Discovery of the Complex Field by Graduated Observation of Plane Rotation

The complex numbers are derived from a single geometric primitive: rigid rotation of the Euclidean plane about a fixed point. The derivation proceeds through a graduated sequence of observer capability levels, each corresponding to a specific physical operation (observation, length comparison, compass-and-straightedge construction, angle comparison, free locomotion). At each level, new algebraic structure becomes visible. The principal result is a necessity theorem: the complex field is the unique topological field whose additive group is a connected, simply connected, locally compact, Hausdorff, two-dimensional topological group, and whose multiplicative group contains a compact connected subgroup acting by isometric rotations and a commuting scaling subgroup. One explicit axiom supplements the primitive: Dedekind completeness of the plane. The graduated observer framework identifies completeness as a boundary condition - a property that no observer capability level can verify in full, and that must therefore be adopted rather than derived. Algebraic closure follows from a winding-number argument exploiting the plane's topology. The derivation inverts the standard construction order: the real numbers are found within the complex field (as the non-rotating similarities) rather than the complex field being constructed from the reals.