A Z3-Ring Parametrisation of the Fermion Masses and a Falsifiable Neutrino Spectrum

Autor: A. Jaster | Datum: 2026/06/14 | DOI: 10.5281/zenodo.20689366

Abstract

We study a Z3-symmetric parametrisation of the Standard-Model fermion masses, mn = A Τnk with Τn = 1+ε cos(α+2πn/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 between families. We separate two layers throughout. The first layer contains the only scheme-independent, data-supported reduction: the charged-lepton Koide value (εℓ=√2), 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 T3 (collapsing the amplitude content to one overall scale Aℓ plus one ratio tan β ≃ 22.7), 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 hypothesis αν=αℓ (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 εν>0, not a dynamical result), ∑ mν ≃ 63–66 meV, lightest mass m1 ≃ 3–5 meV, effective β-decay mass mβ ≃ 9–10 meV, and an upper bound mββ ≲ 8 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 ℏ=c=1 are used.

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\newcommand{\eps}{\varepsilon}
\newcommand{\acol}{\Delta_{\mathrm{c}}}
\newcommand{\Tt}{R}% ring factor R_n (distinct from the causal field Theta of the framework)

\title{\textbf{A $Z_3$-Ring Parametrisation of the Fermion Masses\\
and a Falsifiable Neutrino Spectrum}}
\author{A.\ Jaster}
\date{}

\begin{document}
\maketitle

\begin{abstract}
\noindent
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.
\end{abstract}

\section{Introduction}
The Koide relation for the charged leptons \cite{koide},
\begin{equation}
Q_\ell=\frac{m_e+m_\mu+m_\tau}{\bigl(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\bigr)^2}=\frac23,
\label{eq:koide}
\end{equation}
is reproduced to the current experimental precision and has so far resisted a generally accepted dynamical
explanation. It is algebraically equivalent to a $Z_3$-symmetric form of the square-root
masses, $\sqrt{m_n}\propto 1+\sqrt2\cos(\alpha+2\pi n/3)$, in which the three generations
sit at $120^\circ$ on a discrete ring. The emergent-time framework \cite{jaster1} defines
physical proper time through a scalar field $\Theta(x)\in[0,1]$ built from the retarded matter
distribution in the past lightcone, and its cosmological extension identifies a
four-dimensional causal density $f_B=I_H/\chi_H^4$ as a candidate dark-energy source
\cite{jaster4}. The present paper asks whether the same ring structure, generalised to an
adjustable power, organises the full set of charged-fermion masses, and whether the
scheme-free part of that organisation predicts the neutrino spectrum.

We adopt the parametrisation
\begin{equation}
m_n = A\,\Tt_n^{\,k},\qquad
\Tt_n=1+\eps\cos\!\Bigl(\alpha+\frac{2\pi n}{3}\Bigr),\qquad n=0,1,2 ,
\label{eq:ansatz}
\end{equation}
where $\Tt_n$ is the $Z_3$ ring factor for generation $n$. We deliberately denote it $\Tt_n$
rather than $\Theta$: the symbol $\Theta$ is reserved here, as in the underlying framework
\cite{jaster1}, for the causal time field $\Theta(x)\in[0,1]$. Whether the ring factor $\Tt_n$
is a local realisation of that field is a speculative question taken up only in
Section~\ref{sec:spec}; the parametrisation itself makes no such identification.
We stress at the outset that \eqref{eq:ansatz} with fixed $k$ contains three parameters
$(A,\eps,\alpha)$ for the three masses of a family: it is an exact reparametrisation and,
on its own, neither predicts nor constrains anything. In particular it does not by itself
``contain'' the Koide relation; the Koide constraint corresponds to \emph{imposing}
$\eps=\sqrt2$ rather than fitting it. All physical content resides in relations between
the families' parameters, to which we now turn. Throughout we distinguish clearly between
(i)~scheme-independent, data-supported results, (ii)~scheme-dependent observations, and
(iii)~speculative mechanisms.

