Chapter 12 — Keep the Mathematics Minimal but Sufficient

The big picture

A tiny story

One workshop inspector checks whether the action-conditioned guess matches the latent of what really happened next. A second inspector checks whether the crowd of observation embeddings has shrunk into an easy, degenerate shape.

Listen only to the first and every observation can become one constant answer. Listen only to the second and the cloud can look beautifully spread while forgetting which action reaches the doorway. The two inspectors do not grade the same thing.

The technical backpack

The prediction term is coordinate-wise mean squared error against a connected next-observation embedding:

z = shared_encoder(observation_window)
z_hat = action_conditioned_predictor(z_context, recorded_actions)
L_pred = mean((z_hat - z_next_connected)^2)

Because the target is produced by the shared trainable encoder and is not detached, L_pred can update the predictor, action path, context representation, and target representation.

SIGReg acts on observation embeddings at each relative time position. Random unit directions create one-dimensional shadows; finite cosine/sine fingerprints are compared with the analytic standard-Gaussian fingerprint. It directly updates the visual encoder and encoder projector, not the action encoder or dynamics predictor.

The complete objective is:

L_total = L_pred + lambda * L_SIGReg

lambda is a volume knob for the population pressure. It does not alter what SIGReg measures, choose latent width, set projection count, tune the optimizer, or control planning horizon. Too quiet leaves the trivial shortcut available; too loud can favor global Gaussian shape over transition relationships.

Version labels are part of the mathematics:

The paper describes projection count and the regularization weight as two introduced settings and calls lambda the only effective one to tune after its reported projection ablation. That does not mean the model, optimizer, data, and planner have only one hyperparameter.

Frozen SIGReg uses 1,024 projections and 17 knots from 0 through 3. Batch size controls samples in each empirical shadow; projection count controls how many directions inspect those same samples. One cannot replace the other.

The trick that fools us

Train three matched conditions: no SIGReg, a clearly labelled paper or code weight, and a deliberately overwhelming weight. Record raw L_pred and L_SIGReg separately—not only their weighted sum—plus variance, effective rank, neighbors, action swaps, one-step error, and autoregressive error by horizon.

Expected mechanisms are conditional. Removing SIGReg makes constant agreement compatible with the objective, but one run need not collapse. Overwhelming SIGReg can hurt dynamics, but no universal threshold follows. A spread cloud with action-insensitive predictions is not globally collapsed; it is dynamically weak. Good training gauges with bad planning point to a later link such as rollout, cost, or search.

Evidence receipt

LeWorldModel v3, frozen train.py, frozen SIGReg, and frozen lewm.yaml support the two-term objective and source-specific values. LeJEPA v3 supplies assumption-bounded Gaussian motivation, not a universal physical-state guarantee.

Allowed claim: ideal Gaussian matching excludes the exact constant representation, while prediction supplies temporal/action relationships. Low total loss proves neither physics nor planning.

Quick check

1. Which shortcut does each gauge oppose or constrain?
2. Why can increasing lambda improve population shape while harming dynamics?
3. Why are 0.1 and 0.09 both correct but not interchangeable facts?