The distinction

The fermion/boson distinction is not an additional classification imposed on particles.

It is the distinction between paired and unpaired elements of the record algebra under the involution σ.

Paired (σ-image exists) → ℤ₂ acts trivially → integer spin → boson.

Unpaired (no σ-image — Axiom B) → ℤ₂ acts non-trivially → half-integer spin → fermion.

Axiom S creates the ℤ₂. Axiom B creates the one element that doesn’t fit.

The fermion/boson distinction is S and B read on the manifold.