We study classes of transcendental Horn and Adam maps, which contains the classical horn maps and their quotients coming from the dynamics on simple parabolic basins such as $z^2+1/4$ on the Cauliflower.
We define and describe the structure of the dynamical and parameter “lagoa rays”
and use these rays to design the Yoccoz puzzle machinery in this setting. As a first
consequence, we obtain the precise asymptotics of secondary satellite copies
of “principal elephants” near the cusp of the Mandelbrot set.