Autor: A. Jaster | Datum: 2026/06/16 | DOI: 10.5281/zenodo.21320500
We ask how the masses of the elementary particles of the Standard Model —charged leptons, neutrinos, quarks, the electroweak gauge bosons, the Higgs boson, and the massless photon and gluon—are organised within the model of emergent causal time [1, 2]. The aim is a single, systematic classification in which every mass entry is assigned a role relative to the causal time field Θ(x). The centerpiece is a geometric characterisation: the scheme-free Koide reduction εℓ=√2 for the charged leptons is exactly equivalent to the √m vector lying at 45∘ to the democratic axis, equivalently to equal mean-square power in the monopole (ℓ = 0) and dipole (ℓ = 1) angular sectors of the planar Z3 ring. Whether this sector equipartition is dynamically realised is a separate, testable assumption—the subject of a multipole computation that we carry out here—so we present it as a geometric equivalence, not a derivation. A search—exploratory, over the linear cone statistics and the simplest nonlinear realisations—does not supply the derivation (under the RMS convention the linear statistics give the reference value εRMS=√6); within the tested ansätze it delimits what a mechanism must provide, leaving the geometric characterisation intact. We then interpret the per-fermion scale μf as the momentum scale of the dominant gauge cloud, which organises the lepton–quark dichotomy, and extend the classification to neutrinos, quarks and the boson sector, where the same k-power structure (k = 2, 4) reappears and masslessness sits outside the exponent ladder. A clean separation emerges between the framework-characterised Koide invariant (and the interpretive lepton–quark classification) and framework-external flavour data (the value of αℓ, mixing, the overall Yukawa amplitude); of the latter, the sign of αℓ is inaccessible to the ring by an exact reflection symmetry, while its magnitude is simply not fixed (free parameters). Items beyond established structure are marked speculative throughout.
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\title{\bfseries Particle Masses in Emergent Causal Time:\\[3pt]
the Geometric Content of $\eps=\sqrt2$ and a Map of the\\[2pt]
Standard-Model Mass Spectrum}
\author{A.\ Jaster}
\date{11 July 2026}
\begin{document}
\maketitle
\begin{abstract}
\noindent
We ask how the masses of the elementary particles of the Standard Model
---charged leptons, neutrinos, quarks, the electroweak gauge bosons, the Higgs
boson, and the massless photon and gluon---are organised within the model of
emergent causal time \cite{jaster1,jaster3}. The aim is a single, systematic
classification in which every mass entry is assigned a role relative to the causal
time field $\Theta(x)$. The centerpiece is a geometric characterisation: the
scheme-free Koide reduction $\eps_\ell=\sqrt2$ for the charged leptons is exactly
equivalent to the $\sqrt{m}$ vector lying at $45^\circ$ to the democratic axis,
equivalently to equal mean-square power in the monopole ($\ell{=}0$) and dipole
($\ell{=}1$) angular sectors of the planar $Z_3$ ring. Whether this sector
equipartition is dynamically realised is a separate, testable assumption---the
subject of a multipole computation that we carry out here---so we present it as a
geometric equivalence, not a derivation. A search---exploratory, over the linear cone
statistics and the simplest nonlinear realisations---does not supply the derivation
(under the RMS convention the linear statistics give the reference value
$\eps_{\mathrm{RMS}}=\sqrt6$); within the tested ansätze it delimits what a
mechanism must provide, leaving the geometric characterisation intact. We then interpret the per-fermion scale $\mu_f$ as the momentum scale of
the dominant gauge cloud, which organises the lepton--quark dichotomy,
and extend the classification to neutrinos, quarks and the boson sector, where the
same $k$-power structure ($k=2,4$) reappears and masslessness sits outside the
exponent ladder. A clean separation emerges between the framework-characterised
Koide invariant (and the interpretive lepton--quark classification) and framework-external
flavour data (the value of $\alpha_\ell$, mixing, the overall Yukawa
amplitude); of the latter, the \emph{sign} of $\alpha_\ell$ is inaccessible to the ring by an
exact reflection symmetry, while its magnitude is simply not fixed (free parameters). Items
beyond established structure are marked \emph{speculative} throughout.
\end{abstract}
\section{Introduction}
The base theory \cite{jaster1} defines physical proper time through a scalar field
$\Theta(x)\in[0,1]$ built from the retarded causal matter distribution in the past
light cone, with the clock metric
$\hat g_{\mu\nu}=q_{\mu\nu}+(1-\Theta)\,U_{H\mu}U_{H\nu}/c^2$ replacing the
background metric. Fields propagate on $\hat g_{\mu\nu}$ \cite{jaster3}; the
cosmological sector is treated in \cite{jaster4}. For mechanisms addressing Koide's
relation and the broader context of discrete flavour symmetries and lepton mixing we
refer to \cite{sumino,altarelli}. A $Z_3$-ring parametrisation of the charged-fermion
masses, together with a falsifiable neutrino spectrum, was given in \cite{jasterZ3}.
The present note collects what can and cannot be said about the \emph{full}
elementary mass spectrum within this framework. We organise every Standard-Model
mass entry by its relation to $\Theta$, distinguishing throughout between established
structure and conjecture. The single robust, distinctive result is the geometric
characterisation of the Koide value $\eps_\ell=\sqrt2$ (Sec.~\ref{sec:sqrt2}); the
remainder of the spectrum is mapped onto a framework-characterised invariant and
framework-external flavour data (Secs.~\ref{sec:mu_f}--\ref{sec:sectors}), with explicit
limitations stated in Sec.~\ref{sec:limits}. Speculative extensions covering the
$k$-powers and the boson sector are gathered in Sec.~\ref{sec:speculation}.
\section{Mass generation in emergent causal time}
\label{sec:setup}
In this framework the fermion masses are taken to arise from the local value of the time
field---a modelling ansatz, not a derivation from an action or a self-consistent field
equation,
\begin{equation}
m_f \;=\; \frac{v}{\sqrt2}\,B_f\,F\!\big(\Theta^{f}_{\mathrm{lok}}\big),
\qquad
\Theta^{f}_{\mathrm{lok}} \;=\; \Sinf\!\Big(\frac{\Omega_{\mathrm{lok}}(\mu_f)}
{\Omega_{\mathrm{ref}}}\Big),
\qquad
F(\Theta)=\Theta^{k},
\label{eq:massformula}
\end{equation}
with $v$ the electroweak scale, $\Omega_{\mathrm{ref}}$ the reference
(Higgs-condensate) density, and $\Sinf$ the saturating function of the base theory.
