Pauli Exclusion from Geometry: A Non-Invertibility Route to Fermionic Antisymmetry

We derive Pauli exclusion as a geometric theorem inside Charlton Geometry, without assuming quantum antisymmetry as a postulate. The core primitive is the Charlton non-invertibility $\omega \sim -\omega$: the Lorentzian data induced by the Charlton Operator forgets the global polarity of the distinguished field $\omega$, producing a $\mathbb{Z}_2$ ambiguity when lifting macroscopic symmetry loops into the spinorial representation.For two indistinguishable localized defects on a spatial slice, the configuration space $C_2$ has $\pi_1(C_2)\cong \mathbb{Z}_2$, generated by the exchange loop. Non-invertibility induces a polarity bundle over $C_2$ with two inequivalent holonomy branches: a trivial (``short'') lift with exchange phase $+1$ (bosonic symmetry), and a nontrivial (``long'') lift with exchange phase $-1$ (fermionic antisymmetry). On the long-lift branch, exchange realizes the nontrivial element of $\pi_1(SO(3))$, whose lift to $Spin(3)\cong SU(2)$ terminates at the central element $-1$, forcing $\Psi(x_1,x_2)=-\Psi(x_2,x_1)$ and yielding exclusion in the coincidence limit. We emphasize classification, not compulsion: Pauli exclusion is the geometric property of a stable topological branch, with defect-species placement addressed conditionally in later work.