A Finite Configuration System with Symmetry Group Action and Linear Invariants

This work introduces a fully explicit finite configuration system defined on a 3×3 grid, equipped with a dihedral symmetry group action and a family of linear invariants.Configurations are formalised as functions on a finite position set, with constraints expressed as affine linear conditions over designated line subsets (rows, columns, and diagonals). The solution space is shown to form a low-dimensional affine subspace, whose rank, dimension, and parametrisation are derived explicitly.In addition, a minimal constraint-preserving projection framework is presented, demonstrating how controlled evolution can be introduced while maintaining all linear invariants exactly. The construction is intentionally purely structural: no assumptions are made regarding dynamics, optimisation, stability, or physical interpretation.The 3×3 system serves as a minimal, completely analysable instance intended as a methodological and conceptual baseline for more general symmetry-constrained linear systems.