M13a The Symmetric Core: Foundations

This monograph establishes the Symmetric Core (SC) as a complete, self‑contained operational number system based on the natural logarithm and its iterates, forming a distinct étage within the hyperoperation hierarchy. The work introduces the SC rank coordinate, the symmetric operations Amult, Apow, Atown and the associated DSRT ladder, Hermit units, and conserved quantities. A master structural formula for symmetric operations is developed, showing that the SC requires neither square‑root constructions nor superlogarithms, in contrast to chiral and hyper‑core frameworks. M13a develops the NC plane geometry of the Symmetric Core, including the Minkowski structure underlying non‑commutative coordinates, and establishes the downward extension of the SC to negative and fractional ranks. The Symmetric Core is shown to be algebraically closed and dynamically complete within its own étage. The purpose of this monograph is to provide the definitions, structural results, and conceptual priority of the Symmetric Core as the logarithmic étage of the hyperoperation theory series.