Background I · hypothetical thinking

A world model asks ‘what if?’ inside the mind

The same real state s remains fixed. The model can preview the hypothetical futures of turning left and right before executing either; every ghost route is a short movie that has not happened.

Like thinking two moves ahead in chess: the real piece is still while the imagined board moves.

Background II · a learned simulator

A world model learns what follows from the present plus an action

s is the present, a is the action, and s′ is the next state after that action. A model may output one prediction or a distribution over possible futures. The same car remains; the intuitive question is now written mathematically.

In one sentence: a world model is a learned simulator of the environment.

Background III-A · first candidate

From the same s, try a₀: turning left hits the block

The red hypothetical route grows all the way to the block while the real car stays put. This crash endpoint s′₀ is remembered and will face the other two futures on the same stage.

The first finger-traced maze route hits a wall; record it without moving the real car.

Background III-B · return to the same start

Now try a₁: braking ends exactly at the start

The red a₀ result remains available, but a₁ produces no outward path. The position after braking is the original position, so s′₁=s; it does not move even a little.

Like pressing pause on a remote control: the second candidate endpoint coincides with the start.

Background III-C · third candidate

Finally try a₂: turn right around the block toward the goal

The green route also starts from exactly the same s, fully avoids the obstacle, and stops near the goal. All three actions have now been previewed, while the real car still has not moved.

The third finger-traced route reaches the flag; only now can we compare fairly.

Background III-D · one present, three futures

The world model previews three short movies before the planner chooses one

a₀ hits the block, a₁ brakes in place, and a₂ safely approaches the goal. All hypothetical consequences share the same start s, so their endpoint differences come only from action a.

The world model previews consequences; the planner compares them.

Background IV · from toy car to applications

Predicting action consequences supports four kinds of work

One world-model capability feeds simulation, content generation, route planning, and scientific intervention. All four jobs share one core skill: predict the consequence of an action first.

Like one pair of ‘what-if’ glasses used for both driving and experiments.

Background IV-B · enter the same SCI branch from the four-use overview

Keep the world model and open only its scientific-intervention window

The scientific-intervention window opens from the SCI node: one batch of cells and one dose knob. The dose has not moved yet — first meet the experiment, then turn the knob.

Like circling the laboratory on a large map and magnifying its workbench; the map and laboratory still refer to the same place.

Background V-A · expand a scientific detail inside the same world model

First scientific candidate: a low dose well inside training coverage

The central world model and its four uses do not change. As the separator opens, the lower region expands the SCI branch into the same cells and a dose knob; the first step examines only a low dose within training coverage.

Like giving the same seedlings one small spoon of water and watching the leaves.

Background V-B · same cells, same knob

Change only the dose from low to the target value

The second candidate changes only the action a: the knob moves from low to the target dose, and the green cell response strengthens accordingly.

Like giving the same seedlings two spoonfuls; the pot stays the same and only the amount changes.

Background V-C · third scientific candidate

Turn the same knob to a high dose poorly covered in training

Red OOD means the model steps outside familiar data and deserves caution; it is not a synonym for 'the prediction must be wrong'.

Like a recipe never tested with ten spoons of salt: it is a warning, not proof from tasting.

Background V-D · compare with one synchronized knob

The comparison marker and dose knob use the same selection variable

The knob visits low, target, and high dose in turn; each highlighted cell outcome corresponds to the currently selected dose.

Like a fan knob set to level 2: the level-2 light must be the one illuminated.

Background V-E · return the detail to the same SCI branch

Three turns of one knob — that is the whole meaning of 'actions change the future'

The three-dose comparison stays on record. This small experiment shows the core skill of a world model: same present, different actions, different futures.

Like pulling back a magnifying glass: the specimen remains in the same textbook diagram.

Background VI · formalize the scientific branch

A scientific world model also learns ‘state + intervention → consequence’

The cells and dose return to common notation: s is the scientific system, a is a controllable intervention, and s′ is the predicted result. This can support physics, chemistry, biology, climate, or robotic experiments, while long-horizon reliability, counterfactual actions, and safety remain open problems.

A toy-car steering button and a laboratory dose knob are both actions a in the equation.

Background VII · many observations of one world

Video, audio, sensors, and text can describe the same state

Four data streams connect one by one to central representation zₜ. They are not four unrelated worlds but different eyes and ears on the same environment; the encoder gathers task-relevant information into one internal state card.

Like classmates using a photo, a recording, and a thermometer to describe the same weather.

Background VIII · three world-model routes

Build a 3D world, generate interactive video, or predict only a latent representation

Three cards appear in sequence. The first two emphasize visible scenes; the third compresses the environment into a low-dimensional state and predicts how it changes. We continue along this JEPA-like third route.

Like writing a card for where the character is and where it moves instead of reshooting the whole movie.

Background VIII-B · the research focus among three paradigms

JEPA keeps only predictable structure: what it learns is a transferable ability to predict

A latent is a coordinate system the model invents for itself: it records only what helps predict the future (position, speed, heading) and drops unpredictable detail (cloud shadows, textures). Prediction therefore happens in these internal coordinates rather than in a fixed output space like pixels or tokens — the predictive ability is decoupled from any one task, and transfers. Following the latent route is a research choice, not a verdict on the other two paradigms.

Like a navigator keeping position, speed, and heading without drawing every cloud outside the window onto the map.

Background VIII-C · a sieve metaphor

JEPA is a sieve: it keeps only predictable structure

Everything in the future frame falls. Whatever concerns position, speed, and heading is caught by the mesh and pressed into the z card; cloud shapes and ground textures pass straight through. The model is not memorizing pictures — it is learning a prediction ability it can carry away.

Like panning for gold: sand falls through, gold stays on the mesh.

Background VIII-D · why SIGReg is still needed

The sieved space must stay fair to every readout

LLMs and VLMs learn task-specific prediction; the z card learned by JEPA must serve readout heads of many tasks. SIGReg spreads the cloud into an isotropic Gaussian so that a readout in any direction receives equally much information — fairness is the outcome, the mechanism is only a distribution constraint.

Like spreading sugar flat on a plate: any spoon from any side gets some.

Background IX · ordinary prediction gains an action input

An ordinary model answers y; a world model answers the s′ caused by an action

The upper row is the familiar x→model→y. The lower row keeps the prediction idea but adds action a. Without an action input, the model cannot directly answer what would happen if we acted differently.

‘What is it?’ is recognition; ‘what happens if I push it?’ is an action-conditioned world-model question.

