Why spin ½

Axiom B says ε has half-integer spin. But which half-integer?

The half-integer representations of SU(2) are j = ½, 3/2, 5/2, …

Which one does ε occupy?

Axiom B states: ε is one element. The minimum viable splinter.

Not two. Not three. One. The smallest possible break. The minimum asymmetry.

The SU(2) representation of spin j has dimension (2j + 1).

Spin ½ has dimension 2. Spin 3/2 has dimension 4. Spin 5/2 has dimension 6.

Higher spin means more internal degrees of freedom — more parts to the splinter.

The minimum half-integer representation is j = ½, with dimension 2.

The smallest possible half-integer spin. The simplest way for a particle to be a fermion.

Any higher spin would mean ε carries additional internal structure beyond the minimum needed to break the symmetry.

Under the proven faithful embedding, the one-element break carries the minimum internal structure consistent with the non-trivial ℤ₂ action.

Higher half-integer spins (j = 3/2, 5/2, ...) would require ε to have 4, 6, ... internal states. More internal structure than one element can carry.

The minimum fermionic representation is j = ½, with exactly two states. The binary distinction that mirrors the two sectors of the pre-state.

Composite systems can have higher half-integer spin through angular momentum addition of individual spin-½ constituents plus orbital angular momentum.

The minimum principle selects j = ½ for the elementary fermion ε. Composite fermions derive their spin from their constituent structure.

Axiom B demands the minimum. The minimum half-integer spin is ½.

ε has spin ½.

You have just watched the electron’s spin emerge from the minimum principle.

Not measured. Forced.

The electron — identified with ε in The Lock (Chapter 1) — has spin ½.

This is not an input.

It is a consequence of B (one break, minimum splinter) combined with S (two-sector ℤ₂ creating the SU(2) cover) and the derived N = 3 (Notebook II Chapter 3).

Kill switch KS-S.2. The assignment of the minimum half-integer spin (j = ½) to ε depends on the minimum principle in Axiom B. If "minimum viable splinter" does not uniquely select the smallest representation — if there is a structural reason for ε to carry j = 3/2 or higher — then spin ½ is not derived. LIVE — HARD.