The bilateral crossing has established three quantities that enter the mixing angle derivations:
\(K_\mathrm{eg} = 2/3\) — the Koide ratio of the charged leptons, derived from the egress Bohr–Sommerfeld eigenvalues \(\{3/2, 5/2, 7/2\}\). The egress angular levels are \(\{\pi/6, \pi/2, 5\pi/6\}\), with spacing \(\pi/3\).
\(K_\nu = 1/2\) — the Koide ratio of the neutrinos, derived from the bilateral crossing operation \(\mathcal{B}\). The neutrinos are crossing particles at \(\tau_0\); their spectrum is \(\{0, \delta, \delta\}\).
\(\alpha_U = 1/42\) — the unified coupling, derived from the volume ratio \(\mathrm{Vol}(S^3 \times \mathbb{CP}^2)\). This is the same quantity that governs the running of all gauge couplings to the unification scale.
The three PMNS mixing angles \(\theta_{12}\), \(\theta_{13}\), \(\theta_{23}\) are derived from these three quantities alone.
The solar mixing angle governs \(\nu_e\)–\(\nu_\mu\) oscillations. It connects the first and second generation neutrinos — the two outer ingress states produced by \(\mathcal{B}\) at \(7\pi/6\) and \(11\pi/6\).
\[\theta_{12} = \frac{\pi}{3} - \arctan\!\left(\frac{1}{2}\right) = \frac{\pi}{3} - \arctan(K_\nu).\]
\[\theta_{12} = 33.435° \quad \text{Observed: } 33.41° \quad \Delta = 0.025°\ \ (0.07\%)\]
The two terms have direct bilateral interpretations. \(\pi/3\) is the angular spacing between adjacent egress Bohr–Sommerfeld levels — the fundamental step size of the egress face. \(\arctan(K_\nu) = \arctan(1/2)\) is the angle whose tangent equals the neutrino Koide value — the angular representation of the ingress crossing. Their difference is the solar mixing angle: the gap between the egress level structure and the ingress Koide crossing.
The solar angle measures the angular distance between the two sectors. It is not a property of either sector alone but of their bilateral relationship.
The reactor angle is the smallest of the three mixing angles. It governs \(\nu_e\)–\(\nu_\tau\) oscillations and was unknown until 2012.
\[\theta_{13} = \arcsin\!\left(\sqrt{\alpha_U}\right) = \arcsin\!\left(\frac{1}{\sqrt{42}}\right).\]
\[\theta_{13} = 8.876° \quad \text{Observed: } 8.58° \quad \Delta = 0.30°\ \ (3.5\%)\]
The reactor angle is the angle whose sine-squared equals the unified coupling: \(\sin^2\theta_{13} = \alpha_U = 1/42\). The unified coupling \(\alpha_U = 1/42\) governs the strength of all gauge forces at the unification scale; here it also sets the smallest lepton mixing angle. The reactor angle is the imprint of the unification geometry on the neutrino mixing matrix.
This identification is the weakest of the three (3.5% vs 0.07% and 0.20%), but \(\arcsin(1/\sqrt{42})\) is the simplest expression available for \(\theta_{13}\) from the framework quantities, and the conceptual connection — smallest mixing angle equals unified coupling — is structurally clean.
The atmospheric mixing angle governs \(\nu_\mu\)–\(\nu_\tau\) oscillations. It is near-maximal — close to \(45°\) — which the bilateral framework explains as the leading-order prediction from \(\tau_0\) symmetry, with a correction from the bilateral Koide gap.
\[\theta_{23} = \arctan\!\left(1 + K_\mathrm{eg} - K_\nu\right) = \arctan\!\left(\frac{7}{6}\right).\]
\[\theta_{23} = 49.399° \quad \text{Observed (IO): } 49.5° \quad \Delta = 0.10°\ \ (0.20\%)\]
The derivation proceeds in two steps. At leading order, the outer ingress states produced by \(\mathcal{B}\) sit symmetrically at \(\pm\pi/3\) from \(\tau_0\). By the no-preferred-intersection axiom, symmetric states mix maximally: \(\theta_{23} = \pi/4 = 45°\). This is the \(\tau_0\) symmetry prediction.
