Sector 4 — Up-type Quarks

The \(d=4\) sector has geometry \(\mathbb{CP}^2\) — the complex projective plane — with isometry group SU(3). \(\mathbb{CP}^2\) is a Kähler manifold: it has a complex structure and a natural γ₅ 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 SU(3) — the isometry group of \(\mathbb{CP}^2\) — in the fundamental representation, which is why colour is an SU(3) symmetry. None of this is postulated; it follows from the geometry of the manifold.

The up quark sits +0.77% above PDG — a parameter-free output of the derived \(d=4\) sector scale, well within the sizable PDG up-quark uncertainty. The charm and top masses are quoted bare and overshoot, growing with generation: charm +0.93% (+2.6σ), top +2.20% (+13σ), both open residues.

A former correction (the "Generation Tower Correction") multiplied the charm and top masses by \((1-\varepsilon)^k\), with \(\varepsilon = 1/(280\sqrt{7})\) derived but the per-quark exponent \(k\) a fit. A fitted correction is not a derivation, so it has been removed. The overshoot's sign is supplied by the ℓ=2 component of the sector kernel — a real, correctly-signed second-order self-energy — but its magnitude is not derived, so no correction is applied (an open item).

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 \(\chi(\mathbb{CP}^2)\,\chi(\mathbb{CP}^3)\,\chi(\mathbb{CP}^5)\) — but this is an arithmetic identity in the seed integers, not a derivation. Unlike the additive-tower indices, no condition has been found that selects 72 as a resonance; it is a tier-2 input with an open origin.

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 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.

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 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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