M51b The HCt Coordinate Surface and the TetraEtage

The Theory of Operations asks whether familiar operations such as addition, multiplication, exponentiation, and tetration belong to a larger mathematical hierarchy. The AscentCore moves upward in this hierarchy by repeatedly applying a symmetric operation, while Cpow is the symmetric form of exponentiation. Ascending Cpow produces HCt, the Half-Caterpillar, a balanced rank-four operation positioned between differently associated forms of tetration. HCtf is the special function generated by repeatedly applying HCt, and hlog tells us how many such steps are needed to reach a given value. Using these coordinates, the paper constructs the TetraEtage, a new mathematical setting in which ordinary algebra, calculus, exponentials, and trigonometric functions are rebuilt for this rank-four world. The HCt Coordinate Surface and the TetraEtage develops this proposed rank-four counterpart of exponentiation. It explains how HCt relates to tetration and the AscentCore, studies its values and inverses, and uses its height coordinate to transport familiar mathematical structures into the geometry of iterated rank-four operations. M51b, in particular, develops the functional world generated by the Half-Caterpillar, the self-dual midpoint of the rank-four Master plane. The paper begins by separating three constructions that had previously appeared under different names: the internal Half-Caterpillar orbit U_B(t); its Master-wrapped special function HCtf_B(t) = B^(U_B(t)); the AscentCore channel obtained by iterating Cpow(x,a) at the lifted base B = a^a. The three-chart theorem proves that these are three coordinate readings of one orbit. The channel and the wrapped special function alternate through two elementary strokes: the Tetrasuccessor PYR(x) = x^x and the log-geometric mean LG(x,y) = exp(sqrt(ln(x) ln(y))). Their step maps factor in opposite orders: G_B = PYR o LG(.,B) and g_a = LG(.,B) o PYR. This also yields the identity Cpow(x,y) = LG(PYR(x), PYR(y)) = LG(x^y,y^x). The Half-Caterpillar is therefore a two-stroke protocol whose two visible phases are the channel and the Master-wrapped special function. The wrapped HCt step is locally commutative in the carried state and retained base, although its iteration remains chiral because the base is retained while the state advances. The transform Lambda_H(z) = W(ln(z))^2 turns the step into the product law Lambda_H(G_B(x)) = ln(x) ln(B). This Lambert-square law provides one inverse formula for either argument and allows the retained base to be recovered exactly from any adjacent pair of orbit states. M51b identifies the whole direct line from left- to right-associated tetration as a Krasnosel’skii-Mann relaxation. For f_t(H) = H^(1-t) B^(H^t), the coordinate w = t ln(H) transforms the iteration into w -> (1-t)w + t(ln B)e^w. Ordinary tetration is the fully polarised endpoint t = 1, while HCt is Krasnosel’skii’s equal average at t = 1/2. Above the classical tetration wall, this averaging creates an attraction window in which HCt attracts at the complex fixed point while tetration does not. Two bases receive special roles. The canonical real base sqrt(2) is selected by the unique off-diagonal integer point (2,4) of the classical curve a^b = b^a. Its exponential has fixed points 2 and 4, and the internal and wrapped HCt fixed values coincide at 4. At this base, HCtf maps the real height line onto (1,4). The distinguished complex base is e^(pi/2) = i^(-i), where the exponential fixes i, the internal HCt fixed point is -1, and the Half-Caterpillar has the smallest multiplier modulus among the stated bases above the wall. The special function has two natural continuations below the wall: regular continuation from Koenigs linearisation and basal continuation from inverse iteration toward the multiplicative unit. They differ by the Basal-Regular Comma, a small nonconstant periodic function. M51b adopts the basal continuation as canonical because it extends through the walls and exists for every real base above one. Every non-integer HCt value, height logarithm, flow, and TetraEtage operation must therefore carry a continuation stamp. Three inverse problems are distinguished: hlog_B(y), asking for height; HCr(y,h), asking for the base; G_B^(-1)(y), asking for the previous state. The base inverse is the super-root at height two and a nested Lambert-W expression at height three. At height four it satisfies 2 s^(s+1/2) ln(s) = ln(sroot_2(y)), s = sqrt(B). No reduction of this equation to nested Lambert functions is known, making height four the first candidate for a genuinely new inverse function in this cascade. Around HCt, the paper constructs a real Master plane with two axes. The direct axis runs from left-associated to right-associated tetration, while the diagonal runs from the frozen Tetrasuccessor to self-powering. HCt is their common centre and the only point at which the step is identically commutative, both operands remain recoverable, and both partial inverses use one formula. Complexifying the perpendicular directions produces a four-member phase square with conjugacy, equal-modulus, rotation, and norm-product laws. The TetraEtage is then defined by transporting the Symmetric Core through the height coordinate hlog_B. Its first two corridors form a field isomorphic to the real numbers. At the canonical base sqrt(2), this field lives on (1,4), with zero sqrt(2) and multiplicative unit sqrt(2)^sqrt(2). The TetraEtage is the midpoint of an étage line: at one endpoint the Symmetric Core reappears two rooms higher, while at the other endpoint stands the fully chiral TetraCore built from tetration and slog. Finally, M51b develops the ascent calculus. The native derivative is D_H,B = v_B(y) d/dy, where v_B = 1/hlog_B'(y). It extends the generator ladder d/dx, x d/dx, v_B d/dy from translation, through dilation, to Half-Caterpillar ascent. Exponential, circular, and crate functions composed with hlog_B obey their ordinary derivative and addition laws over the native TetraEtage addition. The Julia equation emerges as the reversed logaderivative of the Abel equation. The regular and basal ends also generate two Schröder eigenfunctions, but the comma prevents a single continuation from making both linearisations exact simultaneously. M51b establishes HCt as a wrapped rank-four special function rather than a disguised form of tetration or a move to rank five. It connects the Half-Caterpillar, the Cpow ascent channel, and the Master Formula through three charts; derives its Lambert-square inversion law; places it at the centre of the rank-four Master plane and étage line; and supplies the TetraEtage with its native field structure, flow, derivative, and lexemic calculus.