What a sector is
A sector is not a separate space. It is a depth — a particular number of spatial directions drawn from the same underlying sector manifold \(\Xi_{10}\). The \(d=2\) sector uses two of those directions. The \(d=6\) sector uses the same two directions, plus four more. The ten directions are shared; what changes from one sector to the next is how many of them a particle occupies.
The sectors are nested: every direction present at depth \(d\) is still present at every depth above \(d\). The \(d=2\) directions are literally inside \(d=3\), inside \(d=4\), all the way up to \(d=10\). This nesting — not any separate rule — is what determines which particles interact with which forces. A force associated with a given sector can only reach particles that live in that sector.
See One Space, Six Depths for a detailed treatment of the nesting structure, and Dimensions Accumulate for how each sector adds to the ones below it.
Why exactly six
The sector set \(D = \{2, 3, 4, 5, 6, 10\}\) is not a choice. It is uniquely determined by two mathematical requirements acting together. The sectors must form a complex Hopf fibration chain — \(S^1\) wrapping around odd spheres and complex projective spaces in sequence. And the coupling coefficient of the terminal sector must reach a precise critical value (the Gegenbauer criticality threshold b = 1/2). These two conditions together admit exactly six sectors and no others. \(d=7, 8, 9\) are excluded because no valid sector coupling forms there; \(d \geq 11\) are excluded because modes there fall below the Jacobi coupling threshold — no particles can form.
The sector set is a theorem, not a parameter.
The six sectors
| Sector | Geometry | Symmetry | Particles | Key property |
|---|---|---|---|---|
| \(d=2\) | \(\mathbb{CP}^1\) | SU(2) | γ, W±, \(Z^0\), H | Universal reference — nested in all sectors above; two polarization handles |
| \(d=3\) | \(S^3\) | SO(4) | d, s, b quarks | Left-handed weak isospin from \(SO(4) = SU(2)_L \times SU(2)_R\) |
| \(d=4\) | \(\mathbb{CP}^2\) | SU(3) | u, c, t quarks | Origin of colour: \(\chi(\mathbb{CP}^2) = 3 = N_c\) |
| \(d=5\) | \(S^5\) | SO(6) | ν₁, ν₂, ν₃ | Majorana spinors forbidden by Clifford algebra; Majorana mass term absent at all orders |
| \(d=6\) | \(\mathbb{CP}^3\) | SU(4) | e, μ | Total colour silence — all QCD interactions geometrically absent |
| \(d=10\) | \(\mathbb{CP}^5\) | SU(6) | τ | Gegenbauer critical point; fractal marginal coupling |
Sectors and forces
Every non-gravitational force in IDWT is associated with a specific sector. Electromagnetism acts through the \(d=2\) directions. The weak force acts through the same \(d=2\) directions (W and Z are \(d=2\) excitations at higher mode indices). The strong force acts through the \(d=3\) and \(d=4\) directions via the \(\mathbb{CP}^2\) kernel contact coupling.
Because the sectors are nested, the \(d=2\) directions are present in every higher sector. This is the geometric reason every charged particle couples electromagnetically — they all contain the \(d=2\) directions in their sector. A particle that lives in \(d=5\) contains \(d=2\), \(d=3\), and \(d=4\) inside it, which is why neutrinos couple to the weak force despite having no colour charge.
Gravity has no sector. It is the curvature of the full sector manifold, spreading through all ten directions at once. This is why gravity couples to everything — it doesn't need sector containment. A 3D observer is uniform across a source's hidden coordinates and reads its curvature integrated over them, which leaves the ordinary Newtonian law; \(G_N = G_\infty/(4\pi)\) is the structural relation, the \(4\pi\) being the observer's 3D Green's-function constant, the same for every sector. See Gravity Is Not a Force.
For a detailed treatment of how sector geometry determines coupling structure, see Quantum Numbers Are Coupling Filters.
The mass formula
Every particle's mass follows from its sector d and mode index n:
\(S(n, d)\) counts the number of ways to distribute \(n\) energy quanta across \(d\) sector directions — the simplex number. Particles in higher sectors have more directions available, so the same mode index n gives a larger count and a heavier mass. Particles with the same d but different n are different excitations of the same sector geometry — they are in the same family, distinguished by their energy level.
The mass scales \(m_{\text{scale},d}\) are not free parameters. They are determined by consistency conditions between the sector coupling constants, which are themselves derived from the seed pair \(\{n_d=1,\, n_u=3\}\) and composite \(n_s=4\). The sole dimensional input is \(m_e = 0.511\) MeV.