The Koide Cone as the Singlet Complement of the ℤ₃ Permutation Symmetry of a Three-Plane Junction

The Koide relation Q = 2/3 for the charged lepton masses holds to better than one part in 10⁵ but has no explanation within the Standard Model. This paper presents a second, structurally independent derivation of this relation from a three-plane junction framework.

The three charged lepton configurations form a triplet of the cyclic group ℤ₃ acting by coordinate permutation x→y→z→x on the junction. This triplet decomposes into a singlet (the democratic axis (1,1,1)/√3 in √m-space) and a complementary doublet. The Koide cone — the surface defined by cos²θ = 2/3 — is precisely the orbit locus at fixed angle from the singlet direction, with the angle fixed by cos²ψ = w/N = 2/3, where N=3 is the number of planes and w=2 is the spinorial double-cover index from the Finkelstein–Rubinstein theorem applied to the Borromean junction.

This derivation is structurally independent of the earlier energy-minimisation derivation (doi:10.5281/zenodo.19174656). The paper also establishes that the junction coupling matrix M = t(U + U⁻¹) is the unique element of the group algebra ℂ[ℤ₃] consistent with the junction's cyclic symmetry, making the equal-tension axiom a consequence of the algebra rather than an independent input. A complementarity relation 1/N + w/N = 1, specific to three spatial dimensions, is derived as a bonus result.

No lepton mass data is used. No parameter is fitted. The two derivations together establish the Koide cone as a structural invariant of the three-plane geometry.