Persistent Homology Analysis of AI-Generated Fractal Patterns: A Mathematical Framework for Evaluating Geometric Authenticity.

We present a mathematical framework for analyzing fractal patterns in AI-generated images using persistent homology. Given a text-to-image mapping M : T → I , we demonstrate that the persistent homology groups H k (t) of sublevel set filtrations { f − 1 ((− ∞ , t ]) } t ∈ R characterize multi-scale geometric structures, where f : M (p) → R is the grayscale intensity function of a generated image. The primary challenge lies in quantifying self-similarity in scales, which we address by analyzing birth–death pairs (b i , d i) in the persistence diagram P D (M (p)) . Our contribution extends beyond applying the stability theorem to AI-generated fractals; we establish how the self-similarity inherent in fractal patterns manifests in the persistence diagrams of generated images. We validate our approach using the Stable Diffusion 3.5 model for four fractal categories: ferns, trees, spirals, and crystals. An analysis of guidance scale effects γ ∈ [ 4.0 , 8.0 ] reveals monotonic relationships between model parameters and topological features. Stability testing confirms robustness under noise perturbations η ≤ 0.2 , with feature count variations Δ μ f < 0.5 . Our framework provides a foundation for enhancing generative models and evaluating their geometric fidelity in fractal pattern synthesis. [ABSTRACT FROM AUTHOR]