Geometric Composition Algebra

Geometric Composition Algebra (GCA) is a combinatorial–geometric framework defined on finite configurations of unit cells arranged in horizontal line segments with an oriented reference.
Configurations are manipulated through a small set of elementary operators, including homothetic expansion, stacking, fusion, geometric complement, maximization, and symmetry operations.

The framework is entirely internal and axiomatic: all objects, operations, and invariants are defined intrinsically, without reliance on external algebraic or number-theoretic structures.
A central feature of GCA is the action of homothetic expansion, which induces a decomposition of configurations into infinite orbits generated from unique primitive configurations. This leads to a natural notion of primitivity, orbit invariants, and an internal counting function associated with primitive generators.

GCA also exhibits non-trivial dynamical behavior. In particular, iterative application of the complement operator displays rapid convergence toward a simple attracting cycle, a phenomenon supported by exhaustive computational experiments for small configurations.

The present work develops the algebraic structure of the operators, establishes a unique primitive decomposition theorem, identifies a minimal set of orbit invariants, and provides exhaustive enumerations for small sizes.
Several results are supported by computational verification or proof sketches, and are presented as part of an ongoing research program.

This preprint is intended as a foundational reference for further theoretical analysis, extensions of the framework, and potential connections with combinatorial geometry and discrete dynamical systems.