The spin-statistics connection

The spin-statistics theorem (Pauli, 1940) states: particles with integer spin obey Bose–Einstein statistics — symmetric wave function under exchange.

Particles with half-integer spin obey Fermi–Dirac statistics — antisymmetric wave function under exchange.

In standard quantum field theory, this is derived from relativistic quantum mechanics plus the requirement of positive-definite energy.

In the 420 Code, it follows directly from the ℤ₂ transformation properties established above.

Paired elements (bosons, integer spin). The ℤ₂ acts trivially. σ completes the swap: σ(a) = a′, σ(a′) = a.

Under exchange of two identical paired elements, the wave function is symmetric. Swapping them is equivalent to applying σ to both, and σ acts trivially on paired representations.

The wave function picks up a factor of +1 under exchange.

Unpaired elements (fermions, half-integer spin). The ℤ₂ acts non-trivially. σ cannot complete the swap for ε — the non-trivial ℤ₂ action produces a phase of −1.

Under exchange of two identical unpaired elements, the wave function is antisymmetric. Swapping them is equivalent to applying the non-trivial ℤ₂ action, which produces a factor of −1.

The wave function picks up a factor of −1 under exchange.

You have just derived the spin-statistics connection — conditional on the three identifications this paper fences: the ℤ₂ ↔ π₁(SO(3)) identification (KS-S.1), the minimum-spin selection (KS-S.2), and the exchange–involution bridge (KS-S.3).

The spin-statistics connection is not an additional law.

It is the ℤ₂ from Axiom S, acting differently on paired and unpaired elements — which is the distinction created by Axiom B — once those three identifications hold.

Kill switch KS-S.3. The derivation requires that exchange symmetry of identical particles on the Hilbert space is correctly represented by the paired/unpaired σ-action inherited from the record algebra. If a consistent construction is shown in which the paired/unpaired classification does not fix the exchange sign (or fixes it differently), the spin-statistics mapping and Pauli derivation fail. LIVE — HARD.