How to read it
A particle in IDWT is fixed by two integers: the sector dimension \(d\) — which of the six geometric depths of \(\Xi_{10}\) it occupies — and the mode index \(n\) — its excitation level within that sector. Its mass is then
The simplex number \(S(n,d)\) counts the microstates of the mode; the sector scale \(m_{\text{scale},d}\) is fixed by the coupling constants. Both trace back to the seed pair \(\{n_d=1,\, n_u=3\}\), composite \(n_s = 4\), and the single mass unit \(m_e\). The mode indices themselves are not free — they follow from the seeds and composite by the generation tower, and the sector geometries follow from the Hopf chain. The table below is therefore not a list of measured inputs; it is the output.
The fifteen
| Particle | \(d\) | Geometry | Isometry | \(n\) | \(S(n,d)\) | Mass |
|---|---|---|---|---|---|---|
| photon γ | 2 | \(\mathbb{CP}^1\) | SU(2) | 0 | 0 | 0 (exact) |
| W | 2 | \(\mathbb{CP}^1\) | SU(2) | 76 | 2,926 | 80.38 GeV |
| Z | 2 | \(\mathbb{CP}^1\) | SU(2) | 81 | 3,321 | 91.23 GeV |
| Higgs | 2 | \(\mathbb{CP}^1\) | SU(2) | 95 | 4,560 | 125.27 GeV |
| down | 3 | \(S^3\) | SO(4) | 1 | 1 | 4.70 MeV |
| strange | 3 | \(S^3\) | SO(4) | 4 | 20 | 94.0 MeV |
| up | 4 | \(\mathbb{CP}^2\) | SU(3) | 3 | 15 | 2.18 MeV |
| charm | 4 | \(\mathbb{CP}^2\) | SU(3) | 20 | 8,855 | 1.27 GeV |
| top | 4 | \(\mathbb{CP}^2\) | SU(3) | 72 | 1,215,450 | 172.6 GeV |
| ν₁ | 5 | \(S^5\) | SO(6) | 10 | 2,002 | 1.49 meV |
| ν₂ | 5 | \(S^5\) | SO(6) | 15 | 11,628 | 8.64 meV |
| ν₃ | 5 | \(S^5\) | SO(6) | 22 | 65,780 | 50.3 meV |
| electron | 6 | \(\mathbb{CP}^3\) | SU(4) | 13 | 18,564 | 0.511 MeV |
| muon | 6 | \(\mathbb{CP}^3\) | SU(4) | 35 | 3,838,380 | 105.7 MeV |
| tau | 10 | \(\mathbb{CP}^5\) | SU(6) | 23 | 64,512,240 | 1776.84 MeV |
Read down the \(d\) column and the six sectors group the spectrum the way nature does: the bosons and Higgs in \(d=2\); the two down-type quarks in \(d=3\); the three up-type quarks in \(d=4\); the three neutrinos in \(d=5\); the electron and muon in \(d=6\) — both genuine 6D objects inhabiting six macroscopic spatial dimensions, whose 3D appearances are what a \(d=3\) observer measures of their six-dimensional sector activity; the tau alone in \(d=10\). Read across a sector and the mode index \(n\) climbs through the generations.
Mode-index map
Every particle at its coordinates in the \((n, d)\) plane. Circle area scales with the base-10 logarithm of the mass, spanning twelve orders of magnitude from the sub-meV neutrinos to the 172.6 GeV top quark. The X axis uses a square-root scale so the low-index particles spread out legibly. Hover a circle for details.
The bottom quark — a beat, not a mode
One familiar particle is missing from the table, and deliberately so. The bottom quark is not a single \((n,d)\) mode: it sits at the resonance site \(k_0 = n_s^2 = 16\) in \(d=3\), where three independent conditions coincide and force a geometric-mean beat between the adjacent modes \(n=16\) and \(n=17\):
Because it is a beat rather than a co-fixed-point pair, the bottom quark is not one of the fifteen \(\Sigma_{\text{pairs}}\); it is a stable \(d=3\) resonance riding between two of them. See Generation Tower Mode Selection for what counts as a selected mode and what does not.
What the map is saying
The Standard Model lists roughly two dozen particles with their masses and quantum numbers as independent measured facts. IDWT replaces that list with this map: two integers per particle, drawn from the seed pair \(\{n_d=1,\, n_u=3\}\) and composite \(n_s=4\), on six geometries that are themselves forced. Nothing in the table is tuned. Every entry is a consequence of the seed pair, \(n_s = 4\), and \(m_e\) — the masses to sub-percent accuracy, the sectors from the Hopf chain, the mode indices from the generation tower.
See also: The Six Sectors · The Simplex Number · The Generation Tower · Why \(n_s = 4\)