The prefactor factorises as $B_f=B_{\mathrm{charge}}(Q)\,B_{\mathrm{gen}}$, where
$B_{\mathrm{gen}}$ is a per-\emph{family} amplitude (one value for the charged leptons, one
for the up-type quarks, one for the down-type quarks, and so on), so that the family
amplitude is $A=(v/\sqrt2)\,B_{\mathrm{charge}}B_{\mathrm{gen}}$ while the three
\emph{generations} within a family are carried entirely by the ring index $n$ in
\eqref{eq:ring}; the two do not double-count. The factor $B_{\mathrm{charge}}$ is a
bookkeeping label that vanishes for states which receive no mass from electroweak symmetry
breaking. It is not a general account of masslessness: the exact masslessness of the photon
and gluon follows from the unbroken $U(1)_{\mathrm{em}}$ and $SU(3)_c$ gauge symmetries,
which forbid a mass term, not from this factor; we use $B_{\mathrm{charge}}=0$ only as
shorthand in the organising map of Sec.~\ref{sec:bosons}. The cloud size $\ell_f=1/\mu_f$ is
the range of the dominant gauge dressing (Sec.~\ref{sec:mu_f}), and $k$ encodes how the mass
term couples to $\Theta$ (Sec.~\ref{sec:speculation}). Per family the masses are parametrised
by a $Z_3$ ring,
\begin{equation}
m_n \;=\; A\,\Reff_n^{\,k},\qquad
\Reff_n \;=\; 1+\eps\cos\!\Big(\alpha+\tfrac{2\pi n}{3}\Big),\qquad n=0,1,2,
\label{eq:ring}
\end{equation}
an exact reparametrisation of the three masses of one family by $(A,\eps,\alpha)$, which is
empty in itself; physical content enters only through the special value $\eps_\ell=\sqrt2$
(the Koide condition, a within-family statement) and through relations across families
\cite{jasterZ3}. Here the (dimensionless) ring factor $\Reff_n$ is the local realisation of
the source modulation, not $\Theta$ itself (Sec.~\ref{sec:Sinf}); it should not be confused
with the cloud-size length $\ell_f$, and the explicit functional map from
$\Omega_{\mathrm{lok}}(\mu_f)$ (or $\Theta^f_{\mathrm{lok}}$) to $\Reff_n$ is a model input,
not derived here. For the charged leptons $k=2$ and the Koide condition
\begin{equation}
Q \;\equiv\; \frac{\sum_n m_n}{\big(\sum_n\sqrt{m_n}\big)^2}=\frac23
\quad\Longleftrightarrow\quad \eps_\ell=\sqrt2
\label{eq:koide}
\end{equation}
holds \cite{koide}, the only scheme-independent reduction.
\section{Charged leptons: the geometric content of $\eps=\sqrt2$}
\label{sec:sqrt2}
\subsection{Koide $\Leftrightarrow$ a $45^\circ$ angle}
For $k=2$ one has $\sqrt{m_n}\propto \Reff_n$. With
$\mathbf s=(\sqrt{m_e},\sqrt{m_\mu},\sqrt{m_\tau})$ and the democratic axis
$\mathbf d=(1,1,1)/\sqrt3$,
\begin{equation}
\cos^2\!\angle(\mathbf s,\mathbf d)=\frac{(\sum_n\sqrt{m_n})^2}{3\sum_n m_n}
=\frac{1}{3Q}.
\end{equation}
Hence \eqref{eq:koide} is \emph{exactly} equivalent to $\cos^2\angle=\tfrac12$, an
angle of $45^\circ$ \cite{foot}. With the PDG masses \cite{pdg} one finds
$Q=0.66666$ and $\angle(\mathbf s,\mathbf d)=45.000^\circ$. This is an algebraic
reformulation of Koide's relation, not by itself a dynamical explanation.
\subsection{Monopole--dipole decomposition and sector equipartition}
Write $\sqrt{m_n}=a+b\cos\theta_n$, $\theta_n=\alpha+2\pi n/3$, splitting the data
into a \emph{monopole} $a$ (the democratic, $Z_3$-symmetric offset, an $\ell{=}0$
mode) and an in-plane \emph{dipole} $b$ (the directional $\ell{=}1$ modulation).
Using $\langle\cos\theta\rangle=0$ and $\langle\cos^2\theta\rangle=\tfrac12$, the
mean-square power carried by each angular sector is
\begin{equation}
P_{\ell=0}=a^2,\qquad P_{\ell=1}=\tfrac12 b^2 .
\end{equation}
The Koide condition is \emph{exactly} equality of the two sector powers,
\begin{equation}
\boxed{\;P_{\ell=0}=P_{\ell=1}\quad\Longleftrightarrow\quad a^2=\tfrac12 b^2
\quad\Longleftrightarrow\quad \eps=\sqrt2\;}
\end{equation}
Numerically (PDG masses, in $\sqrt{\mathrm{MeV}}$) $a^2=313.84$ and $b^2/2=313.84$,
identical to four digits. This is equality between the two angular \emph{sectors},
not a per-mode statement. The $\ell{=}1$ sector is two-dimensional---its $m=\pm1$
components combine into the single amplitude $b$ and phase $\alpha$---so it carries
$b^2/4=156.9$ per real mode, against $a^2=313.84$ for the single monopole mode;
demanding equal power \emph{per mode} would instead give $\eps=2$. The factor $2$ is
the angular-normalisation $\langle\cos^2\rangle^{-1}$ rather than a count of modes (it
persists, for instance, for any $N\ge3$ equally-spaced sampling), so $\eps=\sqrt2$ is
a balance between the $\ell{=}0$ and $\ell{=}1$ sectors.
\subsection{The ring form from $\ell{=}1$ isotropy}
An in-plane dipole is a two-dimensional vector $(b,\alpha)$; its projection onto three
$120^\circ$ axes is exactly $b\cos(\alpha+2\pi n/3)$. \emph{Given} the three-fold, coplanar
sampling (three generations at $120^\circ$ in a chosen plane), statistical isotropy of the
$\ell{=}1$ sector then fixes the functional \emph{form} \eqref{eq:ring} with precisely two
degrees of freedom $(\eps,\alpha)$ and explains why $\alpha$ is a flat direction (isotropy
prefers no $\alpha$). Isotropy alone does \emph{not} explain the three generations or the
three $120^\circ$ points---those are inputs. The
planar geometry is motivated by the $\ell{=}1$ modes of the retarded causal field
$J^-(x)$ on the causal sphere \cite{jaster3}; of the three dipole modes $m=-1,0,+1$,
the one with $m=0$ is orthogonal to the ring plane and projects to zero on the three
coplanar sites, leaving a single in-plane dipole. The ring plane itself---equivalently, which
dipole mode is taken as $m=0$---is a choice not fixed by the framework.
\subsection{What $\Sinf$ does and does not set}
\label{sec:Sinf}
The saturation function does not fix $\eps$. The rigid identification $\Reff_n=\Theta$
(offset exactly $1$) is excluded only on range grounds---at $\alpha=132.733^\circ$ one
has $\Reff_n=(0.04,0.58,2.38)$, exceeding $1$---but a rescaled $\Theta_n\propto\Reff_n$
maps into $[0,1]$, so this is not a general obstruction. The substantive point is that
$\eps$ is a property of the multipole content of the source $\Omega_{\mathrm{lok}}$
(the $\ell{=}0/\ell{=}1$ balance). The saturation function $\Sinf$ supplies only
$\Theta_{\mathrm{lok}}$ itself (the map $\Omega\to\Theta$); the exponent is set
separately by the mass map $F(\Theta)=\Theta^2$, a motivated model choice of the base
theory rather than a derived necessity. The origins thus separate:
\[
\begin{aligned}
k=2 &\;\Leftarrow\; \text{the mass map }F\ \text{(model choice)},\\
\eps=\sqrt2 &\;\Leftarrow\; \ell{=}0/\ell{=}1\ \text{sector balance (assumed; see Sec.~\ref{sec:speculation})}.