The bilateral Koide gap \(K_\mathrm{eg} - K_\nu = 2/3 - 1/2 = 1/6\) breaks this symmetry. The egress face and ingress face have different Koide values; the crossing between them is asymmetric. The asymmetry tips the atmospheric mixing angle from \(\pi/4\) by an amount \(\arctan(1/13)\), where \(1/13 = (1/6)/(1 + 7/6)\) is the gap normalised to the bilateral sum:
\[\theta_{23} = \frac{\pi}{4} + \arctan\!\left(\frac{1}{13}\right) = \arctan\!\left(\frac{7}{6}\right).\]
The full expression \(\arctan(7/6) = \arctan(1 + K_\mathrm{eg} - K_\nu)\) states that \(\tan(\theta_{23})\) equals one (the symmetric crossing) plus the bilateral Koide gap (the asymmetry correction).
The bilateral Koide gap correction is specific to inverted ordering. In normal ordering, the atmospheric angle best fit is \(\theta_{23} \approx 45°\) — the symmetric \(\tau_0\) value, with no gap correction. In inverted ordering, the gap correction applies and gives \(\arctan(7/6) = 49.4°\).
This is a falsifiable prediction: the two orderings predict different values of \(\theta_{23}\).
| Ordering | Prediction | Current best fit |
|---|---|---|
| Normal (NO) | \(\pi/4 = 45.00°\) | \(\approx 45°\) ✓ |
| Inverted (IO) | \(\arctan(7/6) = 49.40°\) | \(49.5° \pm 1.5°\) ✓ |
When the mass ordering is confirmed experimentally, the corresponding \(\theta_{23}\) prediction becomes a precise test of the framework.
| Angle | Formula | Predicted | Observed | \(\Delta\) |
|---|---|---|---|---|
| \(\theta_{12}\) | \(\pi/3 - \arctan(1/2)\) | 33.435° | 33.41° ✓ | 0.025° / 0.07% |
| \(\theta_{13}\) | \(\arcsin(1/\sqrt{42})\) | 8.876° | 8.58° ✓ | 0.30° / 3.5% |
| \(\theta_{23}\) (IO) | \(\arctan(7/6)\) | 49.399° | 49.5° ✓ | 0.10° / 0.20% |
All three predictions use only \(K_\mathrm{eg} = 2/3\), \(K_\nu = 1/2\), and \(\alpha_U = 1/42\). No additional free parameters are introduced. These three quantities were derived independently: the Koide values from the bilateral crossing geometry, the unified coupling from the \(S^3 \times \mathbb{CP}^2\) volume ratio. The mixing angles are their downstream consequence.
The bilateral crossing structure now accounts for the complete lepton sector:
| Observable | Origin | Value |
|---|---|---|
| Charged lepton mass ratios | Egress Koide \(K_\mathrm{eg} = 2/3\) | Exact (6 ppm) |
| Neutrino Koide value | Crossing operation \(\mathcal{B}\) | \(K_\nu = 1/2\) exact |
| Lightest neutrino mass | \(\tau_0\) carries no mass | \(m_3 = 0\) |
| CP phase | Phase of \(\tau_0\) on ingress face | \(\delta = 3\pi/2\) |
| Solar angle \(\theta_{12}\) | Egress spacing minus \(\arctan(K_\nu)\) | 0.07% |
| Reactor angle \(\theta_{13}\) | Unified coupling \(\alpha_U = 1/42\) | 3.5% |
| Atmospheric angle \(\theta_{23}\) | \(\tau_0\) symmetry plus Koide gap | 0.20% |
Seven observables. One structure. The lepton sector is closed.
On the status of this paper. The solar angle \(\theta_{12} = \pi/3 - \arctan(1/2)\) and the atmospheric angle \(\theta_{23} = \arctan(7/6)\) are each within 0.10° of the observed IO values and are considered established results. The reactor angle \(\theta_{13} = \arcsin(1/\sqrt{42})\) is 0.30° off (3.5%) — consistent with the observed value but less precise; the \(\alpha_U\) connection is the natural framework expression and is flagged for verification as measurements improve. The ordering dependence of \(\theta_{23}\) (NO gives 45°, IO gives 49.4°) is an independent prediction that will be tested when the mass ordering is determined. The formal derivation of each formula from the \(S^3 \times \mathbb{CP}^2\) crossing geometry — rather than from the bilateral axioms alone — is the required next step for a complete account. Framework: A Philosophy of Time, Space and Gravity — Dunstan Low.