside a window. For a single task we omit $\tau$. This removes a notational collision between the four colored task routes in `\cref{fig:overview}`{=latex} and the recurrent planning steps in `\cref{fig:planning_pipeline}`{=latex}. For domain $\tau$, let the offline data be pixel--action trajectories $\mathcal D^\tau=\{(\boldsymbol o_{i,s}^\tau,\boldsymbol u_{i,s}^\tau)\}$, where $i$ indexes trajectories and $s$ indexes environment steps. We index model-chunk boundaries by $t$ and form the fixed action block $\boldsymbol a_{i,t}^\tau=[\boldsymbol u_{i,Bt}^\tau;\ldots; \boldsymbol u_{i,Bt+B-1}^\tau]\in\mathbb{R}^{Bd_a}$ with $B=5$. A training window spans $H=5$ such chunks from $o_t$ to $o_{t+H}$, and the shared encoder-projector gives $z_j=E_\theta(o_j)\in\mathbb{R}^d$ and $z_g=z_{t+H}$. At teacher step $k\in\{0,\ldots,H-1\}$, the orange local call in `\cref{fig:overview}`{=latex} uses $z_{t+k+1}-z_{t+k}$, while the green goal call uses $\operatorname{sg}(z_g)-z_{t+k}$. They enter the same four typed input slots of $G_\eta$ and both predict the fixed one-chunk target $a_{t+k}$; the $\widehat A_t^\tau$ box in the figure denotes this conditional action output. The goal endpoint stays fixed while its remaining offset is $r=H-k\in\{5,4,3,2,1\}$ chunks. Thus \`\`multi-horizon" means multiple remaining goal offsets, not multiple output chunk lengths, and the final raw goal policy does not receive $r$ explicitly. At deployment, the lower route of `\cref{fig:planning_pipeline}`{=latex} reuses exactly these symbols: encode the current and goal images as $z_t$ and $z_g$, set $\widetilde z_0=z_t$, then recurrently compute $m_k=z_g-\widetilde z_k$, $\bar a_k=\mu_\eta(\widetilde z_k,m_k,\bar a_{k-1})$, and $\widetilde z_{k+1}=F_\phi(\widetilde z_k,\bar a_k)$. The one-chunk means form the plan $\bar A=(\bar a_0,\ldots,\bar a_{H-1})$ corresponding to the training-time action output in `\cref{fig:overview}`{=latex}. Direct executes its first block and replans from the next observation; Guarded A only verifies this plan locally. The upper route in `\cref{fig:planning_pipeline}`{=latex} instead initializes raw actions independently of these learned intent coordinates. Across both routes, the unchanged LeWM Forward Predictor consumes the last $N=3$ latents and action blocks, $$\widehat z_{t+1}=F_\phi(z_{t-N+1:t},A_\omega(a_{t-N+1:t})), \label{eq:forward}$$ and is trained with prediction plus SIGReg, $$\mathcal L_{\rm world} =\mathbb E[\|\widehat z_{t+1}-z_{t+1}\|_2^2/d] +\lambda_{\rm sig}\mathcal R_{\rm SIG}. \label{eq:worldloss}$$ This learns how a supplied action changes the latent, but not how to obtain an action from a deployable latent request. LeWM consequently samples actions and uses the forward model only to rank them. ## Action-Aligned Intent Coordinates {#sec:single_theory} The center of `\cref{fig:overview}`{=latex} gives the graphical counterpart of this section: its attached local path carries a realized one-chunk successor, while its stop-gradient goal path carries a condition available before acting. In `\cref{fig:planning_pipeline}`{=latex}, only the latter can enter the lower deployment route before the next action is chosen. The endpoint $y$, displacement $m_z(y)$, and conditional law below formalize those two pictured paths and explain why neither one alone closes the forward--action gap. For current latent $z$ and endpoint condition $y$, let $p_E^\star(a\mid z,y)$ be the expert action law. ::: definition **Definition 1** (Conditional action equivalence). *`\label{def:quotient}`{=latex} $$y\sim_z y'\quad\Longleftrightarrow\quad p_E^\star(a\mid z,y)=p_E^\star(a\mid z,y'). \label{eq:quotient}$$* ::: The quotient $\mathcal Y/{\sim_z}$ is in one-to-one correspondence with the realizable action-law image at $z$. ::: proposition **Proposition 1** (Quotient-to-law identification). *`\label{prop:quotient_image}`{=latex} $\Phi_z([y]_z)=p_E^\star(\cdot\mid z,y)$ is well defined and bijective from $\mathcal Y/{\sim_z}$ onto $\{p_E^\star(\cdot\mid z,y):y\in\mathcal Y\}$.* ::: The displacement $$m_z(y)=y-z \label{eq:motion_intent}$$ is a convenient motion-intent coordinate, not a claim that latent dynamics are globally linear or that equivalent endpoints must coincide. A physical inverse likelihood, $$\mathcal L_{z^+}=\mathbb E[-\log p_\eta(a_t\mid z_t,z_{t+1},a_{t-1})], \label{eq:inverse_nll}$$ asks the encoder to preserve action-recoverable transitions, but it is circular at deployment because $z_{t+1}$ is observed only after acting. Conversely, a goal-conditioned likelihood is deployable but, by itself, is GCSL-like hindsight imitation without an attached one-chunk physical anchor. This is the central asymmetry: $$\underbrace{z_{t+1}}_{\substack{\text{physical but unavailable}\\ \text{before acting}}} \qquad\text{versus}\qquad \underbrace{z_g}_{\substack{\text{available but not a}\\ \text{one-step successor}}}.$$ INTACT uses one nonlinear conditional operator for both. This is the single INTACT Predictor drawn twice in `\cref{fig:overview}`{=latex}; its goal invocation is the action-generating operator on the lower route of `\cref{fig:planning_pipeline}`{=latex}: $$G_\eta:(z,m,a_{t-1})\longmapsto p_\eta(a\mid z,m,a_{t-1}), \label{eq:homologous_predictors}$$ while retaining the separate Forward Predictor $F_\phi:(z,a)\mapsto z^+$. The final goal-displacement route constructs $$\boxed{m_t^{\rm local}=z_{t+1}-z_t,\qquad m_t^{\rm goal}=\operatorname{sg}(z_g)-z_t.} \label{eq:local_goal_conditions}$$ The local endpoint remains attached; stop-gradient applies only to the future goal occurrence in the deployment loss. The same $z_g$ is therefore plastic when it appears as a real successor and fixed when it serves as an intent anchor. Current-state latents, $G_\eta$, and the independent forward/physical branches continue to receive gradients. For route $\chi$, the physical and deployable condition families are $$\begin{aligned} \mathcal C_{\rm phy}^{\chi}&=\{(z_t,c_t^{\rm phy,\chi},a_{t-1})\},\\ \mathcal C_{\rm dep}^{\chi}&=\{(z_t,c_t^{\rm dep,\chi},a_{t-1})\}, \label{eq:condition_families} \end{aligned}$$ and one shared action operator serves their union, $$G_\eta:\mathcal C_{\rm phy}^{\chi}\cup\mathcal C_{\rm dep}^{\chi} \rightarrow\mathcal P(\mathcal A). \label{eq:shared_action_operator}$$ Its paired proper likelihood is $$\begin{aligned} \mathcal L_{\rm I2A}^{\chi} ={}&\lambda_{\rm inv}\mathbb E[-\log p_\eta(a_t\mid z_t,c_t^{\rm phy,\chi},a_{t-1})]\notag\\ &+\lambda_{\rm goal}\mathbb E[-\log p_\eta(a_t\mid z_t,c_t^{\rm dep,\chi},a_{t-1})]. \label{eq:family_action_alignment} \end{aligned}$$ The first NLL is the attached physical call in `\cref{fig:overview}`{=latex}, and the second is its stop-gradient deployment-goal call. Thus `\cref{eq:condition_families,eq:shared_action_operator,eq:family_action_alignment}`{=latex} are the set, operator, and loss views of the same two-arrow construction in the figure. No term minimizes $\|m_t^{\rm goal}-m_t^{\rm local}\|^2$. The two calls may have different input scales and output Gaussians. Here **isomorphic** again has two levels: their backbone-input graphs are **isomorphic** by typed slots and shared parameters, while their supported motion-intent families are **isomorphic** through the conditional action-law relation represented by $G_\eta$. Neither level asserts numerical latent equality. ::: proposition **Proposition 2** (Off-support non-identifiability). *`\label{prop:nonidentifiability}`{=latex} If inverse risk is evaluated only on realized successors, any two actors that agree on that support have identical risk even when they disagree on a deployable goal condition outside it.* ::: ::: proposition **Proposition 3** (Supported quotient recovery). *`\label{prop:proper}`{=latex} For a well-specified conditional actor, population proper-NLL minimization recovers $p_E^\star(\cdot\mid z,y)$ on every supported condition. Hence action-equivalent representatives receive the same predicted action law.* ::: Proofs, the horizon-normalized waypoint derivation, the Bernoulli endpoint variant, and the direct-linearity negative control are collected in `\cref{app:proofs,app:method_derivations}`{=latex}. The waypoint request is an ablation; the headline model uses the raw displacement in `\cref{eq:local_goal_conditions}`{=latex} because it preserves direction and remaining distance without imposing equal progress. The single-task objective is $$\boxed{\mathcal L_{\rm ST}=\mathcal L_{\rm world} +\lambda_{\rm inv}\mathcal L_{z^+} +\lambda_{\rm goal}\mathcal L_{\rm goal}.} \label{eq:objective}$$ This end-to-end coupling makes action-law aliasing visible to the encoder: a frozen head cannot recover a controllable distinction already collapsed by $E_\theta$. ```{=latex} \begin{algorithm}[t]\caption{INTACT training} \label{alg:train} \begin{algorithmic}[1] \Require window $(o_{t:t+H},a_{t-1:t+H-1})$; shared $E_\theta,F_\phi,G_\eta$ \State $z_{t:t+H}\gets E_\theta(o_{t:t+H})$; $z_g^{\rm anchor}\gets\operatorname{sg}(z_{t+H})$ \For{$k=0,\ldots,H-1$} \State $j\gets t+k$; $(m_j^{\rm local},m_j^{\rm goal})\gets (z_{j+1}-z_j,z_g^{\rm anchor}-z_j)$ \State accumulate physical and goal NLLs through the same $G_\eta$ \EndFor \State update $(\theta,\phi,\eta)$ with $\mathcal L_{\rm world}+\lambda_{\rm inv}\mathcal L_{z^+} +\lambda_{\rm goal}\mathcal L_{\rm goal}$ \end{algorithmic} \end{algorithm} ``` ## INTACT Predictor and Control `\Cref{fig:overview}`{=latex} depicts the identical input wiring that makes the two INTACT Predictor invocations structurally isomorphic; the four incoming quantities are written explicitly as $$h_t^\Delta(m_t)=[z_t;m_t;z_t\odot m_t;A_\omega(a_{t-1})]. \label{eq:actor_features}$$ The product is a cheap state--intent interaction: $$w^\top(z_t\odot m_t)=z_t^\top\operatorname{diag}(w)m_t, \label{eq:product_interaction}$$ allowing the same displacement to be interpreted differently across contact, pose, and obstacle states. A three-layer MLP outputs a diagonal Gaussian with log standard deviation clipped to $[-5,2]$. Detailed endpoint grammar and the matched A--G product ablation are in `\cref{app:design_selection}`{=latex}. At deployment, the lower INTACT route in `\cref{fig:planning_pipeline}`{=latex} is the recurrent composition of `\cref{eq:actor_features}`{=latex} with the unchanged Forward Predictor: the Gaussian mean generates an $H$-block plan and re-encodes after executing the next real action block. Direct uses no sampled candidate and no terminal cost. Optional Guarded A centers a small $128\times3$ raw-action search at this plan, retains the Direct reference and the globally best candidate, and uses $\sigma_0=0.25$. Pure CEM disables the INTACT Predictor and recovers the upper search-first route in the figure; it is retained only as a compatibility and actor-disabled control. `\Cref{fig:planning_pipeline,alg:control}`{=latex} therefore give the visual and algorithmic views of the same Direct controller and optional local verification. The complete verifier matrix appears in `\cref{app:planner_matrix}`{=latex}. ```{=latex} \begin{algorithm}[t]\caption{INTACT Direct control with optional local verification} \label{alg:control} \begin{algorithmic}[1] \Require latent history, goal, and inference mode $v$ \State $\widetilde z_0\gets z_t$ \For{$k=0,\ldots,H-1$} \State $m_k\gets z_g-\widetilde z_k$; $\bar a_k\gets\mu_\eta(\widetilde z_k,m_k,\bar a_{k-1})$ \State $\widetilde z_{k+1}\gets F_\phi(\widetilde z_k,\bar a_k)$ \EndFor \If{$v=\mathrm{Direct}$} \State \Return $\bar A$ with zero candidates and cost calls \EndIf \If{$v=\mathrm{one\mbox{-}shot}$} \State score $\bar A$ and clipped local perturbations; return the observed minimum \Else \State initialize $(\mu,A_{\rm best})\gets(\bar A,\bar A)$ and $\sigma\gets0.25$ \For{$i=1,\ldots,3$} \State score 128 samples plus $\bar A,A_{\rm best}$; refit on 16 elites \EndFor \State re-score $\mu$; return the best observed reference or candidate \EndIf \end{algorithmic} \end{algorithm} ``` ## Multi-Task Extension {#sec:multitask_theory} For heterogeneous tasks $\tau\in\{1,\ldots,T\}$, INTACT shares the encoder and projector but keeps task-specific Forward, action-embedding, and INTACT heads. The known domain index selects a head; no reward or additional semantic label is