The DADA Fractal: Two-Scale Boundary Lifecycle in the Cathedral Seam

Paper 30 in the "Geometry of the Critical Line" programme. The escape-time Julia set of the direct iteration z → z − exp(−α/z) — a different dynamical system from the Newton map studied in Papers 20–29 — displays a lobe-alternation pattern (informally termed "DADA" in this programme) along the Cathedral seam whose quantitative structure has not previously been measured. This paper probes the seam boundary at six lifecycle landmarks (α = 1/e, 0.42, 0.481, 0.50, 0.60, 1.0) using box-count dimension and curvature-based lobe detection. The results reveal a split between variable global geometry and invariant local ornamentation. At the global scale, box-count dimension decreases from 1.636 at the branch point to a minimum of 1.554 at peak bounce competitiveness (α = 0.60), then rebounds to 1.640 at α = 1.0. The contour length varies by a factor of approximately 4×. At the local scale, however, both the lobe density (~22 detected lobes per 10³ contour pixels) and the mean lobe spacing (~0.032 in z-space) are nearly invariant (