Chiral Coupling Selection

Epistemic status: STRUCTURAL (PRINCIPAL DEBT). The following argument is physically motivated, consistent with all known data, and grounded in the axioms. The coupling theorem is not formally proved. This is the paper’s principal open problem.

The electroweak gauge group SU(2) × U(1)_Y has been derived. But the Standard Model has a remarkable feature: SU(2) does not couple equally to all fermion fields. It couples to left-chiral fields as doublets and to right-chiral fields as singlets. The weak force is maximally parity-violating. Why?

Chirality vs helicity. Chirality is a Lorentz-representation property. A Dirac spinor Ψ decomposes into left-chiral and right-chiral Weyl components via P_L = ½(1 − γ⁵) and P_R = ½(1 + γ⁵). These transform under independent representations of the Lorentz group: Ψ_L ∈ (½, 0) and Ψ_R ∈ (0, ½).

The axioms have one structural asymmetry not yet deployed: Axiom R’s arrow of time.

Proposition 2 (Chiral Coupling Selection). The gauge connection on ℋ_EW couples to left-chiral fermion fields as doublets and to right-chiral fermion fields as singlets.

Argument. The break ε has a direction: it proceeds from the 1:1 symmetry (pre-state) to the manifold (record). This defines a fundamental arrow (Axiom R). On the Lorentzian manifold (AP05), this arrow selects a preferred time-orientation, which selects a preferred decomposition of the Lorentz group into its two chiral half-representations.

The involution σ maps ℒ ↔ 𝒫. By AP22, σ acts as CPT on the manifold. CPT conjugation interchanges P_L and P_R: it maps left-chiral fields to right-chiral fields. Axiom R breaks CPT as a dynamical symmetry — the break goes forward in time, not backward. The two chiral projections are NOT dynamically equivalent.

For a left-chiral field Ψ_L, the arrow of time and the chiral structure are anti-aligned. The two internal components z_ℒ and z_𝒫 are dynamically independent. The field carries the full ℂ² and transforms as a doublet under SU(2).

For a right-chiral field Ψ_R, the arrow of time and the chiral structure are aligned. The alignment locks the two sector-components together. The internal ℂ² collapses to a single effective degree of freedom. The field is a singlet under SU(2).

What this argument needs to become a theorem. (a) Define D_μ on the tensor product bundle (spinor ⊗ ℋ_EW). (b) Show the coupling to P_LΨ is non-trivial (full ℂ²) while the coupling to P_RΨ vanishes (reduces to Abelian U(1)). (c) Derive this from Axiom R, the Lorentz structure (AP05, AP11), and the CPT identification (AP22) without importing the Standard Model’s chiral coupling. Steps (a) and (b) are well-defined mathematical problems. Step (c) is the hard part.

If the coupling theorem cannot be proved, the paper still derives SU(2) × U(1)_Y as the electroweak gauge group (Lemma 1 + Lemma 2 + Proposition 1), but the chiral coupling reverts to structural identification rather than derivation.

You have watched the axiom’s arrow — the irreversibility of records — reach into the internal structure of the gauge field and break the symmetry between left and right. The weak force does not see right-handed particles because the arrow of time has locked their internal doublet into a singlet. Parity violation is not a quirk of nature. It is Axiom R, projected onto the electroweak fibre.