Background X · one future, two learning targets

Generative methods redraw the future; JEPA predicts its task-relevant card

Two pipelines read the same future frame. The generative target demands pixel detail; the joint-embedding target encodes it into zₜ₊₁. Next, each answer is magnified and compared.

If the teacher only asks where the car will be next, you need not redraw every leaf.

Background X-B · enter one retained future from the architecture comparison

First freeze one future frame, then open its two answer formats

One future frame is about to be asked for two answers: the left redraws it pixel by pixel, the right answers only position, speed, and heading — what the task actually needs. No verdict yet.

Like describing the same toy car with either a full picture or a position-plus-heading card.

Background X-C · lay out both answer formats first

The pixel slot wants a full picture; the latent slot wants three chips

No verdict yet — the two answer formats are simply laid side by side: the left waits for cell-by-cell checking, the right waits to read position, speed, heading. The haloed future frame above remains their only source.

One question; one side demands a full transcription, the other just three keywords.

Background XI-A · fix the same future

The pixel target: even irrelevant clouds and lighting count against the error

The pixel branch demands cell-by-cell alignment of color and texture, so drifting clouds, moving leaves, and lighting changes all count against the error — even though none of them matters for driving.

Like a tracing assignment: every little color cell must match.

Background XI-B · same frame, now a representation card

The encoder compresses the same future into zₜ₊₁

The high-level object above remains unchanged. The new green branch below need not preserve every irrelevant pixel; it keeps information such as position, speed, and heading that supports prediction and control.

Like summarizing a whole picture into three cards: position, speed, and heading.

Background XI-C · side-by-side comparison

Both answers come from one future but serve different tasks

The red branch answers what every future pixel is; the green branch answers what the next decision needs to know. JEPA/LeWorldModel follows the green representation-prediction route without denying the value of pixel generation for other tasks. Contrast: an LLM/VLM learns to predict inside a fixed output space (tokens, pixels), coupling predictive ability to one way of reading out; JEPA first learns a predictable space and leaves the readout to each downstream task.

A painter submits the whole picture; a navigator submits a direction card.

Background XI-D · connect the green branch into a full loop

The latent future card gets its official role: the JEPA target card

The lower region holds the final pixel/latent comparison before the separator closes. The green future card does not disappear; it becomes the target card on the right of the full JEPA loop.

Like inserting the answer card we just wrote into the same grading machine.

Background XII · two branches meet in latent space

The predictor guesses a future card; the target encoder reads the real future card

On the left, fθ(oₜ) produces current zₜ and action aₜ enters Pφ to form prediction ẑₜ₊₁. The right side is not another guess: it reads real next observation oₜ₊₁, encodes zₜ₊₁, and stop-gradients the target branch with sg. (The predictor is written Pφ throughout; the paper writes F.)

Like a student submitting a guess before the teacher builds the answer card from what really happened next.

Background XIII-A · inspect a failure under the same JEPA loop

What if the encoder writes c for everything?

The prediction loop above remains fixed. Below, one counterfactual expands: although the cat, dog, and toy car differ, the encoder begins pulling all their representations toward the same constant c.

Like erasing the labels cat, dog, and car and writing only ‘thing’ on every card.

Background XIII-B · distinct inputs exactly coincide

Representation collapse means every latent becomes the same point

The three distinct input cards remain visible below, but z₁, z₂, and z₃ coincide exactly at c. The result is not merely a smaller cloud or slightly less information; internally the model can no longer distinguish cat, dog, and car.

A dictionary that gives every word the same definition can no longer communicate distinctions.

Background XIII-B+ · press each action while retaining the same c

a₀, a₁, a₂, and a₃ change, yet the predictor always returns c

The JEPA loop above and the collapsed point below remain unchanged. Four action buttons light in sequence, yet every line converges to the same endpoint c. Collapse erases not only the identities of cats, dogs, and cars, but also the different consequences of turning, braking, or steering right.

Like wiring four different remote-control buttons to one cable: every press displays the same blank card.

Background XIII-C · prediction loss accepts an empty answer

Three different tasks receive three identical plans

Goals are observations too: charging dock, goal post, and home all pass through the collapsed encoder and become c. Every imagined endpoint is c and every goal is c, so the three tasks share one identical optimal plan — the car can only act at random and hits the block. Training notices nothing: ‖c−c‖² stays exactly 0.

Like a test that rewards matching the answer key when both student and key write the same empty symbol.

Background XIII-D · reject the one-point solution with batch statistics

SIGReg steps in: the cloud spreads out and the three goals separate again

SIGReg pushes the whole batch toward N(0,I): different observations are forced onto different positions, the three goal cards separate, and the three plans differ again. Note it knows nothing about 'tasks' — it only demands that different observations occupy different points; partitioning is a consequence, not the mechanism. The formal cloud expansion arrives in act 049. Put differently: isotropy gives every task’s readout head an equal chance — SIGReg keeps the space fair to all future ways of reading it.

Important: the mean and covariance are intuitive consequences; the full SIGReg objective constrains the aggregate distribution and is not equivalent to checking only these two numbers.

Background XIII-E · return the conclusion to the JEPA loop

Prediction accepts c; SIGReg rejects c

The lower region holds the final verdict before the separator closes. Prediction loss checks whether each prediction matches its target and therefore accepts c; SIGReg sees zero variance across the batch and rejects c. One more thing to remember: SIGReg checks only where points sit, never how distances between them are read — a door left open that someone walks through in act 106.

One teacher checks ‘matches the key’; another checks that the whole class did not submit the same blank card.

Background XIV · complete training-objective overview

LeWorldModel does two jobs at once: predict the future and reject collapse

The complete JEPA loop stays central. Orange ℒpred compares each predicted future with its real-future target; green SIGReg examines the representation distribution across an entire batch Z. No cloud expands here; the two duties simply return to one training diagram.

One teacher grades each answer; another prevents the entire class from handing in the same blank card.

Background XV-A · expand the wiring under the two-job overview

Step one: current observation oₜ passes through fθ to become zₜ

The upper region continuously retains the two-job overview of prediction plus anti-collapse. Below, the first segment shows camera frame oₜ entering encoder fθ to become current latent state zₜ.

Like compressing the whole toy-car photo into one current-state card.

Background XV-B · extend the same wire to the right

Step two: zₜ and action aₜ enter Pφ together

The first segment remains while the lower region adds only the action input and predictor. Pφ uses the current latent and action to produce predicted future ẑₜ₊₁; the hat marks it as a prediction, not the real next state.

