Complex Physical Ontology from Computable Coordinate Frames and Local Helicity

<title>Abstract</title> <p>We propose a complex-valued extension of the computable coordinate system (CCS) by coupling its real geometric frame to a topological phase driven by the recently established local helicity density. The question is simple: why can classical real-valued continuum models describe smooth deformation, yet fail to provide a local dynamical variable for vortex reconnection and other topology-changing events? Our answer is to treat CCS as the real carrier of local pose and scale, and to introduce a phase variable θ that stores topological state. Global helicity H is interpreted as imaginary topological energy, local helicity density LH as the corresponding local phase source, and the complex state C = C exp(iθ) as the minimal object that carries both geometric and topological information. The paper formulates a closed minimal model with two channels: a real geometric evo lution law for the CCS carrier and a phase law ˙ θ = Γ−1LH for the internal sector. A discontinuous topological transition occurs when the accumulated phase reaches a critical threshold tf 0 ˙ θ(t)dt = Θc. We further discuss phase wrapping, dimensional consistency, and a concrete outlook for numerical veri cation. The framework is designed to extend the CCS program rather than replace it: CCS supplies the real kinematic sca old, while local helicity supplies the imaginary topological dynamics.</p>