This monograph establishes horizontal unit cloning as the central new algebraic mechanism of the Symmetric Core (SC), yielding structures that go fundamentally beyond classical flat algebras such as the complex numbers, quaternions, octonions, or the Cayley–Dickson tower. The first part re‑derives the entire Symmetric Core hierarchy from Abel additivity alone. The Dual Representation Theorem shows that every SC operation admits equivalent additive, multiplicative, and Apow‑type forms, all arising from the same Abel structure. This fixes the operational backbone on which cloning acts. The core contribution of the monograph is the theory of horizontal cloning. A single SC Hermit unit, when cloned horizontally under its native operation, generates non‑trivial multi‑unit algebras (“chromatic algebras”). Unlike Cayley–Dickson constructions, which extend dimension by introducing new imaginary directions with fixed multiplication tables, SC cloning produces interaction‑generated algebraic structure: Lie brackets, closure relations, and higher‑order composite algebras emerge from the cloning process itself rather than being imposed axiomatically. These algebras are intrinsically operational and rank‑dependent, and they cannot be reduced to associative or alternative flat algebras. The final part analyses the fiber geometry of the SC rank manifold. Circle fibers over real ranks are described explicitly, the Abel‑Shift Law is proved, and rank gaps are filled via controlled fiber transport, allowing the definition of a family of fractional‑rank operations. This geometric structure explains how cloning algebras persist and deform across rank. M16a thus identifies cloning—not dimensional doubling—as the correct mechanism for generating new algebra in the operational hierarchy. The purpose of this deposit is to document the mathematical foundations, constructions, and conceptual priority of cloning‑based algebra as the genuinely new structure introduced by the Symmetric Core programme.
Paweł Łukasz Garycki (Fri,) studied this question.