Introducing, Eigen-knot, new mathematics to describe an Imaginary transformation on an object. 

The Hype-Shape (or Eigen-knot) is a geometric structure arising from a torus-like parametrization with major radius and minor radius , yielding a self-intersecting spindle torus. Real multipliers preserve the toroidal form in three-dimensional space, while the redefined imaginary unit, CUT-i, acts as a rotate–lift operator: it rotates points in the plane and lifts them along an additional coordinate proportional to the planar radius. Under continuous, damped flow, this produces bounded helical swarms and stable layering in , resolving the instability induced by self-intersection while preserving genus-1 topology.

Hydrogen Atom with i operator

Torus Swarm