Particle Masses in Emergent Causal Time: the Geometric Content of ε=√2 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 {jaster1,jaster3}. 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 ε₌√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 (=0) and dipole (=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 α_, mixing, the overall Yukawa amplitude); of the latter, the sign of α_ is inaccessible to the ring by an exact reflection symmetry, while its magnitude is simply not fixed (free parameters). Item beyond established structure are marked speculative throughout.