M51a The Ascent Depth - A Definitional Paper for the AscentCore: the Depth Coordinate, the Closure of the Lower Endpoint, the Recentred Chirality Plane, and the Abelian Reach of the HyperCore

In short:The AscentCore, within the Theory of Operation - Operational Manifold, is the programme's unification of Addition, Multiplication, and ordinary Exponentiation within one canonical Core: commutativity removes the slot ambiguity at every first ascent, forcing the chain Add -> Mult -> Exp -> HCt, with HCt (Half-Caterpillar Tetration) canonised as the ascent of Cpow (Commucative Exponentiation) and exactly linked to Ctet (Commutative Tetration) through the Half-Caterpillar conjugacy; the analogous selection of a canonical pentation and higher operations is the conjectural frontier. In details: A Definitional Paper for the AscentCore: the Depth Coordinate, the Closure of the Lower Endpoint, the Recentred Chirality Plane, and the Abelian Reach of the HyperCore The HyperCore is a ladder of commutative operations, while ascent freezes one input of an operation and iterates the resulting one-variable map to produce an operation at the next rank. M51a formalises successive ascents, explains when directional choices appear, and determines a limitation of the algebraic curves naturally reached by the construction. Outside the Theory of Operations, the paper concerns iteration, branched coverings, moduli dimensions, and the boundary of what a radical-based construction can represent. The AscentCore is defined as the first-ascent image of the HyperCore: AC_R = Asc_(R,1)(C_(R-1)). Its operations are indexed by: Asc_(R,d,omega), where R is rank, d is ascent depth, and omega records the sequence of operand slots chosen after directionality appears. Depth zero is the HyperCore, depth one the AscentCore, and depths two and above form the branched higher-ascent structure. Ascent is distinct from ordinary iteration. Iteration repeatedly applies one fixed unary map. Ascent starts from a binary operation, fixes one input, and iterates the resulting unary map to create a new operation. Because every HyperCore operation is commutative, its two slot embeddings coincide and the first ascent is canonical. The first ascent creates chirality; only subsequent ascents require directional choices. This gives the production chain: Add -> Mult -> Exp -> HCt -> Asc_(5,1) -> ... The lower endpoint is closed by the uniquely forced commutative rank-zero operation: C_0(a,b) = max(a,b) + 1. Fixing one operand gives X -> max(X,a)+1, whose iteration produces a+n. Its first ascent is therefore Addition, completing the AscentCore from rank one. At depth d >= 2, the directional history becomes a word: omega in {H,B}^(d-1). The two letters represent the height-slot and base-slot ascents. Words of length k form 2^k possible histories. M51a defines two statistics on them, a measure index and a slope multiplier. Both depend only on how many times each letter occurs, not on their order, so the statistics factor through the abelianisation of the free word monoid. The operators themselves are not claimed to commute. The arrival theorem identifies exactly one arriving history among the 2^k words of each length. The all-height history raises the active measure at every step; base-slot choices modify the slope without producing the same arrival. The depth coordinate therefore turns an earlier intuition about “ascent order” into a production hierarchy with explicit addresses. M51a also recentres the rank-three chirality family. Writing: t = 1/2 + tau, the family becomes: L_(1/2+tau)(a,b) = exp(K rho^tau), where: K = sqrt[(b ln(a))(a ln(b))], rho = (a ln(b))/(b ln(a)). Here K contains the symmetric information and rho measures operand asymmetry. Operand exchange acts as: tau -> -tau. Its unique fixed point is tau = 0, corresponding to Cpow. The natural origin of the chirality plane is therefore the commutative midpoint, not ordinary exponentiation. The real and imaginary chirality axes have different effects. Real chirality changes the magnitude of K rho^tau and therefore breaks the rank-three invariant. On the imaginary axis: tau = i theta, rho^(i theta) = exp[i theta ln(rho)], so: |K rho^(i theta)| = K. Imaginary chirality preserves the invariant’s modulus and rotates only its phase. The phase is stationary on the diagonal, where rho = 1, and turns only when the operands are asymmetric. M51a presents this as a structural symmetry candidate, not as a completed dynamical theory. The Chirality Trade connects growth and branching. At rational chirality t = 1/q: rate exponent = 1 - 1/q, cover degree = q. At the commutative midpoint q = 2, commutativity reduces the chiral growth exponent from 1 to 1/2 while introducing a double cover. The growth reading and the algebraic-cover reading are two consequences of the same square root. The paper’s main limitation theorem is the Abelian Reach. Under its definition of native computation, the periods naturally produced by the HyperCore lie on abelian covers of the projective line. A tamer requiring an n-th root corresponds to a cyclic cover of the form: y^n = f(x). Within this reach, the hyperelliptic case n = 2 forms the largest family at fixed genus. Increasing the root denominator produces higher-degree cyclic covers but smaller reachable subfamilies. Thus finer fractional ranks do not enlarge the moduli reach. This is a boundary on the paper’s specific radical-based construction, not a general impossibility theorem for numerical methods or period computation. The arithmetic-geometric mean is the paradigm at genus one. At genus two, the construction can reach a genuine hyperelliptic slice, but its single essential parameter does not cover the full three-dimensional moduli space. Generic curves of genus at least three lie outside the native reach described here. A consequence is recorded for the earlier Hodge programme. The proposed correspondence between codimension p and rank p+1/2 merely reaches top-degree cohomology and is therefore vacuous for the intended purpose. M51a relocates the possible exceptional-class correspondence to: codimension p <-> genus 2p <-> rank 2p + 1/2. Even after that correction, the Abelian Reach prevents the rank coordinate alone from accessing or distinguishing the generic objects relevant to the difficult cases. The linked half-rank analytic tower survives; the claimed native route to the Hodge problem does not. M51a therefore gives the AscentCore its missing coordinate system and simultaneously identifies its boundary. Rank locates an operation, depth counts its production history, chirality resolves its directional deformation, and the Abelian Reach states which algebraic geometry the resulting radical machinery can naturally see.