This monograph develops a rigorous dynamical‑systems framework for right‑caterpillar (RC) hyperoperations, treating addition, multiplication, exponentiation, tetration, and all finite higher ranks within a unified analytic setting. The central objective is to resolve the long‑standing instability and divergence issues associated with iterated exponential constructions by identifying deterministic reset and stabilisation mechanisms that operate across the hyperoperation hierarchy. For complex bases B=a+ibB = a + ibB=a+ib with b≠0b 0b=0 and suitable real part, it is shown that genuinely divergent orbits form a measure‑zero set, while almost all trajectories either converge or undergo a finite reset to rational cycles. These mechanisms propagate to all higher finite ranks, removing divergence as an obstruction to a coherent theory of hyperoperations. The monograph introduces the reset cascade and the associated thread‑point lattice, providing a base‑independent description of post‑reset dynamics. A detailed geometric analysis of the exponential map identifies the real axis as the unique exceptional case where no reset occurs. In addition, a stabilisation law is derived for the rank‑dependent reset height, showing systematic compression of iteration depth as the rank increases, with a distinguished base‑2 anchor exhibiting rank neutrality. The results establish RC hyperoperations as a well‑posed dynamical system across all finite ranks and form a foundational component of a broader theory of operations developed in subsequent monographs. No claims are made regarding classical conjectures beyond the scope of the analysis; the purpose of this deposit is to document the framework, results, and conceptual priority of the RC hyperoperations theory.
Paweł Łukasz Garycki (Fri,) studied this question.