Geometric rigidity and algebraic symmetries of complex transcendental numbers

We propose a structural classification of complex transcendental numbers based on the geometric rigidity of their minimal algebraic locus in the complex projective line. By analyzing the action of the extended algebraic automorphism group Gamma^pm (generated by PGL(2, Q-bar) and complex conjugation), we establish a fundamental dichotomy between flexible and rigid geometries.

The flexible locus consists of transcendental points admitting infinite algebraic symmetries; we prove this locus coincides precisely with the set of points on algebraic generalized circles (real forms of genus zero). While these curves are geometrically rigid under intersection, the dynamics upon them are chaotic: the orbit equivalence relation is shown to be meagre, non-smooth, and admitting perfect antichains.

In contrast, the generic locus is characterized by geometric rigidity and trivial stabilizers. This includes the independence locus (points of transcendence degree 2), which is topologically generic (comeagre) and measure-theoretically dominant. We prove that the group action on this locus is topologically transitive (dense orbits).

Finally, we establish a maximality theorem characterizing Gamma^pm as the unique maximal degree-one transfer semigroup compatible with the algebraic structure of the projective line, identifying it as the natural boundary of algebraic symmetry.