M27b The Heun and AGM Triad Levels: Symmetric-Asymmetric Structure at the Half-Etages R=1.5 and R=2.5

This mini-monograph develops the Triad structure at the half-etage levels R = 1. 5 (AGM) and R = 2. 5 (Heun) of the HC Rank Manifold. It corrects the earlier identification of the genuine AGM with the SC half-rank AddMult operation: these are distinct structures. At R = 1. 5, the Triad consists of HC: AGM (a, b), SC: AddMult (a, b) = sqrt ( (a+b) *ab), a-ONS: CatAGM (a, b) = a*eᵇ. At R = 2. 5, the Triad consists of HC: Heun mean, SC: MultPow, a-ONS: CatHeun (a, b) = a^ ( (b+1) /2) *sqrt (b). The monograph derives the Hermit units and NC equations of these Triad members. Unlike the rank-3 case of M27a, Fluent Invariance generally fails at R = 1. 5 and R = 2. 5: the HC, SC, and a-ONS members do not share one common Hermit unit. This makes the full Rank-3 Triad exceptional. The R = 2. 5 Caterpillar member introduces the Lambert W function through w*eʷ = i*pi, so that tauCatHeun = exp (W₀ (i*pi) ). The work identifies Lambert W as the natural special function of asymmetric half-etage structure, while the genuine AGM connects to elliptic periods and the Heun level connects to the next special-function layer. The monograph therefore builds the bridge from lower half-ranks to the full asymmetric rank-3 exponentiation theory.