Phase transitions, cusp cascades, and a Feigenbaum-type universality in the Aether fractal family
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Abstract
We resolve all three open problems stated in Bird (2026b) concerning the non-holomorphic fractal family z_{n+1} = K z_n exp(Im(z_n)) + c. (OP-A) We compute the cusp-bifurcation locus K_cusp(N) for N = 2,…,7, showing that the locus peaks at N = 4 (K_cusp ≈ 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_{n=0}^{inf} D_n / (ln K)^n, with D_n = (2n-1)!! / 2^n, is a Poincaré asymptotic series derived from the erfi expansion (DLMF 7.12.1), with D_1 = 1/2 proved exactly and verified to 0.66% at K = 10^100. (OP-C) We study the one-dimensional imaginary orbit y_{n+1} = K exp(y_n) y_n + delta…
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Topics
Keywords
- Locus (genetics)
- Universality (dynamical systems)
- Bounded function
- Renormalization group
- Cusp (singularity)
- Fractal
- Absolute continuity
- Sequence (biology)
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