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.

This revision adds two remarks to the published text. No results or conclusions from the original publication have changed.

Group algebra remark. The junction coupling matrix M=t(U+U−1)M = t(U + U^{-1}) M=t(U+U−1) is identified as the unique element of the group algebra ℂ[ℤ₃] consistent with the junction's cyclic symmetry. The equal-tension axiom — previously stated as a physical input — follows as a consequence of the algebra: U+U−1U + U^{-1} U+U−1 is the adjacency matrix of the 3-cycle C₃, the only ℤ₃-symmetric coupling. This is verified with zero reconstruction error. The unified generator U simultaneously produces the inter-generation cyclic permutation and the intra-generation junction coupling, explaining the tau lepton's dual role as a mechanical consequence of the algebra rather than a coincidence.

Koide phase remark. The ratio w/N = 2/3 determines not only the Koide cone angle (cos²ψ = +w/N, exact) but also, at product-ansatz level, the Koide phase (cos θ ≈ −w/N, 0.7% discrepancy from the observed 132.73°). The sign flip between the two reflects the difference between projection onto the singlet direction (positive) and orientation within the doublet plane (negative). The exact phase requires the relaxed junction field φ_exact from the Pontryagin-Thom construction.