used. The objective is $$\mathcal L_{\rm MT}=\sum_{\tau=1}^{T}\rho_\tau \left[\mathcal L_{\rm world}^{\tau} +\lambda_{\rm inv}\mathcal L_{z^+}^{\tau} +\lambda_{\rm goal}\mathcal L_{\rm goal}^{\tau}\right]. \label{eq:multitask_objective}$$ Every task therefore shapes the same visual coordinates without forcing heterogeneous actions into one padded space. The controlled factorial compares forward-only LeWM, physical inverse-only, intent-only, and Full INTACT under matched deterministic Math-SDPA. We test both waypoint and raw-goal intent coordinates. Direct SR measures the deployed conditional; actor-disabled CEM tests the encoder--Forward stack under a fixed optimizer; frozen probes and predicted--expert CKA/kNN diagnose representation and family conversion. None of these diagnostics replaces closed-loop SR. # Experimental Design {#sec:experiments} #### Tasks and evaluation. We use the four official LeWM domains: PushT, OGBench Cube, DMC Reacher, and TwoRoom [@maes2026lewm; @park2025ogbench]. Models receive RGB observations and five-step action blocks; simulator state is used only by probes and success predicates. Headline SR follows the official evaluator for comparability. Unless stated otherwise, every checkpoint is evaluated with seeds $\{0,1,42\}$ and 100 episodes per seed. We separately report the public CLEAR-LeWM v0.5.1 Moderate audit [@sun2026clearlewm], which removes initially solved pairs and repairs known Reacher and TwoRoom evaluator defects. Official and CLEAR scores are never pooled. ## Single-Task Protocol {#sec:single_experiments} Each task-specific model is trained from scratch on its complete dataset for one epoch with batch size 256, AdamW learning rate $5\times10^{-4}$, forward weight 1.0, inverse weight 0.1, goal weight 0.05, and three training seeds. The final goal-displacement model uses SIGReg 0.02 and the shared grammar in `\cref{eq:actor_features}`{=latex}. A matched waypoint chain isolates forward-only LeWM, physical inverse supervision, deployment-intent supervision, and the low-SIGReg setting. Actor sharing is tested against parameter-matched, double-capacity, two-output, and condition-token alternatives. Direct asserts zero candidate sequences and zero terminal-cost calls. Pure CEM disables the INTACT Predictor and evaluates the encoder--Forward stack under a fixed raw-action optimizer; it is a functional control, not an encoder-only statistic. Guarded A evaluates 128 sequences for three iterations around the Direct plan. The complete 576-job planner matrix, coordinate grammar, SIGReg, straightening, optimizer, and search-radius selection studies are deferred to `\cref{app:design_selection,app:planner_matrix}`{=latex} because they select a final configuration rather than establish the principal result. Frozen probes use episode-disjoint train/test episodes. We measure physical state and transition-action $R^2$, effective rank, per-dimension standard deviation, collapsed-dimension fraction, and mean cosine similarity. Together with actor-disabled CEM, these distinguish representation shaping from the benefit of executing the learned action conditional. ## Multi-Task Protocol {#sec:multitask_experiments} One distributed job traverses all four datasets. The ViT-Tiny/14 encoder, projector, and 192-dimensional latent are shared; each task retains a small Forward Predictor, action embedding, and INTACT Predictor. Training uses batch size 256, learning rate $5\times10^{-4}$, weight decay $10^{-3}$, SIGReg 0.03, and five epochs. Encoder and Forward attention are forced to deterministic Math-SDPA and every checkpoint stores backend and data fingerprints. The controlled factorial contains forward-only LeWM, physical inverse-only, intent-only, and Full INTACT. Intent-only and Full are each trained with the waypoint and goal-displacement coordinates, producing six cells. All cells use training seeds $\{0,42,3072\}$ and are evaluated at E1--E5. At each epoch we run native Direct inference where available, actor-disabled pure CEM $300\times30$, Guarded A for actor-equipped cells, frozen probes, and latent diagnostics. The complete coverage contract and E1/epoch tables are reported in `\cref{app:extended_results}`{=latex}. For mechanism analysis, predicted--expert linear CKA, local kNN overlap, action $R^2$, and NLL are computed within each task's native action coordinates and averaged equally over tasks; heterogeneous action spaces are never zero-padded into one embedding. The primary cohort contains both Full INTACT interfaces, three seeds, and E1--E5. A separate task-conditioned gauge audit fits latent coordinate maps on paired, action-free calibration episodes and tests correct pairing, shuffling, calibration size, cross-seed ceilings, and reverse actor swaps under CLEAR Moderate. Full estimator definitions and controls appear in `\cref{app:diagnostics,app:extended_results}`{=latex}. # Results {#sec:results} ## Single-Task Results {#sec:single_results} #### One epoch makes search optional. `\Cref{tab:published_landscape}`{=latex} places the headline on a resource-aware axis. Published rows retain their own protocols and are not paired significance controls. Every row in this single-task comparison trains a separate visual model per domain; the shared-encoder study begins in `\cref{sec:multitask_results}`{=latex}. ```{=latex} \begin{table*}[t] \scriptsize \setlength{\tabcolsep}{3.1pt} \renewcommand{\arraystretch}{0.98} \caption{\textbf{Published landscape and matched task-specific INTACT inference matrix (official SR, \%).} Published values are transcribed from the cited papers and are not paired controls; $\dagger$ marks partial coverage or a distinct reproduction protocol. $\ddagger$ marks our matched-E1 reproduction: one task-specific training seed with mean $\pm$ sample standard deviation over evaluation manifests $\{0,1,42\}$ (100 episodes each), rather than training-seed uncertainty. INTACT values are mean $\pm$ sample standard deviation over three training seeds, each averaged over three 100-episode evaluation seeds. The INTACT block audits all eight exact objective--inference pairings on their matched checkpoint families.} \label{tab:published_landscape} \label{tab:single_direct} \begin{adjustbox}{max width=\textwidth} \begin{tabular}{lllcrrrrrr} \toprule Method / objective & Training & Deployment & Seq. & PushT & Cube & Reacher & TwoRoom & Macro \\ \midrule DINO-WM \citep{zhou2025dinowm} & frozen $E$ & CEM (published) & 9k & $74.0\!\pm\!4.5$ & $86.0\!\pm\!4.7$ & $79.0\!\pm\!5.1$ & $100.0\!\pm\!.0$ & 84.75 \\ LeWM \citep{maes2026lewm} & E2E 10 ep & CEM $300{\times}(30/10)$ & 9k/3k & $96.0\!\pm\!4.0$ & $74.0\!\pm\!3.0$ & $86.0\!\pm\!5.0$ & $87.0\!\pm\!2.5$ & 85.75 \\ Fast-LeWM \citep{gao2026fastlewm} & E2E 10 ep & matched CEM & 9k & 96.0 & 80.0 & 88.0 & 98.0 & 90.50 \\ Fast-LeWM + SC \citep{gao2026fastlewm} & E2E 10 ep & CEM + score & 9k & 98.0 & 82.0 & 90.0 & 98.0 & 92.00 \\ Qantara$^\dagger$ \citep{rakhimov2026qantara} & E2E 10 ep & CEM $300{\times}30$, $K=4$ & 9k & $90.1\!\pm\!1.1$ & $93.7\!\pm\!.7$ & $80.9\!\pm\!1.8$ & $100.0\!\pm\!.0$ & 91.18 \\ GC-IDM \citep{nguyen2026gcidm} & LeWM + 50-ep head & Direct & 0 & $84.7\!\pm\!5.0$ & $99.3\!\pm\!1.2$ & $100.0\!\pm\!.0$ & $100.0\!\pm\!.0$ & 96.00 \\ PRISM$^\dagger$ \citep{wang2026prism} & LeWM + 50-ep head & MPPI $128{\times}30$ & 3,840 & $89\!\pm\!4$ & $79\!\pm\!6$ & -- & -- & -- \\ C-JEPA$^\dagger$ \citep{nam2026cjepa} & frozen $E$ + 30 ep & CEM $300{\times}30$ & 9k & 88.67 & -- & -- & -- & -- \\ \midrule \emph{Matched Math-SDPA E1 reproduction on a frozen task-specific E1 LeWM encoder} \\ GC-IDM (matched E1)$^\ddagger$ & frozen E1 $E$ + 1-ep head & Direct & 0 & $9.33\!\pm\!2.31$ & $69.33\!\pm\!7.02$ & $84.33\!\pm\!2.08$ & $99.33\!\pm\!.58$ & 65.58 \\ PRISM (matched E1)$^\ddagger$ & frozen E1 $E$ + 1-ep head & PoG-MPPI $128{\times}30$ & 3,840 & $18.67\!\pm\!5.51$ & $86.67\!\pm\!1.15$ & $66.33\!\pm\!1.15$ & $88.33\!\pm\!1.53$ & 65.00 \\ \midrule \emph{Matched task-specific INTACT, E2E one epoch} \\ Waypoint INTACT & E2E 1 ep & Direct & 0 & $77.67\!\pm\!.88$ & $99.89\!\pm\!.19$ & $88.11\!\pm\!.69$ & $98.00\!\pm\!.33$ & $90.92\!\pm\!.30$ \\ & & Pure CEM $300{\times}30$ & 9k & $89.67\!\pm\!1.45$ & $67.67\!\pm\!1.33$ & $82.67\!\pm\!1.20$ & $77.33\!\pm\!2.03$ & $79.33\!\pm\!.75$ \\ & & Actor-on CEM $300{\times}30$ & 9k & $92.89\!\pm\!.84$ & $96.56\!\pm\!1.07$ & $80.67\!\pm\!2.08$ & $98.00\!\pm\!.67$ & $92.03\!\pm\!1.06$ \\ & & {\\bfseries\\boldmath Guarded A $128{\times}3$} & {\\bfseries\\boldmath 384} & {\\bfseries\\boldmath $89.89\!\pm\!.38$} & {\\bfseries\\boldmath $99.78\!\pm\!.19$} & {\\bfseries\\boldmath $88.56\!\pm\!1.26$} & {\\bfseries\\boldmath $97.89\!\pm\!.19$} & {\\bfseries\\boldmath $94.03\!\pm\!.25$} \\ \addlinespace[1pt] {\\bfseries\\boldmath INTACT, goal displacement} & {\\bfseries\\boldmath E2E 1 ep} & {\\bfseries\\boldmath Direct} & {\\bfseries\\boldmath 0} & $85.78\!\pm\!1.54$ & {\\bfseries\\boldmath $100.00\!\pm\!.00$} & {\\bfseries\\boldmath $97.67\!\pm\!.00$} & $97.89\!\pm\!1.26$ & {\\bfseries\\boldmath $95.33\!\pm\!.58$} \\ & & Pure CEM $300{\times}30$ & 9k & $88.44\!\pm\!1.17$ & $68.44\!\pm\!.77$ & $83.67\!\pm\!.67$ & $82.89\!\pm\!.84$ & $80.86\!\pm\!.51$ \\ & & Actor-on CEM $300{\times}30$ & 9k & {\\bfseries\\boldmath $93.56\!\pm\!.96$} & $96.89\!\pm\!.19$ & $86.67\!\pm\!.88$ & {\\bfseries\\boldmath $98.00\!\pm\!1.15$} & $93.78\!\pm\!.77$ \\ & & {\\bfseries\\boldmath Guarded A $128{\times}3$} & {\\bfseries\\boldmath 384} & {\\bfseries\\boldmath $92.22\!\pm\!.69$} & {\\bfseries\\boldmath $99.78\!\pm\!.19$} & {\\bfseries\\boldmath $97.44\!\pm\!.77$} & {\\bfseries\\boldmath $98.00\!\pm\!1.15$} & {\\bfseries\\boldmath $96.86\!\pm\!.38$} \\ \bottomrule \end{tabular} \end{adjustbox} \end{table*} ``` ```{=latex} {\\bfseries\\boldmath INTACT uses the fewest reported full-data passes among methods with disclosed epoch schedules: one end-to-end epoch, no frozen-encoder phase, and no separately trained phase-2 controller. Direct reaches 95.33 macro SR with zero candidates and 2.9--5.5\,ms planner-side latency. Guarded A evaluates 384 rather than 9,000 sequences and reaches 96.86 macro, placing INTACT at the state-of-the-art level among currently reported JEPA controllers.} ``` The matched-E1 Math-SDPA rows ask a narrower question than the published headlines: how much control can a one-epoch head amortize from the same frozen, task-specific E1 LeWM representation? GC-IDM uses zero-search Direct control, whereas PRISM uses PoG-MPPI with $128{\times}30=3{,}840$ candidates. Their macros are 65.58 and 65.00, respectively, but the taskwise ordering reverses: PRISM is higher on PushT/Cube, while GC-IDM is higher on Reacher/TwoRoom. Because the objectives and inference budgets differ and each row uses one training seed, this is a controlled E1 stress test rather than a paired compute or significance ranking. Descriptively, the end-to-end one-epoch INTACT Direct row reaches 95.33 macro without candidate search. The complete matched block in `\cref{tab:single_direct}`{=latex} separates training coordinates from deployment algorithms. Goal-displacement INTACT reaches $85.78/100.00/97.67/97.89$% Direct SR on PushT/Cube/Reacher/TwoRoom, averaged over three independently trained models per task. Relative to published LeWM, it uses one tenth as many full-data passes and removes up to 9,000 candidate sequences. Goal displacement outperforms the waypoint coordinate under every matched interface: by 4.41/1.53/1.75/2.83 macro points for Direct, pure CEM, actor-on CEM, and Guarded A, respectively. The actor-disabled macros remain close (80.86 versus 79.33), showing that broad raw-action search largely discards the learned intent interface. More decisively, Goal actor-on CEM spends 9,000 sequences yet falls 1.55 points below zero-search Direct (93.78 versus 95.33), whereas Guarded A raises Direct by 1.53 points to 96.86 with only 384 sequences. The waypoint coordinate shows the same efficiency ordering: Guarded A reaches 94.03 versus 92.03 for actor-on CEM. Search is therefore useful as local verification around the learned plan, while unconstrained iterative search can recreate a train--deployment gap and waste the learned conditional. The separate CLEAR-LeWM Moderate audit preserves the same ordering: Direct and Guarded A reach $83.33\!