Like combining a current-state card with a right-turn button to write the next guess card.

Background XV-C · connect the real next observation

Step three: oₜ₊₁ passes through fθ to form the stop-gradient real-future target

The right branch explicitly begins with real next observation oₜ₊₁ rather than fabricating a target from the prediction. fθ(oₜ₊₁) yields zₜ₊₁, and sg stop-gradients this target branch for the comparison; only then is ẑₜ₊₁ compared with it.

Wait for the real car to finish one step, then build the answer card from its real photo.

Background XV-D · three segments become one training loop

The present, action, and real future are now fully connected

Together the three segments form one complete training step: current encoding, action-conditioned prediction, and the real target — none is optional.

Like powering three segments of an assembly line that remains one machine.

LeWorldModel · complete training objective

Each prediction must approach the real future while the batch must avoid collapse

The orange distance links prediction card ẑₜ₊₁ to stop-gradient target card sg(zₜ₊₁). The green term checks the aggregate distribution of a batch Z. Here SIGReg appears as a batch-statistics certificate; a dedicated 3D coordinate view explains the cloud shape.

First grade every future card, then check that the whole class did not submit identical cards.

LeWorldModel: Stable End-to-End Joint-Embedding Predictive Architecture from Pixels.

Background XVI-B · two observation scopes under the complete objective

ℒpred sees one future pair; SIGReg sees an entire representation batch

The same complete training loop remains above while its orange prediction rope and green batch certificate are highlighted in turn. Below, the left expands the squared distance for one time step and the right expands the batch Z across samples and time. The terms complement one another, but neither directly checks the local ruler later used by the planner.

One teacher grades this answer, another inspects the shape of the whole class's answers; neither has yet checked the ruler drawn on the classroom floor.

Background XVI-C · transition along the same zₜ card

Add no new loss; simply follow the same zₜ into the model's interior

No new loss is added. We follow the encoder's solid zₜ card into the model's interior; the prediction card ẑₜ₊₁ keeps its dashed identity, never confused with the real zₜ.

Like picking up the newly made current-state card from an assembly line and following it to its matching location on the warehouse map.

Background XVII-A · two views of the same state

Outside is frame o; inside is latent point z

The upper region retains the complete observation-to-latent map. Below, the same state crossing fθ is magnified: a pixel frame and a latent point are not the same kind of object, and the arrow denotes encoding only; dashed ẑ is a prediction while solid z is encoded from a real observation.

A photograph is not a map coordinate; the encoder registers the same car on the internal map.

Background XVII-B · enter through the same zₜ

From here on, the story happens inside the model's own coordinates

As the separator closes, the latent map on the right expands to fill the view. Cyan zₜ entering the map is exactly the solid point encoded from the real observation in the preceding view; it is not replaced by a new point.

Like finding ‘you are here’ on a map from a street photo, then unfolding the map across the table.

Background finale · the same map used by the formal story

Encoded real observations join into zₜ→zₜ₊₁→zₜ₊₂→⋯

Background cloud Z is the aggregate distribution over many samples; the bright line is one actual rollout. Orange zᵍ is a separate planning goal rather than a time index, and predicted ẑ is not mixed into this solid path.

The cloud is many samples; the bright line is one real temporal sequence; goal zᵍ is marked separately.

Formal story opening

Begin with one cloud, one rollout, and one goal

The bright line is one actual temporal rollout, the background dots are an aggregate sample distribution, and orange zᵍ is the planning goal. Later acts magnify or transform these fixed objects without silently replacing the map.

Same map · action preview

Changing the action previews another future beside the same state

The current state and goal remain fixed above. Below, left, straight, and right are tried in sequence. All branches are imagined previews; the real toy car has not acted.

Kid example: keep the real car still and sketch three possible routes in your head.

JEPA / LeWorldModel pipeline

Do not redraw every future pixel; predict a useful future latent card

The current frame encodes to zₜ; the predictor combines it with the action into ẑₜ₊₁, compared against the stop-gradient real-future target — from here the training loop and the map coexist side by side.

Kid example: the teacher asks for the toy car's next position and heading, not a redraw of every leaf.

Counterfactual check

The real cloud stays; only a red copy cheats by collapsing to c

Cats, dogs, left, and right are distinct. If the encoder submits the same c card for all of them, prediction error may be small while the representation loses all distinctions.

Collapse cheat · full derivation

The action changes, but prediction and target remain c, so prediction loss stays zero

Above, the cat, dog, left-turn, and right-turn inputs remain collapsed at c. Below, actions a₀…a₃ are tried in sequence. If the predictor also ignores the action and returns c, every case receives a perfect c-versus-c score.

Kid example: the student answers “same” for a cat, a dog, and a toy car. If the answer sheet is also wrongly marked “same,” the score is perfect although nothing was learned.

Continuous view

Every point still crowds the origin: two forces are about to act on this lump of zero

The axes z₁, z₂, z₃ rise while the whole batch is still the single point c. The prediction rope and the SIGReg spring are both about to arrive.

3D batch-shape explanation

The prediction rope checks the future; SIGReg springs reject a one-point cloud

The shortening orange rope means a better prediction; green springs spread the same batch along multiple directions. The jobs complement each other, but a valid aggregate cloud still does not certify every local tile. Another reading of the spring: no direction crushed, none privileged — fairness to every future readout; and this fairness covers only the distribution, never how distances are read.

From distribution to temporal path

Select three consecutive states from the same batch and define two actual velocity arrows

The same batch remains above. Below, only zₜ₋₁, zₜ, and zₜ₊₁ are magnified. Every later straightness value is computed from actual points on this rollout.

Direction ruler 1 / 3

Aligned arrows have straightness 1

Turn the teaching dial parallel to the first arrow. The number is computed live from the two arrows drawn.

Direction ruler 2 / 3

Rotate the second arrow by 90° and straightness falls to 0

This is a teaching ruler attached to the same vₜ, not a fabricated new rollout.

Direction ruler 3 / 3

Opposite arrows have straightness −1

With aligned, perpendicular, and opposite anchors established, the next step returns to the two actual tangents on the trajectory.

Continuous view · return the teaching scale to the real path

The teaching dial retires: from here we measure the real trajectory

The selected consecutive states above never change. The teaching arrows below dock back onto their matching path points, after which the actual vₜ, vₜ₊₁, and cosine are computed from those points; the teaching example never becomes a fabricated rollout.

