A Finite Countermodel to Heisenberg Carrier Selection

What the countermodel proves

Let \(G=S_3\), \(X=(12)\), \(Y=(23)\), and let \(\sigma\) be conjugation by \((13)\). Then \(\sigma\) is an involution exchanging \(X\) and \(Y\), the pair minimally generates \(S_3\), and the standard two-dimensional zero-sum representation is faithful, unitary, and irreducible. Nevertheless,

\[ [X,Y]=(132)\notin Z(S_3), \]

because \(Z(S_3)\) is trivial. The algebraic properties available before the centrality step therefore do not select a Heisenberg group.

Scope. The note does not propose \(S_3\) as the physical carrier. It refutes the published implication from the stated premises to a central commutator, a finite Heisenberg group, and a Weil carrier.

What survives

The ENI no-go theorem is independent of this carrier-selection edge.
Calculations performed on a supplied Heisenberg/Weil carrier retain their internal status.
If a finite Heisenberg group and a non-trivial central character are supplied, the Stone–von Neumann representation-theoretic endpoint remains valid.

What changes is the declared status: the carrier is currently a supplied realization, and any result consuming its selection must be labelled conditional.

Why this matters

The failure is an identification, not a calculation. A non-trivial commutator is not automatically central, and Schur's lemma constrains already-central operators rather than making an arbitrary commutator central. Naming this edge keeps exploratory work possible while preventing a model choice from being promoted to a theorem.