A Z₃-Symmetric Reading of the Charged-Lepton Mass Phase δ = 2/9 — Axioms, Arithmetic Theorems, and Pre-Registered Tests

Abstract. The status of the phase δ in the Z₃-symmetric parametrization of the charged-lepton masses, √mk =μ(1 + √2 cos(2πk/3 + 2π/3 + δ)), is examined. What is registered is not values but conditions and kill clauses.(a) Layer verdict: the Koide relation QK = 2/3 is a constraint and the complex parametrization is the coordinatesystem of that constraint — QK = 2/3 holds automatically for every δ (identity check deposited). (b) Precisionre-determination: the two inputs e and μ alone give δemp = 0.2222220471(4) (pole-mass convention), offsetfrom exact 2/9 by −1.7515×10−7 (about 440σ). Transported to one-loop QED running masses, δ freezes at0.2225527, +3.305×10−4 from 2/9 — the scheme spread is about 1890 times the pole offset, so the proximityaudit (4.4σ) is conditional on the pole convention, and the naive reading “2/9 holds exactly at some cleanscale” is rejected at one loop. (c) τ-mass discrimination protocol: the QK = 2/3 inversion 1776.969027 ±0.000035 MeV versus the exact-2/9 fixing 1776.9675‒1776.985 MeV, separated by 1.4‒16 keV, adjudication eraσ(mτ) ≲ 10 keV. (d) Registered statements: R5 (δν = 13/27, Σmν = 59.32 / 59.43 meV), registered in the deposit of2026-07-04, and its derivatives (mi, mβ, mββ, the ordering lock); the solar-angle branch R6 (TM1 sin2θ12 =0.31834 ± 0.00049 / TM2 0.34083 ± 0.00024 — court: JUNO); the CP-phase sum rule R7 (court: DUNE/Hyper-K);a structural anti-prediction (no mixing of the dark-photon candidates with neutrinos); the production route ofthe ±4/3 slots. (e) Kill conditions K1‒K10 in full and the scoring protocol; four statements of an unfalsifiableform are separated into a non-scoring section. δ = 2/9 remains a fit with no derived mechanism, and survivalproves nothing.