Actual trajectory

As the path straightens, the cosine of its actual adjacent tangents approaches 1

The upper path, lower arrows, and displayed value come from the same array; the score is not detached from the drawing.

Evidence boundary

Straighter after training is observed; monotone straightening at every step is not proved

The red ghost retains the pre-training path and the green curve retains the post-training result; together they separate the observed phenomenon from unproved stepwise monotonicity. Why does it straighten by itself? Three suspects are lined up next; the culprit is revealed in act 103.

From the phenomenon to its dynamical causes

Why does velocity change on the same path? The equation separates three sources of turning

The velocity change of one path splits into three sources: the system’s own A−I, action changes BΔa, and noise changes Δξ. Next each source is isolated in turn.

Kid example: a toy car may turn because the road bends, the driver steers, or a side wind pushes it; the three causes should not be mixed.

Dynamic A comparison · step 1

Fix Δaₜ=0 and Δξₜ=0, then isolate how A≠I turns the arrow

The same vₜ passes through the bent conveyor A=R(35°): each pass twists it by 35°, and repeated passes trace the arc above.

Dynamic A comparison · step 2

Move only A(η): the output arrow and the entire trajectory straighten together

The input velocity, start, action, and noise remain fixed. Only the rotation angle shrinks continuously from 35° to 3°.

Dynamic A comparison · step 3

The comparison freezes: R(35°) arcs, R(3°)≈I goes straight

This comparison shows only that a smaller A−I term favors less turning. It does not mean there is no action, nor does it remove BΔa or noise from the general formula.

Complete dynamics · retain all three terms

A≈I reduces only the first term; action changes and noise can still turn the path

The R(35°)/R(3°) comparison stays on record. The autonomous-turn, steering, and noise vectors grow one by one, laid head to tail — their sum is exactly the true vₜ₊₁−vₜ.

Thus near-identity dynamics are helpful, not a sufficient guarantee that every trajectory is perfectly straight.

Continuous view

Roll the same local dynamics forward K steps and express its endpoint locally as a function of action

The start c₀ stays paired with its K-step endpoint. Near c₀, J describes how each action component pushes endpoint zK in a particular direction.

Rollout-Jacobian workbench

Four action knobs connect to one zK coordinate system

The axes, c₀, four knobs, and four dependency lines remain throughout. All aⱼ are zero, so the shared endpoint is c₀.

Zero-action baseline

With every knob at zero, zK=c₀

First fix the common base point; the next steps measure J column by column.

J column 0

Turn only a₀; the out-and-back zK motion measures J_:0

The knob returns to zero, while the measured column remains on the axes.

J column 1

Retain J_:0 and turn only a₁ to measure J_:1

Base point, endpoint definition, and axes stay comparable: all four measurements share one ruler.

J column 2

Turn only a₂; add the third sensitivity direction to the same axes

Previously measured columns remain while the current column completes an out-and-back motion.

J column 3

Turn only a₃; all four directions remain visible afterward

All four measurements use exactly the same base point and endpoint definition.

Four-column summary

Freeze all four columns of J and return every knob to zero

The four measured sensitivity directions form one fixed matrix J.

Four knobs and four dependency lines

a₀…a₃ move together; four dependency lines track zK

Each knob controls one term J_:j aⱼ, and all four terms add at one endpoint. Every knob completes the motion and returns to zero.

From dynamic test to a fixed candidate

The four knobs settle at canonical a⁺ and form fixed endpoint Φ(a⁺)

Even when a₀ and a₂ are zero, their dependency lines remain and are labeled zero; they are still components of the action vector.

Fix one set of planning objects

The bench is frozen: the next five steps move only the math, never the objects

The next five steps do exactly one thing: translate 'how far the endpoint is from the goal' into 'which way to turn the actions'.

HGN 1 / 5

First define the endpoint residual and squared cost

J*, a⁺, Φ(a⁺), and zᵍ remain fixed above; below, only their difference is introduced.

HGN 2 / 5

The same J* maps a small action change δa to endpoint change δr

The 4D action perturbation and 2D endpoint perturbation remain dimensionally distinct in the scene.

HGN 3 / 5

Pull the endpoint residual back to action space through J*ᵀ

Gradient first — curvature enters only in the next step.

HGN 4 / 5

Differentiate again and highlight the action-side Gauss–Newton curvature

For the current affine endpoint model Φ(a)=c₀+J*a, it is also the exact Hessian. For a general nonlinear world model, it is the Gauss–Newton approximation.

Scientific boundary · separate exact and approximate cases

For an affine action-to-endpoint map, 2J*ᵀJ* is the exact Hessian; generally nonlinear maps use it locally as Gauss–Newton

The same J*, candidate, endpoint, and goal remain fixed above. Below, the left keeps J* constant throughout the region, while the right lets J(a) vary and uses the same probe only for a local tangent approximation.

Kid example: a straight ruler is exact across flat paper; on a curved map, a short ruler is only locally accurate near your feet.

HGN 5 / 5

J*ᵀJ* and J*J*ᵀ have different sizes but share nonzero eigenvalues

The left is action-side curvature; the right object has its own name in control theory: the controllability Gramian. The shared positive spectrum builds the planning–control bridge, though it does not flatten the spectrum by itself.

A new theoretical view of coordinate identifiability

The shared-spectrum certificate retracts as the question expands to the full Gaussian coordinate space

We step into a separate thought-experiment room: the conclusions here hold in any dimension; we return to our 2D numerical example afterwards.

Marginal blind spot

The aggregate Gaussian may remain unchanged while the local grid and same route shear

The same points, grid, and route evolve continuously under Tτ. This counterexample passes only the marginal-Gaussian check; it does not claim to preserve the full temporal or predictive joint structure.

Aggregate check versus local ruler

A healthy aggregate does not certify a healthy tile: aim at one small square

The points may still form a valid Gaussian, while a tiny tile near the probe can be stretched by DT₁ into a skewed quadrilateral.

Kid example: an unchanged class-height histogram does not mean the grid under every child stayed unstretched.

Why prediction adds constraints

The distribution of Z alone is insufficient; temporal prediction sees the pair (Z,Z′)

The marginal-preserving counterexample remains above while a correlated adjacent-time pair is added below. Preserving each Gaussian marginal does not automatically preserve the pairing relation.

Continuous view · marginal → Hermite

The same correlated pair — now we take its 'spectrum' apart

The correlated pair (Z,Z′) is unchanged and never resampled; we read its H₁ and H₂ modes off it order by order.

