The ℤ₂ Representation Theorem

You have followed the structural argument that the universe sorts into two sectors at birth.

This section carries the mathematics behind that claim.

8.1 — Statement

Theorem B.1. The group ℤ₂ = {e, g} where g² = e has exactly two irreducible representations over the real (or complex) numbers: the trivial representation and the sign representation.

8.2 — Proof

Let ρ: ℤ₂ → GL(V) be a representation of ℤ₂ on a vector space V. The generator g satisfies g² = e, so ρ(g)² = I (the identity matrix).

Therefore ρ(g) is an involution: its eigenvalues satisfy λ² = 1, giving λ = +1 or λ = −1.

The vector space V decomposes into eigenspaces:

V = V₊ ⊕ V₋ (8.1)

where V₊ is the +1 eigenspace (ρ(g)v = +v for all v ∈ V₊; vectors invariant under the ℤ₂ action; the trivial representation), and V₋ is the −1 eigenspace (ρ(g)v = −v for all v ∈ V₋; vectors that flip sign under the ℤ₂ action; the sign representation).

Each eigenspace is itself an invariant subspace and carries an irreducible representation of ℤ₂. Since the only eigenvalues are +1 and −1, there are exactly two irreducible representations.

Any representation of ℤ₂ decomposes into copies of these two.

There is no third option.

The mathematics does not negotiate.

Two irreducible representations. Two sectors. Two speed limits. Two fundamental categories.

The universe at birth: sorted into two bins by a theorem proved before electromagnetism was a theory.