M06 Left-Caterpillar Hyperoperations: Angular Dynamics, Stochastic Stabilisation, and the Taming of Recursive Growth

This monograph develops a rigorous theory of left‑caterpillar (LC) hyperoperations, defined by left‑associative bracketing, and analyses their dynamics from tetration through all finite higher ranks. In contrast to right‑caterpillar constructions, LC hyperoperations admit substantial algebraic simplification. At rank 4, LC tetration collapses to a double‑exponential form, leading to dynamics governed primarily by angular behaviour in the complex plane. The monograph provides a complete classification of LC tetration, showing that for generic complex bases the iteration reduces to a binary angular process, producing periodic or equidistributed orbits, with the positive real axis as the unique divergent exception. At higher ranks, LC hyperoperations exhibit a characteristic stochastic reset behaviour: extended excursions through angular‑infinite regimes are followed by rapid collapse to canonical fixed points. The expected reset lengths and stabilisation depths are derived explicitly, revealing universal patterns independent of rank. These results establish that LC hyperoperations are dynamically more tractable than their right‑caterpillar counterparts, while still exhibiting rich recursive structure. The monograph places LC hyperoperations within the broader hyperoperation hierarchy, explaining their role in delayed growth phenomena and their interaction with rank‑dependent stabilisation and zeration limits. The purpose of this deposit is to document the framework, results, and conceptual priority of left‑caterpillar hyperoperations within the hyperoperation theory series.