M27a The Asymmetric R=3 Algebra: Ordinary Exponentiation as an a-ONS on the HC Manifold

This monograph establishes ordinary exponentiation aᵇ as a first-class asymmetric Operational Number System (a-ONS) on the HC Rank Manifold at rank R = 3. The exponentiation a-ONS, or e-ONS, is defined by the asymmetric Abel pair AL (a) = ln a, AR (b) = b, with multiplicative linearisation ln (aᵇ) = b * ln a. This places ordinary exponentiation beside the symmetric rank-3 operations Apow and Cpow, completing the Rank-3 Triad. A central result is the Fluent Invariance Theorem. For the fluent family Lᵣ (a, b) = exp ( (b ln a) ^ (1-r) * (a ln b) ʳ), the diagonal is independent of the chirality parameter r: Lᵣ (u, u) = uᵘ. Therefore the whole fluent family shares the same Hermit condition uᵘ = i*u, or, in NC coordinate w = ln u, w* (eʷ - 1) = i*pi/2. The monograph compares this exact e-ONS Hermit equation with the Symmetric Core approximation w² - w = i*pi/2, showing that the SC equation is the tangent-level shadow of the full exponentiation structure. The work also develops branch loading of powers, e-ONS prime theory, left and right Euler products, Mellin transform as the e-ONS Abel transform, Bernoulli numbers as coefficients of the e-ONS normalising function, and Galois action as branch-circle translation. It connects exponentiation, zeta theory, Kummer extensions, branch microfibres, and the operational foundations of the physical programme.