The unpaired element

Axiom B: one element ε ∈ ℒ with no σ-image in 𝒟.

This is the break. ε is unpaired.

It does not participate in the ℤ₂ symmetry the way paired elements do.

Under the involution: σ acts on ε, but there is no element in 𝒟 to receive it.

The ℤ₂ action cannot complete the swap.

On the Hilbert space, ε transforms under a representation of SU(2) where the ℤ₂ kernel acts non-trivially.

The swap fails. The failure is registered as a phase.

In the classification of SU(2) representations, the ℤ₂ kernel of the covering map SU(2) → SO(3) acts as either +1 or −1 on any irreducible representation.

It acts as +1 if and only if the representation descends to SO(3) — if and only if the ℤ₂ cycle closes.

For paired elements, the cycle closes (σ swaps and returns), so the kernel acts as +1. Integer spin.

For ε, the cycle does not close (no σ-image to return to), so the kernel must act as −1. Half-integer spin.

There is no third option.

The classification is binary, matching the binary nature of ℤ₂ itself.

The unpaired element ε has half-integer spin. It is a fermion.

You have just watched the fermion/boson distinction emerge from the axioms.

Not imposed. Not observed and then encoded. Derived.