The Koide relation is the odd one out among particle-physics coincidences: one part in 105, forty-plus years, no accepted derivation. I came to it from an engineering direction, a coupled solver needed fermion masses as a calculation rather than a lookup table, and stayed for the mystery: Koide published it in 1982 in a now-defunct journal, it went on to predict the tau mass, and it has only sharpened as the lepton masses became precision quantities. Kocik observed that the relation admits a Descartes circle reading, mass square roots as curvatures of mutually tangent circles. This paper pursues the variant in which the circles are mutually tangent and asks the obvious next question: a Descartes configuration of three circles has a fourth curvature, fixed by the other three. What is it?
The answer is a single number fixed by the lepton masses with no free parameter: F² = 95.113 MeV. I computed it fully expecting the fourth circle to land on nothing; instead it landed on the strange-quark mass, evolved to the lepton-sum scale, the same number I had just written down as Σ√mℓ, at +0.06σ: a residual of +0.04 MeV against ±0.69 MeV lattice precision. I was on an airplane at the time and nearly fell out of my seat. That is either a clue or a coincidence, and the paper treats the distinction as the actual subject.
So the second half of the paper is an adversarial exercise against itself. The observation is disclosed as predating the test; the analysis protocols, prior shapes, sample sizes, hit criteria, seeds, cutoffs, were committed to a public, version-pinned repository before any Monte Carlo was executed. Four null tests: random lepton-like spectra under four priors drawn from the flavor-model literature (hit fraction 0.340%, stable within a factor of 2.4 across prior shapes); a measurement-noise parametric bootstrap of the PDG inputs; enumeration of 24 pre-specified lepton functions of comparable algebraic complexity (two land within 1σ of ms, and those two turn out to be algebraically identical under exact Koide); and an exploratory trajectory across the FLAG 2013–2024 strange-mass series. The companion methods note carries the full protocols and complete numerical output.
The paper's position is deliberately modest: the lepton-side quantity is fixed to better than 0.01%, so future lattice improvements will sharpen or refute the agreement. It is a candidate result with its failure modes stated, not a claimed discovery, and it is the empirical anchor for the algebraic sequel that became the MPLA paper.