M51c The Hyperzeuge - Half Ranks Ascended, Symmetrised and Reached by Protocols

Between two integer rungs of the HyperCore, the Theory of Operations can produce objects in three different ways. The normal reading uses a Gauss-compound protocol of the neighbouring storage means. The ascending reading iterates the preceding half-rank operation as an AscentCore step. The symmetrised reading applies the elevator’s symmetrisation to that iterate. The HyperCore is the ladder of symmetric operations generated by repeatedly building an iterative channel and symmetrising it; its rank records the operation’s position on that ladder, with addition at rank one, multiplication at rank two and Cpow at rank three. The aim of this paper is to determine what genuinely occupies the half ranks between these rungs by comparing three routes: Gauss-compound protocols, ascended iterates and their symmetrisations. M51c "The Hyperzeuge - Half Ranks Ascended, Symmetrised and Reached by Protocols" compares these readings at ranks 3/2, 5/2 and 7/2, proves where they agree and where they separate, constructs a canonical continuation of the rank-5/2 ascent, and shows why the half rungs have no basal end. The normal reading contains the HyperCore’s genuine half-rank protocols: at 3/2, the mean AGM and operation AGM-Op(a,b) = AGM(a+b,a*b); at 5/2, the mean SHM and the Symmetric Hyperelliptical Operation SHO(a,b) = SHM(a*b,Cpow(a,b)); at 7/2, the mean ISHE-H and operation ISHE-H-Op. The ascending reading is instead the half rung AC_(R+1/2) = Asc(C_(R-1/2)). It is an iterate, generally chiral, and requires a continuation when its height is non-integer. The symmetrised reading applies Build-and-Symmetrise to this half rung. A midpoint such as CatH is only a formula and belongs to neither reading. A half rank is reached by the protocol. Ascending from a lower half rank and symmetrising the result generally produces another object. The three cells At rank 3/2, the ascending reading degenerates. Above the unit, AC_(3/2)(a,n) = n*a, while below the unit it is a+n-1. Its symmetrisation is ordinary multiplication. The elevator started at rank 1/2 therefore skips the protocolic half rank and closes on rank two rather than on AGM-Op. Rank 5/2 is the first non-degenerate cell. Its three readings share the Absolute Invariant 2#2=4. Height two of the ascent equals the diagonal of AGM-Op, and the normal reading and midpoint formula have the common diagonal slope 4+2*ln(2) at x=2. These agreements do not extend to structural identity. The ascent of AGM-Op, its symmetrisation and SHO differ in growth, absorbers, storage means, floor germs and convergence mechanisms. For fixed base a>1, the ascent of AGM-Op grows geometrically with multiplier AGM(1,a). More generally, ascending the geometric, Gauss and arithmetic constructions formed from the same pair produces the ordered growth bases sqrt(a) < AGM(1,a) < (1+a)/2. On the diagonal, the ascent and its symmetrisation retain the full logarithmic growth class of rank three, whereas SHO and the midpoint formula retain only half of that logarithm. The protocol is genuinely intermediate in growth; the ascent is not. The absorbers also separate. SHO absorbs 1. The symmetrised ascent instead absorbs a_0 = 0.3515259691..., the positive fixed point of the AGM-Op diagonal, defined by AGM(2,a_0) = 1. Its protocol exists above this boundary and numerically fills the quarter-plane (a_0,infinity)^2; below the boundary the diagonal is transversely repelling. The normal protocol and symmetrised protocol are different machines. The SHM compound converges quadratically. The symmetrised ascent generally converges linearly, with a transverse multiplier determined by the base and height derivatives of its continued channel. The multiplier vanishes at the shell m* = 2.85054342..., where the convergence becomes superlinear. M51c proves that the symmetrised ascent is not equal or monotonically conjugate to SHO. For every continuation that is monotone near the corner and differentiable there, its stored mean has the arithmetic floor germ, while SHM has Gauss’s AGM germ. Thus the two objects also differ in storage and first-order floor geometry. The former Ascending Hyperzeug Conjecture fails at every comparison level made precise. The symmetrised ascent is a second natural operation at 5/2, not another construction of SHO. At rank 7/2, the readings align more closely. The normal operation ISHE-H-Op and the symmetrised ascent of SHO both absorb 1. Their diagonals agree to first order near the floor and differ numerically at second order. The floor of the ascending reading is controlled by F(x) = AGM(x+1,sqrt(x)). Starting from Theta_1=1, the sequence Theta_(n+1)=F(Theta_n) converges to Theta* = 2.3579120559.... Thus the floor of the 7/2 iterate is the ascent of the floor of the 5/2 protocol. Below the wall w_(5/2) = 1.2294633745..., the rank-7/2 half rung has an attracting fixed point and a canonical Koenigs continuation. At the wall, two fixed points merge and a Fatou coordinate applies. Above the wall, the orbit escapes in the tower growth class, and no canonical real continuation is currently known. Canonical continuation and the half-rank floor For the rank-5/2 ascent, the slot map is g_a(x) = AGM(x+a,a*x). M51c constructs a canonical continuation at every positive base: for a>1, Koenigs linearisation at infinity; for a=1, a Fatou coordinate at the parabolic wall; for 00+. For the rank-7/2 step, ln(SHO(1+delta,a)) ~ pi*ln(a)/ln(1/delta) as delta->0+. These are logarithmic rather than linear or power germs. Their inverse germs are flat, of order exp(-c/y), and their backward orbits grow as towers. They cannot be linearised at the absorber by a bi-Lipschitz Koenigs or Böttcher coordinate. Neither half rung has a basal end. Their canonical continuations must be constructed from their attracting or escaping ends rather than from the absorber. Gauss’s mean governs both ends: as the growth multiplier at the top and through the logarithmic singularity of the elliptic period at the bottom. M51c establishes the half-rank architecture in its mature form. At 3/2, ascent and symmetrisation skip the protocolic half rank and return multiplication. At 5/2, the normal, ascending and symmetrised readings share selected values but separate in growth, absorber, storage, conjugacy, floor germ and convergence machine. At 7/2, the absorbers and first-order floor behaviour realign, although the operations remain distinct. The paper supplies the canonical continuation and floor of the 5/2 ascent, proves the crawl law and regularity at its wall, and shows that the logarithmic ends of both half rungs prevent a basal continuation.