Hermite 1 / 3

First draw the actual H₁(x)=x and H₂(x)=x²−1

Vertical normalization only makes the shapes legible; it does not change the exact weights shown later.

Hermite 2 / 3

Adding H₃ and H₄ reveals more bends at higher degree

These are probabilists' Hermite polynomials; sine waves may illustrate complexity but cannot replace these basis functions.

Hermite 3 / 3

Multiply the same modes by ρ^|α|: higher degree decays faster across time

The degree-one mode is line-like, while higher modes resemble increasingly complex ripples. Every opacity is driven by the exact weight ρ^k.

Theorem continuous view

Higher-degree ghosts retract as the retained degree-one structure expands into global orthogonal axes

This step holds only under the paper's stated Gaussian, predictability, and zero-error or optimality assumptions; it is not presented as an unconditional fact about every world model.

Global rigid coordinates

Rotate the 3D axes together: the whole map undergoes one shared Q

The cloud, route, and all three axes rotate together. Lengths, angles, straightness, and J's singular values remain. Different positions cannot use different rotations.

Fixed reference for global Q

Fix this orthogonal coordinate Q* as the reference frame

The finite-error discussion that follows is measured against this single θ*.

Practical conclusion

Real errors are not exactly zero, so the conclusion is only conditional closeness to some Qz

The exact Q* reference remains fixed above. The lower schematic approaches it as prediction and Gaussian-matching errors shrink. It is not an equality plot of a theorem bound.

From representation to controlled identifiability

Even with near-Q coordinates, an unturned action knob cannot be identified from data

The near-Q conclusion remains above while the controlled transition appears below, changing only action coverage during training. The conditions are complementary but not identical.

Continuous view · near-Q → action exploration

The same controlled transition — now we ask about its action coverage

A, B, and J* stay fixed. First see what happens when only a₀ is exercised: a narrow slit in the 4D action space and its 2D endpoint projection.

Persistent excitation · turning the first knob

Rotation finally means something: turn a₀ and its direction enters the data

The knob panel at the upper right is this chapter's console. Turning a₀ sends a streak of light simultaneously into the transition formula, the coverage bars, and the projected ellipse: three cells respond in the same frame. The three untouched knobs stay under gray glass.

Like a radio where only the volume knob was ever turned: nobody yet knows what the others do.

Persistent excitation 1 / 3

Turn only a₀: the 4D action covariance has rank 1 and endpoints move through a narrow slit

Lower left is 4D action coverage; lower right is the 2D endpoint projection obtained from the same covariance through fixed J*. The views are linked but do not have the same dimension or rank.

Persistent excitation 2 / 3

B and B′ may agree on seen actions yet predict different futures for unseen e₂

If training covers only span(e₀), data can check only Be₀. A model may be correct on-policy yet wrong for a counterfactual action that was never tried.

Kid example: if only the first of four knobs was tested, no one knows what the third knob truly controls.

Persistent excitation · the counterfactual magnifier

Where data cannot reach, a model may answer anything

Magnify the two forked endpoints from the previous act, then light up every training point: each lies inside span(e₀), with no line connecting it to either forked endpoint. The problem is not too little data — the direction simply never occurred.

Chapter three was never on any exam, so two students with different chapter-three answers both keep perfect scores.

Persistent excitation 3 / 3

Actually turn a₀…a₃ one by one until the 4D λmin becomes positive

Action colors stay aligned with the four columns of the same J*. The left asks whether every action direction was tried; the right shows only the 2D endpoint reachable projection.

This shares the necessary idea behind persistent excitation in control theory: an unexcited direction cannot be identified from data. It does not claim the full theorems are equivalent.

Continuous view

Near-Q representation conditions and action-excitation conditions enter one evidence spine

The condition families are distinct but complementary: one constrains coordinates, while the other asks whether control actions were sufficiently observed.

Evidence status attaches directly to the spine

✓ established, △ conditional, and ? open claims form one logical spine

The three nodes remain visible while each view expands one node's local derivation.

Evidence audit 1 / 3 · green light

First mark the parts supported by experiments or standard linear algebra

The three-node spine above remains. Below, the before/after straightness comparison and the color-matched spectra of J*ᵀJ* and J*J*ᵀ light up in sequence, representing an empirical observation and an exact shared-positive-spectrum identity.

Evidence audit 2 / 3 · amber light

Next switch on the conditions required for near-Q and controlled identifiability

Gaussian population, predictability, global optimum, finite-error stability, and action excitation appear one by one. An arrow may reach near-Q or identified action mechanisms only when the corresponding switches are satisfied.

Amber does not mean wrong; it warns that the conclusion has premises and the conditional arrow is not an unconditional equality.

Evidence audit 3 / 3 · red light

Finally state three things these results do not yet prove

Below, finite-sample behavior, whether nonlinear SGD selects the straightest coordinates, and whether deployment preserves the model-to-planner chart are marked in sequence. Red denotes open gaps, not a rejection of earlier evidence.

This is the break LatentGauge magnifies next: the internal map can be good while the ruler delivered to the planner is replaced.

Keep the three-step logic visible

The three-step logic takes its place: now check it item by item

Nodes, status badges, and arrows retain their correspondence.

Unified chain 1 / 3

Prediction constrains temporal futures, SIGReg rejects collapse, and theorem assumptions are required for near-Q

The three cards appear and remain in sequence; the conditional arrow is not omitted. Controlled settings additionally require sufficiently rich actions.

Unified chain 1 · lookback

First link checked: why training walks into near-Q coordinates

The upper left freezes chain 1's final state. The lower right marks its place on the three-chain map: first cell lit. The lower left lists the assumption chips this chain spent: Gaussian target, stationarity, the zero-error limit.

After the first flight of stairs, glance back before the second.

Unified chain 2 / 3

A≈I, smooth actions, and moderate noise reduce three sources of turning

The three boxes correspond to autonomous bending, action changes, and noise changes.

Unified chain 2 · lookback

Second link checked: how three turning sources are suppressed

Chain 2's final state stays upper left: each of the three turning-source boxes now carries its price. The second progress cell lights; lower left shows its assumption chips — linearization neighborhood, smooth actions, moderate noise.

Second flight done; its conditions of use are taped to the rail.

Unified chain 3 / 3

Global Q preserves straightness and J singular values; shared spectra link planning and control

Shared spectra only establish a conditioning correspondence; they do not automatically balance the spectrum. The final question is whether the deployed planner receives the same chart.

Handoff: training theory may certify a good internal map, but it has not certified the deployed model-to-planner interface.

