4 Fermion Spin Structure

The winding number m determines the symmetry of the fermion’s compression residual. The half-integer spin of the fermion itself is encoded separately, in the universal phase structure established in Paper VII.

For any single fermion hop, the off-diagonal entries of B carry a universal −π∕2 phase. Four successive hops accumulate a total phase of 4 × (−π∕2) = −2π, returning the system to its original state. This is precisely the spinor double-cover structure of spin-1∕2: a 2π rotation returns a fermion to its state only up to a sign, while a 4π rotation is the identity.

The conservation law therefore unifies two structures in a single codec object:

• The diagonal F carries half-integer spin, inherited from the −π∕2 universal phase of the fermion hop.
• The off-diagonal B carries the internal symmetry class determined by the winding number m.

This is not a postulate; it follows from the imaginary encoding ψ = f₀ + if₁ and the definition of B, as shown in Paper VII.