M03 Iterator Perturbations and Nestrootation: A Unied Framework for Modied Hyperoperations

This monograph develops a systematic framework for perturbing hyperoperation iterators through rank‑specific modifiers applied to the recursive step, with primary focus on right‑caterpillar constructions. The analysis introduces kkk-modified tetration, defined by the recurrence zn+1=b^ (znᵏ), k∈C which unifies standard tetration (k=1) and nestrootation (k=−1) within a single family. The resulting dynamics are studied via conjugation to the exponential family, yielding an explicit classification of parameter space into attracting basins, period‑doubling cascades, and chaotic regimes. Inverse operators, including the nest superlog and nest superroot, are defined by swapping functional equations relative to classical tetration inverses. At the tetration (rank‑4) level, the monograph introduces Pyramidation, based on balanced bracketing that minimises effective iteration depth. The geometry of such bracketings is formalised using associahedra, whose vertices are classified into structural narratives (caterpillars, pyramids, snakes, and concatenations). Analogous perturbations are examined at lower ranks, including nested division at the multiplication level and nested subtraction at the addition level. Analytic continuations to fractional heights are obtained via conjugation, producing complex‑valued oscillatory behaviour even for real bases. The purpose of this deposit is to document the framework and results concerning iterator perturbations as a standalone component of the hyperoperation theory series. No claims are made beyond the scope of the constructions and analyses presented.