M36c The HyperSymmetric Core - The h-Étage Species, the Trivial Freezing Theorem, and the Lambert W Bridge at R = 3

M36c introduces the HyperSymmetric Core (HSC), the h-étage companion to the classical Symmetric Core. The central idea is that every operational rank has two possible étage species: the classical s-étage, generated by freezing the asymmetric a-ONS member of the rank Triad, and the h-étage, generated by freezing the HyperCore symmetric member. At R = 3 the rank Triad consists of Cpow, Apow, and Exp. Freezing Exp at base e gives the usual exponential generator and hence the classical Symmetric Core. Freezing Cpow at base e gives a genuinely new generator, FC (x) = Cpow (e, x) = exp (sqrt (e x ln x) ), which defines the HyperSymmetric Core. The key structural result is the Trivial Freezing Theorem. The SC member Apow is doubly rigid: freezing Apow at its identity e gives the identity map, while freezing Apow at any other admissible base produces no new étage and regenerates ordinary multiplication. Thus the SC member is not a new generator. It is self-referential: it either trivializes or returns the multiplication already present in the classical s-étage. The HSC is therefore generated by the HyperCore member Cpow. Its Room 1 operation is C-Multiplication, a odotC b = FC (AC (a) +AC (b) ), where AC = FC^-1. This operation is commutative, associative, has identity 1, and is distinct from ordinary multiplication. The HSC Abel coordinate is explicitly expressed through the Lambert W function: AC (y) = exp (W ( (ln y) ²/e) ). This produces the Lambert W Bridge: although the HSC is anchored at rank R = 3, its Abel coordinate contains the special function associated with the R = 2. 5 half-étage. Thus the HSC is a rank-3 object internally coloured by R = 2. 5 structure. M36c also reformulates the rank-3 Triad as a self-completing system. Exp generates the s-étage, Apow is the s-étage member itself and is doubly rigid, while Cpow generates the h-étage. The Triad therefore contains both generators and their classical output in one closed structure. The monograph concludes with the general principle that every rank carries exactly two étage species: the s-étage and the h-étage. The h-étage at R = 3 is the HyperSymmetric Core; the analogous h-étage at R = 4 is proposed as the HyperTetration Core. A half-rank bridge conjecture is stated: the h-étage at rank R carries, in its Abel coordinate, the special-function structure of the half-rank R - 1/2.