Prime-Height Tetration as an Analytic Generator: From x x to p-Towers via Abel Linearization, Mode Coordinates, and Modular Phase Extraction

We develop a unified analytic framework that extends the "2-tetration" generator x x to prime-height p-tetration in the user-defined sense:with right-associative towers.The first contribution is an exact tower calculus: a closed, recursion-based Jacobian ladder for d dx ( p x) and its log-derivative.The second contribution is an Abel-linearization construction in which the nonlinear tower-building operator is conjugated to translation in a continuous height coordinate, yielding a continuous-height super-exponential T (x, h) satisfying T (x, h + 1) = x T (x,h) and interpolating integer heights, hence capturing prime heights p as samples T (x, p) = p x.The third contribution is a mode-coordinate representation for tower values compatible with discrete spectral sums and contour-residue "selector" kernels, together with a modular phase extraction layer that can be applied either to tower values or to linearized height coordinates.Throughout, derivations are exact (non-asymptotic): no numerical approximations are invoked.