\end{aligned}
\]
\section{The screening interpretation of $\mu_f$ and the lepton--quark dichotomy}
\label{sec:mu_f}
The per-fermion scale $\mu_f$ admits a framework-compatible interpretation: $\mu_f$ is the
characteristic momentum (inverse-length) scale of the dominant \emph{gauge cloud} of the
fermion, so that $\ell_f=1/\mu_f$ is the spatial cloud size. The causal density couples to the
energy--momentum tensor, which is smeared over a cloud of range $1/\mu_{\mathrm{gauge}}$;
the cloud size $\ell_f=1/\mu_f$ is therefore the integration radius. Through the
momentum-dependent mass function $\Sigma(p^2)$ the assignments follow:
\begin{itemize}[leftmargin=1.3em,itemsep=2pt]
\item coloured light quarks ($u,d,s$): $\mu_f\sim\Lambda_{\mathrm{QCD}}$ (dynamical
chiral symmetry breaking);
\item coloured heavy quarks ($c,b$): $\mu_f\sim m_q$ (Coulombic regime);
\item the top: $\mu_f\sim m_t$ (electroweak/Yukawa scale; the top does not hadronise, but
still carries a perturbative QCD self-energy);
\item colourless leptons: $\mu_\ell\sim\mu_{\mathrm{EW}}$, the electroweak cloud scale
(the lepton mass itself is the Yukawa coupling to the Higgs, not this dressing),
\emph{universal} over the three generations.
\end{itemize}
This organises the central dichotomy of \cite{jasterZ3}. Leptons share one universal cloud
scale $\mu_\ell$, so the $Z_3$ structure of the ring is preserved (a clean ring); quarks have
generation-dependent $\mu_f$, so $Z_3$ is broken and the ring becomes scheme-dependent. (The
scheme-independence of the Koide relation itself is a property of the pole-mass definition,
not a consequence of the equal cloud scale.) We note only, and do not rely on, the
dimensional observation that a four-dimensional volume factor
$(m_W/\Lambda_{\mathrm{QCD}})^4\sim10^{10}$ is a large number of the kind that any account of
the wide spread of Standard-Model masses must ultimately produce; we do not identify it with
the ratio of any two specific observed masses (and indeed $m_W$ itself exceeds the hadronic
scale, so the phrase should not be read as ``hadrons above leptons'').
It remains open whether $\Omega_{\mathrm{lok}}(\mu_f)$ becomes generation-universal despite the
different $\mu_f$ (which would protect $Z_3$), and the light-quark case is intrinsically
non-perturbative (lattice/DSE). The time-bubble scenario of the base series
\cite{jasterBubble} fixes only global constants and cannot deliver $\mu_f$, which is
Standard-Model gauge dynamics.
\paragraph{The dressing scale made quantitative.}
The qualitative assignments above can be made numerical for the coloured fermions. For the
light quarks the mass is largely dynamical (non-perturbative); for $c,b,t$ it is dominated by
the explicit Yukawa value, but the same momentum-dependent self-energy still defines a QCD
dressing scale. That scale is the momentum at which the dressing turns over; two
related---but not identical---operational proxies are
the inflection of the momentum-dependent mass function $M_f(p^2)$,
$\mu_f^{(1)}=\arg\max_p|d\ln M_f/d\ln p^2|$, and the euclidean constituent scale
$\mu_f^{(2)}=M_f^{E}$ defined by $p^2=M_f^2(p^2)$; we use the latter below and note that the
two need not coincide. This QCD diagnostic does not apply to the colourless leptons, whose
mass is the electroweak Yukawa coupling; for them the relevant cloud scale is the electroweak
threshold $\mu_\ell\sim\mu_{\mathrm{EW}}$, universal over the generations, and the euclidean
QCD scale is simply not the applicable quantity---so the lepton and quark entries below are
scales of somewhat different type, compared only for their sector ordering. Taking the
euclidean scales $M_f^{E}$ of a unified
Dyson--Schwinger analysis \cite{serna} for $u,d,s,c,b$ as an independent, non-refitted input
($M^{E}_{u,d}\simeq0.39$, $M^{E}_s\simeq0.53$, $M^{E}_c\simeq1.54$, $M^{E}_b\simeq4.68$~GeV;
these are Landau-gauge, kernel-dependent quantities, not gauge-invariant cloud sizes), and
assigning the top its electroweak/Yukawa scale ($\mu_t\simeq m_t\simeq173$~GeV, the source
not covering the top), a representative sector scale---the geometric mean over the three
generations, used only as a sector proxy and not a derived weighting---gives
$\mu_{\mathrm{down}}\simeq(0.39\cdot0.53\cdot4.68)^{1/3}\simeq1.0$~GeV against
$\mu_{\mathrm{up}}\simeq(0.39\cdot1.54\cdot173)^{1/3}\simeq4.7$~GeV. The ordering
$\mu_{\mathrm{down}}<\mu_{\mathrm{up}}$ is robust to the top assignment (using $m_H$ or $m_W$
in place of $m_t$ gives $4.2$ or $3.6$~GeV, still well above $\mu_{\mathrm{down}}$). Under the
(assumed, not here computed) monotonicity ``lower dressing scale $\Rightarrow$ larger cloud
$\Rightarrow$ more strongly smeared dipole,'' this is consistent with the observed depletion
ordering $\eps_{\mathrm{down}}<\eps_{\mathrm{up}}$ (a smaller $P_{\ell=1}/P_{\ell=0}$ for the
down type). We do not claim more: without a derived map $\eps(\mu_f)$ only the ordering is
exhibited, not the magnitude of the depletion, and $\mu_f$ itself remains Standard-Model gauge
dynamics external to the emergent-time construction, with $\eps_\ell=\sqrt2$ the assumed sector
balance of Sec.~\ref{sec:sqrt2}.
\section{The complete fermion spectrum}
\label{sec:fermions}
\paragraph{Neutrinos.}
Applying the ring with the hypothesis $\alpha_\nu=\alpha_\ell$ and a canonical
(normal) ordering yields \cite{jasterZ3}, for the two variants:
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l c c c c c}
\toprule
variant & $\eps_\nu$ & $A_\nu$ [meV] & $(m_1,m_2,m_3)$ [meV] & $\sum m_\nu$ [meV] & $m_{\beta\beta}^{\max}$ [meV]\\
\midrule
direct ($k=2$) & $0.796$ & $15.9$ & $(3.4,\,9.3,\,50.2)$ & $62.9$ & $\lesssim 6$\\
seesaw ($k=4$) & $0.346$ & $15.7$ & $(5.4,\,10.2,\,50.4)$ & $66.0$ & $\lesssim 8$\\
\bottomrule
\end{tabular}
\end{center}
with $m_\beta=9.5$ and $10.4$~meV respectively. The $m_{\beta\beta}^{\max}$ entries are
\emph{upper} bounds: through Majorana-phase cancellation the allowed value reaches down
to ${\sim}0$, so non-detection in $0\nu\beta\beta$ does not falsify the spectrum. In the
equipartition language $\eps_\nu<\sqrt2$ means an \emph{under-equipartitioned} dipole:
the dipole-to-monopole sector-power ratio is $P_{\ell=1}/P_{\ell=0}=\eps_\nu^2/2$, equal
to $0.32$ (direct) and $0.06$ (seesaw), versus $1$ for the charged leptons.
Interpretatively, neutral fermions couple more weakly to the directional ($\ell{=}1$)
part of the causal field. The mass ordering here follows from the adopted canonical
assignment rather than being an independent output; the cosmological
$\sum m_\nu$ must be recomputed in the model's own non-$\Lambda$CDM background
\cite{jaster4} before comparison.