\pm\!1.00$ and $84.83\!\pm\!1.32$ macro, versus $65.47\!\pm\!.77$ for pure CEM. Reacher drops from 97.67 official to 49.56 Moderate after angle-topology correction, illustrating why CLEAR is reported separately rather than pooled. The complete taskwise table and protocol contract are in `\cref{tab:clear_main,app:clear_scope}`{=latex}. The matched actor-sharing audit is deferred to `\cref{tab:actor_sharing,app:actor_sharing}`{=latex}: separating the two calls loses 5.67 PushT points at matched capacity, and even doubling actor capacity remains 4.00 points below the fully shared conditional. ```{=latex} \begin{figure*}[t] \includegraphics[width=0.92\textwidth]{figures/query_representation.pdf} \caption{\textbf{Single-task action likelihoods shape the representation.} Actor-disabled CEM evaluates the encoder--Forward stack under a fixed optimizer; episode-disjoint probes measure readable state and transition action. Physical inverse supplies attached successor shaping, whereas the goal branch updates current-state coordinates and the shared actor with a stop-gradient goal anchor. Error bars are sample standard deviations over three PushT training seeds.} \label{fig:query_representation} \end{figure*} ``` #### The encoder improves before the actor is used. On PushT, adding physical inverse supervision raises actor-disabled CEM $30\times10$ from $42.22\!\pm\!9.50$ to $57.67\!\pm\!2.52$% and frozen endpoint-action $R^2$ from $.673\!\pm\!.062$ to $.791\!\pm\!.003$. Adding matched waypoint supervision raises CEM to $61.44\!\pm\!2.04$%; the larger Direct improvement primarily identifies the deployable conditional. The low-SIGReg waypoint model reaches $69.44\!\pm\!1.64$% actor-off CEM and $77.67\!\pm\!.88$% Direct. Its lower rank is not collapse: per-dimension standard deviation remains near one and no dimension collapses. #### The taskwise gains isolate the learned interface. Replacing the waypoint coordinate by goal displacement changes Direct SR by $+8.11/+0.11/+9.56/-0.11$ points on PushT/Cube/Reacher/TwoRoom. The gain is therefore concentrated in contact-sensitive PushT and continuously actuated Reacher, while Cube and TwoRoom are already near saturation. Under actor-disabled pure CEM, the same replacement improves macro SR by only 1.53 points (79.33 to 80.86); after the learned actor is restored, the Direct gain is 4.41 points and Guarded A reaches 96.86. This separation matters: the objective improves the world representation, but its larger benefit comes from exposing a deployment condition that the shared actor can interpret directly. Coordinate grammar, SIGReg, straightening, Huber, extra-epoch, interaction, and planner-radius controls are reported together in `\cref{app:design_selection,app:extended_results}`{=latex}. They select the final model but are not repeated as headline experiments. ```{=latex} \FloatBarrier ``` ## Multi-Task Results {#sec:multitask_results} We test the linked representation, control, and stability consequences on matched checkpoints below. #### One shared encoder improves all four domains. At E5, goal-displacement INTACT reaches $89.39\!\pm\!.77$ macro Direct SR, versus $66.17\!\pm\!2.67$ for matched shared-encoder LeWM with CEM $300\times30$. It improves LeWM by 5.66/32.23/12.56/42.44 points on PushT/Cube/Reacher/TwoRoom and exceeds the 85.75 published task-specific LeWM macro. Against the matched intent-only cell, adding the attached physical likelihood raises macro SR by $8.78\!\pm\!1.63$ points; the gain is positive on three tasks and $-0.44$ on the saturated Cube task. This is the cleanest evidence that Full INTACT is not merely a goal-conditioned behavior-cloning head. ```{=latex} \begin{figure*}[p] \includegraphics[width=0.95\textwidth]{figures/multitask_e5_sr_summary.pdf} \caption{\textbf{Controlled E5 success across four tasks.} Bars visualize the native rows of \cref{tab:multitask_e5}: LeWM uses CEM $300{\times}30$, learned heads use Direct, and the final bar uses Guarded A $128{\times}3$. Values are means $\pm$ sample standard deviations over three training seeds. The ordinate starts at 30\% to expose high-SR differences and is not a zero-based effect size.} \label{fig:multitask_e5_sr_summary} \scriptsize \setlength{\tabcolsep}{4.5pt} \renewcommand{\arraystretch}{1.20} \captionof{table}{\textbf{Controlled Math-SDPA shared-encoder E5 result.} Native interfaces, actor-disabled pure CEM, and optional verification are evaluated on the same checkpoints. Values are mean $\pm$ sample standard deviation over three training seeds; each seed averages official evaluation seeds $\{0,1,42\}$ with 100 episodes each.} \label{tab:multitask_e5} \begin{adjustbox}{max width=\textwidth} \begin{tabular}{llrrrrr} \toprule Training cell & Evaluation & PushT & Cube & Reacher & TwoRoom & Macro \\ \midrule \emph{Native interface (Direct where available)} \\ LeWM & CEM $300{\times}30$ & $74.56\!\pm\!3.67$ & $67.33\!\pm\!1.86$ & $83.11\!\pm\!.96$ & $39.67\!\pm\!8.39$ & $66.17\!\pm\!2.67$ \\ Inverse only & Direct & $35.00\!\pm\!1.45$ & $61.89\!\pm\!1.26$ & $61.00\!\pm\!.58$ & $68.78\!\pm\!3.83$ & $56.67\!\pm\!.22$ \\ Waypoint intent only & Direct & $58.89\!\pm\!.69$ & {\\bfseries\\boldmath $100.00\!\pm\!.00$} & $65.89\!\pm\!6.71$ & $72.11\!\pm\!9.83$ & $74.22\!\pm\!3.23$ \\ Goal intent only & Direct & $70.67\!\pm\!1.76$ & {\\bfseries\\boldmath $100.00\!\pm\!.00$} & $82.33\!\pm\!3.18$ & $69.44\!\pm\!6.75$ & $80.61\!\pm\!2.11$ \\ Waypoint INTACT & Direct & $71.22\!\pm\!3.36$ & $99.00\!\pm\!.67$ & $58.22\!\pm\!9.10$ & $77.22\!\pm\!2.52$ & $76.42\!\pm\!2.32$ \\ {\\bfseries\\boldmath Goal-displacement INTACT} & {\\bfseries\\boldmath Direct} & {\\bfseries\\boldmath $80.22\!\pm\!1.26$} & {\\bfseries\\boldmath $99.56\!\pm\!.77$} & {\\bfseries\\boldmath $95.67\!\pm\!1.76$} & {\\bfseries\\boldmath $82.11\!\pm\!4.11$} & {\\bfseries\\boldmath $89.39\!\pm\!.77$} \\ \midrule \emph{Actor-disabled pure CEM $300{\times}30$} \\ LeWM & Pure CEM & $74.56\!\pm\!3.67$ & $67.33\!\pm\!1.86$ & $83.11\!\pm\!.96$ & $39.67\!\pm\!8.39$ & $66.17\!\pm\!2.67$ \\ Inverse only & Pure CEM & $75.78\!\pm\!1.02$ & $66.78\!\pm\!2.36$ & $81.78\!\pm\!1.64$ & $51.44\!\pm\!1.07$ & $68.94\!\pm\!.43$ \\ Waypoint intent only & Pure CEM & $78.44\!\pm\!4.72$ & $68.67\!\pm\!1.76$ & $83.22\!\pm\!2.69$ & $39.11\!\pm\!8.17$ & $67.36\!\pm\!4.06$ \\ Goal intent only & Pure CEM & {\\bfseries\\boldmath $80.78\!\pm\!2.36$} & $68.00\!\pm\!3.18$ & {\\bfseries\\boldmath $84.67\!\pm\!1.33$} & $40.11\!\pm\!8.30$ & $68.39\!\pm\!2.22$ \\ Waypoint INTACT & Pure CEM & $78.22\!\pm\!3.02$ & $69.56\!\pm\!2.80$ & $83.00\!\pm\!2.19$ & $46.44\!\pm\!3.34$ & $69.31\!\pm\!.68$ \\ {\\bfseries\\boldmath Goal-displacement INTACT} & {\\bfseries\\boldmath Pure CEM} & $78.11\!\pm\!1.17$ & {\\bfseries\\boldmath $70.33\!\pm\!3.00$} & $80.33\!\pm\!2.03$ & {\\bfseries\\boldmath $51.56\!\pm\!4.53$} & {\\bfseries\\boldmath $70.08\!\pm\!1.13$} \\ \midrule \emph{Optional local verification} \\ {\\bfseries\\boldmath Goal-displacement INTACT} & {\\bfseries\\boldmath Guarded A $128{\times}3$} & {\\bfseries\\boldmath $85.00\!\pm\!2.85$} & {\\bfseries\\boldmath $99.00\!\pm\!.00$} & {\\bfseries\\boldmath $97.89\!\pm\!.69$} & {\\bfseries\\boldmath $80.00\!\pm\!4.41$} & {\\bfseries\\boldmath $90.47\!\pm\!.84$} \\ \bottomrule \end{tabular} \end{adjustbox} \par\footnotesize \noindent\textbf{Conclusion.} Preserving and organizing the state-conditioned intent--action family relation jointly determines whether actions can be read out directly, whether the actor-disabled world model remains plannable, whether low-budget search is effective, and whether shared multi-task training is stable. \normalsize \noindent\textbf{The preferred coordinate changes during shared training.} At E1, Waypoint INTACT is the strongest first-pass cell at $42.69\!\pm\!7.19$ macro, compared with $36.83\!\pm\!5.63$ for Goal INTACT; its advantage is concentrated in PushT, Cube, and high-variance TwoRoom. By E5, the ordering reverses: goal displacement reaches $89.39\!\pm\!.77$ versus $76.42\!\pm\!2.32$ for waypoint, with per-task differences of $+9.00/+0.56/+37.45/+4.89$ points. Thus E1 measures optimization speed, whereas E5 reveals the stronger converged intent coordinate. \hfill \normalsize \noindent\textbf{Actor-disabled controls separate representation from readout.} Goal INTACT reaches $70.08\!\pm\!1.13$ pure-CEM macro versus $66.17\!\pm\!2.67$ for LeWM. Its native Guarded A result is $90.47\!\pm\!.84$, only $1.08\!\pm\!.14$ above Direct, again supporting search as a verifier. E1 matrices, all inference modes, CLEAR Moderate scores, and controlled E1/E5 probes are in \cref{tab:multitask_core,tab:multitask_inference_decomp,tab:multitask_diagnostics}. The higher effective rank of E5 waypoint intent-only (93.87 versus 89.26) but lower SR (74.22 versus 89.39) confirms that rank is not a semantic certificate. \end{figure*} ``` ```{=latex} \clearpage ``` ```{=latex} \begin{table*}[p] \small \setlength{\tabcolsep}{4.5pt} \renewcommand{\arraystretch}{1.45} \caption{\textbf{Theory-linked evidence for shared four-task INTACT.} Correlations use official Direct SR; the actor-disabled and gauge rows are matched interventions that test representation and correspondence directly.} \label{tab:multitask_theory_evidence} \begin{tabularx}{\textwidth}{@{}>{\arraybackslash}p{3.0cm}>{\arraybackslash}p{7.15cm}>{\arraybackslash}X@{}} \toprule Evidence & Measured result & Interpretation \\ \midrule Pred.--expert kNN\newline (45 eligible E1--E5 checkpoints) & \makecell[l]{Pooled {\\bfseries\\boldmath $r=.954$} $[.928,.969]$;\\ adjusted {\\bfseries\\boldmath $r=.902$}.\\ Within Waypoint/Goal: {\\bfseries\\boldmath $.968/.981$}.} & Local action-family neighborhoods track deployable control. \\ Pred.--expert linear CKA\newline (same 45 checkpoints) & \makecell[l]{Pooled {\\bfseries\\boldmath $r=.897$} $[.837,.930]$.\\ Within Waypoint/Goal: {\\bfseries\\boldmath $.979/.986$}.\\ Leave-one-epoch-out: {\\bfseries\\boldmath $[.888,.914]$}.} & Global family geometry remains predictive across interfaces and epochs. \\ Pointwise action $R^2$\newline (same 45 checkpoints) & $r=.815$. & Recovering one expert action is weaker than preserving the family relation. \\ Effective-rank inversion\newline (controlled E5 cells) & Rank $93.87>89.26$, while SR $74.22<89.39$. & Latent spread is a capacity check, not a semantic certificate. \\ Actor-disabled planning\newline (matched E5 checkpoints) & Pure-CEM macro $66.17\rightarrow\mathbf{70.08}$. & Action losses improve the Encoder--Forward stack before actor execution. \\ Paired gauge intervention\newline (21,600 CLEAR Moderate episodes) & Shuffled $9.46\%\rightarrow\mathbf{68.04\%}$ paired ($+58.58$ pp). & Correct task-local correspondence, rather than coordinate identity, restores control. \\ \bottomrule \end{tabularx} \normalsize \noindent\textbf{Matched interfaces separate representation, readout, and search.} With the actor disabled, Goal INTACT changes LeWM by $+3.55/+3.00/-2.78/+11.89$ points on PushT/Cube/Reacher/TwoRoom: representation shaping is positive overall but not uniformly sufficient. Restoring the native Direct interface raises macro SR from 70.08 to 89.39, and adding Guarded A contributes only another $1.08\!\pm\!.14$ points. Conversely, Full Goal INTACT exceeds goal-intent-only Direct by 9.55/$-0.44$/13.34/12.67 points. The three comparisons jointly rule out a single-factor explanation: physical likelihood improves the Encoder--Forward stack, shared intent readout supplies most closed-loop control, and low-budget search performs a final local check. \medskip \noindent\textbf{Summary.} Physical inversion improves action-recoverable world features; deployment intent identifies a zero-search action interface; sharing forces both condition families to be interpreted by one conditional law. \hfill \normalsize Goal displacement is the strongest final coordinate without forcing latent trajectories to be linear. The single- and multi-task results agree: Direct is the native interface, actor-disabled planning remains useful, and small search is best understood as optional verification. \medskip \noindent\textbf{Why these measurements follow the theory.