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 number of ways to distribute up to \(n-1\) quanta across \(d\) sector directions; the sector scale \(m_{\text{scale},d}\) is fixed by the coupling constants. Both trace back to the seed pair \(\{n_d=1,\, n_u=3\}\), \(n_s = 4\), and the single mass unit \(m_e\). The mode indices themselves are not free — they follow from the seeds 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 — \(\Sigma_{\rm pairs}\)
\(\Sigma_{\rm pairs}\) contains exactly 15 tower-derived stable objects: 14 single-mode pairs \((n,d)\) and the bottom beat at \(k_0=n_s^2=16\). The photon (\(n=0\)) is the massless \(d=2\) sector ground state — physically real, but \(n=0\) is not a generation-tower output and is therefore not in \(\Sigma_{\rm pairs}\); it is shown below the table.
| Particle | \(d\) | Geometry | Isometry | \(n\) | \(S(n,d)\) | Mass |
|---|---|---|---|---|---|---|
| W | 2 | \(\mathbb{CP}^1\) | \(\mathrm{SU}(2)\) | 76 | 2,926 | 80.38 GeV |
| Z | 2 | \(\mathbb{CP}^1\) | \(\mathrm{SU}(2)\) | 81 | 3,321 | 91.23 GeV |
| Higgs | 2 | \(\mathbb{CP}^1\) | \(\mathrm{SU}(2)\) | 95 | 4,560 | 125.27 GeV |
| down | 3 | \(S^3\) | \(\mathrm{SO}(4)\) | 1 | 1 | 4.70 MeV |
| strange | 3 | \(S^3\) | \(\mathrm{SO}(4)\) | 4 | 20 | 94.0 MeV |
| bottom (beat) | 3 | \(S^3\) | \(\mathrm{SO}(4)\) | beat \(k_0=16\) | \(\sqrt{S(16,3)S(17,3)}\) | 4181 MeV |
| up | 4 | \(\mathbb{CP}^2\) | \(\mathrm{SU}(3)\) | 3 | 15 | 2.18 MeV |
| charm | 4 | \(\mathbb{CP}^2\) | \(\mathrm{SU}(3)\) | 20 | 8,855 | 1.28 GeV |
| top | 4 | \(\mathbb{CP}^2\) | \(\mathrm{SU}(3)\) | 72 | 1,215,450 | 172.5 GeV |
| \(\nu_1\) | 5 | \(S^5\) | \(\mathrm{SO}(6)\) | 10 | 2,002 | 1.49 meV |
| \(\nu_2\) | 5 | \(S^5\) | \(\mathrm{SO}(6)\) | 15 | 11,628 | 8.64 meV |
| \(\nu_3\) | 5 | \(S^5\) | \(\mathrm{SO}(6)\) | 22 | 65,780 | 50.3 meV |
| electron | 6 | \(\mathbb{CP}^3\) | \(\mathrm{SU}(4)\) | 13 | 18,564 | 0.511 MeV |
| muon | 6 | \(\mathbb{CP}^3\) | \(\mathrm{SU}(4)\) | 35 | 3,838,380 | 105.7 MeV |
| tau | 10 | \(\mathbb{CP}^5\) | \(\mathrm{SU}(6)\) | 23 | 64,512,240 | 1776.84 MeV |
Photon \(\gamma\) (\(d=2\), \(n=0\), \(S(0,2)=0\), \(m=0\) exact): the \(d=2\) sector ground state. Massless because it carries zero excitations. Physically real and stable, but \(n=0\) is not produced by any generation-tower operation — not a hockey-stick output, additive edge, \(g\)-rule output, or beat site. Not in \(\Sigma_{\rm pairs}\).
Read down the \(d\) column and the six sectors group the spectrum the way nature does: the bosons and Higgs in \(d=2\); the three down-type quarks (including the bottom beat) in \(d=3\); the three up-type quarks in \(d=4\); the three neutrinos in \(d=5\); the electron and muon in \(d=6\); 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 \(n=17\):
The bottom quark is a member of \(\Sigma_{\text{pairs}}\): its beat site \(k_0 = n_s^2 = 16\) is a named generation-tower output (the unique triple-coincidence site \(k_0 = n_s^2 = n_e + n_u = S(n_s,3) - S(2,3)\)), so it is tower-derived and in the set. It enters as a composite beat resonance — the geometric mean of the two adjacent \(d=3\) modes at \(n=16\) and \(n=17\) — rather than as a single \((n,d)\) pair. See Generation Tower Mode Selection for the tower operations.
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 \(n_s=4\), 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 \(n_s = 4\)