Particle Masses in Emergent Causal Time: the Geometric Content of $\varepsilon=\sqrt2$ and a Map of the Standard-Model Mass Spectrum

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 $\varepsilon_\ell=\sqrt2$ 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 an open multipole computation—so we present it as a geometric equivalence, not a derivation. We then interpret the per-fermion scale $\mu_f$ as the size 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 a framework-native \emph{eigenvalue sector} (the Koide quotient and the dichotomy) and a generic \emph{orientation sector} (the value of $\alpha_\ell$, mixing, the overall Yukawa amplitude); the latter is shown to be inaccessible to the ring by an exact reflection symmetry. Items beyond established structure are marked \emph{speculative} throughout.