} The quotient-to-law view in \cref{prop:quotient_image,prop:proper} predicts that control should follow agreement between the predicted and expert action-law families. kNN and CKA operationalize its local neighborhoods and global centered geometry; pointwise $R^2$ deliberately asks the weaker single-action question. Effective rank then serves as a capacity falsifier, actor-disabled planning tests representation before readout, and the paired gauge intervention tests correspondence while allowing coordinates to change. Their observed ordering matches these roles: family metrics are strongest, higher rank can accompany lower SR, physical supervision improves pure CEM, and correct pairing restores 58.58 points without requiring coordinate identity. \end{table*} ``` ```{=latex} \clearpage ``` ```{=latex} \begin{figure*}[p] \includegraphics[width=0.96\textwidth]{figures/intent_vs_action_alignment_modes.png} \caption{\textbf{Intent--action relation, rather than task clustering, tracks control.} Columns separate waypoint and goal-displacement interfaces; rows compare task purity, pointwise action $R^2$, predicted--expert CKA, and local kNN overlap. Each point is one checkpoint and each annotation is a pooled checkpoint-level Pearson correlation with official four-task Direct SR. Historical runs are lower opacity and never rank the controlled objectives.} \label{fig:intent_action_alignment} \small \noindent\textbf{Family conversion predicts closed-loop control.} Across the 30 primary Math-SDPA Full-INTACT checkpoints, predicted--expert CKA and local kNN overlap correlate with official Direct SR at $r=.861$ and $.926$. Within complete waypoint/goal learning trajectories, CKA reaches $.979/.986$ and kNN $.968/.981$. Pooling the separately labeled strict replication cohort gives CKA $r=.897$ (95\% run-cluster bootstrap CI $[.837,.930]$) and kNN $r=.954$ ($[.928,.969]$); adjusted kNN remains $.902$. By contrast, intent task purity is negatively associated with SR. Cleaner task clusters are therefore not the desired relation: what matters is whether the state-conditioned intent family is carried into the corresponding action family. \hfill \small \noindent\textbf{Gauge intervention separates correspondence from actor function.} Correctly paired task-local calibration recovers 68.04\% CLEAR Moderate SR, whereas identity and shuffled-pair maps obtain 7.96\% and 9.46\%. Same- objective swaps define a high alignment ceiling, and LeWM/inverse backbones can also align into a trained Full actor. The reverse direction is decisive: a Full backbone mapped into a LeWM actor remains at 5.08\%, while an inverse actor recovers only inverse-level functionality. A coordinate map can restore a learned conditional but cannot create one absent from the target actor. The 72-cell, 21,600-episode audit and calibration curves are in \cref{fig:task_conditioned_gauge,app:extended_results}. \par \small \noindent\textbf{The eight panels reject a clustering-only explanation.} Across both deployment coordinates, task purity has the wrong sign, pointwise action recovery is informative but weaker, and relation-aware CKA/kNN give the most consistent monotone trends. The agreement between current Math-SDPA runs and the separately marked historical cohort further shows that the result is not produced by one checkpoint family. Together with the gauge intervention, the figure supports a narrower claim than global latent isomorphism: successful control requires each state-conditioned intent neighborhood to retain the action-family correspondence that its task-specific head can read. \end{figure*} ``` ```{=latex} \clearpage ``` # Limitations {#sec:limitations} #### Three seeds remain a coarse scaling estimate. The controlled multi-task matrix is complete for all six objective cells, three training seeds, three evaluation seeds per checkpoint, and E1--E5 under Math-SDPA. Official Direct, pure CEM, Guarded A, and CLEAR Moderate have full applicable coverage. The primary conversion audit is likewise complete for both Full INTACT interfaces, three seeds, and E1--E5. The separately marked historical replication cohort predates the backend audit and is used only for mechanism robustness, not objective ranking. The final single-task goal-displacement study now uses three training seeds on all four tasks, but three seeds still provide only a coarse estimate of training variability. #### The action quotient is identified only on demonstrated support. `\Cref{prop:proper}`{=latex} applies to endpoint conditions with positive sampling probability. It does not identify arbitrary counterfactual goals, nor does a single expert action reveal whether another action would have been equally valid. The full goal displacement preserves direction and distance but remains an approximate request around obstacles, contact switches, and multimodal demonstrations. The local-intent call physically anchors the representation, while the goal-intent call relies on demonstration support; neither construction proves correct behavior outside that support. Using the same goal-intent construction and INTACT Predictor at training and deployment removes the forward-model/action-proposal interface mismatch, not every train--test gap: after the first direct step, autoregressive control conditions on predicted latents whose distribution can drift from encoded demonstration states. #### The current diagnostics are necessary checks, not certificates. Effective rank and SRS can reward isotropic noise. $G_{\mathrm{sep}}$ measures response to deterministic goal changes, but two different goals can require the same current action, so it is only a proxy for an action-distinct quotient. Expert NLL shares the INTACT Predictor used by Direct and is therefore not an independent causal variable. A stronger test would pair action-equivalent and action-distinct counterfactual conditions, report normalized Jacobian spectra, and predict held-out checkpoint failures without refitting thresholds. The t-SNE geometry is qualitative and is not used for model selection. #### Direct Gaussian control can hide multimodality. The deployed action is the mean of a diagonal Gaussian. At junctions or contact transitions, this mean can lie between valid modes. Search can verify a proposal against forward dynamics, but then loses the zero-candidate identity. Mixture actors, sampling from the INTACT Predictor's learned conditional uncertainty instead of a unit-variance raw-action distribution, uncertainty-triggered verification, and quotient-aware mode selection are natural extensions; our current evidence supports a simple cross-task default rather than a universal action distribution. #### Gauge equivalence is task-manifold conditioned. The base Full--goal-only swap uses two training seeds; the expanded 72-cell audit strengthens causal controls but does not establish exact conjugacy or a single global affine chart. Correct pairing is necessary, and same-objective cross-seed swaps define a high alignment ceiling. However, LeWM and inverse-only backbones can also be aligned into a trained Full actor, so alignability is not an INTACT-exclusive representation property. Conversely, reverse swaps show that a gauge cannot create a deployment map absent from the target actor. Cross-task and leave-one-task-out transfer still fail. This task-local scope is sufficient for task-specific control heads; a single universal actor would require explicit cross-task alignment, adapters, or stronger joint coverage. #### Scope and evaluation. Experiments use four simulated tasks, fixed image goals, task-specific action heads, and offline expert trajectories. We have not established real-robot, out-of-distribution goal, distractor, or cross-embodiment generalization. The present headline uses the official LeWM benchmark to preserve direct comparability. Our stricter legality-aware audit is reported separately and should be validated independently. Finally, PRISM, GC-IDM, SMWM, QuoVLA, and related concurrent systems use different data, training schedules, and evaluators. We therefore compare their interfaces and assumptions, not unmatched success-rate numbers. ```{=latex} \FloatBarrier ``` # Conclusion INTACT starts from a mismatch that is easy to overlook: a physical successor is a valid inverse-dynamics condition but is unavailable before acting, whereas a deployable goal intent is available but is not itself an observed transition. Their useful commonality is neither endpoint type nor Euclidean proximity. At a fixed world state, it is the expert action law they induce. This conditional action quotient leads to a deliberately small statistical unit: each supervised conditional consists of one demonstrated state, one supported endpoint condition, and one correct action. The forward JEPA and SIGReg remain responsible for preserving the richer world representation. In the full objective, the two action losses also remain distinct: local-intent likelihood shapes the encoder through the attached real successor, whereas goal-intent likelihood shapes the current-state encoding toward the deployable conditional and has no successor-gradient path. This is a branch-local stop-gradient, not a detached batch or encoder: the same future latent is plastic when used as a physical successor and fixed when reused as a deployment anchor. Taken separately, these are familiar objectives: local inverse dynamics and a GCSL-like hindsight action likelihood. INTACT begins at their irreducible coupling: one weight-shared, graph-isomorphic conditional operator must explain both families in action-law space, while asymmetric endpoint gradients preserve their distinct physical and deployment roles and the Forward Predictor retains richer world information. The method therefore does not reduce to placing two losses beside each other. The controlled evidence supports the resulting predictions. In single-task training, the final goal-displacement model reaches 85.78%, 100.00%, 97.67%, and 97.89% official Direct SR after one epoch with no candidate search, with three training seeds on every task. The completed $2\times4$ task-specific matrix further shows that goal displacement exceeds the waypoint coordinate under Direct, pure CEM, actor-on CEM, and Guarded A. On the final coordinate, actor-on CEM spends 9,000 sequences yet falls from 95.33% Direct macro to 93.78%, whereas plan-centered Guarded A reaches 96.86% with 384 sequences. Controlled head-grammar interventions show a 7.00-point A$\rightarrow$E gain along the matched-product path. The new G control improves A by only 0.89 points when centering the product alone, whereas matching the centered first-order intent with its product yields a 3.89-point residual--product interaction; under the final goal coordinate, removing that matched product costs 2.56 points. Earlier matched PushT attribution shows that the waypoint encoder reaches $69.44\!\pm\!1.64$% with the actor removed under pure CEM $30{\times}10$, separating representation shaping from proposal quality. In shared four-task training, the deterministic Math-SDPA factorial gives Full INTACT the strongest E1 Direct macro, $42.69\!\pm\!7.19$%, and lower TwoRoom variation than waypoint-intent-only. At E5, Goal-displacement INTACT reaches $89.39\!\pm\!0.77$% Direct macro and exceeds matched three-seed goal-intent-only by $8.78\!\pm\!1.63$ points. Actor-disabled pure CEM improves from $66.17\!\pm\!2.67$% for LeWM to $70.08\!\pm\!1.13$%, while Guarded A raises the Full model to $90.47\!