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.
The two pieces of quantum mechanics the framework used to import are now derived down to two named assumptions.
Every displacement makes a bubble and its void, and the cell fixes what the void is: it sits on the opposite face (the antipodal map \(V\)) and carries the opposite twist (conjugation \(K\)). The twin map \(\Theta = V \circ K\) is antiunitary, the same species of operation Wigner proved time reversal has to be. A pair state that mirrors itself in every measurement basis at once exists if and only if the twin map is antiunitary, and given \(\Theta\) that state is unique. It is maximally entangled, carries exactly zero total torsion charge, sits exactly on the Tsirelson bound (\(\mathrm{CHSH} = 2\sqrt{2}\)), and is one local rotation from the textbook singlet. The void coupling \(\eta\) sets how often pairs form, never how strongly they correlate, which is why entanglement is fragile but always full strength when it arrives.
Outcome statistics come from the same audit. On a discrete substrate the counting measure is an identity, not a postulate: a detector channel with amplitude \(c_k\) holds exactly \(N|c_k|^2\) micro-quanta. Two structural properties of the imprint event (it cannot read phase, and a local refinement of channels cannot change coarse probabilities) then force \(p_k = |c_k|^2\) as the only consistent rule. The second property turns out to be no-signalling in disguise, and the framework already derives no-signalling from the incompressible bulk (\(P = \rho c^2\)), so the Born exponent traces to the foam's equation of state. The phase property is itself a theorem: a channel's phase is a time offset of its carrier, and an equilibrated substrate triggers imprints with statistics that cannot depend on a time offset. Even the choice of qubit is forced: a lab that cannot resolve Planck-cell orientation can only route on the two paired sectors Schur's lemma allows, one of which is the gauge direction, leaving the chiral doublet alone. Still open, and said plainly: the continuum bridge, showing a real detector realises the doublet the symmetry singles out.