Method Notes

1. Problem

Let an encoder map an observation to a latent state, $z_t=E_\theta(o_t)$. A conventional latent world model learns a forward conditional

$$F_\psi(z_t,a_t)\approx z_{t+1},$$

but obtains actions for a desired endpoint by numerically inverting $F_\psi$ at deployment. INTACT instead learns a deployable conditional action law jointly with the representation.

2. One Input Grammar, Two Intent Instances

For an observed transition and a future goal from the same demonstration, INTACT constructs

$$m_t^{\mathrm{local}}=z_{t+1}-z_t, \qquad m_t^{\mathrm{goal}}=\mathrm{sg}(z_g)-z_t.$$

Both are instances of exactly one predictor input grammar:

$$x_t(m_t)=[z_t,m_t,z_t\odot m_t,A(a_{t-1})], \qquad G_\eta(x_t(m_t))\mapsto p_\eta(a_t\mid x_t(m_t)).$$

The first-order intent and its state-intent interaction form a matched input grammar within each call. The interaction is a fixed bilinear feature, not a learned second-order dynamics model. The local and goal cases are not different input types or separate actors; only the value and gradient role of $m_t$ change.

3. Gradient Contract

The two calls are parameter-isomorphic but not gradient-symmetric.

Path	Endpoint	Gradient behavior	Purpose
Physical intent	observed $z_{t+1}$	current and successor latents attached	ground reachability and preserve action-recoverable change
Goal intent	future $z_g$	current latent attached, future goal detached	train the condition available before acting without moving the goal anchor

There is no pointwise $\lVert m_t^{\mathrm{goal}}-m_t^{\mathrm{local}}\rVert_2$ penalty. Two conditions may be action-equivalent without being close in latent coordinates.

The statistical unit is one supported triplet $(z_t,m_t,a_t)$: one endpoint condition and one proper NLL. Local and goal triplets may share the same demonstrated action, but INTACT never treats endpoint type itself as an equivalence label.

4. Objective

The complete objective combines the unchanged forward prediction term, representation regularization, and two calls to one proper conditional action likelihood:

$$\mathcal L = \mathcal L_{\mathrm{forward}} +\lambda_{\mathrm{reg}}\mathcal L_{\mathrm{SIGReg}} +\lambda_{\mathrm{local}}\mathcal L_{\mathrm{NLL}}^{\mathrm{local}} +\lambda_{\mathrm{goal}}\mathcal L_{\mathrm{NLL}}^{\mathrm{goal}}.$$

The exact task-specific and shared-encoder configurations will be released with the code. Published result tables should always identify which configuration, checkpoint epoch, and inference interface generated each number.

5. Conditional Action Quotient

At fixed current state $z$, define

$$y\sim_z y' \quad\Longleftrightarrow\quad p_E^\star(a\mid z,y)=p_E^\star(a\mid z,y').$$

The quotient groups conditions by the expert action law they induce. Its image under the conditional action operator is canonically isomorphic to the set of realizable action-law families. This is a behavioral correspondence, not a claim that every latent coordinate is unique or globally identifiable.

For finite data and continuous control, the operational version is local: with a task-appropriate action metric and tolerance $\epsilon$, predictions inside the same neighborhood $[a_t]_\epsilon$ may be behaviorally equivalent. The proper likelihood learns probability mass over that neighborhood instead of requiring an exact pointwise match. This gives direct readout tolerance to small prediction errors; it does not identify unsupported goals or guarantee robustness outside the demonstrated action manifold.

6. Deployment Interfaces

Direct. Encode the current observation and goal, predict one action chunk, roll it through the forward model, and replan from the next real observation. No candidate sequence or terminal latent cost is evaluated.

Guarded local verification. Center a small raw-action CEM distribution on the coherent Direct plan. The world model verifies a local neighborhood; it no longer discovers the plan from a broad random proposal.

Actor-disabled CEM. Disable all INTACT action heads and run the original CEM planner. This isolates whether action supervision improved the encoder-forward stack before direct readout.

7. What INTACT Is Not

• It is not inverse dynamics alone: the deployable goal call is essential.
• It is not goal-only behavior cloning: the attached physical call and forward JEPA jointly shape the representation.
• It is not a post-hoc sampling prior: the encoder and action interface train end to end in one stage.
• It does not force a goal endpoint to imitate a one-step successor.
• It does not require globally straight or linear latent trajectories.

8. Released Implementation Contract

The clean release uses a ViT-Tiny/14 encoder with a 192-dimensional projected latent, a depth-6 Forward Predictor, and a depth-3 diagonal-Gaussian intent Actor. Both intent calls have the exact four-slot grammar

[z_t, m_t, z_t * m_t, A(a_{t-1})].

A(a_{t-1}) follows the boundary-aware previous-action contract. An interior clip uses the real primitive action block immediately before its start. At a true episode boundary, only unavailable history is left-padded with raw zero commands and then normalized. Physical and goal supervision both cover all seven adjacent transitions of the eight-frame window. On every index, the same demonstrated action contributes a physical and a goal NLL through the shared Actor. The implementation folds all seven-local/seven-goal rows into one vectorized Actor call.

The task-specific baseline objective is

L = 1.0 * L_forward + 0.02 * L_SIGReg
  + 0.10 * L_local_NLL + 0.05 * L_goal_NLL.

Shared E5 training changes only the SIGReg coefficient to 0.03 and uses fused AdamW with constant learning rate 5e-4, weight decay 1e-3, batch 256 per task, five epochs, and Math SDPA. The Forward loss covers all seven adjacent transitions in the eight-frame window.

Direct and Guarded A require a real action history. Actor-disabled CEM does not call the Actor; its zero-mean proposal is only a search distribution and is never inserted into an Actor input or stored as a checkpoint ablation option.