\pm\!0.84$%; search therefore remains a compatible verifier rather than the learned interface. Across 45 eligible simultaneous-dual-likelihood E1--E5 checkpoints, predicted--expert CKA/kNN correlate with official Direct SR at $r=0.897/0.954$; kNN remains $r=0.902$ after epoch/cohort adjustment. This mechanism audit complements, rather than replaces, the controlled performance comparison. On matched PushT waypoint training, parameter-matched independent actors lose 5.67 Direct points, and even double capacity remains 4.00 points below the fully shared operator. This directly supports sharing as a semantic coupling, not a parameter-count accident. The A--G head grammar supplies a complementary intra-call constraint: the first-order intent and its state--intent bilinear interaction must carry matched semantics inside each call. The deployment loss is still a goal-conditioned imitation likelihood; these controls show why the complete method is not reducible to goal-only or post-hoc imitation: it couples the GCSL-like deployment likelihood end to end with JEPA dynamics and the same shared conditional operator on attached one-chunk physical successors. The 21,600-episode gauge audit supplies a direct causal check. Correct pairing recovers 68.04% CLEAR Moderate SR, versus 9.46% after pair shuffling, and the calibration curve saturates near 64/full episodes. Same-objective swaps set a high alignment ceiling, while LeWM/inverse backbones also align into a trained Full actor; alignability is therefore not unique to INTACT. In the reverse direction, Full-backbone-to-LeWM-actor remains at 5.08%, and an inverse actor recovers only inverse-level functionality. A coordinate gauge can restore a learned map but cannot invent a deployment conditional absent from the actor. The complete 576-job planner audit adds a compatible efficiency result. Guarded A centers a $128{\times}3$ raw-action residual search on the deterministic INTACT plan and reaches 96.86% macro and 92.22% worst-task SR, exceeding matched pure CEM $300{\times}30$ by 16.00 macro points with 23.44$\times$ fewer sampled candidate sequences. Ordinary $\sigma_0=0.25$ is only 0.08 macro point lower. A diagnostic L2-trust variant raises PushT to 92.78% but is less uniform across tasks, so it is not part of the recommended inference interface. The finite correction region improves contact-sensitive PushT without discarding the precision already present in Reacher, whereas iterative actor-particle sampling compounds mixture and model error. The broader lesson is that amortized control need not be attached after world model learning or reduced to a planner-side sampling trick. A single JEPA can retain predictive world state, learn which conditions are equivalent for the current action, and expose that conditional directly at deployment. Search then becomes an optional verifier rather than the mandatory interface to the model. At a more general level, the construction is not tied to JEPA: whenever two condition families share supervised behavioral labels, a shared proper conditional operator can align their quotient semantics while asymmetric gradient routing preserves the role of each family. INTACT therefore offers a template for relation-preserving family mappings beyond world models, without requiring pointwise coordinate collapse. ```{=latex} \bibliographystyle{unsrtnat} ``` ```{=latex} \clearpage ``` ```{=latex} \appendix ``` # Appendix {#appendix .unnumbered} ```{=latex} \addcontentsline{toc}{section}{Appendix} ``` ```{=latex} \setcounter{figure}{0} ``` ```{=latex} \renewcommand{\thefigure}{A\arabic{figure}} ``` ```{=latex} \renewcommand{\theHfigure}{appendix.\arabic{figure}} ``` ```{=latex} \setcounter{table}{0} ``` ```{=latex} \renewcommand{\thetable}{A\arabic{table}} ``` ```{=latex} \renewcommand{\theHtable}{appendix.\arabic{table}} ``` # Proofs and Statistical Interpretation {#app:proofs} ```{=latex} \renewcommand{\qedsymbol}{} ``` ::: proof *Proof of `\cref{prop:nonidentifiability}`{=latex}.* Let $S=\operatorname{supp}(z,z^+)$. The successor-only risk is an integral of $-\log p_\eta(a\mid z,y)$ over triples with $(z,y)\in S$. If two actors agree on $S$, their integrands and therefore their risks are equal almost everywhere. Changing either actor on a goal intent $m$ outside $S$ cannot change that integral. Thus the successor-only objective does not identify deployment behavior at $m$. ◻ ::: ::: proof *Proof of `\cref{prop:proper}`{=latex}.* Condition on an endpoint pair $(z,y)$ with positive probability. Its expected negative log likelihood decomposes as $$\begin{aligned} &\mathbb E_{a\sim p_E^\star(\cdot\mid z,y)} [-\log p_\eta(a\mid z,y)]\\[-1pt] &\qquad=H[p_E^\star(\cdot\mid z,y)] +D_{\mathrm{KL}}(p_E^\star\|p_\eta). \end{aligned}$$ The entropy is constant in $\eta$ and the KL term is nonnegative, with equality only when the modeled and expert conditional laws agree almost everywhere. Applying this argument to every supported endpoint proves the result. If $y\sim_z y'$, both recovered laws equal the same expert law; if the expert laws differ, a proper likelihood preserves that distinction. ◻ ::: For a Bernoulli endpoint sample, linearity of expectation gives `\cref{eq:endpoint_nll}`{=latex}. A paired estimator of $(1-p)\ell(a;z^+)+p\ell(a;z+m^{\rm goal})$ can have the same population expectation. Our claim is consequently not that sampling changes the population optimum by itself. It keeps one conditional observation per transition, fixes the action-loss scale, and changes finite-batch gradient covariance under adaptive optimization. The paired controls test this finite-sample prediction. # Implementation and Evaluation Identities {#app:implementation} The encoder is ViT-Tiny/14 followed by a two-layer projection to 192 dimensions. The forward model is a six-layer causal transformer with 16 attention heads and history length three. One model action contains five environment actions. The shared INTACT Predictor is a three-layer width-1024 MLP with LayerNorm and GELU; it emits a diagonal-Gaussian mean and log standard deviation clipped to $[-5,2]$. Images use ImageNet normalization and actions use training-split statistics. The backbone has 18.03M trainable parameters and the shared INTACT Predictor adds 3.11M. Task heads are separate; the encoder and projector are shared across all four tasks. Training calls `seed_everything(seed, workers=True)` before dataset and model construction. Checkpoints are loaded in strict mode and every evaluation records the checkpoint hash, manifest hash, evaluator fingerprint, task, protocol, solver identity, and seed. ```{=latex} \scriptsize ``` ```{=latex} \begin{adjustbox}{max width=\columnwidth} \begin{tabular}{lccc} \toprule Mode & Actor & Candidates/iter. & Cost calls \\ \midrule Direct & ✓ & 0 & 0 \\ Pure CEM $5{\times}2$ & ✗ & 5 per iteration & $>0$ \\ Pure CEM $30{\times}10$ & ✗ & 30 per iteration & $>0$ \\ Pure CEM $300{\times}30$ & ✗ & 300 per iteration & $>0$ \\ \bottomrule \end{tabular} \end{adjustbox} ``` # Extended Method Derivations and Control Pseudocode {#app:method_derivations} ## Waypoint route and equal-progress assumption The waypoint attribution route solves, at teacher step $k$ with $r=H-k$, $$\min_{\delta_k,\ldots,\delta_{H-1}} \sum_{j=k}^{H-1}\|\delta_j\|_2^2 \quad\text{s.t.}\quad \sum_{j=k}^{H-1}\delta_j=z_g-z_{t+k}. \label{eq:query_variational}$$ Strict convexity gives the equal-progress request $$m_{t+k,H-k}^{\rm waypoint} =\frac{\operatorname{sg}(z_g)-z_{t+k}}{H-k}, \label{eq:query}$$ implemented in the endpoint grammar as $$q_{t+k,H-k}=z_{t+k}+m_{t+k,H-k}^{\rm waypoint}. \label{eq:waypoint_endpoint}$$ It is paired with the demonstrated one-chunk action, $$\mathcal L_{\rm goal}^{\rm waypoint} =\mathbb E\!\left[-\log p_\eta\!\left( a_{t+k}\,\middle|\, \substack{z_{t+k},q_{t+k,H-k},\\a_{t+k-1}} \right)\right]. \label{eq:query_nll}$$ This route narrows train--deployment support but divides away remaining-distance magnitude. The final raw-goal route instead uses $m_t^{\rm goal}=\operatorname{sg}(z_g)-z_t$ and lets the nonlinear actor choose state-dependent progress. The two routes share the same branch-local gradient rule. The future endpoint is detached only in the deployment occurrence; it remains attached when it is the real successor of the last physical transition. Neither a complete batch nor the encoder is detached. ## Why intent coordinates do not straighten trajectories A realized transition under the goal-conditioned actor may be written descriptively as $$\Delta z_t^{\rm real} =\alpha_t(z_t,m_t^{\rm goal},a_{t-1})m_t^{\rm goal}+r_t, \label{eq:adaptive_progress}$$ where $r_t$ can contain detours, contact changes, and multimodal residuals. The Forward Predictor updates reachability and the goal residual, $$\begin{aligned} &a_t\sim\pi_\eta(\cdot\mid z_t,m_t^{\rm goal},a_{t-1}),\\ &\widehat z_{t+1}=F_\phi(\widehat z_t,a_t),\\ &m_{t+1}^{\rm goal}=z_g-\widehat z_{t+1}. \label{eq:adaptive_rollout} \end{aligned}$$ This does not impose the Euclidean trajectory penalty $$\begin{aligned} \mathcal L_{\rm lin} &=\sum_{k=1}^{H-1} \left\|z_{t+k}-[(1-\alpha_k)z_t+\alpha_kz_g]\right\|_2^2,\\ \alpha_k&=k/H, \end{aligned} \label{eq:direct_linearization}$$ whose matched negative control appears in `\cref{app:linearity_control}`{=latex}. ## Alternative endpoint sampling and end-to-end necessity One may draw $b\sim\operatorname{Bernoulli}(p)$ and use one endpoint per transition, $$\begin{aligned} &y=(1-b)z^++b(z+m^{\rm goal}),\\ &\mathcal L_{\rm end} =\mathbb E[-\log p_\eta(a\mid z,y,a_{t-1})]. \label{eq:endpoint_nll} \end{aligned}$$ Its expectation is $(1-p)\mathcal L_{z^+}+p\mathcal L_{\rm goal}$, although paired evaluation of both calls changes finite-batch gradient covariance. More importantly, if $E(o)=E(o')$ for observations requiring different action laws under the same intent, no frozen downstream actor can undo that aliasing. Backpropagating the proper likelihood through $E$ exposes the missing controllable distinction during representation learning. ## Endpoint grammar and interaction prediction The waypoint experiments use $$h_t^{\rm end}(y_t)=[z_t;y_t;z_t\odot y_t;A_\omega(a_{t-1})]. \label{eq:actor_features_endpoint}$$ For an endpoint $q_t=z_t+\delta_t$, $z_t\odot q_t=z_t\odot z_t+z_t\odot\delta_t$. The controlled A--G grammar study distinguishes a matched displacement/product pair from changing only the product slot. If the product acts as a state--intent interaction, paired Direct scores obey the finite-capacity prediction $$\begin{aligned} &\Delta_{\rm match}=(S_C-S_D)-(S_A-S_B)>0,\\ &S_E-S_F>0, \end{aligned} \label{eq:product_prediction}$$ while $S_C>S_G$ rejects the explanation that centering the product alone is sufficient. The complete experiment is in `\cref{app:design_selection}`{=latex}. ## General endpoint conditioning For heterogeneous tasks, a more general endpoint design may expose an optional relation $r$, $$\mathcal L_{\rm act}^{\tau}(\nu_\tau) =\mathbb E_{(z,a),(y,r)\sim\nu_\tau} [-\log p_{\eta_\tau}(a\mid z,y,r,a_{t-1})]. \label{eq:general_endpoint}$$ The final raw-goal policy is not explicitly horizon-conditioned: training spans remaining offsets $r=1,\ldots,5$, but its input contains only the current state, raw goal displacement, interaction feature, and previous action. # Design Selection and Ablation Logic {#app:design_selection} This appendix separates model definition from design selection. Each study is organized as a question, a matched intervention, and the conclusion used to fix the final configuration; none changes the headline evaluation protocol. ## Which intent coordinate should the shared operator receive? The waypoint endpoint $q_t$, centered waypoint displacement $q_t-z_t$, and full goal displacement $z_g-z_t$ successively remove an unnecessary absolute coordinate and the hand-set equal-progress scale. Product-off controls test whether the gain comes from the coordinate or its state-dependent interaction; G centers only the interaction while leaving the endpoint main effect intact. All cells use one full-data pass, three training seeds, the same evaluation episodes, and deterministic Math-SDPA. ```{=latex} \scriptsize ``` ```{=latex} \begin{adjustbox}{max width=\columnwidth} \begin{tabular}{clcr} \toprule ID & Main condition & Interaction & Direct SR \\ \midrule A & $q_t$ & $z_t\odot q_t$ & $78.78\!\pm\!0.96$ \\ B & $q_t$ & omitted & $80.56\!\pm\!0.96$ \\ G & $q_t$ & $z_t\odot(q_t-z_t)$ & $79.67\!\pm\!1.76$ \\ C & $q_t-z_t$ & $z_t\odot(q_t-z_t)$ & $83.11\!\pm\!1.39$ \\ D & $q_t-z_t$ & omitted & $81.00\!\pm\!1.45$ \\ F & $z_g-z_t$ & omitted & $83.22\!\pm\!0.51$ \\ E & $z_g-z_t$ & $z_t\odot(z_g-z_t)$ & {\\bfseries\\boldmath $85.78\!