Real-time rendering of complex fractals

The paper informally derives the distance estimator d(z)=|G(z)|/|∇G(z)| from the Böttcher potential and states it as an “upper bound estimation,” yet its own inequality reads |ε| ≥ |G(z)|/|∇G(z)| (which would make d a lower bound), creating a sign inconsistency. It does correctly present the limit formula d(z)=lim_{n→∞} |f_n(z)| log|f_n(z)| / |(f_n)'(z)| and the recurrences for f_n and (f_n)' as well as the tetrahedral gradient estimator, all in a non-rigorous, implementation-oriented style . The candidate solution reproduces these formulas, but asserts a proof that the Euclidean distance satisfies dist(z,K) ≤ d(z) based on stepwise descent along −∇G, which requires unproven regularity/curvature assumptions and is not generally valid. Moreover, the paper’s implementation caps the marching step by min(d, 0.2), implicitly acknowledging a lack of a rigorous lower-bound guarantee for safe sphere tracing . In short, both the paper and the model capture the right formulas and algorithms, but neither provides a complete, consistent, and rigorous treatment of the distance bound or its domain of validity.