The construction sits within a large literature on the Koide relation and on discrete-symmetry
($Z_3$ and related) approaches to flavour \cite{koide,flavour}; what distinguishes the present
treatment is less the ring form itself than the explicit separation of scheme-free from
scheme-dependent statements and the resulting falsifiable neutrino spectrum. We do not attempt a
full comparison with that literature here.

\section{Parametrisation and fit}
With $x_n\equiv m_n^{1/k}=A^{1/k}\Tt_n$ and the identities $\sum_n\cos(\cdot)=0$,
$\sum_n\cos^2(\cdot)=3/2$, the fit of \eqref{eq:ansatz} for one family is closed in terms
of the first three moments of $x_n$,
\begin{equation}
B\equiv A^{1/k}=\tfrac13\textstyle\sum_n x_n,\qquad
(\eps B)^2=\tfrac23\textstyle\sum_n(x_n-B)^2,\qquad
\tan\alpha=\frac{-\sum_n(x_n-B)\sin(2\pi n/3)}{\sum_n(x_n-B)\cos(2\pi n/3)} .
\label{eq:closed}
\end{equation}
We use the PDG~2024 quark masses \cite{pdg} ($u,d,s$ as $\overline{\mathrm{MS}}$ at
$\mu=2~\mathrm{GeV}$; $c,b$ as $\overline{\mathrm{MS}}$ at $\mu=m_q$; $t$ from the direct
measurement) and the charged-lepton masses from CODATA/PDG \cite{codata,pdg}. The
generation index is $n=0,1,2$ for the first to third family. Quoted uncertainties are
propagated from the input-mass errors by Monte Carlo.

\section{Layer 1: scheme-independent results}

\subsection{The charged-lepton sector and the Koide condition}
For the charged leptons we adopt the power $k=2$ (i.e.\ $m_\ell\propto\Tt_\ell^2$). This
choice is not the outcome of a goodness-of-fit comparison --- with three masses and the
three-parameter form \eqref{eq:ansatz} the fit is exact for \emph{any} $k$ --- but is singled
out because at $k=2$ the modulation depth comes out as $\eps_\ell=\sqrt2$ to one part in
$10^5$, which is precisely the Koide condition \eqref{eq:koide}. Imposing $\eps_\ell=\sqrt2$
(rather than fitting it) removes one parameter from the lepton family; this is the single
scheme-independent reduction of the construction, lowering the count for the nine
charged-fermion masses from nine to eight. The phase that accompanies the fit is
$\alpha_\ell=132.733^\circ$. Because the lepton masses are precisely known and
renormalisation-convention-free, $\alpha_\ell$ is scheme-independent; it depends, however, on
the generation-assignment convention (here $n=0$ is the lightest generation), and it will
serve as the sole phase input to the neutrino prediction, under the same assignment.

\subsection{Beyond Koide}
All further regularities --- an amplitude grouping consistent with weak isospin, a common quark
phase, and the structural power $k=4$ --- involve the quark masses and are therefore
scheme-dependent. They constitute Layer~2 and are presented in Section~\ref{sec:layer2},
together with the resulting parameter count.

\section{Layer 2: scheme-dependent observations}
\label{sec:layer2}
The regularities of this section involve the quark masses, which are running parameters whose
values depend on the renormalisation scheme and scale. A ratio of two running masses is not in
general scheme-invariant; dimensionlessness removes only the unit of mass, not the scheme
dependence. We therefore report the following as observations that hold in a stated convention
(the PDG mixed convention: $u,d,s$ as $\overline{\mathrm{MS}}$ at $\mu=2~\mathrm{GeV}$, $c,b$
at $\mu=m_q$, $t$ from the direct measurement, which is not strictly an
$\overline{\mathrm{MS}}$ mass and may itself influence the pattern), not as scheme-free results.
We also note that no principle singles out this convention as fundamental; it is the convention
in which the regularities take their cleanest form. This is a selection effect --- the Layer-2
relations are exhibited in precisely the scheme that makes them sharpest --- and it weakens
``to six'' as evidence accordingly; we report the reduction as a feature of this convention, not
as a scheme-free count.