\pm\!1.84$} \\ \bottomrule \end{tabular} \end{adjustbox} ``` The matched endpoint-to-centered change A$\rightarrow$C is +4.33 points and the centered-to-full-goal change C$\rightarrow$E is +2.67 points; A$\rightarrow$E is +7.00 points (exact $p=4.17\times10^{-9}$). The interaction prediction also closes quantitatively: the matched difference-in-differences is +3.89 points, $S_E-S_F=+2.56$, and $S_C-S_G=+3.44$. G shows that merely deleting the $z_t\odot z_t$ component is insufficient. The supported inductive bias is the matched intent-dependent sub-grammar $[z_t,m_t,z_t\odot m_t]$ within the full four-slot input; the shared action-history slot $A_\omega(a_{t-1})$ remains unchanged. Here $m_t$ explicitly carries the intended displacement. ```{=latex} \begin{figure*}[t] \includegraphics[width=0.99\textwidth]{figures/single_task_condition_geometry.pdf} \caption{\textbf{Controls used to select the intent grammar.} (a) Coordinate and interaction interventions. (b) Re-encoding every 1, 2, or 5 action blocks. (c) Detaching the goal anchor and explicitly reducing chord error separate action-field learning from Euclidean latent straightening. Error bars are training-seed standard deviations.} \label{fig:single_condition_geometry} \end{figure*} ``` ## Does the result require trajectory straightening or a particular optimizer? For the selected full-goal variants, re-encoding intervals 1/2/5 give 85.67/86.00/86.67% for E and 81.50/81.67/83.00% for F; the endpoint differences are not significant ($p=0.392/0.253$). The waypoint curve is not a clean drift test because every reset also restarts its $1/h$ phase. Joint goal-intent gradients reach 78.78%; detaching that branch reaches 79.67%. Adding a weight-0.01 strict-dynamic SmoothL1 chord penalty lowers validation chord MSE from 0.2173 to 0.2064 but gives 79.22% SR ($p=0.764$). A stronger flexible linearity weight of 0.3 collapses Direct SR to 14.11%. Geometry can therefore become visibly straighter without becoming a better controller. The waypoint SIGReg sweep gives 78.56/77.67/76.56% at 0.02/0.03/0.04 and 68.22% at 0.09. We use the stable low-regularization region rather than claim monotonic benefit. A single-seed Huber screening run obtains 55.0%, whereas the formal probabilistic NLL settings obtain $68.22\!\pm\!1.26$% and $77.67\!\pm\!0.88$%; protocols are not paired, so this only rejects the claim that pointwise regression is already known to be preferable. Forward weights 1.0/0.3/0.1/0 give 77.67/73.11/59.56/25.22%, confirming that the action interface complements rather than replaces world prediction. ## Why is Guarded A the optional verifier? `\Cref{fig:planner_matrix}`{=latex} visualizes the parameter-selection audit; the full 576-job table is in `\cref{tab:planner_matrix_full}`{=latex}. PushT benefits from a finite correction radius, rising from 85.78% Direct to 92.33% at $\sigma_0=0.25$, whereas Reacher degrades at larger radii. Guarded A retains the Direct reference and global best, reaches 96.86% macro with 384 sampled sequences, and is statistically near the simpler unguarded $\sigma_0=0.25$ rule at 96.78%. Doubling to $K=256$ reaches 96.83% and provides no resolved gain. Hence Direct is the primary interface and Guarded A is a bounded verifier, not a replacement controller. ```{=latex} \begin{figure*}[t] \includegraphics[width=0.99\textwidth]{figures/planner_matrix.pdf} \caption{\textbf{Verifier-selection controls.} The complete three-seed audit compares Direct, guarded and unguarded plan-centered search, trust penalties, actor particles, and actor-disabled CEM.} \label{fig:planner_matrix} \end{figure*} ``` ## Qualitative multi-task geometry The fixed-sample projections below illustrate collapse, partial expansion, and healthy SIGReg-supported spread. A spherical or shell-like t-SNE projection is not a success criterion: it is a nonlinear view of a high-dimensional cloud and is reported only to diagnose gross failure morphology. ```{=latex} \begin{figure*}[t] \includegraphics[width=0.99\textwidth]{figures/multitask_geometry_e5.pdf} \caption{Exploratory E5 collapse morphologies from fixed samples. These legacy non-Math-SDPA runs are qualitative diagnostics and do not enter performance ranking or headline correlations.} \label{fig:multitask_geometry} \end{figure*} ``` Across these controls, the selection rule is consistent: preserve the world objective, expose a distance-preserving intent coordinate and its matched state interaction, learn both condition families through one probabilistic operator, and add search only as bounded local verification. # Extended Controlled Results {#app:extended_results} This section preserves the complete protocol tables and selection evidence summarized in the main text. It contains no additional model selection based on test outcomes: the single-task grammar and verifier were locked before the final matrices, and every multi-task row uses the audited deterministic Math-SDPA contract. ## Single-task evaluator and attribution details ```{=latex} \begin{table*}[t] \scriptsize \caption{\textbf{Official versus CLEAR-LeWM v0.5.1 Moderate single-task SR (\%).} Every row evaluates the same three one-epoch goal-displacement checkpoints per task; each checkpoint averages evaluation seeds 0/1/42 with 100 episodes per seed. Values are mean $\pm$ sample standard deviation across training seeds. Protocol scores are never pooled.} \label{tab:clear_main} \begin{adjustbox}{max width=\textwidth} \begin{tabular}{llrrrrr} \toprule Protocol & Deployment & PushT & Cube & Reacher & TwoRoom & Macro \\ \midrule Official & Pure CEM $300{\times}30$ & $88.44\!\pm\!1.17$ & $68.44\!\pm\!.77$ & $83.67\!\pm\!.67$ & $82.89\!\pm\!.84$ & $80.86\!\pm\!.51$ \\ Official & Direct & $85.78\!\pm\!1.54$ & $100.00\!\pm\!.00$ & $97.67\!\pm\!.00$ & $97.89\!\pm\!1.26$ & $95.33\!\pm\!.58$ \\ {\\bfseries\\boldmath Official} & {\\bfseries\\boldmath Guarded A $128{\times}3$} & {\\bfseries\\boldmath $92.22\!\pm\!.69$} & {\\bfseries\\boldmath $99.78\!\pm\!.19$} & {\\bfseries\\boldmath $97.44\!\pm\!.77$} & {\\bfseries\\boldmath $98.00\!\pm\!1.15$} & {\\bfseries\\boldmath $96.86\!\pm\!.38$} \\ \midrule CLEAR Moderate & Pure CEM $300{\times}30$ & $90.22\!\pm\!2.27$ & $49.44\!\pm\!3.60$ & $50.11\!\pm\!1.17$ & $72.11\!\pm\!2.79$ & $65.47\!\pm\!.77$ \\ CLEAR Moderate & Direct & $88.22\!\pm\!.39$ & $99.89\!\pm\!.19$ & $49.56\!\pm\!2.52$ & $95.67\!\pm\!2.33$ & $83.33\!\pm\!1.00$ \\ {\\bfseries\\boldmath CLEAR Moderate} & {\\bfseries\\boldmath Guarded A $128{\times}3$} & {\\bfseries\\boldmath $92.78\!\pm\!1.71$} & {\\bfseries\\boldmath $99.44\!\pm\!.20$} & {\\bfseries\\boldmath $51.67\!\pm\!3.18$} & {\\bfseries\\boldmath $95.44\!\pm\!2.22$} & {\\bfseries\\boldmath $84.83\!\pm\!1.32$} \\ \bottomrule \end{tabular} \end{adjustbox} \end{table*} ``` ```{=latex} \begin{figure*}[t] \includegraphics[width=0.98\textwidth]{figures/query_direct_main.pdf} \caption{\textbf{Search-free control and optional verification.} Left: physical-successor inversion versus goal-displacement INTACT after one epoch. Right: Direct uses no candidate sequence, Guarded A locally verifies 384, and broad actor-disabled CEM evaluates 9,000. More search is not automatically better once the action conditional is aligned.} \label{fig:query_direct} \end{figure*} ``` ```{=latex} \small ``` ```{=latex} \begin{adjustbox}{max width=\columnwidth} \begin{tabular}{lrrrrr} \toprule Training & Direct & CEM $30{\times}10$ & State $R^2$ & Action $R^2$ & Rank \\ \midrule LeWM & -- & $42.22\!\pm\!9.50$ & $.616\!\pm\!.008$ & $.673\!\pm\!.062$ & $63.7\!\pm\!.9$ \\ $+$ physical inverse & $35.78\!\pm\!.69$ & $57.67\!\pm\!2.52$ & $.652\!\pm\!.010$ & $.791\!\pm\!.003$ & $67.0\!\pm\!.8$ \\ $+$ matched waypoint intent & $68.22\!\pm\!1.26$ & $61.44\!\pm\!2.04$ & $.658\!\pm\!.003$ & $.786\!\pm\!.003$ & $67.0\!\pm\!.4$ \\ Waypoint INTACT, SIGReg $0.03$ & {\\bfseries\\boldmath $77.67\!\pm\!.88$} & {\\bfseries\\boldmath $69.44\!\pm\!1.64$} & {\\bfseries\\boldmath $.680\!\pm\!.005$} & {\\bfseries\\boldmath $.804\!\pm\!.002$} & $43.7\!\pm\!.6$ \\ \bottomrule \end{tabular} \end{adjustbox} ``` The Forward Predictor remains necessary: weights 1.0/0.3/0.1/0 give PushT Direct SR 77.67/73.11/59.56/25.22%. Increasing the one-epoch peak learning rate to $7.5\times10^{-4}$ gives $77.00\!\pm\!2.03$%; a second epoch gives $77.22\!\pm\!1.07$%, versus $77.67\!\pm\!.88$ at the selected setting. The second epoch raises effective rank from about 43.7 to 58.8 without raising SR, directly separating geometric spread from action-aligned representation quality. ## Complete shared-encoder low-budget result ```{=latex} \begin{table*}[t] \small \caption{\textbf{Controlled Math-SDPA shared-encoder E1 result.} Values are mean $\pm$ sample standard deviation over three training seeds; each seed averages official evaluation seeds $\{0,1,42\}$ with 100 episodes each. LeWM has no INTACT Predictor and uses CEM $300{\times}30$; other rows use Direct.} \label{tab:multitask_core} \begin{adjustbox}{max width=\textwidth} \begin{tabular}{llcrrrrr} \toprule Training cell & Evaluation & $n$ & PushT & Cube & Reacher & TwoRoom & Macro \\ \midrule LeWM & CEM $300{\times}30$ & 3 & $17.00\!\pm\!7.17$ & $52.89\!\pm\!3.95$ & {\\bfseries\\boldmath $25.00\!\pm\!4.26$} & $34.33\!\pm\!1.76$ & $32.31\!\pm\!2.34$ \\ Inverse only & Direct & 3 & $24.89\!\pm\!5.97$ & $48.22\!\pm\!3.83$ & $9.44\!\pm\!3.01$ & $51.33\!\pm\!13.32$ & $33.47\!\pm\!3.98$ \\ Waypoint intent only & Direct & 3 & $17.67\!\pm\!2.40$ & $68.78\!\pm\!5.50$ & $11.89\!\pm\!2.78$ & $44.67\!\pm\!31.13$ & $35.75\!\pm\!7.58$ \\ Goal intent only & Direct & 3 & $24.00\!\pm\!4.36$ & {\\bfseries\\boldmath $73.22\!\pm\!3.67$} & $11.56\!\pm\!2.04$ & $48.22\!\pm\!17.20$ & $39.25\!\pm\!5.40$ \\ {\\bfseries\\boldmath Waypoint INTACT} & {\\bfseries\\boldmath Direct} & 3 & {\\bfseries\\boldmath $30.56\!\pm\!9.34$} & $69.89\!\pm\!10.99$ & $10.33\!\pm\!5.03$ & {\\bfseries\\boldmath $60.00\!\pm\!10.58$} & {\\bfseries\\boldmath $42.69\!\pm\!7.19$} \\ Goal INTACT & Direct & 3 & $26.89\!\pm\!6.26$ & $61.44\!\pm\!10.46$ & $10.33\!\pm\!3.18$ & $48.67\!\pm\!21.31$ & $36.83\!\pm\!5.63$ \\ \bottomrule \end{tabular} \end{adjustbox} \end{table*} ``` Waypoint INTACT is the strongest first-pass cell, whereas goal-displacement INTACT becomes the strongest E5 cell. This reversal is why the E1 table is a low-budget optimization diagnostic rather than the final objective ranking. ## Matched E5 inference decomposition ```{=latex} \begin{table*}[t] \scriptsize \setlength{\tabcolsep}{4.2pt} \renewcommand{\arraystretch}{0.96} \caption{\textbf{Matched E5 inference decomposition.} Values are mean $\pm$ sample standard deviation over three training seeds. Panel (a) removes all learned action heads; panel (b) locally verifies the learned plan; panel (c) uses CLEAR Moderate with each cell's native interface.} \label{tab:multitask_inference_decomp} \textbf{(a) Official actor-disabled pure CEM $300{\times}30$}\par \begin{adjustbox}{max width=\textwidth} \begin{tabular}{lrrrrr} \toprule Training cell & PushT & Cube & Reacher & TwoRoom & Macro \\ \midrule LeWM & $74.56\!\pm\!3.67$ & $67.33\!\pm\!1.86$ & $83.11\!\pm\!.96$ & $39.67\!\pm\!8.39$ & $66.17\!\pm\!2.67$ \\ Inverse only & $75.78\!\pm\!1.02$ & $66.78\!\pm\!2.36$ & $81.78\!\pm\!1.64$ & $51.44\!\pm\!1.07$ & $68.94\!\pm\!.43$ \\ Waypoint intent only & $78.44\!\pm\!4.72$ & $68.67\!\pm\!1.76$ & $83.22\!\pm\!2.69$ & $39.11\!\pm\!8.17$ & $67.36\!\pm\!4.06$ \\ Waypoint INTACT & $78.22\!\pm\!3.02$ & $69.56\!\pm\!2.80$ & $83.00\!\pm\!2.19$ & $46.44\!\pm\!3.34$ & $69.31\!\pm\!.68$ \\ Goal intent only & {\\bfseries\\boldmath $80.78\!\pm\!2.36$} & $68.00\!\pm\!3.18$ & {\\bfseries\\boldmath $84.67\!\pm\!1.33$} & $40.11\!\pm\!8.30$ & $68.39\!\pm\!2.22$ \\ {\\bfseries\\boldmath Goal INTACT} & $78.11\!\pm\!1.17$ & {\\bfseries\\boldmath $70.33\!\pm\!3.00$} & $80.33\!\pm\!2.03$ & {\\bfseries\\boldmath $51.56\!\pm\!4.53$} & {\\bfseries\\boldmath $70.08\!\pm\!1.13$} \\ \bottomrule \end{tabular} \end{adjustbox} \textbf{(b) Official Guarded A $128{\times}3$}\par \begin{adjustbox}{max width=\textwidth} \begin{tabular}{lrrrrr} \toprule Training cell & PushT & Cube & Reacher & TwoRoom & Macro \\ \midrule Inverse only & $55.67\!\pm\!4.10$ & $65.89\!\pm\!1.02$ & $78.67\!\pm\!1.20$ & $66.33\!\pm\!6.08$ & $66.64\!\pm\!.88$ \\ Waypoint intent only & $74.67\!\pm\!3.79$ & {\\bfseries\\boldmath $99.78\!\pm\!.19$} & $87.78\!\pm\!1.07$ & $74.78\!\pm\!10.63$ & $84.25\!\pm\!3.58$ \\ Waypoint INTACT & $83.22\!\pm\!2.36$ & $99.11\!\pm\!.51$ & $85.33\!\pm\!.58$ & $74.89\!\pm\!7.15$ & $85.64\!\pm\!2.04$ \\ Goal intent only & $80.44\!