The Internal State Space

Axiom S states that reality has two sectors, ℒ and 𝒫, connected by the involution σ: ℒ ↔ 𝒫. The break ε is the boundary between them — the object that connects the two sectors and, through Axiom R, writes records on the manifold.

The break is not purely in ℒ. It is not purely in 𝒫. It is the boundary: the thing that sits between and connects them. When ε propagates on the manifold as a quantum field, it necessarily carries information about its relationship to both sectors.

This relationship is an internal degree of freedom. It is not spatial position (governed by Axiom C). It is not spin (the Lorentz spinor structure, AP11). It is not colour (spatial orientation freedom, AP19). It is not overall phase (the U(1) of AP15). It is the one remaining structural feature: the break’s relationship to the two sectors.

Lemma 1 (Internal Electroweak State Space). The break ε, as the boundary between sectors ℒ and 𝒫 (Axiom S), carries an internal state on the manifold encoding its relationship to each sector. The space of internal states is ℋ_EW ≅ ℂ².

Proof. (i) Axiom S provides exactly two sectors: ℒ and 𝒫. No more, no fewer. The break connects them. On the manifold, a propagating field arising from ε carries a state that encodes how ε relates to each sector. Since there are two sectors, the state has two components: one for ε’s relationship to ℒ, and one for ε’s relationship to 𝒫.

(ii) Why is this a complex linear vector space? The components must support superposition: the pre-state is a complex Hilbert space (AP09), and every quantum degree of freedom is represented as a vector in a Hilbert space. The sector-relationship components are quantum degrees of freedom — they participate in interference, entanglement, and the Born rule (AP25). A discrete label (e.g., “ℒ or 𝒫” with no superposition) would not support interference. A real vector space would not support the complex phase structure required by AP09. Therefore the internal state space is a two-dimensional complex Hilbert space.

(iii) The state is a vector (z_ℒ, z_𝒫) ∈ ℂ². The inner product on ℋ_EW is physical: it determines transition amplitudes via the Born rule (AP25). The internal state space is ℋ_EW ≅ ℂ².

(iv) This space is internal: not spacetime, not spin, not colour, not phase. It is the space of the break’s internal orientation relative to the two-sector structure.

Note: why exactly two components and not more? Because Axiom S provides exactly two sectors. Three sectors would give ℂ³. n sectors would give ℂⁿ. Two sectors give ℂ². The same logic as AP19: three spatial dimensions give ℂ³ for colour.

Note: states in ℋ_EW are not merely classical labels “ℒ or 𝒫”. The break can exist in coherent superpositions of |ℒ⟩ and |𝒫⟩. A general state α|ℒ⟩ + β|𝒫⟩ is physically meaningful. Such superpositions exhibit interference. This is what gives rise to the full continuous U(2) gauge freedom rather than a discrete ℤ₂ swap.

Note: ℋ_EW is distinct from all previously derived spaces. In fibre bundle language: the spinor bundle, the colour bundle, and the electroweak bundle are three distinct fibre bundles over the manifold, with structure groups SU(2)_spin, SU(3), and U(2) respectively. Their fibres are different vector spaces. Their connections are independent.