The Coupling Grid

It is natural to picture an interaction as two objects reaching across a gap and trading something between them. That picture is not how IDWT works, and holding onto it makes the grid below look mysterious instead of obvious.

There is one wave, \(\Psi_\infty\). What we call particles are excitations of it in different sectors — different numbers of the manifold's directions (see One Wave). When two of them "interact," nothing crosses a gap and nothing is exchanged. The single wave couples to itself, through its own kernel \(g_{d,d'}(\xi_d\cdot\xi_{d'})^2\), on the coordinates the two sectors share. An interaction is one object folding back on itself where two of its depths overlap.

So the only question for any pair of particles is: which coordinates do they share, and what lives on those coordinates? That is the whole grid.

IDWT has four coupling structures, and each one resides in particular directions of the manifold. A structure can act on a pair only if it lives on the coordinates that pair shares — and only if neither particle's own geometry switches it off (the coupling filter).

• Electric charge — \(U(1)\). The charge \(Q = T_3 + Y\) is read from the \(T_8\) direction of the colour symmetry \(\mathrm{SU}(3)=\mathrm{Isom}(\mathbb{CP}^2)\). A particle takes part only if its charge is nonzero — which leaves out the neutrinos.
• Weak isospin — \(\mathrm{SU}(2)_L\). The \(\mathrm{SU}(2)\) part of the \(U(2)\) holonomy of \(\mathbb{CP}^2\). The Kähler geometry makes it act only on the left-handed (holomorphic) half of each fermion. Every matter particle has such a half, so every fermion takes part.
• Colour — \(\mathrm{SU}(3)\). The isometry of \(\mathbb{CP}^2\), with \(N_c = \chi(\mathbb{CP}^2) = 3\) colours. Only quarks carry it: the neutrino's \(S^5\) geometry averages colour down to a single neutral state, and the charged leptons' \(\mathbb{CP}^3\) and \(\mathbb{CP}^5\) geometries cancel it outright.
• Gravity. Curvature of the whole manifold \(M_\infty\). It has no sector boundary and no filter, so it acts on every particle and every pair (see Gravity Is Not a Force).

Because these structures attach to sectors, every particle in a given sector carries exactly the same set. The five matter sectors:

The \(d=2\) sector — the home of γ, W, Z and H — is not a row here. Those are excitations of the one wave like any other particle, not a matter family with a single charge. The \(U(1)\) and \(\mathrm{SU}(2)_L\) structures are the geometry of \(\mathbb{CP}^2\) itself, not something the \(d=2\) modes carry between particles. The grid below is the matter particles.

Twelve matter particles on each axis. Every cell is just the overlap of the two rows above: a structure appears only when both particles carry it. Nothing else is computed — the grid is an intersection.

E W C G — electromagnetic, weak, colour, gravity  ·  E W G — electromagnetic, weak, gravity  ·  W G — weak, gravity

Gravity is in every cell. It is the curvature of the whole manifold, sourced by whatever mass is present, with no direction it cannot reach. There is no pair of particles it leaves out.

The neutrino rows and columns lose electromagnetism. A neutrino is neutral and colourless — its \(S^5\) geometry keeps only the weak structure (and gravity). So every pairing that includes a neutrino collapses to W·G, the sparse stripe running through the grid.

Colour survives only in the quark block. The top-left 6×6 corner — every down-type with every up-type, and each among themselves — is the only place all four structures act at once, because colour needs both partners to be quarks. The moment one partner is a lepton or a neutrino, colour drops.

The grid is not 144 independent facts. It is five rows of information — one per sector — tiled out across the twelve particles. The down quark and the bottom quark have identical rows because they are the same sector excited at different depths; what couples to them is fixed by the sector, not by the individual particle. Mass is the one thing that separates a down from a bottom, and mass is not on this grid.

That tiling is itself the result. In IDWT a particle's interactions are written into the geometry of the directions it occupies, so two particles sharing a sector cannot differ in what couples to them. The blocks are the visible shape of that fact: coupling is a property of sectors, and the particles are just where the sectors are occupied.

The kernel self-coupling \(g_{dd'}\) between any two sectors is not independent — it equals the product of the two sector coupling amplitudes \(v_d\,v_{d'}\), giving a rank-1 matrix. The heatmap below shows all 21 distinct entries of \(G_{dd'}\) on a logarithmic colour scale. The \(d=2\) row and column dominate because \(v_2 = \sqrt{g_{22}} \approx 26.9\) — the photon/W/Z/Higgs sector couples most strongly to everything.