\pm\!.96$ & $99.33\!\pm\!.33$ & $94.67\!\pm\!.33$ & $69.33\!\pm\!5.93$ & $85.94\!\pm\!1.57$ \\ {\\bfseries\\boldmath Goal INTACT} & {\\bfseries\\boldmath $85.00\!\pm\!2.85$} & $99.00\!\pm\!.00$ & {\\bfseries\\boldmath $97.89\!\pm\!.69$} & {\\bfseries\\boldmath $80.00\!\pm\!4.41$} & {\\bfseries\\boldmath $90.47\!\pm\!.84$} \\ \bottomrule \end{tabular} \end{adjustbox} \textbf{(c) CLEAR-LeWM v0.5.1 Moderate}\par \begin{adjustbox}{max width=\textwidth} \begin{tabular}{lrrrrr} \toprule Training cell & PushT & Cube & Reacher & TwoRoom & Macro \\ \midrule LeWM (pure CEM) & $78.78\!\pm\!2.52$ & $51.33\!\pm\!1.20$ & $46.00\!\pm\!4.06$ & $33.33\!\pm\!13.86$ & $52.36\!\pm\!2.97$ \\ Inverse only & $32.67\!\pm\!.88$ & $42.67\!\pm\!1.67$ & $31.67\!\pm\!2.33$ & $70.11\!\pm\!2.17$ & $44.28\!\pm\!1.33$ \\ Waypoint intent only & $60.89\!\pm\!2.91$ & {\\bfseries\\boldmath $99.78\!\pm\!.19$} & $35.78\!\pm\!7.58$ & $67.11\!\pm\!12.01$ & $65.89\!\pm\!2.79$ \\ Waypoint INTACT & $74.78\!\pm\!4.34$ & $99.11\!\pm\!.84$ & $32.22\!\pm\!5.43$ & $73.33\!\pm\!3.28$ & $69.86\!\pm\!1.06$ \\ Goal intent only & $73.00\!\pm\!3.18$ & {\\bfseries\\boldmath $100.00\!\pm\!.00$} & $39.44\!\pm\!2.71$ & $60.67\!\pm\!5.51$ & $68.28\!\pm\!1.17$ \\ {\\bfseries\\boldmath Goal INTACT} & {\\bfseries\\boldmath $84.33\!\pm\!2.00$} & $99.11\!\pm\!1.54$ & {\\bfseries\\boldmath $47.44\!\pm\!2.91$} & {\\bfseries\\boldmath $79.33\!\pm\!5.03$} & {\\bfseries\\boldmath $77.56\!\pm\!.39$} \\ \bottomrule \end{tabular} \end{adjustbox} \end{table*} ``` ## Representation and conversion diagnostics ```{=latex} \begin{table*}[t] \scriptsize \caption{\textbf{Controlled shared-encoder representation snapshots.} $\bar\sigma$ is mean per-dimension latent standard deviation, cosine is mean pairwise cosine, and action $R^2$ is an episode-disjoint frozen probe.} \label{tab:multitask_diagnostics} \label{tab:multitask_alignment} \textbf{E1}\par \begin{tabular}{lcrrrr} \toprule Training cell & $n$ & Rank & $\bar\sigma$ & Cos. & Act. $R^2$ \\ \midrule LeWM & 3 & 25.53 & .830 & .554 & .309 \\ Inverse only & 3 & 24.03 & .902 & .590 & .372 \\ Waypoint intent only & 3 & 24.83 & .846 & .582 & .328 \\ Goal intent only & 3 & 24.45 & .839 & .604 & .349 \\ Waypoint INTACT & 3 & {\\bfseries\\boldmath 25.73} & .877 & .587 & {\\bfseries\\boldmath .383} \\ Goal INTACT & 3 & 25.34 & .888 & .616 & .382 \\ \bottomrule \end{tabular} \hfill \textbf{E5}\par \begin{tabular}{lcrrrr} \toprule Training cell & $n$ & Rank & $\bar\sigma$ & Cos. & Act. $R^2$ \\ \midrule LeWM & 3 & {\\bfseries\\boldmath 94.06} & .964 & .039 & .366 \\ Inverse only & 3 & 85.24 & .956 & .087 & .401 \\ Waypoint intent only & 3 & 93.87 & .959 & .049 & .393 \\ Goal intent only & 3 & 93.47 & .965 & .047 & .390 \\ Waypoint INTACT & 3 & 87.17 & .960 & .080 & .406 \\ Goal INTACT & 3 & 89.26 & .958 & .081 & {\\bfseries\\boldmath .409} \\ \bottomrule \end{tabular} \end{table*} ``` Across 45 formula-eligible checkpoints, pooled predicted--expert CKA/kNN correlations with official Direct SR are $.897/.954$; pointwise action $R^2$ is $.815$ and NLL is $-.786$. Leave-one-epoch-out values remain $[.888,.914]$ for CKA and $[.946,.963]$ for kNN. These analyses use one point per checkpoint, not one point per transition. Intent-only, inverse-only, and LeWM cells are excluded because a transition must evaluate both local and goal likelihoods through the same actor to instantiate the claimed family map. Fixed-sample t-SNE is retained only as a qualitative diagnostic. Under SIGReg, healthy high-dimensional clouds can project to an interleaved shell; this appearance neither computes nor ranks SR. The quantitative analyses use original-space rank, standard deviation, forward error, probes, and predicted--expert action-family relations. ## Task-conditioned gauge controls Correct pairing raises CLEAR Moderate SR from 9.46% shuffled to 68.04%, a $+58.58$ point gain (8-cluster bootstrap 95% CI $[48.83,67.58]$), positive for all 24 protocol units. Calibration with 4/16/64/full episodes gives 26.67/56.00/64.42/64.83% SR. Same-objective cross-seed swaps define the alignment ceiling; LeWM and inverse backbones aligned into a trained Full actor are positive alignability controls. In reverse, Full-backbone-to-LeWM-actor stays at 5.08%, and the inverse actor recovers only inverse-level function. ```{=latex} \begin{figure*}[t] \includegraphics[width=0.995\textwidth]{figures/gauge_controls.pdf} \caption{\textbf{Gauge controls isolate correspondence from actor functionality.} Correctly paired calibration restores control; recovery saturates near 64/full episodes; same-objective swaps define a ceiling; reverse swaps show that a coordinate map cannot create an untrained deployment conditional. The 72-cell audit contains 21,600 CLEAR Moderate episodes and excludes all evaluation episodes from calibration.} \label{fig:task_conditioned_gauge} \end{figure*} ``` Cross-task and leave-one-task-out actor-output transfer remain negative, while pooled calibration reaches only $.484/.539$ $R^2$. The shared visual backbone can therefore support several task-local charts; the task-specific heads do not require one global canonical action space. # Actor-Sharing Mechanism Audit {#app:actor_sharing} ```{=latex} \begin{table*}[t] \small \caption{\textbf{Sharing is the operative inductive bias.} Matched PushT waypoint Direct SR (\%). Except for the explicit double-capacity control, parameter counts match within 0.12\%. $p$ is an episode-paired exact McNemar comparison with the fully shared model.} \label{tab:actor_sharing} \begin{adjustbox}{max width=\textwidth} \begin{tabular}{lrrrr} \toprule Action-conditional design & Params (M) & Direct SR & $\Delta$ vs shared & Paired $p$ \\ \midrule Waypoint intent only & 3.110 & $65.56\!\pm\!1.35$ & $-11.33$ & $4.23\!\times\!10^{-15}$ \\ Independent, parameter matched & 3.114 & $71.22\!\pm\!2.27$ & $-5.67$ & $1.15\!\times\!10^{-5}$ \\ Independent, double capacity & 6.220 & $72.89\!\pm\!2.17$ & $-4.00$ & 0.00362 \\ Shared trunk, two outputs & 3.110 & $74.67\!\pm\!1.53$ & $-2.22$ & 0.0927 \\ Condition-token shared & 3.111 & $77.11\!\pm\!2.12$ & $+0.22$ & 0.923 \\ {\\bfseries\\boldmath Fully shared (default)} & 3.110 & $76.89\!\pm\!.69$ & 0 & -- \\ \bottomrule \end{tabular} \end{adjustbox} \end{table*} ``` The closed-loop result is paired with mechanism measurements on fixed PushT waypoint batches. \`\`Shared fraction" is the fraction of action-network parameters touched by both local and deployment losses. Gradient cosine is computed in the complete action-network parameter coordinates; it is zero by construction when the two actors are disjoint. Shuffle response is the normalized-action MAE between the original goal-conditioned output and the output after permuting goals across the batch. Consistency is $\|F_\phi(z_t,\mu_\eta(z_t,q_t))-q_t\|_2^2$. ```{=latex} \begin{table*}[t] \small \caption{Mechanism diagnostics for the matched PushT actor-sharing ablation. These measurements explain the closed-loop table but are not substitutes for SR.} \label{tab:actor_sharing_diagnostics} \begin{adjustbox}{max width=\textwidth} \begin{tabular}{lrrrr} \toprule Action-conditional design & Shared fraction & Grad. cosine & Shuffle-goal MAE & Consistency MSE \\ \midrule Condition-token shared & 1.000 & 0.506 & 1.256 & 0.1320 \\ Fully shared & 1.000 & {\\bfseries\\boldmath 0.575} & {\\bfseries\\boldmath 1.263} & 0.1275 \\ Shared trunk, two outputs & 0.987 & 0.138 & 1.200 & 0.1309 \\ Independent, double capacity & 0 & 0 & 1.030 & 0.1382 \\ Independent, parameter matched & 0 & 0 & 1.029 & 0.1370 \\ Goal-intent only & 1.000 & 0.321 & 0.889 & {\\bfseries\\boldmath 0.1133} \\ \bottomrule \end{tabular} \end{adjustbox} \end{table*} ``` The fully shared model receives positively aligned physical/deployment gradients (cosine 0.575). As sharing is weakened, gradient coordination and SR decline together. Across the six designs, shuffle-goal response has exploratory Spearman $\rho=0.943$ with SR ($n=6$), indicating that stronger controllers use the goal condition more actively; the small design count precludes a universal claim. Two negative controls are especially informative. Goal-intent-only has the lowest consistency MSE but the worst SR, so self-consistency with the Forward Predictor does not establish physical correctness. Conversely, the double-capacity independent model obtains real/deployment NLLs of $-1.347/-1.160$ yet reaches only 72.89% SR. Separately fitting both likelihoods is therefore not equivalent to learning one deployable inverse map. # Exploratory Diagnostic Definitions and Correlations {#app:diagnostics} For centered latent samples $Z\in\mathbb{R}^{n\times D}$ with normalized covariance eigenvalues $\bar\lambda_j$, entropy effective rank is $$d_{\mathrm{eff}}=\exp\!\left(-\sum_j\bar\lambda_j \log(\bar\lambda_j+\epsilon)\right).$$ Let $r_i=\lVert Z_i-\bar Z\rVert_2$, let $\mathrm{CV}_r$ be the coefficient of variation of these radii, and let $\mathrm{CV}_{\chi(d)}$ be that of a chi distribution with $d$ degrees of freedom. We define $$\begin{aligned} S_{\rm rank}&=d_{\rm eff}/D,\\ S_{\rm shell}&=\exp\!\left[-\left|\log \frac{\mathrm{CV}_r}{\mathrm{CV}_{\chi(d_{\rm eff})}}\right|\right],\\ \mathrm{SRS}&=100\sqrt{S_{\rm rank}S_{\rm shell}}. \label{eq:srs_definition} \end{aligned}$$ SRS is high when the spectrum uses many dimensions and sample radii resemble the Gaussian shell expected at that effective dimension; it does not measure what information those dimensions encode. For the actor diagnostic, write $(\mu_t^{\rm dep},\sigma_t^{\rm dep})$ for the Gaussian under the deployed goal condition and $(\widetilde\mu_t,\widetilde\sigma_t)$ for a deterministic different-goal condition on the same current state. The reported condition response is the per-sample RMS standardized mean shift $$G_{\rm sep}=\mathbb E_t\!\left[ \left(\frac{1}{d_a}\sum_{j=1}^{d_a} \frac{(\mu_{t,j}^{\rm dep}-\widetilde\mu_{t,j})^2} {\tfrac12[(\sigma_{t,j}^{\rm dep})^2+ \widetilde\sigma_{t,j}^{2}]+\epsilon}\right)^{1/2}\right]. \label{eq:gsep_definition}$$ Thus $G_{\rm sep}$ asks whether the actor uses its condition relative to its own predictive uncertainty. It does not ask whether the changed goal requires a different expert action, nor whether the response has the correct direction. We pair rank, SRS, and $G_{\rm sep}$ with forward MSE, deployment NLL/RMSE, symmetric KL between action conditionals, and shuffled-goal $\Delta$NLL. Every method uses the same 64 sequences per task. Correlations first subtract the mean within each task and then pool the 56 method--task cells. These endpoint runs predate the controlled Math-SDPA factorial. Their matched sampling makes the geometric and conversion metrics informative as exploratory mechanism diagnostics, but their SR values are not used to rank training objectives in the main performance comparison. ```{=latex} \small ``` ::: {#tab:metric_correlations} Metric Moderate Strict -------------------- ---------- ---------- Effective rank 0.888 0.846 $G_{\mathrm{sep}}$ 0.819 0.731 Deployment NLL $-0.640$ $-0.766$ : Task-centered Pearson correlation with E5 Direct SR. These are diagnostic associations, not independent causal effects. ::: SRS correlates with CLEAR v0.3 E5 Moderate Direct SR at Pearson $r=0.978$ and Spearman $\rho=0.652$, and with Strict Direct SR at $r=0.977$ and $\rho=0.809$, over the ten semantic endpoint configurations. This smaller analysis is useful for geometry auditing but is not treated as a universal threshold. # Single-Task Latent-Linearity Stress Test {#app:linearity_control} `\Cref{tab:linearity_control}`{=latex} is a separately trained four-frame PushT isolation, not a row from the controlled multi-task factorial. Every model uses full data for one epoch, learning rate $10^{-4}$, batch size 256, physical-inverse weight 0.1, and SIGReg 0.09; only the latent-linearity term changes. Each cell pools three training seeds (3072--3074), three evaluation seeds (0, 1, 42), and 100 episodes per pair, for 900 episodes. The flexible penalty permits a learned local adjustment, whereas the strict penalty directly targets Euclidean waypoints. Machine-readable values come from experiment `linearity4_exact_isolation_pusht_e1`, specifically the Direct and