Sector 4 — Up-type Quarks

The \(d=4\) sector has geometry \(\mathbb{CP}^2\) — the complex projective plane — with isometry group \(\mathrm{SU}(3)\). \(\mathbb{CP}^2\) is a Kähler manifold: it has a complex structure and a natural \(\gamma_5\) operator from that structure, which provides the chirality of up-type quarks. This is why up quarks have left-handed weak coupling — not by assumption, but because the Kähler geometry of \(\mathbb{CP}^2\) selects a chirality.

\(\mathbb{CP}^2\) is also the colour sector. Its Euler characteristic \(\chi(\mathbb{CP}^2) = 3\) directly gives the number of quark colours: \(N_c = 3\). The three independent modes of the \(d=4\) sector geometry are exactly the three colour states of a quark. They transform under \(\mathrm{SU}(3)\) — the isometry group of \(\mathbb{CP}^2\) — in the fundamental representation, which is why colour is an \(\mathrm{SU}(3)\) symmetry. None of this is postulated; it follows from the geometry of the manifold.

The table lists the bare combinatorial mass \(m = m_{\text{scale},4}\,S(n,4)\) — the free-quark count, before confinement. The up quark sits +0.77% above PDG, well within the sizable up-quark uncertainty. The bare charm and top overshoot, growing with generation (+0.93%, +2.20%), because a coloured quark is never free: part of that bare count is colour-field binding energy that is not present in the physical mass.

The derived confinement correction supplies exactly that: \(m_{\text{phys}} = m\,(1 - x_e\langle k\rangle)\), applied only in the two colour sectors (\(d=3, 4\)), with \(\langle k\rangle = d(n-1)/(d+1)\) the degeneracy-weighted mean excitation level and \(x_e = 3/(16 N_b)\) the per-state softening of a finite confining well — the \(d=4\) well occupation \(N_b\) fixed by the colour energy law. It brings every up-type quark within \(\pm1\sigma\) of PDG: up +0.70% (+0.2σ), charm +0.34% (+0.9σ), top −0.04% (−0.2σ).

The top mode index \(n = 72 = N_c \cdot n_s \cdot N_f = 3 \cdot 4 \cdot 6\) equals the product of the three Kähler sector Euler characteristics \(\chi(\mathbb{CP}^2)\,\chi(\mathbb{CP}^3)\,\chi(\mathbb{CP}^5)\) — an arithmetic identity in the seed integers. The condition that selects it is the electroweak spectral closure: the shares \(\sum_i (m_i/v)^2\) of the electroweak channel's one conserved quantum cannot oversubscribe it, and 72 is the unique integer at the subscription boundary given the boson chain \(n_W, n_Z, n_H\) hung off it.

In the Standard Model, \(N_c = 3\) is an experimental input. In IDWT it is a theorem: \(N_c = \chi(\mathbb{CP}^2) = 3\). The Euler characteristic of \(\mathbb{CP}^2\) is 3 because \(\mathbb{CP}^2\) has one cell in each of the dimensions 0, 2, and 4 — three cells total, alternating in sign — giving \(\chi = 1 - 0 + 1 - 0 + 1 = 3\).

Those three independent modes of the \(\mathbb{CP}^2\) geometry are the three colour states per quark. They are not three copies of something; they are the three geometrically independent directions of the sector space, and they transform into each other under the \(\mathrm{SU}(3)\) isometry of \(\mathbb{CP}^2\). Colour is not a property attached to quarks from outside — it is what it means to be a mode in the \(d=4\) sector.

\(\mathrm{SU}(3)\) colour symmetry follows from the requirement that physics not depend on the local orientation of the colour frame in the \(d=4\) sector. Different orientations of the three colour directions are physically equivalent — this requirement forces local \(\mathrm{SU}(3)\) invariance, which is what we call colour gauge symmetry. It is a consistency requirement of the geometry, not a postulate.

Because \(N_c = \chi(\mathbb{CP}^2) = 3 = n_u\), and \(n_s = n_u + 1 = 4 = \chi(\mathbb{CP}^3)\), all sector coupling constants are functions of \(N_c\) alone.

The \(d=4\) sector self-coupling is \(g_{44} = 12/\sqrt{7}\), derived from \(n_s\) and \(n_u\):

T9a: \(g_{33} \times g_{44} = 8\sqrt{7} \times 12/\sqrt{7} = 96\) exactly. This product is the Hopf universality product for sectors \(d=3\) and \(d=4\), equal to \(N_c(N_c+1)^3/2 = 96\).

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