M21c Asymmetric ONS: Olympus Programme, Paper III: LC and RC Narratives as a-ONS, Pyramidation and the Half-Caterpillar, The Associahedron of Tetration, The HC as the Unique ONS Elevator, and the Elevenfold Propagation

M21c introduces Asymmetric Operational Number Systems (a‑ONS): operational number systems generated by non‑commutative hyperoperations. Unlike symmetric ONS, an a‑ONS admits distinct left and right factorizations, producing two prime sets and two Euler products. While the Operational Theta function and the critical strip persist, the self‑dual functional equation fails unless commutativity holds. The monograph shows that within the Elevenfold parameter space, exactly one operation is commutative: the Hyper Core (HC). All others generate a‑ONS. At rank 3, LC and RC powering are related by an exact chirality swap, with the symmetric midpoint (Cpow) as the unique commutative case. From rank 4 onward, this symmetry breaks: LC and RC become independent a‑ONS families, quantified by the Quadratic Delay Law, which measures their intrinsic asymmetry . Two additional a‑ONS elevators are analyzed: Pyramidation (balanced parenthesization with logarithmic tower growth) and the Half‑Caterpillar (the LC–RC midpoint). Both are chirally balanced yet provably non‑commutative and distinct from the HC operation . Interpreting the associahedron of tetration, M21c shows that each parenthesization corresponds to a distinct a‑ONS, while the HC occupies the unique symmetric center. This leads to the HC Uniqueness Theorem: the Hyper Core is the only ONS elevator through the hierarchy; all others necessarily generate a‑ONS. The monograph concludes by formulating the open problems addressed in M21d, including split spectral theory, chiral zero births, and generalized ONS unification .