Structural Compatibility II: Recurrent Compact Internal Symmetry and the Emergence of Complex Projective Kinematics
Alexandre Ramakers
This preprint develops the second step of the Structural Compatibility series. Starting from the affine persistence framework established in Structural Compatibility I, it examines the consequences of recurrent compact internal symmetry acting within an admissible persistence regime. The article shows that recurrent internal affine flows have compact closure, and that under a rank-one assumption the connected compact internal symmetry reduces to a circle. After fixed-point reduction and linearization, this symmetry acts on the underlying real state space through orthogonal rotation sectors. On the rotational sector, the action induces a canonical complex structure, an invariant Hermitian comparison form, and a unitary realization of the internal symmetry. Under an additional phase-uniformity assumption, the circle action becomes a genuine global phase acting on the whole state space. This makes it possible to pass to the associated complex projective state space, which then inherits the standard Fubini–Study Kähler geometry. The scope of the result is strictly kinematical. No probabilistic axiom, measurement postulate, or dynamical equation is assumed. The paper does not derive full quantum theory, but identifies a structural route from an affine real persistence regime to the minimal complex-projective kinematical architecture usually taken as one of the starting points of quantum formalism. This article is part of the Ranesis framework, developed by Alexandre Ramakers.
•Mar 18, 2026•0 citations