\subsection{An amplitude grouping consistent with weak isospin (in this convention)}
Fitting the quark families at the structural power $k=4$ (Section~\ref{sec:powers}) and the
leptons at $k=2$, the amplitude factors $A$ in \eqref{eq:ansatz} group, in this convention, not
by charge or colour but by the weak-isospin component $T_3$:
\begin{equation}
\frac{A_{\text{down}}}{A_\ell}=0.9991\pm0.0029\qquad(0.3\sigma\ \text{from }1),
\label{eq:AdAl}
\end{equation}
although down-type quarks (colour triplet, $Q=-\tfrac13$) and charged leptons (singlet,
$Q=-1$) share only $T_3=-\tfrac12$. The quoted $0.3\sigma$ propagates only the input-mass
errors; it does \emph{not} include the scheme and scale systematics, which for the quark side
are the conceptually dominant uncertainty and are not quantified here. Moreover the
``$T_3$-only'' statement rests on a single near-coincidence \eqref{eq:AdAl}: the $T_3=+\tfrac12$
charged sector has only one member (the up-type quarks), so the grouping is an extrapolation
from one data point, not an established regularity --- ``consistent with'' a weak-isospin
grouping, not demonstrated to be controlled by it. With these caveats, the four amplitude
values $\{A_u,A_d,A_\ell\}$ collapse to an overall scale plus one ratio,
\begin{equation}
\tan\beta\equiv\frac{A(T_3=+\tfrac12)}{A(T_3=-\tfrac12)}=22.72\pm0.05
\qquad(\text{input-error only}).
\label{eq:tanb}
\end{equation}
The label $\tan\beta$ and the phrase ``type-II two-doublet'' are used only as a mnemonic for
this pattern (one scale feeding $d$-type quarks and charged leptons, the other feeding $u$-type
quarks); no two-Higgs-doublet model is constructed here, and the analogy carries no derivation.

\begin{table}[h]
\centering
\caption{Fitted parameters by weak isospin, in the PDG mixed convention. Only the
charged-lepton phase $\alpha_\ell$ is scheme-free; the quark amplitudes and phases, and hence
$\tan\beta$ and $A_d/A_\ell$, are scheme-dependent. Errors propagate input-mass uncertainties
only and exclude scheme/scale systematics. The two quark phases coincide within these errors,
$\alpha_{\text{up}}=\alpha_{\text{down}}$.}
\label{tab:A}
{\small
\begin{tabular}{lcccc}
\toprule
Sector & $T_3$ & $k$ & $A$ [GeV] & $\alpha$ \\
\midrule
Up-quarks $(u,c,t)$    & $+\tfrac12$ & 4 & $7.130\pm0.015$   & $133.798^\circ\pm0.030^\circ$ \\
Down-quarks $(d,s,b)$  & $-\tfrac12$ & 4 & $0.3136\pm0.0009$ & $133.849^\circ\pm0.073^\circ$ \\
Charged leptons $(e,\mu,\tau)$ & $-\tfrac12$ & 2 & $0.31384$ & $132.733^\circ$ (scheme-free) \\
\midrule
\multicolumn{5}{c}{$\tan\beta=A(+\tfrac12)/A(-\tfrac12)=22.72\pm0.05$;\quad
$A_{\text{down}}=A_\ell$ at $0.3\sigma$ (scheme-dependent, input-error only)}\\
\bottomrule
\end{tabular}
}
\end{table}

