A Finite Countermodel to Heisenberg Carrier Selection from Admissibility Axioms

Two companion papers in the Cosmochrony programme claim that admissibility axioms select the finite Heisenberg group and its Weil carrier. Their load-bearing algebraic implication is that a finite faithful irreducible unitary carrier, equipped with a minimally generating pair exchanged by an involution and having a non-trivial commutator, must have a central commutator and hence be of Heisenberg type. We disprove that implication by an explicit six-element countermodel. Let \(G={\mathfrak{S}}_3\), \(X=(12)\), \(Y=(23)\), and let \(\sigma\) be conjugation by \((13)\). Then \(\sigma\) is involutive and exchanges \(X\) and \(Y\); the pair minimally generates \(G\); and the standard action on the two-dimensional zero-sum subspace of \(\mathbb{C}^3\) is faithful, unitary, and irreducible. Nevertheless, \([X,Y]\) is a non-trivial three-cycle and is not central, since \(Z({\mathfrak{S}}_3)=\{1\}\). We also show independently that the published prime-order step reverses the content of Schur's lemma: a central subgroup acts through one character on an irreducible complex representation and therefore does not yield the asserted non-trivial eigenspace decomposition. The algebraic consequences extracted from the admissibility inputs in the published proof do not select class-two nilpotence, a finite Heisenberg group, or a Weil carrier. Interpretive status. The countermodel changes the status of the carrier-selection edge from theorem to postulate; it does not invalidate results proved after a Heisenberg carrier is supplied.