Five-Step Descent Certificates for the Juggler Map: Parity Statistics of Nested Floor Powers

The Juggler map applies the integer part of the square root at even positive integers and of the three-halves power at odd positive integers. We prove that the starting values admitting a power-envelope descent certificate within five operations have natural density 7/8. More precisely, their count up to N is 7N/8 + O_(ε)(N^(127/128 + ε)) for every ε > 0. The four-step subfamily has density 13/16. The proof uses exact carry identities, centered Fourier expansions, and van der Corput differencing, with all floor exceptions and partition endpoints counted. The two five-letter classes OOEOE and OOOEE are treated through their distinct formal chains. For OOOEE, the full mixed phase is retained and its signed curvature is recomputed after each frequency center is frozen. The complete analytic argument for this density is included in Appendices A–C. Fair-share densities at every fixed depth would imply density-one finite certificates; that hypothesis and general decorated estimates remain open. For the two five-letter words, the starts reaching a given fifth iterate lie in a window of about n^(5/32) integers near a start n; all but a power-saving proportion of these windows carry their fair share (Theorem 6.3). Localization to a prescribed short interval remains open. No result asserts universal arrival at 1.