Definition: A space with constant positive curvature.

Condition: Finite in volume, compact, and simply connected. All geodesics (straight lines) are closed loops. Values cycle within a boundary-less finite system.

Definition: A space with zero curvature (flat space).

Condition: Infinite and uniform. It follows standard XYZ coordinates where parallel lines never meet. Ideal for predictable, linear journey mapping.

Definition: A space with constant negative curvature.

Condition: Exponential divergence of parallel lines. Volume increases exponentially with distance, allowing for infinite expansion and rapid exploration paths.

Definition: Cartesian product of a 2-sphere and a real line.

Condition: Cylindrical topology. Combines a repeating circular cross-section with an infinite linear progression along a time-axis.

Definition: Cartesian product of a hyperbolic plane and a real line.

Condition: Combines exponential expansion in the base plane with linear vertical stacking. Ideal for growing platform architectures.

Definition: Geometry of the Heisenberg group.

Condition: Twisted manifold where horizontal movement causes vertical displacement. Non-linear, algorithmic linkage between different dimensions of the system.

Definition: Universal cover of the special linear group SL(2, ℝ).

Condition: Spiraling structure over a hyperbolic base. Represents multi-layered trust or credit cycles that return to similar points at higher energy levels.

Definition: A solvable Lie group geometry.

Condition: Highly anisotropic. Simultanously stretches along one axis and shrinks along another. Used for global arbitrage and dynamic equilibrium between opposing forces.