\paragraph{Quarks.}
With $k=4$ the up- and down-type rings have $\alpha_{\rm up}=133.80^\circ\approx
\alpha_{\rm down}=133.85^\circ$ (a $0.65\sigma$ coincidence) and amplitudes related by
$\tan\beta=A_u/A_d=22.7$ (with $A_d=A_\ell$, the down-type amplitude set equal to the
lepton one), $\eta=\tfrac12\ln\tan\beta=1.56$ \cite{jasterZ3}. These quark
statements---the phase $\alpha$, $\tan\beta$, and the exponent $k=4$---are
\emph{scheme-dependent}, unlike the scheme-free lepton Koide relation. Koide is
violated ($Q\neq\tfrac23$), consistent with the broken $Z_3$ of the
generation-dependent $\mu_f$.
\section{The boson sector}
\label{sec:bosons}
Bosons carry no generation index and do not fit the ring. They form the
$\Tthree$-\emph{symmetric} reference against which the fermion asymmetry is measured.
\paragraph{Massive gauge bosons.}
The $W^\pm$ masses are exactly equal, $m_{W^+}=m_{W^-}$, fixed by
$|\langle\varphi\rangle|^2$, the opposite of the fermion $\Tthree$ asymmetry; the
physical relation is $m_Z=m_W/\cos\theta_W$ (Weinberg, at tree level), not a ring. Custodial $SU(2)$
keeps $\rho=1$ at tree level, broken principally by the fermion $\Tthree$ splitting;
the leading top--bottom contribution is
$\Delta\rho\simeq 3G_F m_t^2/(8\sqrt2\,\pi^2)\approx0.0093$ \cite{pdg} (the hypercharge
coupling contributes more weakly). The amplitude exponent $\eta$ of the fermion sector
is \emph{related to} (not identical with) this custodial breaking:
$\eta=\tfrac12\ln(A_u/A_d)$ measures the overall up/down Yukawa-scale ratio, of which
the custodial $t$--$b$ splitting is one manifestation. Thus $\Tthree$ plays two distinct
roles---a custodial Lie index for the bosons and a $\varphi/\tilde\varphi$ selector for
the fermions.
\paragraph{Massless gauge bosons.}
The photon and gluon are massless because their gauge symmetries ($U(1)_{\rm EM}$,
$SU(3)_C$) are unbroken: there is no charged condensate to source a mass, i.e.\
$B_{\mathrm{charge}}=0$ in \eqref{eq:massformula}. This is \emph{not} a value $k=0$ of
the exponent (which would give $F=\Theta^0=1$ and a non-zero mass); masslessness lies
outside the $\Theta$-exponent scheme. Gluon confinement is an infrared effect, not a
mass.
\paragraph{Higgs.}
The Higgs mass arises from the scalar potential after symmetry breaking,
$m_H^2=2\lambda v^2$ (for the standard normalisation $V\supset\lambda|\varphi|^4$); its
quartic origin is \emph{analogous to} (not identical with) the $k=4$ ring exponent of
the quarks, in contrast to the gauge masses $\sim|D_\mu\varphi|^2$ ($k=2$). This
$k$-power analogy across sectors motivates the speculation of Sec.~\ref{sec:speculation}.
\begin{table}[t]
\centering
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c l l}
\toprule
Particle class & $k$ & mass origin & framework status\\
\midrule
charged leptons & $2$ & ring eigenvalue, $\eps=\sqrt2$ & eigenvalue sector (clean)\\
quarks & $4$ & ring eigenvalue, $Z_3$ broken & eigenvalue $+$ $\mu_f$-dependent\\
neutrinos & $2/4$ & ring, $\eps_\nu<\sqrt2$ & eigenvalue (under-equipart.)\\
$W,Z$ & $2$ & $|D\varphi|^2$, $m_Z=m_W/\cos\theta_W$ (tree level) & $\Tthree$-symmetric reference\\
Higgs & $4$ & $\lambda\varphi^4$, $m_H^2=2\lambda v^2$ & quartic (analogy), not a ring\\
photon, gluon & --- & unbroken gauge symmetry, $B_{\mathrm{charge}}=0$ & massless\\
\bottomrule
\end{tabular}
\caption{Classification of the Standard-Model mass spectrum relative to the
emergent-time structure. The $k$ column is the $\Theta$-exponent where applicable;
gauge-boson masslessness follows from unbroken gauge symmetry and is only \emph{tagged}
$B_{\mathrm{charge}}=0$ here, not caused by it (Sec.~\ref{sec:setup}). The
$k$-assignments and their unification are discussed---as speculation---in
Sec.~\ref{sec:speculation}.}
\label{tab:map}
\end{table}
\section{Framework-characterised versus framework-external
(the eigenvalue/orientation split)}
\label{sec:sectors}
The mass sector splits by \emph{status}, not by mathematical type, into a part the
framework characterises and a part it does not. A circulant is diagonalised by the same
discrete-Fourier matrix irrespective of $\eps$ and $\alpha$, so within one family there is no
eigenvector-orientation freedom at all; the masses (eigenvalues) are set by $(A,\eps,\alpha)$.
The \emph{only} $\alpha$-independent invariant the framework fixes is the Koide quotient
$Q=2/3$ ($\eps_\ell=\sqrt2$); everything else is framework-external. That remainder groups
objects of two different kinds. The ring phase $\alpha_\ell$ is a reflection-odd
\emph{spectral} phase that shifts the individual mass ratios $m_\mu/m_e$, $m_\tau/m_e$---so
the ratios, though eigenvalue data, are \emph{not} framework-fixed, being carried by
$\alpha_\ell$ at fixed $\eps=\sqrt2$. The inter-family PMNS mixing, by contrast, is of a
genuinely different type: a basis rotation \emph{between} families, not a spectral phase. We
collect these under one heading---the \emph{orientation sector}---only because they share the
status of not being fixed by the ring; the term denotes that common status, not a common
mathematical structure.
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{>{\raggedright\arraybackslash}p{0.17\textwidth}
>{\raggedright\arraybackslash}p{0.37\textwidth}
>{\raggedright\arraybackslash}p{0.37\textwidth}}
\toprule
& \textbf{Framework-characterised} (the eigenvalue sector)
& \textbf{Framework-external} (the orientation sector)\\
\midrule
content & Koide invariant $Q=2/3$; lepton/quark dichotomy (an interpretive $\mu_f$
classification, not itself an invariant)
& value of $\alpha_\ell$; mass ratios; PMNS mixing; overall amplitude $A$ / Yukawa scale\\
mechanism & $Z_3$-ring \emph{eigenvalue} balance; $\mu_f$ screening
& ring phase $\alpha_\ell$ (a spectral phase); inter-family basis rotation (mixing)\\
status & geometrically characterised (equipartition assumed)
& free input; not fixed by the ring\\
\bottomrule
\end{tabular}
\end{center}
\section{Limitations: what the framework does not fix}
\label{sec:limits}
Three results delimit the orientation sector as framework-external.
