Sector 3 — Down-type Quarks

The \(d=3\) sector has geometry \(S^3\) — the three-sphere — with isometry group SO(4). Unlike the complex projective spaces \(\mathbb{CP}^n\) that appear at even sectors, \(S^3\) is a real manifold. This absence of a complex (Kähler) structure has direct physical consequences: the \(d=3\) sector is vector-like rather than chiral, meaning the left and right handedness of quarks are not distinguished by the sector geometry itself.

SO(4) decomposes as \(SU(2)_L \times SU(2)_R\), a left-right symmetric group. The left-handed component \(SU(2)_L\) is what provides down-type quarks with their weak isospin — their coupling to the W boson.

The \(d=3\) sector is also our observable 3D space. Down-type quarks therefore live at the same dimensional depth as we do. They are not hidden in any additional direction — they occupy exactly the three directions we already occupy.

Three particles live in the \(d=3\) sector: the down quark, the strange quark, and the bottom quark.

The down and strange masses are parameter-free outputs of the derived \(d=3\) sector scale (+0.04% and +0.57% vs PDG 2024), both within the sizable PDG light-quark mass uncertainties.

The bottom quark is special. It does not arise from a regular mode index. Instead, it appears as a quartic bifurcation beat at the resonance site \(k_0 = 16 = n_s^2\), the kernel algebraic fixed-point. This is the unique mode that satisfies three independent conditions simultaneously and is therefore structurally distinct from the down and strange quarks.

The \(S^3\) geometry provides down-type quarks with their weak isospin — the left-handed coupling to the W boson. The \(SU(2)_L\) factor of SO(4) is the geometric handle. The right-handed \(SU(2)_R\) factor is present in the geometry but does not generate a coupling in the observed weak interaction; it is latent.

Down-type quarks also carry colour charge, but they do not get it from the \(S^3\) sector geometry itself. They get it derivatively: \(d=3 \subset d=4\), so every \(d=3\) particle also lives in \(d=4\), and \(\mathbb{CP}^2\) is where colour originates. Down quarks couple to the strong force through those shared coordinates. The colour charge is a consequence of containment, not of the \(S^3\) geometry directly.

This distinction matters: the \(d=3\) sector's own coupling contribution is the weak isospin structure. The colour coupling is an inheritance from \(d=4\) above it.

The \(d=3\) sector self-coupling is \(g_{33} = 8\sqrt{7}\), derived from \(n_s = 4\) and \(n_u = 3\):

Together with \(g_{44} = 12/\sqrt{7}\) from the \(d=4\) sector, this satisfies T9a exactly: \(g_{33} \times g_{44} = 96\). This product is fixed by the geometry and cannot be altered without cascading changes through the PMNS angles, the ℓ=2 kernel scale ε, and the neutrino mass scale.

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