Energy lens

Make the objective feel like a surface the runtime is descending.

The point of this chapter is not to dump equations. It is to show which terms are doing conceptual work and why the relaxation step looks like a principled descent rather than a black box.

Edge and self terms stabilize the local geometry.
Hub, Hopfield, and slow-memory terms pull the state toward useful structure.
The free-energy objective ties those forces into one interpretable update.

Free-energy lens

The objective balances total energy against entropy.

Free-energy objective F_{t o t a l} = E_{t o t a l} - T S_{t o t a l}

The entropy and temperature terms are explicit here so this page matches the overview equation.

E_{t o t a l} = E_{e d g e} + E_{se l f} + E_{h u b} + E_{h o p f i e l d} + E_{s l o w} + E_{c l am p}

Energy-term breakdown

Relative contribution

Interactive term view

Edge energy

E_{e d g e} = (i, j) \in E \sum a_{ij} ∥ x_{i} - x_{j} ∥^{2}

Encodes the cost of neighboring states disagreeing on the tape graph.

Relaxation step

Inference is a descent, not a single pass.

x_{t + 1} = x_{t} - η \nabla_{x} F_{t o t a l}

The point of the energy route is to make the update rule feel tangible: the model is not merely predicting a token, it is settling toward a lower free-energy state that is compatible with what was already observed.