\begin{enumerate}[leftmargin=1.6em,itemsep=3pt]
\item \textbf{The framework fixes $\alpha_\ell$ only up to reflection.} Assuming $Z_3$
generation symmetry, every $Z_3$-symmetric function of the ring \emph{eigenvalues} is
exactly even under $\alpha\to-\alpha$ (which merely permutes the eigenvalues) and
contains only $\cos(3j\alpha)$ ($j=1,2,\dots$), never $\sin(3\alpha)$
(App.~\ref{app:kernel}). We stress what this does and does not forbid. It does \emph{not}
forbid an interior magnitude: a reflection-even potential with two or more harmonics, e.g.\
$V(\alpha)=a\cos3\alpha+c\cos6\alpha$, has extrema at $\cos3\alpha=-a/(4c)$ and can place its
minimum at an interior value---for $c>0$ and $a/c\simeq-3.14$ (so that the interior extremum
is a minimum, not a maximum) the global minimum sits exactly at
$\alpha=12.733^\circ$ ($\equiv132.733^\circ$), with the three masses non-degenerate. What the
even symmetry \emph{does} forbid is distinguishing $+\alpha_\ell$ from $-\alpha_\ell$: the
reflected partner ($107.267^\circ$ here) is an exactly degenerate minimum. Hence a
reflection-odd ($\sin3\alpha$) source is needed only to lift this reflection degeneracy
(select the sign), not to generate the magnitude; and no ring-internal potential provides
it. The magnitude itself is likewise not \emph{derived}---it is set by the free ratio of
even harmonics---so $\alpha_\ell$ is framework-external on two counts: an un-fixed sign
(symmetry) and an un-derived magnitude (free parameters). We add one caveat: in the
\emph{isolated} charged-lepton spectrum $\alpha\to-\alpha$ is merely a generation
relabelling, so the sign of $\alpha_\ell$ becomes a physical observable only relative to an
externally fixed flavour or mixing frame.
\item \textbf{No ring-internal kernel helps.} Since $\Theta$ is real, the retarded
response obeys $G(-\omega)=G(\omega)^*$ analytically, so its phase is odd in $\omega$;
an odd phase contributes only even harmonics and hence nothing to $\sin(3\alpha)$
(derived in App.~\ref{app:kernel}; also confirmed numerically). For a kernel whose
spectral weight sits at the Planck scale, the dissipative-to-reactive ratio
$\chi_I/\chi_R$ of the response at the lepton scale $\omega\sim m_W$ is moreover
suppressed by $\sim(m_W/M_P)$. This closes only \emph{ring-internal} (eigenvalue-only)
kernels; a kernel carrying an external flavour frame is not excluded (and is exactly the
loophole of App.~\ref{app:kernel}).
\item \textbf{$\alpha_\ell$ and the PMNS matrix are distinct.} The phase that fixes
$\alpha_\ell$ is Cartan-like (it commutes with the circulant and does not mix); mixing
requires a non-commuting transverse rotation. Identical rings in both sectors give the
unit PMNS (no mixing); a maximally misaligned choice gives a democratic (magic) matrix with
$|U_{ij}|^2=1/3$ (one such maximally-unbiased option, not unique; distinct from tri-bimaximal
mixing \cite{hps}), hence
$\theta_{13}=\arcsin(1/\sqrt3)=35.3^\circ$, excluded by data \cite{nufit}. The observed
mixing lies between and requires a non-distinguished intermediate orientation as free
input.
\end{enumerate}
\section{Speculative outlook}
\label{sec:speculation}
\paragraph{Origin of the $k$-powers (speculative).}
The exponent $k$ in \eqref{eq:massformula}--\eqref{eq:ring} reflects how the mass term
couples to the local time field, and the massive sectors populate two values: $k=2$
(a $|D_\mu\varphi|^2$/kinetic structure: gauge-boson masses and charged leptons, with
$F(\Theta)=\Theta^2$) and $k=4$ (a quartic $\lambda\varphi^4$ structure: the Higgs and
the quarks). Masslessness (photon, gluon) is not a third exponent but the separate
statement $B_{\mathrm{charge}}=0$. Speculatively, the quartic $k=4$ echoes the
four-dimensional normalisation $n=4$ of the cosmological causal density
$f_B=I_H/\chi_H^4$ \cite{jaster4}, suggesting that the same 4D spacetime measure
underlies the Higgs quartic, the quark $k=4$, and the constraint-pressure exponent. If
correct, the ladder $2\to4$ would be a single statement about how matter couples to the
accumulated causal content rather than two independent choices. This is a conjecture; no
mechanism currently fixes which sector sits at which $k$.
\paragraph{The amplitude and the Yukawa hierarchy (speculative).}
The overall amplitude $A$, equivalently the absolute Yukawa scale, is not
framework-fixed. Empirically $A(\Tthree)=A_0\,e^{\eta\,2\Tthree}$ with
$\eta=\tfrac12\ln\tan\beta$ \cite{jasterZ3}. Since $m_{W^+}=m_{W^-}$ is exact at tree
level, the causal $\varphi/\tilde\varphi$ asymmetry ($\tan\beta$) must be a property of
the Yukawa line, not of the vacuum expectation value. The numerical proximity
$\eta\approx m_H/m_W$ is suggestive but lacks a mechanism; an attempt via a gauged curved
scalar manifold (HEFT) needs an \emph{anisotropic} (hypercharge) curvature, which the
past-directed first moment of the retarded kernel (T-odd) can seed but at a curvature
scale $\sim1/\mu_f^2$ that returns to the $\mu_f$ wall. The top is special:
$m_t\approx v/\sqrt2$, Yukawa $\approx1$, saturates the vacuum expectation value---the one
place where the amplitude docks onto a framework-near scale.
\paragraph{Deepening $\eps=\sqrt2$ (speculative, but testable).}
The geometric characterisation of Sec.~\ref{sec:sqrt2} leaves exactly one physical
assumption open: equality of the $\ell{=}0$ and $\ell{=}1$ sector powers. Isotropy fixes
only the form of the $\ell{=}1$ modulation, not its magnitude relative to the monopole.
A framework-native test is the multipole power spectrum of the retarded integral
$J^-(x)$: compute the $\ell$-resolved variance of the causal density at the lepton scale
$\mu_f\simeq m_W$ and check whether the $\ell{=}0$ and $\ell{=}1$ sector powers coincide.
Agreement would turn the present equivalence into a \emph{derivation} of $\eps=\sqrt2$; a
mismatch would predict the deviation. The same computation for the neutral coupling should
yield the dipole suppression $\eps_\nu<\sqrt2$. This computation has since been carried out;
its outcome, and the exploratory search it prompted, are reported in Sec.~\ref{sec:search}.
\paragraph{Reconnecting the orientation sector (speculative).}
The value of $\alpha_\ell$ and the PMNS matrix are generic flavour
(Sec.~\ref{sec:limits}). One conjectural path would recouple them to emergent time if the
causal structure fixed the orientation axis---e.g.\ to the preferred direction $U_H$ or to
$\Tthree$---and if that single axis carried both a Cartan component (for $\alpha_\ell$,
through an external CP phase) and a transverse tilt (for PMNS) in the right ratio. A direct
test is whether an axis tied to $U_H$ or $\Tthree$ reproduces the observed angle hierarchy
($\theta_{23}$ large, $\theta_{13}$ small). First estimates argue against it---a single
axis gives the wrong ordering---so this remains an open and pessimistic conjecture.
\paragraph{A single missing ingredient (a unifying conjecture).}
The reductions left open above may not be independent, though the connection is a
conjecture, not a result. The quark depletions $\eps_{\mathrm{up}},\eps_{\mathrm{down}}$ and
the exponent $k=4$ turn on the sector-dependent dressing scale $\mu_f$; the lepton phase
$\alpha_\ell$ turns, in addition, on the coefficient $\zeta$ of the causal-gradient operator
$\delta\mathcal L=b_\mu\,\bar\psi\gamma^\mu\gamma_5\psi$ with
$b_\mu=\zeta\,\partial_\mu\Theta_{\mathrm{lok}}$ of the quantum-field companion \cite{jaster3}.
