M15 The Parameter Frontier Multi-Element Operations and the Angular Calculus of Recursive Depth

This monograph develops the concept of the parameter frontier, treating parameters in recursive operations as dynamical objects rather than fixed constants. The work introduces an angular calculus of recursive depth, in which multipliers, Schwarzian derivatives, and rotation data jointly classify the local and global behaviour of hyperoperation iterators. A complete fixed‑point classification is established by combining the Schwarzian sign with a spiral‑pitch invariant, yielding a uniform taxonomy of monotone, oscillatory, parabolic, and superattracting regimes across multi‑element operations. The monograph analyses how parameters propagate through right‑ and left‑caterpillar constructions, pyramid‑type operators, and loaded dynamics, showing that qualitative changes in behaviour arise from controlled motion in parameter space rather than from changes of operation rank. This provides a precise language for bifurcation, stability loss, and regime transitions in higher‑rank recursion. M15 positions the parameter frontier as the boundary between algebraic definition and dynamical behaviour, supplying the tools needed to navigate continuous deformations of recursive systems while preserving structural invariants. The purpose of this deposit is to document the definitions, classification results, and conceptual priority of the parameter frontier within the hyperoperation theory series.