\subsection{Sector powers and the common quark phase}
\label{sec:powers}
We \emph{adopt} the power $k=4$ for the quarks as a structural assumption (motivated, in the
speculative reading of Section~\ref{sec:spec}, by the four-dimensional causal density). This is
a genuine assumption, not a fit: with three masses the family fit is exact for any $k$. It is
accompanied by a supporting consistency check --- leaving the power $p$ in $m^p=B\Tt$ free and
asking which value makes the two quark families share a phase gives $1/p^\ast=4.04\pm0.06$,
consistent with the adopted $k=4$. This check is not independent of the phase equality counted
below as a reduction: $p^\ast$ is by construction the exponent that equalises the two quark
phases, so it restates the same fact rather than confirming $k=4$ from a separate direction. An
independent, non-empirical motivation for $k=4$ appears only in the speculative reading of
Section~\ref{sec:spec}. Having adopted $k=4$, the empirical content is that the two quark
families then share a common phase:
$\alpha_{\text{up}}=133.798^\circ\pm0.030^\circ$ and
$\alpha_{\text{down}}=133.849^\circ\pm0.073^\circ$, so that
$\alpha_{\text{up}}-\alpha_{\text{down}}=-0.05^\circ\pm0.08^\circ$, consistent with zero at
$0.65\sigma$. This phase equality is the genuine, non-trivial relation between the two quark
families that removes one phase parameter (see Section~\ref{sec:count}); we note that the
agreement rests on a mixed scheme --- $\alpha_{\text{up}}$ in particular combines
$\overline{\mathrm{MS}}$ masses for $u,c$ with a directly measured top mass --- which makes it
especially convention-fragile. The common
quark phase lies above the lepton phase by
$\acol=\alpha_{\text{quark}}-\alpha_\ell=+1.07^\circ=0.0187~\mathrm{rad}$. All of this is
convention-dependent: running the six quark masses to a common scale, or evaluating each at its
own scale, shifts the up--down phase agreement by $1.6^\circ$ to $8.7^\circ$, and changes both
the value and the sign of $\acol$. Making these statements scheme-free would require the
framework to fix the mass-evaluation scale $\mu_f$ for each quark, which it does not yet do
(see open problems).

\paragraph{The colour shift $\acol$.}
The shift is a rotation of the ring argument, not a rescaling of masses. For a small
multiplicative mass perturbation $m_n\to m_n(1+\delta_n)$ \emph{with a common generation pattern
$\delta_n$ across the two families}, the ring phase responds as
$\delta\alpha=-(6\eps B)^{-1}\sum_n x_n\delta_n\sin(\alpha+2\pi n/3)$ (verified numerically), so
$\delta\alpha\propto1/\eps$. Such a common mass-level dressing would therefore predict the ratio
of the two families' shifts to track $\eps_{\text{up}}/\eps_{\text{down}}=1.34$, whereas the
data give $1.05\pm0.08$, disfavouring the mass-level reading at $3.8\sigma$ (with this caveat,
and on input errors only) and favouring a shift of the phase argument itself. Numerically
$\acol$ is of order $\alpha_s/2\pi$ at the electroweak scale
($\alpha_s/2\pi\approx0.0187~\mathrm{rad}$, compared in radians, not degrees), but we claim no
derivation, and we stress that this numerical proximity holds only in the present convention:
matching a convention-dependent quantity (whose value and sign vary by $1.6^\circ$--$8.7^\circ$
across mass schemes) against a physical constant is intrinsically unstable, and we record it as
numerology, not evidence.