This operator is a candidate for the external-frame, reflection-odd source that
App.~\ref{app:kernel} and Sec.~\ref{sec:limits} identify as what is needed to lift the
reflection degeneracy of $\alpha_\ell$ (to select $+\alpha_\ell$ over $-\alpha_\ell$): it is
not a function of the ring eigenvalues alone but couples the ring
to the preferred frame $U_H$ through $\partial_\mu\Theta_{\mathrm{lok}}$. We stress that it is
only a candidate: that a spacetime operator $\partial_\mu\Theta_{\mathrm{lok}}\,
\bar\psi\gamma^\mu\gamma_5\psi$ actually projects onto the reflection-odd ring harmonic
$\sin3\alpha$---rather than averaging away or generating other harmonics---is itself not
established here. In the \emph{minimal} truncation to the first harmonics,
$V(\alpha)=a\cos3\alpha+b\sin3\alpha$, the observed angle corresponds to $b/a\simeq0.787$
(higher even harmonics such as $\cos6\alpha$ would shift this and can themselves carry the
magnitude, cf.\ Sec.~\ref{sec:limits}, so $0.787$ is illustrative, not a unique condition).
Within that truncation the reflection-even coefficient
$a$ comes out \emph{negative} for a dissipative self-energy dressing $\propto
m\ln(\mu_{\mathrm{EW}}^2/m^2)$ (checked numerically; simple power-law dressings give $a>0$ and
a boundary minimum), as an interior minimum requires; and a non-zero $b$ is what breaks the
$\pm\alpha_\ell$ degeneracy---at $b=0$ the truncated even minimum sits at the symmetric point
$\alpha\equiv0$, where two charged leptons coincide, whereas the physical interior value
requires either the odd $b$ or the higher even harmonics of Sec.~\ref{sec:limits}.
Two cautions attach to the odd term. First, $b_\mu$ is a pure
gradient, so any mechanism built on its curl (a ``flux'' through the generation triangle)
vanishes identically ($\partial_{[\mu}\partial_{\nu]}\Theta=0$); the direct axial coupling
itself does \emph{not} vanish, but---integrated by parts, up to boundary terms and with the
properly quantised axial anomaly---it is equivalent to
$\Theta_{\mathrm{lok}}$ multiplying $\partial_\mu(\bar\psi\gamma^\mu\gamma_5\psi)$, i.e.\ to the
pseudoscalar mass term $2m\,\bar\psi i\gamma_5\psi$ plus the chiral anomaly, so its physical
imprint on $V(\alpha)$ is tied to the fermion mass (hence to the ring) and is non-zero only
where $\partial_\mu\Theta_{\mathrm{lok}}$ itself is appreciable. Second, that last requirement
favours a sharp transition front (as in a first-order electroweak transition---the same
condition invoked for electroweak baryogenesis through the same operator \cite{jaster3}) over
the Standard-Model crossover, whose slow gradient $\sim H_{\mathrm{EW}}/m_H$ is plausibly, but
not here demonstrably, too small; this is an estimate, not a computation. Implicit throughout
is a freeze-in: any $V(\alpha)$ generated at the transition epoch must survive as a frozen
relic to set the present-day lepton masses, since today $\partial_\mu\Theta_{\mathrm{lok}}
\simeq0$ and the tilt is no longer sourced. Matching
$b/a\simeq0.787$ then points, very roughly, to $\zeta\sim10^{-2}$ with an $O(1)$ transition
depth, the hierarchy $m_H/m_\ell$ offsetting part of the radiative smallness of $\zeta$; we
give no equation for $\zeta$, and the map from the operator coefficient $\zeta$ to the
effective ratio $b/a$ is itself not derived here. The unifying point is only this: the value
of $\mu_f$ is Standard-Model gauge dynamics (external), but \emph{how} the ring responds to it,
the coefficient $\zeta$, and the causal (advanced-free) form of the matter back-reaction are
all facets of the self-consistent constraint response $\delta\Theta/\delta\phi\neq0$ and its
Schwinger--Keldysh (in-in) closure, the central open problem of \cite{jaster3}. Whether the
remaining reductions genuinely reduce to that one construction is a hypothesis we record, not a
result we establish.
\section{The multipole computation: an exploratory search}
\label{sec:search}
The test proposed in Sec.~\ref{sec:speculation}---the $\ell$-resolved power spectrum of
the retarded causal density---has since been carried out, together with the natural
follow-up questions it raises. The outcome is negative for the simplest dynamical
realisations but sharpens what a successful mechanism must supply; we record it here
because a map of what does \emph{not} work is itself a result, and because it
leaves the geometric characterisation of Sec.~\ref{sec:sqrt2} untouched. Throughout, the
target is the sector balance $\eps_{\mathrm{pred}}=\sqrt{6\,C_1/C_0}$, with $C_\ell$ the
$\ell$-sector power of the directional causal density built from the Hadamard
normal-ordered energy density (normal-ordered relative to an unspecified Hadamard reference
state; the choice, together with the lightcone regulator, affects the ratio below) on the
past lightcone at the lepton scale $\mu_f\simeq m_W$.
The $C_\ell$ are ensemble sector powers of the causal field; they are confronted with the
single observed ring realisation $(a,b)$ read off from the lepton masses. Accordingly
$\eps_{\mathrm{pred}}=\sqrt{6\,C_1/C_0}$ is a ratio of root-mean-square amplitudes, which is
\emph{not} the mean of the ring value $\eps=b/a$ (in general
$\langle b/a\rangle\neq\sqrt{\langle b^2\rangle/\langle a^2\rangle}$): it does not by itself
fix a sharp $b/a$ for a single realisation, and the realisation scatter---the distribution
$p(\eps\mid C_0,C_1)$---is not modelled here.
\paragraph{The factor $6$, and the bridge to Sec.~\ref{sec:sqrt2}.} Here $C_\ell$ denotes
the \emph{per-mode} power of the field multipoles $a_{\ell m}$ (the convention in which a
white field has a flat spectrum, $C_\ell=\text{const}$). The prefactor $6$ then follows from
how the three coplanar ring sites sample the field on a great circle. The monopole
contributes $a=a_{00}Y_{00}$, while on the equator the in-plane dipole is
$\sqrt{3/4\pi}\,(a_{1x}\cos\varphi+a_{1y}\sin\varphi)$, giving a ring amplitude
$b=\sqrt{3/4\pi}\,(a_{1x}^2+a_{1y}^2)^{1/2}$. With $\langle a_{00}^2\rangle=C_0$ and
$\langle a_{1x}^2\rangle=\langle a_{1y}^2\rangle=C_1$ this gives
\begin{equation}
\eps_{\mathrm{RMS}}^2=\frac{\langle b^2\rangle}{\langle a^2\rangle}=6\,\frac{C_1}{C_0},\qquad
6=\underbrace{3}_{N_{1}^2/N_{0}^2}\times\underbrace{2}_{\text{in-plane modes}} ,
\end{equation}
where $N_\ell^2$ is the squared normalisation constant of the (real) degree-$\ell$ harmonic
on the great circle, $N_1^2/N_0^2=(3/4\pi)/(1/4\pi)=3$. Two conventions are implicit here and
should be stated: $C_0=\langle a_{00}^2\rangle$ is used as a total monopole power, so that for
a monopole with a non-zero mean $\langle a_{00}\rangle\neq0$ one has
$C_0=\langle a_{00}\rangle^2+\operatorname{Var}(a_{00})$ (mean and fluctuation are not
separated); and the ``white field'' baseline $C_1=C_0$ is the equal-per-mode reference in this
zero-separation convention, not a claim about a physical spectrum.
