Phase transitions, cusp cascades, and a Feigenbaum-type universality in the Aether fractal family

We resolve all three open problems stated in Bird (2026b) concerning the non-holomorphic fractal family z₍+₁ = K zₙ exp (Im (zₙ) ) + c. (OP-A) We compute the cusp-bifurcation locus Kcusp (N) for N = 2, …, 7, showing that the locus peaks at N = 4 (Kcusp ≈ 11. 047) and ceases to exist at N ≥ 8: the cusp is a finite-depth phenomenon with no N → ∞ limit. (OP-B) We prove that the Class C subleading expansion A (K) sqrt (ln K) = 2R sum₍=₀^inf Dₙ / (ln K) ⁿ, with Dₙ = (2n-1) !! / 2ⁿ, is a Poincaré asymptotic series derived from the erfi expansion (DLMF 7. 12. 1), with D₁ = 1/2 proved exactly and verified to 0. 66% at K = 10¹00. (OP-C) We study the one-dimensional imaginary orbit y₍+₁ = K exp (yₙ) yₙ + delta as a parametric map in K and locate its period-doubling bifurcation sequence K₁, …, K₉. The ratio sequence rₙ = (Kₙ - K₍-₁) / (K₍+₁ - Kₙ) converges to 4. 671 at n = 8, within 0. 03% of the Feigenbaum constant deltaF = 4. 6692…. The cascade accumulates near Kₐcc ≈ 14. 9048, well below the phase-transition threshold K* ≈ 25 of Papers 2 and 3. This two-stage structure identifies K* as a secondary geometric event inside an already-chaotic regime, placing the Aether family within the universality class of one-dimensional maps governed by Feigenbaum renormalisation. An analytically exact non-holomorphic distance estimator is also derived, exploiting the Exact Decoupling Lemma, and produces publication-quality boundary renders at any resolution.