M04 Commutative Hyperoperations: The Cpow–Ctet Ladder, the Fluent Protocol, and the Symmetric Ascent Through All Ranks

This monograph develops a rigorous construction of commutative hyperoperations beyond multiplication, addressing the fundamental non‑commutativity of classical higher hyperoperations such as exponentiation, tetration, and beyond. At each finite rank nnn, a commutative binary operation Cn(a,b)=Cn(b,a) is constructed, forming a parallel hierarchy—referred to as the Hyper Core—that ascends from commutative addition through Commutative Powering (Cpow), Commutative Tetration (Ctet), Commutative Pentation, and higher symmetric ranks. The construction follows a uniform two‑step protocol at every rank: (i) a central iterative operation is built by iterating the symmetric rank‑nnn operation, yielding a unique narrative with no bracketing ambiguity; (ii) a diagonal symmetrisation iteration is applied, producing a fully symmetric binary operation at rank n+1. At rank 3, the symmetrisation admits a closed‑form expression, Cpow(a,b)=exp⁡⁣(sqrt(a*b*ln⁡a*ln⁡b)) derived via an exactly conserved quantity preserved by the iteration. Rigorous convergence results are established, with monotone, superlinear, and oscillatory regimes governed by an explicit contraction rate expressed in terms of the Lambert W function. At rank 4, iterating Cpow yields Commutative Tetration (Ctet). The dynamics of the Ctet iterator are analysed in detail, including existence regions, fixed points, and multipliers. In particular, the value of Ctet at height three is shown to coincide with the geometric mean of left‑ and right‑caterpillar tetrations when expressed in tower‑logarithmic coordinates. The monograph establishes that the commutative ladder is well‑defined and dynamically controlled at each finite rank, providing a systematic symmetric counterpart to classical hyperoperations. The present work focuses exclusively on finite‑rank constructions and their analytic properties.