Autor: A. Jaster | Datum: 2026/06/14 | DOI: 10.5281/zenodo.20689366
We study a $Z_3$-symmetric parametrisation of the Standard-Model fermion masses, $m_n = A\,\Tt_n^{\,k}$ with $\Tt_n = 1+\eps\cos(\alpha+2\pi n/3)$ over the three generations $n=0,1,2$, motivated by the Koide relation and by a possible link between fermion mass and the four-dimensional causal density of the emergent-time framework. For each family the three-parameter form is an exact reparametrisation and carries no content by itself; all physical statements concern relations \emph{between} families. We separate two layers throughout. The first layer contains the only scheme-independent, data-supported reduction: the charged-lepton Koide value ($\eps_\ell=\sqrt2$), which lowers the parameter count for the nine charged-fermion masses from nine to eight. The second layer contains regularities that hold in a fixed mass-renormalisation convention and are therefore scheme-dependent: an amplitude factor that, in that convention, depends only on the weak isospin $T_3$ (collapsing the amplitude content to one overall scale $A_\ell$ plus one ratio $\tan\beta\simeq22.7$), a common quark phase, and the selection of the fourth power; taken together these bring the count to six. Applied to the neutrinos under the explicit \emph{hypothesis} $\alpha_\nu=\alpha_\ell$ (the scheme-free charged-lepton phase), together with the ring ansatz and the canonical generation assignment, the construction predicts, from the two measured mass-squared splittings under these assumptions, a complete spectrum: normal ordering (a built-in consequence of the assignment and $\eps_\nu>0$, not a dynamical result), $\sum m_\nu \simeq 63$--$66~\mathrm{meV}$, lightest mass $m_1\simeq 3$--$5~\mathrm{meV}$, effective $\beta$-decay mass $m_\beta\simeq 9$--$10~\mathrm{meV}$, and an upper bound $m_{\beta\beta}\lesssim 8~\mathrm{meV}$ (maximised over the unknown Majorana phases, with the allowed value reaching near zero). This prediction is sharply falsifiable within the stated assumptions, the live near-term discriminators being the mass ordering and the cosmological mass sum. Natural units $\hbar=c=1$ are used.