Structural Origin of Neutrino Mass Relations

This work presents a zero-parameter structural formulation of neutrino mass relations.Using the fixed ratios\[R=\frac{13}{6}, \qquad S=\frac{31}{24},\]together with the inherited electron-sector quantities \(\alpha\) and \(m_e\), the neutrino mass spectrum is represented by\[\frac{m_{\nu,n}}{m_e}=B_\nu(n+\Phi_n),\qquadB_\nu=12S\alpha^2_{[2\to0]}=12S\alpha^4,\qquadn=0,1,2.\]The branch factors are\[\Phi_0=1,\qquad\Phi_1=\frac{1}{3R^2},\qquad\Phi_2=\frac{1}{R},\]and \(\alpha^2_{[i\to j]}\equiv\alpha^{2(i-j)}\) is used as a compact structural-transfer notation. The resulting absolute masses are\[m_{\nu,0}\approx0.02246\,\mathrm{eV},\quadm_{\nu,1}\approx0.02405\,\mathrm{eV},\quadm_{\nu,2}\approx0.05529\,\mathrm{eV},\]with\[\sum m_\nu\approx0.10180\,\mathrm{eV}.\]The corresponding mass-squared differences are\[\Delta m_{21}^2\approx7.418\times10^{-5}\,\mathrm{eV}^2,\qquad\Delta m_{32}^2\approx2.478\times10^{-3}\,\mathrm{eV}^2.\] The formulation introduces no neutrino-sector fitting parameters. The inputs \(\alpha\) and \(m_e\) are inherited from the electron-sector structure, while the neutrino hierarchy is determined by the fixed ratios \(R\) and \(S\). A minimal Python implementation is provided for full reproducibility:https://github.com/yasuotanakaresearch/zero-parameter-structure