pure-CEM summary artifacts. ```{=latex} \small ``` ```{=latex} \begin{adjustbox}{max width=\columnwidth} \begin{tabular}{lrrr} \toprule Linearity term & Weight & Direct & Pure CEM $30{\times}10$ \\ \midrule Inverse control (no $\mathcal L_{\rm lin}$) & 0 & 34.00 & {\\bfseries\\boldmath 43.78} \\ Flexible & 0.01 & {\\bfseries\\boldmath 35.89} & 41.11 \\ Strict & 0.01 & 35.11 & 41.22 \\ Strict & 0.03 & 35.11 & 41.78 \\ Flexible & 0.30 & 14.11 & -- \\ \bottomrule \end{tabular} \end{adjustbox} ``` The strict losses reduce latent waypoint error but do not improve either Direct or actor-disabled planning. At flexible weight 0.30, effective rank falls to 41.96 and the frozen action probe falls to $R^2=0.111$. This negative control is why INTACT uses the Euclidean construction only to form an actor condition and never as a target for the encoder trajectory. Its 34.00% control should not be compared numerically to the 35.78% single-task physical-inverse row in `\cref{tab:single_direct}`{=latex}: the stress test uses an independently trained seed set and a four-frame isolation protocol. # Historical Latent-Perturbation Diagnostic {#app:latent_diffusion} Before INTACT, we tested whether replacing the deterministic latent predictor with five-step flow matching improved LeWM. On PushT, the deterministic model reached 26%, 86%, 94%, and 96% official SR at E1, E5, E10, and E15. The best x-prediction/loss-x flow model reached 4%, 76%, 88%, and 92%; a second flow run with matched SIGReg 0.09 reached 4%, 44%, 72%, and 76%. At matched 139,319 optimizer steps, the strongest flow run used 0.568 seconds/step versus 0.479 for the deterministic predictor, an 18.4% increase. This historical control is deliberately not a headline ablation: the strongest flow run changes SIGReg from 0.09 to 0.03, and there is no matched RGB-noise control. It supports only the narrower observation that perturbing an already compressed latent did not improve low-budget learning in these runs. This is consistent with GC-IDM's independent latent-noise ablation, whose taskwise optimum occurs at zero noise [@nguyen2026gcidm]. It does not establish that latent diffusion is universally harmful or that it is directly comparable to pixel-space diffusion. # Complete Goal-Displacement Planner Matrix {#app:planner_matrix} `\Cref{tab:planner_matrix_full}`{=latex} reports every completed planner cell used in `\cref{fig:planner_matrix}`{=latex}. Each method contains 36 jobs: three training checkpoints per task, each evaluated with seeds $\{0,1,42\}$ and 100 episodes. Macro is the unweighted task mean. The 12 base rules contribute 432 jobs and the four guarded rules contribute 144, each under a separate PASS audit. ```{=latex} \begin{table*}[t] \scriptsize \caption{Complete 576-job MATH-only planner matrix, official SR (\%). ``Plan'' means that the deterministic goal-displacement action sequence initializes the factorized raw-action distribution; trust variants use coefficient one. Guarded rules preserve the Direct reference and globally best observed sequence across refits.} \label{tab:planner_matrix_full} \begin{adjustbox}{max width=\textwidth} \begin{tabular}{llrrrrrr} \toprule Planner family & Setting & PushT & Cube & Reacher & TwoRoom & Macro & Worst \\ \midrule Direct & Goal-displacement mean & 85.78 & {\\bfseries\\boldmath 100.00} & {\\bfseries\\boldmath 97.67} & 97.89 & 95.33 & 85.78 \\ Actor particles & K128-I1 & 91.44 & 99.67 & 82.78 & 98.00 & 92.97 & 82.78 \\ Actor particles & K128-I3 & 91.89 & 99.67 & 78.44 & 97.44 & 91.86 & 78.44 \\ \midrule Plan + raw CEM & $\sigma_0=0.05$ & 87.89 & {\\bfseries\\boldmath 100.00} & 96.33 & 97.78 & 95.50 & 87.89 \\ Plan + raw CEM & $\sigma_0=0.10$ & 89.78 & {\\bfseries\\boldmath 100.00} & 97.11 & 97.67 & 96.14 & 89.78 \\ Plan + raw CEM & $\sigma_0=0.25$ & 92.33 & 99.78 & 97.11 & 97.89 & 96.78 & 92.33 \\ Plan + raw CEM & $\sigma_0=0.50$ & 92.11 & 99.78 & 93.78 & 97.89 & 95.89 & 92.11 \\ Plan + raw CEM & $\sigma_0=1.00$ & 88.78 & 99.56 & 84.78 & {\\bfseries\\boldmath 98.33} & 92.86 & 84.78 \\ Plan + L2 trust & $\sigma_0=0.50$ & {\\bfseries\\boldmath 92.78} & 99.78 & 94.11 & 97.89 & 96.14 & {\\bfseries\\boldmath 92.78} \\ Plan + actor-NLL trust & $\sigma_0=0.50$ & 92.11 & 99.89 & 94.22 & 97.89 & 96.03 & 92.11 \\ \midrule Guarded B & K256-I3, $\sigma_0=0.25$ & 92.00 & 99.89 & {\\bfseries\\boldmath 97.67} & 97.78 & 96.83 & 92.00 \\ Guarded C & K128-I5, smooth & 91.56 & 99.89 & 97.56 & 97.89 & 96.72 & 91.56 \\ Guarded D & K128-I3, actor-cov + L2 & 91.67 & 99.89 & 97.22 & 98.00 & 96.69 & 91.67 \\ {\\bfseries\\boldmath Guarded A} & {\\bfseries\\boldmath K128-I3, $\sigma_0=0.25$} & {\\bfseries\\boldmath 92.22} & {\\bfseries\\boldmath 99.78} & {\\bfseries\\boldmath 97.44} & {\\bfseries\\boldmath 98.00} & {\\bfseries\\boldmath 96.86} & {\\bfseries\\boldmath 92.22} \\ \midrule Pure raw CEM & K128-I3 & 58.00 & 65.44 & 73.44 & 80.56 & 69.36 & 58.00 \\ Pure raw CEM & K300-I30 & 88.44 & 68.44 & 83.67 & 82.89 & 80.86 & 68.44 \\ \bottomrule \end{tabular} \end{adjustbox} \end{table*} ``` # Episode-Disjoint Anti-Retrieval Audit {#app:episode_disjoint} The fixed PushT manifest contains 14,948/1,868/1,869 train/validation/test episodes with zero overlap. Normalization, optimization, and checkpoint selection exclude the test episodes. We retrain transition-only and full goal-intent models, then evaluate the same 100 test-only start--goal pairs in each of three training-seed by three evaluation-seed cells. ```{=latex} \small ``` ::: {#tab:episode_disjoint} Training objective eval 0 eval 1 eval 42 pooled --------------------------- -------- -------- --------- ----------- Transition inverse 35.67 44.33 40.67 40.22 \+ deployment goal intent 75.33 81.33 76.00 **77.56** : PushT Direct SR (%) on the held-out 10% episode split. Each cell contains 100 paired test-only episodes. ::: The full model reaches `77.56{}`{=latex}%, only 0.12 point below the 77.67% standard-protocol estimate. Among 900 paired episodes, 358 are solved only by the goal-intent model and 22 only by transition inverse (descriptive exact McNemar $p=2.36\times10^{-79}$; checkpoint clustering precludes treating all episodes as independent). Every training-seed mean improves: 39.33$\rightarrow$``{=html}77.00, 40.33$\rightarrow$``{=html}80.33, and 41.00$\rightarrow$``{=html}75.33. Thus the Direct gain survives both unseen expert episodes and independent optimization runs rather than retrieving a training episode. ```{=latex} \small ``` ::: {#tab:episode_disjoint_cells} Transition inverse Goal intent --------------------------------------------- -------------------- ------------- ----- ---- ---- ----- 2-4`\cmidrule`{=latex}(lr)5-7 Training seed e0 e1 e42 e0 e1 e42 3072 35 42 41 74 83 74 3073 36 44 41 81 81 79 3074 36 47 40 71 80 75 : PushT successes on held-out expert episodes (100 per cell). ::: # Evaluation Protocol Scope {#app:clear_scope} The official LeWM benchmark remains the headline protocol because it preserves comparability with prior work. CLEAR-LeWM is a separate, versioned, MIT-licensed software benchmark rather than a contemporaneously reviewed paper; we cite the public v0.5.1 release and immutable commit as a software artifact [@sun2026clearlewm]. Official and CLEAR scores are never pooled. The INTACT audit pins v0.5.1 commit `32f4416c`, canonical manifests, batch one, Math-only SDPA, deterministic algorithms, full checkpoint hashes, and complete episode outcomes. ## Why a corrected protocol is informative The historical sampler draws rows uniformly and retains initially successful start--goal pairs. This creates both episode-length bias and automatic-success mass. Across the released datasets, 38.38% of valid Cube pairs and 8.82% of TwoRoom $+25$ pairs are already within the historical endpoint threshold. Random control consequently reaches 49% and 26%, respectively, in the three-run historical audit. Moderate removes initially solved pairs, samples episodes uniformly, restores the final valid row, and fixes only demonstrated evaluator defects. This lowers the Cube and TwoRoom random floors to 15.67% and 6.67% while retaining strong official-checkpoint performance (`\cref{tab:clear_floor}`{=latex}). A lower floor increases usable dynamic range; it does not guarantee that every model score must decrease because the sampled pair distribution also changes. ```{=latex} \scriptsize ``` ::: {#tab:clear_floor} Task Initially solved Historical random Moderate random --------- ------------------ ------------------- ----------------- PushT 0.22 4.00 4.00 Cube 38.38 49.00 15.67 Reacher 0.57 13.00 4.33 TwoRoom 8.82 26.00 6.67 : Historical automatic-success audit and CLEAR Moderate calibration. Historical random is the mean of three runs under the released stack. CLEAR values use canonical seeds 0/1/42 and 100 episodes per seed. ::: ## Moderate and Strict are different claims Moderate is the closest corrected continuation of the released task. Strict builds on the same data hygiene and physics but asks for precise task-semantic completion. They are reported separately rather than averaged (`\cref{tab:clear_contract}`{=latex}). ```{=latex} \tiny ``` ```{=latex} \setlength{\tabcolsep}{2pt} ``` ::: {#tab:clear_contract} Task Moderate: minimal compatibility correction Strict: task-semantic precision --------- ------------------------------------------------------------------------------------- ------------------------------------------------------------------------------- PushT Released pusher + T position $<20$ px and T angle $<20^\circ$; first hit. T object only, $<10$ px and $<10^\circ$; hold 3. Cube Initially unsolved, cube-center distance $\leq4$ cm; first hit. Cube center $\leq3$ cm and symmetry-aware orientation $\leq15^\circ$; hold 3. Reacher Periodic unbounded shoulder, raw bounded wrist, joint error $<0.05$ rad; first hit. Physical fingertip endpoint $\leq1$ cm; hold 2. TwoRoom Clean cross-room pair, continuous swept-disk collision, endpoint $<16$ px. Legal doorway crossing, valid route, goal-side arrival, endpoint $<8$ px. : CLEAR-LeWM v0.5.1 task contracts. \`\`Hold" is the number of consecutive successful environment steps. The 24-fold Cube comparison minimizes orientation error over the proper rotational symmetry group of a cube. ::: The task-specific repairs matter. Reacher's released all-periodic comparison can wrap its physically bounded wrist coordinate; Moderate wraps only the unbounded shoulder. The rewritten TwoRoom runtime previously checked requested segment endpoints and could admit motion through a wall. Both modes now use continuous swept-disk collision for the full agent radius; Strict additionally requires a legal cross-room route and goal-side arrival. Cube Moderate retains the OGBench 4 cm position task so that compatibility is explicit, whereas Strict adds object orientation modulo 24 proper cube rotations. PushT Moderate retains the released full pusher--object goal state; Strict removes the task-irrelevant terminal pusher pose and scores precise T-object completion. ## Pinned references and public checkpoint bundles `\Cref{tab:clear_public}`{=latex} summarizes the publicly auditable reference set; INTACT has completed Moderate only, so no Strict score is imputed. ```{=latex} \scriptsize ``` ```{=latex} \begin{adjustbox}{max width=\columnwidth} \begin{tabular}{lrrrr} \toprule Model & PushT & Cube & Reacher & TwoRoom \\ \midrule INTACT goal, Direct & 88.22/-- & 99.89/-- & 49.56/-- & 95.67/-- \\ Official LeWM & 86.33/70.33 & 50.33/26.33 & 46.00/43.00 & 84.00/58.33 \\ Paired random & 4.00/5.00 & 15.67/6.00 & 4.33/5.00 & 6.67/1.67 \\ DINOv2 No-Proprio & 8/7 & 43/17 & --/-- & 55/26 \\ GCBC Joint & 9/9 & 16/3 & --/-- & 15/9 \\ \bottomrule \end{tabular} \end{adjustbox} ``` The comparison illustrates the division of labor between protocols. The official benchmark supplies historical continuity. Moderate removes automatic success and known implementation defects while changing as little task semantics as possible. Strict supports a stronger physical-completion claim. For single-task INTACT Direct, Moderate preserves high PushT/Cube/TwoRoom performance but changes Reacher from 97.67% official SR to 49.56%; this is exactly the kind of near-ceiling ambiguity that the corrected topology is intended to reveal.