The Koide Relation: A 40-Year Mystery Resolved by Register Geometry
A Formula That Should Not Exist
In 1981, physicist Yoshio Koide discovered something strange. If you take the three charged leptons — the electron (\(m_e = 0.511\) MeV), the muon (\(m_\mu = 105.66\) MeV), and the tau (\(m_\tau = 1776.9\) MeV) — and compute:
\[K = \frac{m_e + m_\mu + m_\tau}{\left(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau}\right)^2}\]
you get, to extraordinary precision:
\[K = \frac{2}{3}\]
This holds to better than 1 part in \(10^5\). Three masses spanning four orders of magnitude conspire to produce a clean fraction. No theory predicted it before Koide found it. None has fully explained it since.
Until PLVS.
What the Register Geometry Says
In the PLVS framework, the lepton mass pattern is encoded in a specific defect configuration of the \(Y_9\) register. The key quantity is the ratio of two lobe radii:
- The outer lobe \(Y_9\) has radius \(b\)
- The inner lobe \(Y_7\) has radius \(a\)
For the defect pattern D=(2,0,0) — the configuration associated with the lepton sector — the geometry forces:
\[\frac{b}{a} = \frac{R_9}{R_7} = \sqrt{2}\]
This is a pure geometric result. Now the Koide parametrisation for three masses satisfying \(K=2/3\) is:
\[\sqrt{m_k} = M\left(1 + \sqrt{2}\,\rho\cos\!\left(\delta + \frac{2\pi k}{3}\right)\right), \quad k=0,1,2\]
The \(\sqrt{2}\) factor in this formula — the one that sets the amplitude of variation around the mean — is exactly the lobe radius ratio \(b/a = \sqrt{2}\) from the register geometry. The three lepton masses are the three orientational readouts of the D=(2,0,0) defect, spaced \(120°\) apart around the phase orbit.
The Koide ratio \(K = 2/3\) follows directly. It is not fitted; it is the geometric consequence of the \(\sqrt{2}\) lobe ratio.
Why Two Thirds?
The underlying geometry is the decomposition of the tetrahedral register into parallel and perpendicular components relative to the body diagonal \((1,1,1)/\sqrt{3}\).
For three equal-weight vectors on a sphere arranged symmetrically at \(120°\) intervals in the equatorial plane:
- \(\frac{1}{3}\) of each vector’s squared length lies along the body diagonal
- \(\frac{2}{3}\) lies perpendicular to it
This 1/3–2/3 split is a pure fact about tetrahedral geometry. The Koide formula says exactly that the lepton \(\sqrt{m}\) vectors are in this configuration. PLVS explains why they are: it is because the D=(2,0,0) defect forces the three phase-readout directions into the equatorial plane of the tetrahedral register.
The Precision
The PLVS derivation gives \(K = 2/3\) exactly. The measured value is \(K = 0.666659...\) — consistent with \(2/3\) to the precision of the lepton mass measurements. No residual.
The Two-Path Confirmation
What makes this result especially robust is that it was derived two independent ways:
Path A (Boolean-Clifford lift): Starting from the AND-carry structure of the \(\mathrm{Cl}(0,3)\) algebra, the Frasor defect configuration is constructed from first principles. The Koide parametrisation emerges from the three-fold phase readout of this object.
Path B (Inverse DFT): Taking the Koide mass relation as the target and applying an inverse three-point discrete Fourier transform, the resulting object has exactly the structure of the PLVS Frasor.
Two entirely different mathematical starting points produce the same geometric object. That convergence is not coincidental.
What This Means for the Mass Hierarchy
The Koide derivation does not yet give the absolute mass scale — why the electron has mass \(0.511\) MeV rather than \(5.11\) MeV or \(0.0511\) MeV. That is the open problem of deriving the \(V_{\rm AND}\) surface tension \(\sigma\) from the substrate dynamics (see the post on the phase-bubble mass formula). But it does give the exact ratios between the three lepton masses — and it gives them from geometry, not from fitting.
Full derivation: Preprint v2.9 — doi:10.5281/zenodo.20484732