An Algebraic Reformulation of the Koide Relation via the Cube Roots of Unity

Abstract. The Koide formula Q = (me + mμ + mτ)/(√me + √mμ + √mτ)² takes the value 2/3 tofive decimal places when current measured charged-lepton masses are substituted. This paperdoes not derive that value from a dynamical model; it presents an algebraic reformulationtogether with one clearly labelled hypothesis. Writing the mass-square-root amplitudes as aphase-symmetric triple along the cube roots of unity (1, ω, ω²), Q is shown to be algebraicallyidentical to the ratio of complex-conjugate-pair weight to total weight in the triple, equal to 2/3,and equivalently to the phase-weighted-sum identity |L|² = (3/4)Σm. In the one-parameter (η, δ)form Q = (1 + 2η²)/3 the Koide condition is η² = 1/2, independent of scale and of δ, so that δ isthe single residual degree of freedom that sets the spectrum; the charged leptons select δ =2/9. At hypothesis level, if a single carrier law |L|² = (3/4)Σm holds on symmetric configurations,positive semidefiniteness of the Gram matrix forces Q ≤ 2/3, with equality exactly at cube-rootalignment — so any family established with Q > 2/3 refutes the hypothesis. Changes in Version 2. (i) The single most precise τ-mass measurement (1777.09 ± 0.14 MeV)is incorporated. With that input Q = 0.666673, which is above 2/3 — the direction that refutesthe hypothesis of Section 5 (currently 0.86σ, not significant). The test that Version 1 stated inthe future tense is now live and pointing the unfavourable way; this is the principal reason forVersion 2. (ii) The neutrino subsection now carries two values side by side (the m₁ → 0 limit, Q ≈0.585, and the author's separate framework's own prediction Σmν = 59.3 meV, giving Q ≈0.527), with the sensitivity of Qν to m₁ made explicit. (iii) The pole-mass / QED self-energyproblem (Section 7.4) is expanded. (iv) An observation from the author's separate latticeframework — that δ = 2/9 sits at a combinatorial seat 6/27 — is recorded at footnote level withan explicit statement that it is not a derivation. (v) The status of every falsification handle isconsolidated in a new Section 8.