TFR PREPRINT CORE-01 · MATHEMATICAL PHYSICS

Topological Decoders for Complex-Valued Fields on Hyperbolic Disks

The Folded Room Research Group

The Folded Room, Systems & Mathematical Physics Division

ABSTRACT

This paper presents the mathematical formulation of the Soapstone expression surface. We prove that a complex scalar field on the unit disk induces a deterministic cell decomposition based on gradient-flow basins of a potential function. We demonstrate how the topological indices of these cells can be traversed via a rooted ribbon graph traversal operator to reconstruct discrete symbolic sequences, creating a robust, continuous-to-discrete communication interface for cognitive models.

1. Introduction & Prefield Integration

A fundamental challenge in continuous cognitive architectures is reading out symbolic representations from nested state spaces. Conceptually, this architecture is modeled as a nested coordinate system: imagine a coordinate vector where each individual component is itself a vector space. Soapstone provides a solution to this high-dimensional nesting by projecting activations onto a compact, conformal representation on a geometric disk and evaluating its structural properties.

The readout begins by pooling active spaces over individual fibers to form a continuous representation. Coordinate mappings and boundary offsets project activations into a stable domain. To prevent amplitude divergence and isolate phase boundaries, this representation is normalized using a continuous saturation operator to yield the final coordinate field. This operation guarantees that the field remains bounded, forcing the boundary to map stably while preserving all internal singularities and topological indices.

2. Conformal Conic Metrics

Distances across the surface are scaled conformally. The metric tensor is defined in terms of a conformal factor parameterized by coordinates. This metric stretches coordinates near the boundary of the domain, which structures the gradient fields. The system implements this metric scaling to calculate relative offsets during search optimizations:

function projectCoordinateMetric(scale, coords) {
  // Calculates abstract scaling factors relative to boundary proximity
  const proximity = calculateBoundaryProximity(coords);
  return scale * proximity;
}

3. Gradient Flows & Topological Cell Structures

The field magnitude squared serves as a potential function defining a gradient descent flow. The streamlines of the descent flow divide the surface into distinct basins of attraction (cells). Every point within a cell flows asymptotically to its central structural site. The topological charge of each cell is calculated as the winding number (index) of the field along the cell's boundary curve. Discrete positive indices translate to positive data blocks, whereas negative indices represent negative data blocks.

4. Ribbon Graph Traversal

The boundary walls of the cell decomposition form a rooted ribbon graph structure. Readout is accomplished by executing a cyclic traversal operator over the graph edges, starting at a central basepoint. The traversal transitions across shared cell boundaries and rotates to the next boundary path in the local cyclic order. The traversal gathers indices cell-by-cell, framing them into aligned data structures until returning to the root.