The Peres-Mermin magic square demonstrates quantum contextuality: there's no way to assign definite values to a set of observables consistent with all the constraints simultaneously. This is “state-independent” — it doesn't depend on which quantum state you prepare. The standard proofs are case-by-case, working with specific operator arrangements.
Commutation groups algebraize the argument (arXiv:2603.12197). Define a group by generators (observables) and relations (commutation and product constraints). A “contextual word” in this group is an algebraic witness of contextuality — an element that equals the identity when evaluated through quantum representations but cannot equal the identity under any classical (non-contextual) value assignment. String rewriting determines which words are contextual.
The structural insight: contextuality becomes a property of group words, not operator arrangements. The Peres-Mermin square is a specific contextual word in a specific commutation group. The algebraic framework answers: given a set of commutation relations, does a contextual word exist? The answer depends on the group structure, not the Hilbert space representation. Contextuality is in the algebra, not the physics. The physics (unitary representations of commutation groups as generalized Pauli groups) is how the algebra manifests, but the obstruction to classical value assignments is purely algebraic.