Zeration as the Floor and Attractor of the Hyperoperation Hierarchy

This paper develops a rigorous foundation for zeration, identified as the rank‑0 hyperoperation H0 (B;z) =z+1H₀ (B;z) = z + 1H0 (B;z) =z+1, and establishes it as the absorbing floor and global attractor of the hyperoperation hierarchy. Zeration is derived directly from the standard hyperoperation recurrence without auxiliary postulates by extending the height parameter to all integers and invoking a density argument for uniqueness over the complex plane. Within this framework, it is shown that all negative‑rank hyperoperations collapse identically to zeration, demonstrating that the hierarchy does not descend indefinitely but instead reflects at a well‑defined floor. The monograph further proves that addition emerges from zeration through a shift in initialisation height, rather than being taken as an independent axiom. This yields a unified view in which succession, addition, and higher operations arise as rank‑dependent manifestations of a single recursive structure. Connecting zeration to the right‑caterpillar (RC) hyperoperations developed in earlier work, the analysis introduces the thread‑point lattice: the locus in parameter space where every finite‑rank hyperoperation already acts as zeration. It is shown that this lattice expands monotonically with rank and, at the ordinal limit n=ωn = =ω (Infinitation), forces Hω (B;z) =z+1H_ (B;z) = z + 1Hω (B;z) =z+1 everywhere. The number 222, appearing as the universal output of zeration at height one, is identified as the canonical anchor of counting and the terminal value of the hierarchy. The present monograph treats only the RC convention; the left‑caterpillar (LC) extension from zeration to Infinitation is substantially more delicate and is deferred to future work. No claims are made beyond the scope of the analysis presented. The purpose of this deposit is to document the framework, results, and conceptual priority of zeration as the foundational floor of the hyperoperation hierarchy.