Unified chain 3 · lookback

Three links closed: the whole chain from objective to easy planning is lit

All three progress cells shine. Upper left keeps chain 3's final state: Q preserves straightness and the shared positive spectrum wires planning to control. From the next chapter on, a newly inserted module will pinch this glowing chain in the middle — remember how it looks now.

The whole string of lights is on — which is exactly why the coming cut will be visible.

Camera handoff · from the map to the delivery line

Zoom out: the whole coordinate map is the interior of the clean-z node

The camera pulls back for the first time. The coordinate map we studied for three chapters — axes, cloud, straight route — shrinks and settles into the third node of a five-stage delivery line. The space is not replaced; we finally see where it sits inside the machine. The next chapter walks this line segment by segment.

Like looking up from an atlas: the page you were reading is one small window on a dashboard.

Deployment break · clean interface

Fix the world model first: the planner directly receives clean z

The upper-left retains the trained f, P, and the same clean latents. The lower-left follows one z along the unchanged handoff to the planner. The moving light only tracks that same z; it does not generate a new future.

Kid analogy: first verify both the treasure map and the route used to deliver it.

Deployment interface · sole modification

Keep the internal map fixed and insert Tτ only between clean z and the planner

The upper-left is the same frozen pipeline. Below, only Tτ enters the original handoff. Candidates, rollouts, and world dynamics are not recomputed; only the coordinates used by the planner to measure distance change. The door left open in act 031 is pushed open here.

Continuous view · I → M

Expand the T₀=I interface card into the same latent coordinate map

The card and the map are two scales of one object: the interface card reading T₀=I unfolds into an identity coordinate map that changes nothing.

Coordinate map · safe negative control

Global Q₀=R(θ₀) rotates the whole map together while the local ruler remains I

The same points, route, and grid share one constant angle θ₀, so every tiny square remains a square.

Kid analogy: turning the whole graph paper does not change cell lengths.

Continuous view · global rotation to shell comparison

At τ=0, first verify: no relative twist between adjacent shells

Both shells come from one coordinate system. At τ=0 the map is exactly Tθ₀,0(z)=R(θ₀)z: every radius shares the same θ₀ and the relative angle is Δθ=0 — this is what “a global rotation is harmless” looks like.

Coordinate map · from Q to position-dependent T

Adjacent shells begin to rotate relative to each other: the α′≠0 shear is born here

As τ turns from 0 to 1, the global angle θ₀ stays put while the r=1 shell of zᵍ and its neighbor receive different α(r²) angles. The short spokes joining the shells shear — the first visible evidence of position-dependent twisting.

Continuous view · shell comparison to the full T₁ map

The shear is now written into the map: this is the shape of canonical T₁

θ₀ is the global base rotation; on top of it sits the τ=1 inter-shell shear — their composition is this already-deformed coordinate map.

Visual intuition · once only

Twist the same grid strongly once so coordinate warping becomes visible

This is an illustrative τdisplay:1→5; it is not used in the attack budget, condition number, or ranking. The map remains invertible and does not merge points.

Coordinate map · lock the numerical version

All exaggeration returns to zero: this canonical T₁ is the world of every act to come

Substituting θ₀=0 into Q₀=R(θ₀) gives Q₀=R(0)=I, the identity matrix. This canonical T₁ is inherited by the metric, the optimizer, the candidate ranking, and the curvature spaceship to come.

Continuous view · full T₁ map to a local probe

Aim at the goal point zᵍ: the next four acts happen in its neighborhood

The same zᵍ=[1,0] serves as the shared probe for DT, G, H, and the optimizer.

Coordinate map · same probe

zᵍ has exactly zero displacement while a neighboring-shell point rotates relative to it

The canonical T₁ remains above. Below, the same zᵍ and a neighboring-shell point show how α(1)=0 and α(r²)≠0 can both hold.

Kid analogy: a thumbtack stays fixed while nearby graph-paper cells can still shear.

Coordinate map · same samples

The same 160 samples, binned by radius: rotation changes no radius

Every N is a direct count over the same samples. Within a shell the map only rotates, so every radius is unchanged.

Coordinate map · from bins to neighbors

Four neighboring points testify: the inter-shell shear lives on real samples

The gray dashed line shows original positions; the red line shows the same four points after Tτ. Different radii accumulate different angles, shearing the local neighbor line.

Local geometry · shared zᵍ

Separate two facts: zᵍ moves zero, yet its local square shears strongly

The two lower-left labels occupy the same pixel because T₁(zᵍ)=zᵍ. The lower-right square becomes a parallelogram around that same center because the derivative differs.

Local geometry · one edge of the tile

DT₁(zᵍ) tells how the same tiny arrow changes under T

The lower-left shows clean dz, and the lower-right shows DT₁dz for that same arrow. Both grow from the same origin and remain complete.

Local geometry · from one arrow to the whole ruler

Combine all tiny arrows to obtain G₁=DT₁ᵀDT₁

An equal-distance circle in scoring y-space becomes a local ellipse in clean δ-space; its axis ratio is √κ.

Local geometry · no skipped derivation

The same action perturbation δa first becomes an endpoint displacement through J*, then G₁ measures its length

The same zᵍ, J*, and G₁ remain above. Lower left combines four action knobs into δz=J*δa; lower right places δz in the planner's new ruler and expands the quadratic form step by step.

Kid analogy: four control strings first decide how the car moves; a skewed floor ruler then decides how expensive that move looks to the planner.

Local geometry · central formula

Frozen J* transports the planner's latent ruler back to action space

The lower detail connects δa→J*δa→G₁→J*ᵀ, yielding Gauss–Newton curvature at the same probe zᵍ.

Local geometry · control

The control experiment: dial τ back to 0 and the ellipse relaxes into the circle

The control copy returns to T₀ only for side-by-side comparison; the attack itself never lets go. The ellipse relaxing back into a circle is the clean optimizer's starting point.

Continuous view · local ruler to optimization slice

The clean baseline: circular contours — no route wastes a step

The plot shows the reachable slice u=J*a∈ℝ²: the optimum, start, viewing direction, and scale remain fixed. The full action-side H₀∈ℝ⁴ˣ⁴ has rank 2, so it also has two unshown null directions.

Local geometry · dynamic optimizer trajectory 1

Inspect the clean 2D reachable slice first: circular contours and balanced positive spectrum

The left plot shows optimization in u=J*a∈ℝ²; J*, the start, and optimum u* are fixed. The corresponding full 4D H₀ has rank 2 and two unshown null directions. The green probe demonstrates the one active-slice update.

