The bilateral crossing divides the frontier into an egress face (\(s_0 = 2/3\)), an ingress face (\(1-s_0 = 1/3\)), and a crossing point \(\tau_0\) — the Present. The egress face spans angles \([0,\pi]\). By the no-preferred-intersection axiom, three fermionic states fill it at equal thirds:
\[\theta_0 = \frac{\pi}{6}, \qquad \theta_1 = \frac{\pi}{2}, \qquad \theta_2 = \frac{5\pi}{6}.\]
These correspond to Bohr–Sommerfeld eigenvalues \(y_n = n + \tfrac{3}{2}\) for \(n = 0,1,2\). The Koide formula applied to these eigenvalues gives:
\[K_\mathrm{egress} = \frac{y_0+y_1+y_2}{(\sqrt{y_0}+\sqrt{y_1}+\sqrt{y_2})^2} = \frac{2}{3},\]
in agreement with the charged lepton masses to \(6\,\mathrm{ppm}\). This is the empirical anchor of the framework. The question is what happens when the same bilateral geometry acts on the ingress face.
The natural map from egress to ingress respecting bilateral symmetry is the composition of two elementary operations:
Step 1 — Reflect about the egress midpoint \(\pi/2\): \(\;\theta \mapsto \pi - \theta.\)
Step 2 — Rotate by \(\pi\) into the ingress face \([\pi,2\pi]\): \(\;\theta \mapsto \theta + \pi.\)
No other composition of these operations is geometrically natural: reflection preserves the internal symmetry of the face, rotation changes face. Applied to the three egress levels:
The middle egress level maps to the ingress face midpoint \(3\pi/2\) — exactly \(\tau_0\). The outer two map to \(3\pi/2 \pm \pi/3\), symmetric about \(\tau_0\).
The mass of a state is proportional to the squared angular distance from \(\tau_0\). This is the unique choice consistent with \(\tau_0\) carrying no mass (it belongs to neither face) and mass growing with displacement from the crossing.
Under \(\mathcal{B}\) and the mass–angle correspondence, the neutrino mass spectrum is \(\{m_3, m_1, m_2\} = \{0,\,\delta,\,\delta\}\) with \(\delta = (\pi/3)^2\) [natural units], and \(m_3 < m_1 \approx m_2\): inverted ordering.
The neutrino Koide ratio is exactly \(1/2\).
\(K_\nu = 1/2\) is the arithmetic mean of the egress and ingress fractions:
\[K_\nu = \frac{1}{2} = \frac{s_0 + (1-s_0)}{2} = \frac{2/3 + 1/3}{2}.\]
Neutrinos are not face particles. They are crossing particles defined by \(\tau_0\), straddling both faces. \(K = 1/2\) is the only Koide value consistent with this position. The complete bilateral duality:
Setting \(m_3 = 0\) and using PDG 2024 inverted-ordering values (\(\Delta m^2_{21} = 7.53\times10^{-5}\,\mathrm{eV}^2\), \(\Delta m^2_{31} = 2.453\times10^{-3}\,\mathrm{eV}^2\)):
\[m_1 = \sqrt{\Delta m^2_{31}} = 49.5\,\mathrm{meV}, \qquad m_2 = \sqrt{\Delta m^2_{31}+\Delta m^2_{21}} = 50.3\,\mathrm{meV}.\]
| Quantity | Predicted | Observed / Bound |
|---|---|---|
| \(K_\nu\) | 1/2 (exact) | 0.500007 (IO) ✓ |
| Mass ordering | Inverted | Slight NO preference (not decisive) |
| \(m_3\) | 0 (exact) | <0.45 eV (KATRIN) ✓ |
| \(m_1/m_2\) | 1 (leading order) | 0.985 (1.5% correction) ✓ |
| \(\Sigma m_i\) | \(\approx 99\,\mathrm{meV}\) | <120 meV (Planck) ✓ |
| \(K_\nu\) deviation from 1/2 | \(\sim\)1.5% at next order | \(7\times10^{-6}\) ✓ |
The observed \(K_\nu\) deviates from \(1/2\) by \(7\,\mathrm{ppm}\), comparable to the charged lepton deviation from \(2/3\) (\(6\,\mathrm{ppm}\)). Both are second-order frontier corrections: QED in the lepton sector, the solar splitting in the neutrino sector.
1. Inverted mass ordering. The lightest neutrino is \(m_3\). Current data show a slight preference for normal ordering but are not decisive. JUNO and Hyper-Kamiokande will determine the ordering at \(>3\sigma\). Confirmed normal ordering falsifies this derivation.
2. One massless neutrino: \(m_3 = 0\). For IO with \(m_3 = 0\), the effective Majorana mass is \(m_{\beta\beta} \approx 15\text{–}50\,\mathrm{meV}\), testable by nEXO and LEGEND.
3. Sum of masses \(\Sigma m_i \approx 99\,\mathrm{meV}\). Inside current Planck bounds (\(<120\,\mathrm{meV}\)). CMB-S4 and DESI will constrain to \(\sim20\,\mathrm{meV}\) precision. Established \(\Sigma m_i < 90\,\mathrm{meV}\) falsifies this prediction.
4. Solar splitting is second-order. The \(m_1 \neq m_2\) splitting (\(\Delta m^2_{21}/\Delta m^2_{31} \approx 3\%\)) is a frontier correction to the leading-order degeneracy — the neutrino analog of the \(6\,\mathrm{ppm}\) QED correction in the charged lepton sector. Its derivation from the frontier correction formalism is an open problem.
The Koide formula gives \(K = 2/3\) for the egress face and \(K = 1/2\) for the crossing. The degenerate limit of any three equal masses gives \(K = 1/3\). These three values — \(1/3\), \(1/2\), \(2/3\) — are the ingress fraction, the crossing midpoint, and the egress fraction respectively. They are the complete set of bilaterally natural Koide values. The framework predicts no other fermion sector will have a Koide value outside this set.
Both the Weinberg angle and the neutrino Koide value follow from the same bilateral crossing geometry. The Weinberg angle is the fixed point of the electroweak crossing; \(K_\nu = 1/2\) is the fixed point of the ingress face under \(\mathcal{B}\). Both are self-consistency conditions on the same structure. Both are exact.
On the status of this paper. The derivation of \(K_\nu = 1/2\) from the bilateral crossing operation \(\mathcal{B}\) is complete and verified numerically: \(K_\nu = 0.500007\) from current IO oscillation data, deviation \(7\,\mathrm{ppm}\). The spectrum \(\{0, \delta, \delta\}\) follows by construction from \(\mathcal{B}\) applied to the egress eigenvalues. What is not yet established is the derivation of the solar/atmospheric splitting ratio \(\Delta m^2_{21}/\Delta m^2_{31} \approx 3\%\) from the frontier correction formalism — this is the neutrino analog of the QED correction to the charged lepton Koide value and requires the frontier perturbation theory to be developed. The ordering prediction (inverted) is the sharpest near-term test. Framework: A Philosophy of Time, Space and Gravity — Dunstan Low.