M15b Native Calculus for the Symmetric Core: Derivatives, Transforms, and Special Functions in the ln-Based Operational Hierarchy

This monograph develops the native calculus of the Symmetric Core (SC): derivatives, measures, transforms, and special functions that are intrinsic to the logarithm‑based operational hierarchy. The Symmetric Core consists of symmetric operationsOpₙ (x, y) = expⁿ (lnⁿ (x) + lnⁿ (y) ), built only from iterated natural logarithms and exponentials. Unlike the Hyper Core, it uses no superlogarithms, tetration, or Koenigs machinery. The key idea is the native coordinateuₙ (x) = ln^ (n-1) (x), in which every SC operation becomes ordinary multiplication. In these coordinates, the native derivativeDₙ f (x) = x * ln (x) * ln ln (x) *. . . * ln^ (n-2) (x) * f' (x) obeys an exact SC power ruleDₙ (uˢ) = s * u^ (s-1), mirroring classical calculus with no approximation. The monograph develops the native Fourier transform on SC lattices and shows that it encodes the Euler product of the SC zeta function prime by prime. Native special functions are constructed systematically, including the SC Gamma function (with exact recurrence and reflection), native trigonometric functions, and a depth‑independent native Theta function. The Tower Circle and Trig Core give a trigonometric reformulation of SC‑2 arithmetic. The final part formulates the SC Riemann Programme forzetaSC (s) = prodₚ (1 - exp (-p*s) ) ^ (-1), introducing its functional equation, LP curve, zero symmetries, and unit‑cascade dynamics. A concise dictionary translates classical analytic objects exactly into their SC counterparts. The purpose of this deposit is to document the native analytic framework of the Symmetric Core and establish its conceptual priority as a parallel, logarithm‑based operational calculus.