A white field $C_1=C_0$ therefore gives the RMS reference value $\eps_{\mathrm{RMS}}=\sqrt6$
(not a sharp $\eps$), not $\sqrt2$; the Koide value requires the dipole \emph{per mode} to be
suppressed by exactly $C_1/C_0=1/3$. This does not
contradict the sector balance of Sec.~\ref{sec:sqrt2}: the ring sector powers there are
$P_\ell=C_\ell^{\mathrm{tot}}/4\pi$ with $C_\ell^{\mathrm{tot}}=(2\ell+1)C_\ell$, so
$P_1/P_0=3\,C_1/C_0$ and $P_0=P_1\Leftrightarrow C_1/C_0=1/3\Leftrightarrow\eps=\sqrt2$. The
$(2\ell+1)$ multiplicity is the entire difference between ``equal sectors'' ($\eps=\sqrt2$)
and ``equal modes'' ($\eps=\sqrt6$); equivalently $\eps^2=2\,C_1^{\mathrm{tot}}/C_0^{\mathrm{tot}}$
in the total-power convention.
\paragraph{Four negative results.} Four routes were examined; three bear on the
\emph{amplitude} $\eps=\sqrt2$ (results 1, 2, 4) and one---the tilt---on the \emph{phase}
$\alpha_\ell$ (result 3), which are logically separate problems (Sec.~\ref{sec:limits}) and are
kept apart here. Within the ansätze and parameter ranges tested none delivers a derivation,
the three amplitude routes closing in those ranges and the phase route leaving only a
contingent, non-derivational possibility noted below. The results below are exploratory: the
numerical scans and the one-loop estimate are not documented to reproduction standard here.
\begin{enumerate}[leftmargin=1.6em,itemsep=3pt]
\item \textbf{Linear cone statistics give the RMS value $\eps_{\mathrm{RMS}}=\sqrt6$.} For a
free source the
four-dimensional correlator $\langle{:}T_{00}{:}\,{:}T_{00}{:}\rangle\propto r^{-8}$ has a
spacetime spectral density of the schematic form $k^4\ln k^2$ plus scheme-dependent local
contact terms (the naive $\propto k^4$ is incomplete); with a lightcone regulator the
resulting sector ratio $C_1/C_0$ is
cutoff-dominated and tends to unity---a regulator-dependent statement, sensitive to the
window and to the contact terms---i.e.\ $\eps_{\mathrm{RMS}}\to\sqrt6$, the white-noise value
at which every angular mode is equally populated. This requires a falling aperture to be
tamed. A one-loop $T_{00}$ form factor of the dressed lepton, from a preliminary estimate
whose integrals, counterterms and scheme are not reproduced here (and which we therefore do
not treat as establishing the result, only as indicative), appears to
provide none: its finite part grows as $F_{\mathrm{tree}}(k)\,[\,c_0+c_1\ln(k/m_B)\,]$, a
logarithmically running point coupling with no normalisable cloud size. The leading
divergence is generation-universal; this alone does not exclude $Z_3$ breaking, however,
since the generation-dependent finite parts (through $m_B$ in the logarithm) are not
controlled by the universal divergence.
\item \textbf{Static $S_3$-symmetric attractors give $\eps^\star\in\{0,1,2\}$.} Driving
the ring by a saturating mass gain $\sum_n S(\lambda\Reff_n)^2$ and seeking the most
probable configuration yields, for either sigmoid tested and throughout the scanned
$(\beta,\lambda)$ domain, only the maximally symmetric strata: the symmetric condensate
($\eps^\star=0$) in the condensed region and the edge/corner values ($\eps^\star=1,2$) on
a narrow onset. In the saturated regime $S$ is concave, so equal sharing maximises the
gain (a Jensen argument); $\eps^\star=\sqrt2$ occurs in no cell.
\item \textbf{The saturating-gain tilt gives boundary minima, not the interior angle.}
At fixed $\eps=\sqrt2$ the specific double-well $V(\alpha)$ produced by the saturating
mass-gain driver has its minima at the boundaries of the fundamental cell (the $Z_3$ lattice
point and the positivity wall $\Reff_{\min}=0$), not at the interior
$\alpha_\ell=132.733^\circ\equiv12.733^\circ$ (in the cell $[120^\circ,135^\circ]\equiv
[0^\circ,15^\circ]$). This is a property of \emph{that particular driver}, not a general
obstruction: as shown in Sec.~\ref{sec:limits}, a reflection-even potential carrying a second
harmonic ($\cos6\alpha$) can place its minimum at the interior magnitude. What no
ring-internal even potential can fix is the reflection \emph{sign} of $\alpha_\ell$; that
requires a reflection-odd ($\sin3\alpha$) term. Consistent with App.~\ref{app:kernel},
every framework-native CP-odd source examined (the curl/flux of a pure-gradient $b_\mu$
coupling, which vanishes by $\partial_{[\mu}\partial_{\nu]}\Theta=0$; a
$\Box\Theta_{\mathrm{lok}}$ term times the ring discriminant; a CKM import through
lepton--quark loops) is either identically zero or smaller than the generic CP-even term
by many orders of magnitude. Not excluded---though equally not a derivation---is the direct
(mass- and anomaly-mediated) part of the $b_\mu$ axial coupling at a \emph{first-order}
electroweak transition, where $\partial_\mu\Theta_{\mathrm{lok}}$ is not slow-roll
suppressed; this contingent possibility is discussed in Sec.~\ref{sec:speculation}.
\item \textbf{Colour does not supply the factor $3$.} One might hope the $3$ in
$\eps^2=6=2\times3$ is the colour number $N_c$ rather than the dipole multiplicity. It is
not: for any constant (momentum- and generation-independent) colour correlation matrix the
colour sum is a global prefactor
common to every $C_\ell$, so it cancels exactly in $C_1/C_0$ (verified numerically to
machine precision under variation of $N_c$ and of an artificial cross-colour coherence).
Neither the colour number nor the fractional electric charge enters the sector ratio.
\end{enumerate}
\paragraph{What the search does establish.} The exercise is not empty. The four results
above are explicit (partly exploratory) computations---with the exception of the one-loop $T_{00}$ form factor of
result~1, which is only a preliminary estimate as noted there; the classification that follows
is a characterisation, not a further derivation. The three candidate values are cleanly classified as the
equipartition outcomes of three distinct norms: the field (Parseval) norm gives $\sqrt6$, the ring norm $2$, and the coupling norm
$u_0^2+|\vec d\,|^2$ gives exactly $\sqrt2$, where $u_0$ is the monopole and
$\vec d=(d_x,d_y)$ the in-plane dipole coupling vector ($u_0\propto a$, $|\vec d\,|\propto b$).