Local geometry · dynamic optimizer trajectory 2

One τ jointly changes Gτ, Hτ, κ₊, and the 2D active trajectory

The circle stretches into an ellipse while the condition number rises from 1 to 7.30; the red-square path in the u=J*a slice becomes zig-zagged. Here J*J*ᵀ=I, so κ(H̄τ)=κ₊(Hτ)=κ(Gτ); the full Hτ remains a 4D rank-2 matrix.

Local geometry · dynamic optimizer trajectory 3

Step-by-step check: every one of the 16 zig-zag steps comes from the same H₁

The orange ring is only explanatory: the u-space squares, H̄₁, and guide stay fixed. The two null directions of full H₁ remain unshown.

Local geometry · dynamic optimizer trajectory 4

Five steps already expose the cost: crossing a long narrow valley, direction flapping back and forth

This is the 2D active update in u=J*a. Highlighted squares remain fixed while later iterates continue from k=5. Null directions of the full 4D H₁ do not contribute curvature to this plot.

Local geometry · dynamic optimizer trajectory 5

Steps crowd tighter near the optimum — and still flap back and forth

As the error shrinks near u*, consecutive red squares become denser; all sixteen positions from u⁽⁰⁾ through u⁽¹⁵⁾ remain visible at the end.

Local geometry · step comparison

One optimizer: sixteen clean vs sixteen attacked steps replayed side by side

Two windows step from 0 to 16 together: the left strides straight over circular contours, while the right is forced into short zigzags by the κ(G₁)=7.30 ellipse. Every landing aligns frame by frame; the gap accumulates into the conclusion.

Two students copy one map; one walks flat ground, the other is rerouted into a slanted corridor.

Local geometry · complete final result

Keep all 16 active-slice steps and add only the meaning of ρGD

ρGD is the worst-case error fraction retained per active-slice step. Nearer 1 means slower shrinkage. Here κ(H̄₁)=κ₊(H₁); the two null directions of full 4D H₁ are excluded from κ₊.

Local geometry · why the zig-zag appears

Decompose the same error into stiff and soft modes: equal-magnitude contraction, different sign behavior

With this demo's optimal constant step η*=2/(λmax+λmin), the two multipliers are exactly −ρ and +ρ, so both magnitudes contract by the same factor each step. The large-eigenvalue component flips sign and causes the zig-zag, while the small-eigenvalue component keeps its sign. Their sum is the same trajectory above.

Kid analogy: both arrows shrink to ρ times their size each step; one alternates left and right, while the other keeps pointing the same way.

★ The tidiest symmetry in the film

Making planning easy and making planning hard share one formula channel

Left, straightening: minimize 1−𝒞 and flatten the condition number of J⊤J. Right, LatentGauge: never touch J — only swap the metric in the middle to G_T. The two skeletons connect node by node: one Gauss–Newton skeleton, run forward and in reverse.

One corridor: a handrail installed at one end, the floor tiles tilted at the other.

Continuous view · action curvature to candidate rollouts

From 'how fast' to 'which route': the two candidates enter

The red squares above are optimizer iterates and the two curves below are action rollouts; both are linked by the same frozen J* and planning problem.

Candidate comparison · candidates never reset

Draw the complete frozen rollouts a⁺ and a⁻ from one common start

The optimizer remains above. The lower curves are schematic guides for following the two action sequences. Their action vectors, clean endpoints zK(a±), costs, and later ranking are computed exactly from one numerical example and remain fixed.

Candidate comparison · two action sequences

a⁺ and a⁻ are two fixed action sequences, not signs of numbers

The focus circles a⁺ and then a⁻, but the other path never disappears. Curve shapes are schematic; each action vector and clean endpoint from zK(a)=c₀+J*a is exact and fixed, and later costs use only those endpoints.

Continuous view · cost view of the candidate actions

Cost has its own center zᵍ, while the twist pivots at the origin 0

The goal zᵍ and latent origin 0 are explicitly distinct: the twist is radial about 0, while cost is centered at zᵍ.

Candidate comparison · clean boundary

A clean equal-cost boundary must be a circle centered at zᵍ

The lower circles are drawn once and remain in place next. Every point on one circle has the same squared Euclidean distance to zᵍ.

Kid analogy: a rope of fixed length around a flagpole traces a circle.

Candidate comparison · circles do not reset

Keep the same circles and measure only two radii: a⁺ is cheaper

Lower C is better. The squared distance from a⁺ to zᵍ is 0.25, below 0.4096 for a⁻, so the clean winner is clearly a⁺.

Candidate comparison · the same rollouts

Send the same clean endpoints and goal through the same Tτ

Hollow points always mark clean coordinates and filled points mark scoring coordinates. As τ moves from 0 to 1, only coordinate readings move; actions and clean rollouts stay fixed.

Continuous view · two readings of the same candidates

One set of candidates, two ledgers

The two columns share the same a⁺, a⁻, zᵍ, and rollouts; only the scoring coordinates differ.

Candidate comparison · two rulers for the same rollouts

Same candidates; the right-hand ruler turns — watch the costs follow

The right-side costs, radii, and winner change together until a⁻ becomes cheaper.

Continuous view · two costs to one margin

Subtract the two costs into one needle that can cross zero: the margin

The two costs combine into one decision quantity that can cross zero: positive selects a⁺ and negative selects a⁻.

Candidate comparison · success criterion

The margin crosses from positive through zero to negative, flipping the planner from a⁺ to a⁻

候选、终点、目标从头到尾没动；唯一在变的是评分参数 τ。mτ<0 就是成功攻击状态。

Kid analogy: the two toy cars do not change; skew the ruler near the finish and the judge changes the ranking.

Candidate comparison · three boundaries 1/3

Clean exact: a z-space circle centered at zᵍ

The successful attack context stays above. Below, one complete circle is traced around the same zᵍ and held at terminal.

Candidate comparison · three boundaries 2/3

Attacked local: near the goal, the second-order approximation is a G₁ ellipse

The faint clean circle remains as reference. The orange ellipse explains exact cost only for small δ and is not presented as a global exact boundary.

Candidate comparison · three boundaries 3/3

The exact y-space circle becomes a curved exact z-space preimage under T₁⁻¹

The two windows are two different spaces: planner-scoring y on the left, clean z on the right. The red contour is the exact nonlinear boundary.

