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We study classes of transcendental Horn and Adam maps, which contain 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.