Orbit-Induced Redundancy in Finite Symbolic Systems

We study how the action of a finite symmetry group $G$ over a symbolic grid $\Sigma^{n^2}$ induces structural redundancy, dimensional compression, reading-direction invariance, and passive recovery under noise. Motivated by the Sator Square, we formalize these mechanisms across finite group actions on square grids. We prove an exact identity for the information action $A[M,G]$, determined by the number of orbits of $G$, and introduce transition entropy as a non-trivial diagnostic separating weakly constrained systems from structurally rigid ones.

Computational experiments over representative finite groups provide evidence that baseline-corrected recovery is better predicted by $d_{min}^{orb}(G)$, the size of the smallest non-trivial orbit, than by the group order $|G|$. The Sator Square is thereby reframed as a minimum-complexity historical example of an orbit-constrained symbolic system rather than an isolated mathematical anomaly.