The identification

The rotation group of the spatial manifold is SO(3). Its fundamental group is ℤ₂.

The record algebra has a ℤ₂ symmetry — from Axiom S.

The ℤ₂ from Axiom S IS the fundamental group of SO(3).

The identification is not arbitrary. It is forced.

EH is proven (Notebook I Chapter 3). The discrete record algebra embeds faithfully into the smooth manifold.

Axiom S’s ℤ₂ symmetry is part of the algebra. Under the proven faithful embedding, it maps to a ℤ₂ symmetry of the manifold.

On a 3+1 dimensional Lorentzian manifold with spatial rotation group SO(3), the only intrinsic ℤ₂ associated with the rotation structure is π₁(SO(3)).

There is nowhere else for the algebraic ℤ₂ to go.

You are watching the axiom find its place on the manifold.

More precisely: σ reverses order (Axiom S).

On the Lorentzian manifold, this is the time-reversal operation. Established in Chapter 1 — σ ↔ complex conjugation, which is the time-reversal operator on the Hilbert space.

The time-reversal operator, combined with spatial rotations, generates the full Pin group — the extension of the rotation group by the ℤ₂ that includes time reversal.

The ℤ₂ in the Pin group structure is exactly π₁(SO(3)).

Axiom S’s ℤ₂ embeds into the manifold’s rotation structure as its fundamental group.

The double cover SU(2) → SO(3) is therefore a consequence of the axioms.

Axiom S creates the ℤ₂. N = 3 creates SO(3). Together they create SU(2).

The group under which quantum systems on the spatial manifold can transform.

You have just watched the double cover emerge from the axioms.

The possibility of half-integer spin is not an additional feature of nature.

It is the two-sector structure of the record algebra, read on the manifold.

Kill switch KS-S.1. The identification of the algebraic ℤ₂ with π₁(SO(3)) is the load-bearing step. It rests on the argument that under faithful embedding, the algebraic ℤ₂ maps to the only intrinsic ℤ₂ in the rotation structure. If a different embedding of the algebraic ℤ₂ is shown to be equally consistent — one that does not correspond to π₁(SO(3)) — then the derivation of the double cover from the axioms fails. LIVE — HARD.