What Held Spin Owes the Leptons

This program has a mass account on its record. It says mass is held spin: a thing weighs what its held spin weighs, spin held at a center that could not pair off, read through one coupling — the Einstein–Cartan response of torsion to spin. The account named its own test when it was stated. The charged leptons, the electron, the muon, and the tau, obey a precise pattern in their masses, the Koide relation, and the account either writes that pattern in its own terms or it does not. This paper runs the test. The pattern is real. From the Particle Data Group's 2024 masses, the Koide ratio comes out Q = 0.666661 ± 0.000007, inside one standard deviation of the claimed 2/3. The bill is that precise, and it is owed. I then walked every route the account itself offers toward paying it — the held spin the three leptons share, the pairing ledger's counts, the one coupling run honestly, the numbers the wider framework owns, and the square-root shape of the relation itself. Every route stops at the same place. The account holds nothing that differs among the three leptons. Each one carries spin one-half. The coupling is one and the same for all three. A ledger with no rule for assigning its counts, a coupling with no density to work on, and a mass read straight off held spin cannot produce three different masses, let alone these three. Without a fitted number, the only masses the account can give the three leptons are equal masses, and equal masses give Q = 1/3, not 2/3. So the verdict is plain: the account does not pay the Koide bill, and its claim on the charged leptons' masses ends here, at exactly that size. What the account says elsewhere — about the vacuum, about the crossing — this paper does not touch. In the lepton sector, it is done.