Historical excerpt: IDWT_03_Forces.md

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0.8b Sector Properties at a Glance

The table below consolidates, for each sector $d \in D$ , the geometric, coupling, and scale properties discussed separately in §0.7–§0.8 and Parts 1–2. One row per sector; read across for the complete coupling profile of each particle family.

$d$	Manifold	Isometry	$\chi$	$g_{dd}$ (exact)	$m_{\rm scale}$	Hosted particles	Coupling filter consequence
2	$\mathbb{CP}^1$	$SU(2)$	2	$722.5$	27.47 MeV	$\gamma,\;W^\pm,\;Z^0,\;H$	Orientation filter: two helicity states = complete internal geometry; coupling $\propto \varepsilon_\mu j^\mu$ , zero for $j^\mu \perp \varepsilon_\mu$
3	$S^3$	$\mathrm{SO}(4)$	0	$8\sqrt{7}$	4.702 MeV	$d,\;s,\;b$	No intrinsic chirality: $S^3$ is odd-dimensional, so $\gamma^1\gamma^2\gamma^3 \propto \mathbb{1}$ furnishes no chirality operator and its Dirac spinors are vector-like. The down-type left-handed weak structure is not produced here — it is inherited from the $d=4$ $\mathbb{CP}^2$ Kähler chirality through $d{=}3\subset d{=}4$ and the cross-coupling $g_{3,4}$ ; the $(u_L,d_L)$ doublet is a $\mathbb{CP}^2$ object (§6, §7)
4	$\mathbb{CP}^2$	$\mathrm{SU}(3)$	3	$12/\sqrt{7}$	0.145 MeV	$u,\;c,\;t$	Colour filter: $\chi(\mathbb{CP}^2) = N_c = 3$ colour handles from index theorem (§2, Part 8); colour-nonsinglet asymptotic states geometrically impossible
5	$S^5$	$\mathrm{SO}(6)$	0	$96/g_{22}\approx 0.133$	$7.4\times10^{-13}$ MeV	$\nu_1,\;\nu_2,\;\nu_3$	Majorana/LNV filter: no $C$ on $S^5$ spinor bundle ( $d \bmod 8 = 5$ ); Hopf $S^1\!\to\!S^5\!\to\!\mathbb{CP}^2$ projects colour to singlet
6	$\mathbb{CP}^3$	$\mathrm{SU}(4)$	4	$1/4$	$2.75\times10^{-5}$ MeV	$e^-,\;\mu^-$	Colour-silence filter: $\chi(\mathbb{CP}^3)=4\neq N_c$ ; colour contributions cancel in $\mathrm{SU}(4)$ representation; zero strong coupling at all energies
10	$\mathbb{CP}^5$	$\mathrm{SU}(6)$	6	$1/4$	$2.75\times10^{-5}$ MeV	$\tau^-$	Gegenbauer-critical filter: $b_{k_0}(10)=1/2$ exactly (Part 9 T5); coupling weight distributed with no dominant decay channel; back-reaction correction $+1/1680$ required

Column notes. $\chi$ is the Euler characteristic of the compact local $\mathbb{CP}^k$ model; for odd-sphere sectors ( $d=3,5$ ) the sphere has $\chi(S^d)=0$ and structural counting roles are carried by the CP sector in the Hopf pair. Specifically: $\chi(\mathbb{CP}^2)=N_c=3$ , $\chi(\mathbb{CP}^3)=n_s=4$ , $\chi(\mathbb{CP}^5)=N_f=6$ (Part 9 T15). The $g_{dd}$ values are exact from the seed $n_u=3$ (primitive); derivations in §0.7 and Part 2 §9. The $m_{\rm scale}$ values use $m_e=0.511$ MeV as the unit reference (Part 2 §10).

Coordinate containment vs coupling filter. These are two distinct necessary conditions. Coordinate containment (§0.8) governs which forces can reach a particle: a force couples to a particle only if the particle's sector contains that force's sector. The coupling filter governs what happens within that sector: even with support in the sector, the particle's own sector geometry may project the relevant representation to zero. Both must hold for coupling to occur. Neutrinos ( $d=5$ ): coordinates are nested inside $\Xi_4$ (colour sector), but the $S^5$ Hopf projection selects only the colour singlet — they are colour-neutral despite spatial overlap with the colour sector. Electrons ( $d=6$ ): $\chi(\mathbb{CP}^3)=4\neq 3$ ; colour contributions cancel in the $\mathrm{SU}(4)$ representation regardless of spatial overlap with $\Xi_4$ .

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3a. Explicit Colour Connection Construction 🔶

The argument above establishes that a connection must exist. This subsection constructs it explicitly from the IDWT sector field.

The colour triplet state. $\chi(\mathbb{CP}^2) = 3$ (see Part 8 §2 for explicit construction) gives three left-chiral modes $\phi_a\in L^2(\mathbb{CP}^2)$ , $a = 1, 2, 3$ . At spacetime point $x$ , the $d=4$ sector component of $\Psi_\infty$ decomposes as:

$\psi_{\rm color}(x) = \bigl(\psi^1(x), \psi^2(x), \psi^3(x)\bigr) \in \mathbb{C}^3$

where

$\psi^a(x) = \int_{\mathbb{CP}^2} \phi_a^*(\xi)\,\Psi_\infty^{(d=4)}(x,\xi)\,d\mu_{\mathbb{CP}^2}$

This integral extracts the three colour amplitudes from the full sector-space field at each spacetime point $x$ . It is the IDWT definition of the colour state at $x$ .

Colour connection formula. The colour frame can be chosen independently at each spacetime point. The unique torsion-free $\mathrm{SU}(3)$ connection preserving the $L^2(\mathbb{CP}^2)$ inner product on $H_{\rm colour}$ is:

$A_\mu(x) = i\,\psi_{\rm color}^\dagger(x)\,\partial_\mu\psi_{\rm color}(x) \;\in\; \mathfrak{su}(3)$

This is the colour connection — the $\mathrm{SU}(3)$ -valued coupling encoding local colour frame freedom in the $d=4$ sector.

Coupling transformation check. Under $U(x)\in\mathrm{SU}(3)$ : $\psi_{\rm color}(x)\to U(x)\psi_{\rm color}(x)$ . Direct computation: ✅

$A_\mu \;\to\; i(U\psi)^\dagger\partial_\mu(U\psi) = U^\dagger A_\mu U + U^\dagger\partial_\mu U$

This is the correct $\mathrm{SU}(3)$ transformation law. The curvature

$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + i[A_\mu, A_\nu] \;\in\; \mathfrak{su}(3)$

is the colour curvature 2-form. This is a mathematical object encoding the $\mathrm{SU}(3)$ holonomy of the colour frame; it is not a propagating field. The IDWT action (§0.2) contains no kinetic term $F_{\mu\nu}F^{\mu\nu}$ for this connection — the colour interaction is entirely in the kernel contact term.

Status. The colour connection is defined and transforms correctly for the zero-mode sector. What remains is constructing $\psi_{\rm color}(x)$ explicitly for propagating quark modes — that is, extending the colour projection $P_{\rm color}$ : $\Psi_\infty^{(d=4)} \to \psi_{\rm color}$ beyond the three $\mathbb{CP}^2$ zero modes to the full occupied spectrum ( $n=3$ up, $n=20$ charm, $n=72$ top). The zero-mode construction is complete (Part 8 §2); the propagating-mode projection operator is the remaining step.