A Finite Obstruction to Heisenberg Carrier Selection from Admissibility Constraints

Abstract

We test whether the admissibility axioms of the Cosmochrony framework, together with Born–Infeld parity, select a finite Heisenberg group and its irreducible carrier. The 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. We disprove that implication by an explicit six-element countermodel. Let $G_n={\mathfrak{S}}_3$, $X=(12)$, $Y=(23)$, and let $\sigma$ be conjugation by $(13)$. The involution exchanges $X$ and $Y$, the pair minimally generates $G_n$, 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. We also show that irreducibility does not force the centre to have prime order: a central subgroup acts through one character, not through several non-zero character eigenspaces. Stone–von Neumann remains a valid conditional endpoint once a finite Heisenberg group and a non-trivial central character are supplied. The associated Weil action is a further representation of the symplectic automorphism group, not the Heisenberg representation itself. Interpretive status. The result obstructs the unique-selection bridge but leaves intact the mathematics proved internally on a supplied Heisenberg carrier or associated Weil module.