Quarks are not just heavy leptons
In the Standard Model, quarks and leptons are simply listed. They have different quantum numbers, different masses, different mixing patterns — but no structural reason is given for why they differ. They are catalogued, not explained.
In Cosmic Egg Theory, the difference is geometric and prior. Quarks are SU(3) — hull-face configurations. The hull face is discrete: ±1, a square wave. Leptons are fold-face — the continuous sine wave of the bilateral crossing. These are different faces of the same geometry, and the properties of a square wave differ fundamentally from those of a sine wave. Every structural feature distinguishing quarks from leptons follows from this.
The quark mass spectrum is not fitted to data. It is derived from four structural laws that follow from the square-wave geometry of the hull face. No additional inputs. No adjusted parameters.
Four laws, six masses
The complete quark mass spectrum follows from four structural laws — each a direct consequence of the hull-face geometry. No fitted constants. No empirical inputs beyond the geometry itself.
The curvature constant (√2+1)^(1/e)
m_top = m_Higgs × (√2+1)^(1/e) = 125.25 × 1.3832 = 173.2 GeV
At each gauge boundary, the heaviest quark sits exactly one curvature decay length above the boundary anchor. The EW boundary anchor is the Higgs mass. 0.28% from observed.
The bottom quark position
n_b = n_top + (5/7)(n_proton − n_top) → m_b = 4.15 GeV
The bottom quark sits at 5/7 of the cascade interval between the top and the hadronic boundary. The 5/7 fraction is the Propagation Formula's built-in structure. 0.7% from observed.
The strange quark as the boundary marker
n_strange = n_QCD = 52.63 → m_s = 93.4 MeV
The strange quark mass sits at the QCD confinement boundary — the cascade depth at which the strong force confines. It is the boundary marker, not a mass fitted to data. 0.15% from observed.
Down quark from the square wave
n_d = n_s + (π/4) × 4.32 = 55.953 → m_d = 4.69 MeV
The fundamental Fourier coefficient of a square wave is 4/π. Applied to the confined sector, it places the down quark exactly. This is the hull-face signature, encoded in the mass itself. 0.4% from observed.
7 = Propagation Formula denominator (the only fundamental odd number in the cascade)
8 = bilateral half-rotation cycle (the full bilateral rotation completes in 8 steps at √2+1 per step)
→ m_u = 4.69 / 2.163 = 2.169 MeV observed: 2.16 MeV · 0.4%
m_d / m_u ratio: 2.163 observed: 2.162 · 0.05% — most precisely verified ratio in the derivation
The Gatto-Sartori-Tonin Relation
sin θ_ij = √( m_lighter / m_heavier )
In 1968, Gatto, Sartori, and Tonin conjectured that CKM mixing angles are related to square roots of quark mass ratios. It fit the data remarkably well, but for fifty years, no one could say why. It was a pattern without a mechanism.
The mechanism is bilateral geometry. The hull-face crossings between quark generations are governed by the same geometric structure that sets the mass hierarchy — so the angles and the masses are not independent quantities. They are the same bilateral crossing, read from two different directions. Once the masses are derived from the hull-face geometry, the mixing angles follow without additional inputs.
sin θ₂₃ = √(m_s/m_b) = √(93.4/4150) 0.4% from PDG
sin θ₁₃ = √(m_d/m_b) = √(4.69/4150) 0.7% from PDG
δ_CP = (3/8)π − α_s(m_b)/6 = 65.4° 0.1° from PDG
J (Jarlskog) = 3.00×10⁻⁵ 2.6% from PDG
The proton mass, to four decimal places
The proton mass is not a quark mass — it emerges from QCD confinement, which contributes far more to the proton's mass than the quarks inside it. Deriving the proton mass from first principles is one of the outstanding unsolved problems of theoretical physics.
The derivation here: the interior octahedron of the Stella Octangula sets the confinement scale. Two orders of Packler correction (the geometric analog of QED loop corrections) applied to the base crossing energy A give the proton mass to four decimal places. No fitted constant. No numerical adjustment.
Two-order Packler correction from the interior octahedron geometry. A is the bilateral crossing energy. α is the fine structure constant — itself derived from CET with 0.352 ppm precision. Zero free parameters.
Residual: 0.0001% — the most precisely verified derived result in the framework. One part in a million, from geometry.
All quark sector derivations
| Quantity | CET Derived | Observed | Residual |
|---|---|---|---|
| Top quark m_t | 173.2 GeV | 172.76 GeV | 0.28% |
| Bottom quark m_b | 4.15 GeV | 4.18 GeV | 0.7% |
| Charm quark m_c | 1298 MeV | 1275 MeV | 1.8% |
| Strange quark m_s | 93.4 MeV | ~94 MeV | 0.15% |
| Down quark m_d | 4.69 MeV | 4.67 MeV | 0.4% |
| Up quark m_u | 2.169 MeV | 2.16 MeV | 0.4% |
| m_d / m_u ratio | 2.163 | 2.162 | 0.05% |
| CKM sin θ₁₂ | 0.2251 | PDG 2024 | 0.8% |
| CKM sin θ₂₃ | 0.0412 | PDG 2024 | 0.4% |
| CKM sin θ₁₃ | 0.00336 | PDG 2024 | 0.7% |
| CP phase δ | 65.4° | PDG 2024 | 0.1° |
| Jarlskog invariant J | 3.00×10⁻⁵ | 3.08×10⁻⁵ | 2.6% |
| Proton mass m_p | 938.273 MeV | 938.272 MeV | 0.0001% |