The Sator Square as a Zero-Entropy Symbolic Structure: Symmetry, Information Theory, and the Klein Group

We present a formal investigation of the Sator Square as a symmetry-constrained symbolic information structure. We prove that its generating symmetries form the Klein four-group \(G Z_2 × Z_2\), whose action partitions the 25 matrix positions into 9 independent orbits. This reduces the unconstrained symbolic space from \(26²⁵\) to \(26^9\), yielding a compression factor of \(26⁻¹⁶ ≈ 2.29 × 10⁻²³\). Shannon entropy measurements show omnidirectional entropy parity under canonical reading operations, with numerical deviations bounded by machine precision. We further model the square as a constraint satisfaction problem and as a non-linear, symmetry-constrained symbolic structure with effective rate \(R = 9/25\). Error-recovery experiments demonstrate high robustness under single-position corruption. Finally, we generalize the construction across curated lexicons and identify non-trivial Sator-like families in Portuguese, suggesting that the Sator Square belongs to a broader combinatorial class rather than being an isolated artifact.