Spin-Statistics from Collision Topology

Derived — 90% confidence

Claim: The spin-statistics theorem derives from the monodromy of line bundles around collision points on the shape sphere, not from relativistic QFT.

Phase: ψ → ψ

Watch the yellow tracer loop around the red collision point. The phase flips sign for k=odd (fermion).

The Construction

The N=3 shape sphere S² has three collision points (B₁₂, B₂₃, B₃₁) where two particles coincide. Each collision point carries a monopole-like singularity. Line bundles L^k over the punctured sphere have a topological invariant: the holonomy around any collision loop.

L^k over S² → monopole charge g_k = k/2 → spin j = k/2

Holonomy around collision: e^iΓ_k = (−1)^k = (−1)^2j

k even (integer spin): ψ → +ψ (boson)
k odd (half-integer spin): ψ → −ψ (fermion)

What This Means

Exchanging two particles is a loop on shape space that encircles a collision point. The phase picked up by the wavefunction is determined by the bundle’s monopole charge, which is a topological integer — not a dynamical parameter. Spin is the monopole charge. Exchange statistics is the holonomy. The connection between them is forced by the topology of the collision locus, not by any axiom about identical particles.

Why this matters for Barbour’s programme: On the call, Julian was reaching toward the idea that collision structure could explain exclusion. This derivation confirms the instinct: collision monodromy supplies the exchange sign behind exclusion. The topology of the collision locus on shape space is the spin-statistics theorem.

Open: The N=3 case is exact. The general-N version requires the full collision stratification of the higher Kendall shape space.