TFR PREPRINT CORE-02 · DYNAMICAL SYSTEMS

Recurrent Update Laws and Gaze Dynamics in Gated Complex Vector Surfaces

The Folded Room Research Group

The Folded Room, Systems & Mathematical Physics Division

ABSTRACT

We detail the dynamical cognitive update laws of the Virgil controller. Each cognitive domain is modeled as a structural coordinate space updating under local dynamics, external inputs, and gated interference. We analyze the convergence properties of this system as the active family settles into stable window assignments. We demonstrate how credit assignment is backpropagated from Soapstone readout errors to guide learning.

1. Cognitive Bundles and Windows

The Virgil controller maps continuous trajectories through hierarchical spaces. At the core of the Virgil controller is the concept of nested coordinate spaces—analogous to a coordinate vector whose components are themselves vectors. The controller maintains sections of active spaces representing these nested cognitive domains. Each domain is restricted to a compact region using local gating functions. These gating functions map the underlying standing state space into compact regions. Windows represent gates that isolate local dynamics and prevent interference. To build complex coalitions, separate windows are coupled dynamically via directed relationships.

2. Gaze Law Trajectories on Manifolds

Window parameters (coordinates) flow along a trajectory on a multi-dimensional state manifold. The dynamics of this trajectory are defined under a gradient flow field of the total system energy. These velocity fields are driven by internal velocities and connection coupling. As the system evolves, trajectories settle toward stable attractors representing cognitive coalitions.

3. The Universal Energy Functional

Learning and self-update dynamics are driven by gradient descent over a unified optimization functional under external source terms. The functional consists of terms that govern different aspects of field behavior:

Objective Component	Functional Focus	Structural Role
Field Regularity	Regularity Focus	Enforces consistency and boundary constraints.
Phase Alignment	Coherence Focus	Controls structural alignment, preventing boundary slips.
Attractor Relaxation	Attractor Focus	Relaxes dynamics into periodic paths or attractor states.
Task Objective Alignment	Target Focus	Aligns representations with predicted states and target outputs.

4. Credit Assignment Loops

To keep predictive models from becoming epiphenomenal, Virgil utilizes a closed-loop credit assignment. Readout failures on the Soapstone disk increase the error functional. The feedback is backpropagated to adjust the connection values, ensuring that only structurally consistent states settle into stable attractor basins.