A Monte Carlo Null-Model Test of an Outer-Soddy Completion of the Koide Lepton Triple

The Koide relation \( Q = (\sum m_\ell)/(\sum \sqrt{m_\ell})^2 = 2/3 \) for the charged leptons has held to one part in \( 10^5 \) for over forty years without an accepted derivation and is widely regarded as numerology. This paper takes the relation as a clue rather than an endpoint. Treating lepton mass square roots as Descartes-circle curvatures, the outer root of the Descartes quadratic equals the closed form \( \mathcal{F} = e_1 - \sqrt{p_2} \) when Koide holds exactly (Proposition 1); equivalently, \( \mathcal{F}^2 = \alpha_K^2\,\mu_\star \) with \( \alpha_K^2 = 5/2 - \sqrt{6} \) and \( \mu_\star = \sum_\ell m_\ell \) the lepton-sum scale. The three-input symmetric-polynomial identity thus collapses to one dimensionless Koide-determined constant times the lepton-sum scale. Kocik [1] first observed a Descartes-like reading of Koide; our mutually-tangent variant is mathematically distinct but follows the same geometric spirit. The four-curvature completion carries a testable consequence absent from the bare three-mass relation: evaluating the squared fourth curvature numerically, \( \mathcal{F}^2 = 95.113 \) MeV, and comparing against the strange-quark \( \bar{MS} \) mass at \( \mu_\star \) within current lattice precision yields a residual of \( +0.04 \) MeV against \( \pm 0.69 \) MeV, about \( +0.06\sigma \). The lepton-side quantity is fixed to better than \( 0.01\% \); future lattice improvements will sharpen or refute the present numerical agreement. To our knowledge this paper implements the first Monte Carlo null test of the Koide relation under a random-spectrum prior; a Koide-conditioned null-model calibration across four prior shapes pre-registered for the analysis gives hit fractions at the sub-percent level — model-conditional frequencies, not \( p \)-values. Scale, input, prior, and filter sensitivities, together with the error budget, are reported; full Monte Carlo protocols, numerical output, and pre-registration are in a companion methods note [2].