The observed value would therefore correspond to a dynamics that
equipartitions the \emph{coupling} norm rather than the field norm---a sharp,
if still unrealised (and here only named, not dynamically derived), target. Two exact
identities emerge alongside. First, the positivity boundary $\Reff_{\min}=0$ is a curve in the
$(\eps,\alpha)$ plane; its intersection with the Koide line $\eps=\sqrt2$ sits at
$\alpha=135^\circ$, and the observed phase $\alpha_\ell=132.733^\circ$ (which comes from the
individual mass ratios, not from the Koide relation, whereas $\eps=\sqrt2$ is the Koide datum)
is a $2.27^\circ$ standoff from it. This is equivalent to the smallness of the dimensionless
ring factor $\Reff_{\min}\simeq0.04$, which together with $A$ and $k$ yields the small electron
mass---a coincidence of the phase with the positivity edge, not an explanation of the absolute
electron mass. Second, the discriminant $\mathcal D=(\Reff_0-\Reff_1)
(\Reff_1-\Reff_2)(\Reff_2-\Reff_0)=\tfrac{3\sqrt3}{4}\eps^3\sin 3\alpha$ is the natural
reflection-odd order parameter: any $\sin 3\alpha$ dependence is proportional to it, which
restates the reflection identity of App.~\ref{app:kernel} in a single object.
\paragraph{What remains open.} Two logically distinct problems remain, and should not be
conflated. \emph{(i) The amplitude} $\eps=\sqrt2$: realising it dynamically now has a definite
specification---coupling-norm rather than field-norm equipartition---with, at present, no
framework candidate; the logical possibilities are a non-equilibrium (frozen) asymmetric
realisation of a symmetric dynamics, or a genuine $O(1)$ source that biases the coupling norm.
\emph{(ii) The phase} $\alpha_\ell$: this is a separate question, not addressed by the
equipartition of (i). Its magnitude can be carried by the free ratio of even ring harmonics
($\cos3\alpha,\cos6\alpha,\dots$; Sec.~\ref{sec:limits}) and is therefore un-derived rather
than forbidden; its \emph{sign} additionally requires a reflection-odd ($\sin3\alpha$) source,
for which the $b_\mu$ operator is a contingent candidate (Sec.~\ref{sec:speculation}). A tilt
potential does not by itself explain the amplitude $\eps=\sqrt2$. As before, this leaves the
geometric equivalence of Sec.~\ref{sec:sqrt2} intact: the equipartition behind
$\eps=\sqrt2$ is characterised, and now more precisely delimited, but not derived.
\section{Conclusion}
Within emergent causal time the elementary mass spectrum organises into a small
framework-characterised core and a larger interpretive remainder. The robust,
distinctive result is geometric: $\eps_\ell=\sqrt2$ is equivalent to a $45^\circ$ angle and
to equality of the monopole ($\ell{=}0$) and dipole ($\ell{=}1$) sector powers of the causal
field, so the Koide value is characterised as a sector-power balance, with its dynamical
realisation (cross-$\ell$ equipartition) an open, testable assumption rather than a settled
derivation. The saturation function supplies $\Theta_{\mathrm{lok}}$ and the mass map $F$
fixes the exponent $k=2$ as a model choice, neither of them $\eps$. The screening reading of
$\mu_f$ as a gauge-cloud scale suggests an organising map (Table~\ref{tab:map}) covering
neutrinos, quarks, the massive and massless gauge bosons, and the Higgs through the $k=2,4$
structure, with masslessness handled through the bookkeeping label $B_{\mathrm{charge}}=0$
(the gauge bosons' exact masslessness following, properly, from unbroken gauge symmetry;
Sec.~\ref{sec:setup}). What the framework does
\emph{not} fix---the value of $\alpha_\ell$, the mixing, and the overall Yukawa amplitude---is
framework-external data; under $Z_3$ symmetry an exact reflection identity places the
\emph{sign} of $\alpha_\ell$ beyond the reach of the ring, while its magnitude is left
un-derived (free even-harmonic ratios) rather than forbidden. The most direct probe of the
amplitude question is the
multipole power spectrum of
$J^-(x)$, which would turn the RMS characterisation behind $\eps=\sqrt2$ into a
derivation of that RMS value (a sharp single-realisation value would in addition require the
distribution $p(\eps\mid C_0,C_1)$); that computation has now been performed
(Sec.~\ref{sec:search}): within the tested
ansätze it does not deliver the derivation, but it delimits what a
successful mechanism must supply---coupling-norm rather than field-norm equipartition---while
leaving the geometric characterisation untouched. The screening reading of $\mu_f$, in turn,
now carries a modest quantitative anchor: euclidean Dyson--Schwinger dressing scales are
consistent with the observed depletion ordering $\eps_{\mathrm{down}}<\eps_{\mathrm{up}}$
without refitting the dressing scales, though the sector average and the assumed monotonic
$\mu_f\!\to\!\eps$ link are additional modelling choices, and only the ordering, not its
magnitude, is exhibited
(Sec.~\ref{sec:mu_f}). We finally record---as a conjecture, not a result---that the reductions
which remain open (the quark $\eps$'s, the exponent $k=4$, and the phase $\alpha_\ell$) may
localise to a single missing input, the self-consistent constraint response
$\delta\Theta/\delta\phi$ of the quantum-field companion \cite{jaster3}, rather than to several
independent mechanisms (Sec.~\ref{sec:speculation}). Any such phase-selecting effect would,
moreover, have to be a frozen relic of the electroweak epoch, since today
$\partial_\mu\Theta_{\mathrm{lok}}\simeq0$ and the reflection-odd source is no longer active.
\appendix
\section{Reality of the retarded kernel and the absence of $\sin(3\alpha)$}
\label{app:kernel}
The retarded response of a real field is real in the time domain, $G(t)\in\mathbb R$, so in
frequency space $G(-\omega)=G(\omega)^*$: the modulus is even and the phase $\arg G(\omega)$ is
odd in $\omega$. A dissipative correction to a ring level
$\lambda_n=2|t|\cos(\alpha+2\pi n/3)$ enters through this phase evaluated at the level energy,
$\psi(\lambda_n)$, an odd function of $\lambda_n$.
Independently of the detailed dynamics, any basis-independent (spectral) invariant of the ring
is a symmetric function of the eigenvalue set $\{\lambda_n\}$. Under $\alpha\to-\alpha$ one has
$\lambda_n(-\alpha)=\cos(-\alpha+2\pi n/3)=\lambda_{-n}(\alpha)$, so the set maps to itself (the
labels $1$ and $2$ are exchanged) and every symmetric function is invariant---hence even in
$\alpha$, with a Fourier expansion in $\cos(3j\alpha)$ only. A $\sin(3\alpha)$ term, odd under
$\alpha\to-\alpha$, is therefore absent from every spectral invariant, whether reactive (from
$|t|$) or dissipative (from the odd kernel phase). The same conclusion holds for the physical
masses $m_n=A\,\Reff_n^{\,k}$, since the set $\{\Reff_n\}$ permutes under $\alpha\to-\alpha$ in
exactly the same way; any symmetric function of the masses is likewise even in $\alpha$.
Producing a $\sin(3\alpha)$ term requires a reflection-odd structure that is not a function of
the eigenvalues alone---for instance a coupling to an external frame carrying its own
phase---consistent with the assignment of $\alpha_\ell$ to the framework-external data
(Sec.~\ref{sec:sectors}). This evenness obstructs only the \emph{sign} of $\alpha_\ell$: an
even potential with several $\cos(3j\alpha)$ harmonics can still place its minimum at an
interior magnitude (Sec.~\ref{sec:limits}), so the magnitude is left undetermined by the free
harmonic ratios, not forbidden by the symmetry.
\begin{thebibliography}{9}
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