Visualizing Discrete and Continuous Structures: Daisy Trees, Phase-State Surfaces, Exotic Loops, and Curvature Anomalies A Computational Study with Mathematical Guarantees

We present and analyze a Python program that visualizes four mathematical constructions: (i) a q-regular rooted "daisy tree"; (ii) a truncated phase-state function H τ over the complex plane; (iii) a non-holomorphic iterative system that generates fractal "exotic loops"; and (iv) a curvature field with angular modulation and a divergence-free flow. We provide precise definitions, structural properties, and convergence conditions in theorems, lemmas, and propositions. We also include guidance and placeholders for figures rendered by the program and the complete source code verbatim for reproducibility. Contents