The whole framework hangs on a single eigenvalue spectrum. Every step below is integer-rigid: nothing is fitted, and changing a cell integer changes the answer.
Space is an infinite pre-existing foam at the Planck scale. Every event is a displacement in that foam: a bubble plus a void equals a displacement. There is a lowest level, and it is geometric — particles are modes of the cell, not objects inside it.
Partition space into equal-volume cells at minimum interfacial area and the cell is not a choice — it is the truncated octahedron (Kelvin's solution). Among the five Fedorov parallelohedra it is singled out by its face-Laplacian spectrum, a uniqueness shown in the framework's spectral-uniqueness result.
Everything downstream is built from the cell's topology — its symmetry order, vertices, edges and faces. These are the table below; there are no other free quantities.
Treat the 14 faces as nodes. Two faces are adjacent when they share an edge: each square touches the four hexagons around it, each hexagon touches three squares and three hexagons. The face Laplacian is \(L = D - A\). Its spectrum is fixed, and it is the engine of the whole theory.
The two \(T_{1u}\) eigenvalues — the left- and right-handed fermion modes — are the roots of \(\lambda^2 - 9\lambda + 16 = 0\). Its discriminant is \(\Delta = 17\) (prime), and \(\sqrt{17}\) threads through nearly every Standard Model ratio that follows.
The fine-structure constant comes from \(O_h\) representation theory; the weak mixing angle, the Higgs-to-Z ratio, the lepton masses (via Koide), and the CKM/PMNS mixing sectors all follow from the same spectrum. See Predictions.
Every input to every formula on this site is one of these. They are properties of the truncated octahedron, not adjustable constants.
| Symbol | Value | Meaning |
|---|---|---|
| |O_h| | 48 | Order of the octahedral symmetry group |
| V | 24 | Vertices |
| E | 36 | Edges |
| F | 14 | Faces (8 hexagonal + 6 square) |
| d | 3 | Spatial dimensions |
| Δ | 17 | Discriminant of the master equation (prime) |
| C_A | 3 | Colour number (= F_hx/F − 1, natural normalisation) |
| r₁ | (9−√17)/2 ≈ 2.438 | Lower T₁u eigenvalue — left-handed fermions |
| r₂ | (9+√17)/2 ≈ 6.562 | Upper T₁u eigenvalue — right-handed fermions |
Fourteen faces give fourteen eigenvalues, grouped by \(O_h\) irrep. Bar height is multiplicity. This is exactly what the twelve-line script on the Verify page prints — hover any bar for its role.
The \(T_{1u}\) doublet sits at the roots of one quadratic. Sum of roots \(r_1 + r_2 = 9\); product \(r_1 r_2 = 16\); discriminant \(\Delta = 81 - 64 = 17\). The prime 17 is not chosen — it falls out of the face counts.
Because \(\sqrt{17}\) enters here, it reappears in \(\sin^2\theta_W = (17-3\sqrt{17})/20\), in \(m_H/M_Z = 18/(9+\sqrt{17})\), in the Cabibbo angle, and across the mixing sector. One quadratic, threaded through the whole Standard Model.
The spectrum isn't abstract: each eigenvalue is a standing wave on the cell's 14 faces, and the framework identifies each with a Standard Model sector. Pick a mode — outward motion is amber, inward is teal, nodes stay dark. Drag to rotate, scroll to zoom.