\subsection{Parameter count}
\label{sec:count}
For the nine charged-fermion masses the naive fit uses nine parameters: three families, each
with $(A,\eps,\alpha)$ at the sector-fixed power $k$ (the powers $k_\ell=2$, $k_{\text{quark}}=4$
are structural assumptions, not fitted; they cost no parameter). The only
\emph{scheme-independent} reduction is the Koide condition $\eps_\ell=\sqrt2$, which lowers the
count to eight; this is the sole Layer-1 statement. The remaining reductions are
scheme-dependent and hold in the fixed convention:
\begin{enumerate}[label=(\roman*)]
\item the common quark phase $\alpha_{\text{up}}=\alpha_{\text{down}}$ (one genuine reduction,
$8\to7$): two independent quark phases collapse to one;
\item the isospin amplitude equality $A_{\text{down}}=A_\ell$ (one genuine reduction, $7\to6$).
\end{enumerate}
The single common quark phase may then be written as $\alpha_\ell+\acol$; this is a
\emph{relabelling} of that one phase in terms of the lepton phase and the offset $\acol$, not a
further reduction. The resulting count, in the fixed convention, is
\begin{equation}
\boxed{\;9\xrightarrow[\text{(SI)}]{\text{Koide}}8
\xrightarrow[\text{(SD)}]{\alpha_{\text{up}}=\alpha_{\text{down}}}7
\xrightarrow[\text{(SD)}]{A_d=A_\ell}6\;}
\qquad
\bigl\{A_\ell,\ \tan\beta,\ \alpha_\ell,\ \acol,\ \eps_{\text{up}},\ \eps_{\text{down}}\bigr\}.
\end{equation}
For comparison, the Standard-Model Yukawa sector uses nine parameters for these masses (and,
together with the mixing angles and phases, encodes the full flavour structure, which the
present construction does not address); the Koide relation alone --- the only scheme-free step
--- reduces the mass count to eight. The further reduction to six holds only in the fixed mass
convention (Table~\ref{tab:count}), and treats $k=4$ as a structural input. Of the six, two
($\acol$ and possibly $\tan\beta$) are candidates for further derivation; both attempts are at
present unsuccessful and are discussed under open problems.

\begin{table}[h]
\centering
\caption{Parameter reduction for the nine charged-fermion mass \emph{eigenvalues} (mixing not
included). ``SI/SD'' marks whether the reducing condition is scheme-independent or
scheme-dependent. Only the first step (Koide) is scheme-free; the reduction below eight is
convention-dependent and assumes the structural power $k=4$ for the quarks.}
\label{tab:count}
\begin{tabular}{llc}
\toprule
Step & Free parameters & Reducing condition \\
\midrule
Standard-Model Yukawa (masses) & 9 & --- \\
\;+ Koide & 8 & $\eps_\ell=\sqrt2$ (SI) \\
\;+ common quark phase & 7 & $\alpha_{\text{up}}=\alpha_{\text{down}}$ (SD) \\
\;+ isospin amplitude & 6 & $A_{\text{down}}=A_\ell$ (SD) \\
\bottomrule
\end{tabular}
\end{table}

\section{The neutrino prediction}
\label{sec:nu}
We now apply the construction to the neutrinos. The prediction rests on one scheme-free input
--- the charged-lepton phase $\alpha_\ell$ --- together with a set of explicit assumptions that
we state plainly, since they carry the result:
\begin{enumerate}[label=(\roman*)]
\item the \emph{hypothesis} $\alpha_\nu=\alpha_\ell$, i.e.\ the neutrinos sit on the same ring
phase as the charged leptons under the same generation assignment ($n=0$ lightest). This is a
model assumption, not a measured or derived quantity;
\item the ring ansatz \eqref{eq:ansatz} applies to the neutrinos, with a power $k\in\{2,4\}$;
\item a real ring with $\eps_\nu>0$ (no extra ring phase).
\end{enumerate}
Given these, the two free parameters $(A_\nu,\eps_\nu)$ are fixed completely by the two
measured mass-squared splittings (NuFIT~6.0, normal ordering:
$\Delta m^2_{21}=7.49\times10^{-5}~\mathrm{eV}^2$,
$\Delta m^2_{31}=2.513\times10^{-3}~\mathrm{eV}^2$ \cite{nufit}). Two structural variants are
considered:
\begin{itemize}
\item \textbf{Variant A (direct):} $m_\nu=A_\nu\Tt_\nu^2$, as for the charged leptons.
\item \textbf{Variant B (seesaw):} $m_\nu=A_\nu\Tt_\nu^4$, the form obtained when a
$k=2$ Dirac mass enters a seesaw with a degenerate heavy sector.
\end{itemize}
Under assumptions (i)--(iii) both variants give normal ordering: with the canonical assignment
and $\eps_\nu>0$ the ring yields $m_1    
    

Index