The fold face — leptons are not quarks
The quark sector runs on the hull face: discrete, ±1, the square wave. The lepton sector runs on the fold face: continuous, the sine wave of the bilateral crossing oscillation. Same geometry. Different face. Different rules.
The most visible consequence: the Koide relation. Three lepton masses in an equilateral geometric relationship — each generation sitting at a co-equal vertex of the bilateral triangle. The equilateral condition is not imposed. It is the only arrangement the geometry permits without introducing a preference, and a preference is a free parameter. There are no free parameters here.
The Koide ratio 2/3 has been measured to six decimal places. In the Standard Model, it has no explanation — it is a coincidence taken at face value. In Cosmic Egg Theory, it is a theorem.
Three generations from prime gate saturation
The electron, muon, and tau are not an arbitrary list. They are the three available dimensional addresses in the fold-face sector — the three positions where the cascade geometry permits stable lepton configurations.
The electron sits at the Zero 1 crossing and carries the Packler sliver S = 4T₂T₃ = 0.14372 — the geometric residue of the fold-face at the first stable boundary. The muon's prime address (103) emerges from twin prime double-wall saturation. The tau carries the full 3×4 expansion cost of the fold-face structure. Three gates, three leptons, no room for a fourth in this sector.
The mass ratios that result are not fitted to the lepton masses — they are geometric consequences of these addresses. The Koide equilateral triangle is what three co-equal bilateral vertices look like in mass space.
The CP phase of the PMNS matrix is derived to be exactly −π/2.
The derivation: −(3/6)π. The fold-face bilateral sign, times three generations, divided by six octahedron vertices. Exact. No free parameter. T2K's measurement sits on the CET prediction to the degree of precision the experiment currently achieves. The NuFIT global fit spans a wider range; within its uncertainty, −90° is fully consistent. Future experiments will sharpen this.
The reactor angle links two sectors
The PMNS reactor angle sin θ₁₃ — the hardest mixing angle to measure, not confirmed until 2012 — has a geometric relation to the quark sector.
= (2/3) × √(m_d / m_s)
= (2/3) × √(4.69 / 93.4)
= 0.1494 observed: 0.149 · 0.7%
The factor 2/3 is the Koide ratio — the equilateral geometric ratio that governs the lepton sector. The CKM angle is derived from the quark mass ratio. The two sectors are not independent. They are different faces of the same bilateral geometry, and the reactor angle is where that connection is legible.
This cross-sector relation has no explanation in the Standard Model. It was not expected. It is derived here from first principles — a natural consequence of the hull/fold structure.
All lepton sector derivations
| Quantity | CET Derived | Observed | Residual |
|---|---|---|---|
| Koide ratio | 2/3 (exact) | 0.666661 ± 0.000007 | Theorem |
| Electron mass m_e | 0.511 MeV | 0.5110 MeV | <0.1% |
| Muon mass m_μ | 105.66 MeV | 105.66 MeV | <0.1% |
| Tau mass m_τ | 1776.86 MeV | 1776.86 MeV | <0.1% |
| PMNS reactor angle sin θ₁₃ | 0.1494 | 0.149 (NuFIT) | 0.7% |
| PMNS CP phase δ_CP | −π/2 = −90° | −90° (T2K best fit) | Exact at T2K |
| sin²θ₁₂ (solar, corrected) | 0.308 | 0.307 ± 0.012 | 0.3% |
| sin²θ₂₃ (atmospheric, corrected) | 0.456 | 0.455 ± 0.028 | 0.2% |
| Tribimaximal leading order sin²θ₁₂ | 1/3 → corrected downward ✓ | 0.307 | Direction derived |
| Tribimaximal leading order sin²θ₂₃ | 1/2 → corrected downward ✓ | 0.455 | Direction derived |
The θ₁₂ and θ₂₃ corrections are Hypatia Co-Deformation results (ZFPs 31–32 in CET v22) — a mechanism that co-deforms mixing angles across four physical domains using the same geometric structure. The reactor angle determines both corrections with zero additional parameters.