Continuous view · equal-cost contour to parameter feasibility

A new question: which τ flip the ranking while staying inside the displacement budget?

The spatial contour still corresponds to the same Tτ; the parameter chart asks which τ both flips the ranking and satisfies the displacement budget.

Feasible region · feasible set 1/4

The margin is a continuous function of τ: it crosses zero at τ=0.766

marker 从 τ=0 走过真正的 mτ=0 决策边界。穿过后 mτ<0，a⁻ 严格获胜。

Feasible region · feasible set 2/4

The attack cannot move points arbitrarily: add D(τ)≤ε²

The construction of Tτ already preserves Gaussian measure exactly; the extra budget limits mean squared displacement. The upper and lower markers still share one τ.

Feasible region · feasible set 3/4

真正可行区是 mτ<0 与 D(τ)≤ε² 的交集

mτ=0 is only a boundary and is not a successful attack. The green region begins strictly to its right and ends at the budget boundary.

Feasible region · feasible set 4/4

From the point just past the boundary to the final τ=1 we actually use

Boundaries, curves, and candidates do not reset. The terminal marker is exactly τ=1, not an intermediate value.

Feasible region · controls and defenses

When the attack is exactly powerless: three precise negative controls

A global Q preserves every distance. If a candidate endpoint and goal zᵍ lie on the same origin-centered shell, that pair also shares one rotation. When α′=0 there is no inter-shell shear. Detection and blocking mechanisms are separated into the next scene.

Negative control · precise same-radius statement

Only when a candidate endpoint and goal zᵍ lie on the same origin-centered shell do they receive one common rotation

The successful attack remains above. Below, a pair satisfying ‖zK(a)‖=‖zᵍ‖ lies on one circle. Tτ applies the same orthogonal matrix to both, so their Euclidean distance is exactly preserved. Merely putting a⁺ and a⁻ on the same radius is not enough to preserve each cost to the goal.

Defense directions · not claimed as completed experiments

Three protections inspect different objects: the local ruler, trusted anchors, and the deployment interface

Below, the left gauge measures ‖G−I‖, the center compares trusted anchors before and after the interface, and the right verifies the chart actually called by the planner. These are mechanism-derived defense directions, not a completed defense benchmark in the current numerical demo.

Continuous view · parameter feasibility to evidence summary

Before counting the win, check three premises: invertible, measure-preserving, within budget

The upper view retains the same Gaussian sample, T₁, trajectory, and candidates; the lower view checks invertibility, measure preservation, and the displacement budget.

Coordinate map · evidence chain 1/4

Prediction and anti-collapse certificates remain valid but do not certify the deployment interface

The canonical map remains above while the lower view checks the two training guarantees: prediction and anti-collapse.

Coordinate map · evidence chain 2/4

The same J* gives matching positive spectra for J*ᵀJ* and J*J*ᵀ

Shared spectra only connect conditioning on the action and latent sides; they do not automatically flatten the spectrum. Optimization becomes easier only under extra conditions, such as balanced positive eigenvalues and a valid local quadratic model.

Surprising similarity · central derived insight

Insert the planner's new ruler G₁ into the clean bridge to obtain a weighted spectral bridge between attacked planning and control

The same frozen J* remains above. Below, define K=G₁¹ᐟ²J*. Then KᵀK=J*ᵀG₁J* on the action side and KKᵀ=G₁¹ᐟ²J*J*ᵀG₁¹ᐟ² on the endpoint side; linear algebra gives them the same positive eigenvalues.

Meaning: T₁ does not change J*, yet G₁ changes both action-side optimizer curvature and endpoint-side weighted controllability geometry. Shared spectra establish correspondence, not balance.

Coordinate map · evidence chain 3/4

没有重跑世界模型，名次却翻了：m₀>0 → m₁<0

The clean margin is positive and the T₁ margin negative, showing that changing scoring geometry is sufficient to flip the ranking.

Coordinate map · evidence chain 4/4

训练证书、共享正谱、canonical T₁ 与 m₁<0 汇到同一张地图

The four checks correspond in order to prediction and anti-collapse, positive Gram spectra, canonical T₁, and the negative margin of the same candidates.

Continuous view · coordinate map to curvature spaceship

Fly the planned route yourself: is the paper bent, or the road?

Four pieces of evidence are on record. One question is left to intuition: in coordinates twisted by T₁, what does it feel like to fly a route that is actually straight?

Curvature spaceship · curvature spaceship 1/4

Before takeoff, look down: the route is straight, the grid beneath is twisted

The green tangent arrow points along the true route. Note it is not parallel to the grid lines underfoot — the grid is coordinate paper twisted by T₁.

Curvature spaceship · ignition

Before liftoff, three gauges come alive

The route map freezes upper left: a straight path over a twisted grid. The instrument panel magnifies upper right: displacement D(s) zeroed, angle Δθ zeroed, budget full at ε²=0.0049. The tail flame lights and the hull lifts a few pixels — the voyage starts next act.

Zero the odometer, fill the tank, then start the engine.

Curvature spaceship · curvature spaceship 2/4

The first half is uneventful: a straight road stays straight

The flight itself is perfectly normal — straightness is a property of the dynamics, and no coordinate twist changes it. The anomaly waits at the terminal.

Curvature spaceship · curvature spaceship 3/4

The anomaly appears at the terminal: the moment distances are measured

On arrival the cost ruler comes out — and it reads in the twisted units. The flight was fine; the billing is not. That is the entire attack.

Curvature spaceship · curvature spaceship 4/4

This analogy borrows exactly one thing: coordinates can bend

T₁ is invertible and only the latent coordinate grid bends. This is not physical spacetime, and no geodesic is solved.

Continuous view · curvature spaceship to demonstrated contribution

Analogy filed; back to the real numbers: the same candidate pair, margin still −0.0558

The upper view retains the same canonical map, while the lower view reconnects the computed a⁺, a⁻, and m₁.

Contribution boundary · demonstrated

World-model inference-time latent-coordinate modification attack

The real canonical T₁, route, and ship remain above; the real candidates remain below. f, P, J*, actions, and clean rollouts are frozen. The only modification is Tτ before the planner.

Finale · demonstrated and future work separated

Preserve marginal distributional stealth while degrading planning reliability

The real canonical map, endpoint ship, candidates, and successful margin remain throughout. The green box is the demonstrated coordinate-stealth planning attack. Blue arrows unfold unlearning, sparsity, and repair/defense, explicitly marked as future research.

The attack is the first demonstration in a broader latent-geometric modification space, not the endpoint.