The Koide relation explained: the most constraining result of TQNT

Q = 2/3 satisfied to 3.3 ppm · the tau mass predicted to +22 ppm · and since July 2026, the electron mass computed to ±3% with no parameters.

A coincidence at 10 parts per million, with no explanation

In 1981, the Japanese physicist Yoshio Koide noticed that the masses of the three charged leptons — the electron, the muon and the tau, spread over four orders of magnitude — satisfy a relation of extraordinary precision:

Q ≡ (m_e + m_μ + m_τ) / (√m_e + √m_μ + √m_τ)² = 2/3

On the measured values, Q = 0.666 664 5 — a deviation of only 3.3 ppm from exactly 2/3. The Standard Model offers no explanation: the three masses are three free constants, with no relation between them. Forty-five years later, this coincidence remains one of the sharpest numerical riddles of particle physics.

3.3 ppmprecision of Q = 2/3 on the measured masses

+22 ppmm_τ predicted (1776.97 MeV) vs measured (1776.93)

±3%m_e computed with no parameters (v2, July 2026)

The TQNT reading: three generations = three modes of the trefoil knot

In the Topological Knot Quantum Theory, the charged lepton is the trefoil knot 3₁ — the simplest of all knots, three crossings. Now the trefoil possesses an order-three symmetry (Z₃): it is invariant under a one-third turn. The three generations (e, μ, τ) are the three eigenmodes of this symmetry — which is why there are exactly three generations, neither two nor four. This structure enforces exactly the Koide parametrisation (the cube-root-of-unity identities), and the amplitude √2 then yields Q = 2/3 algebraically.

The relation being a single equation among three masses, it predicts the tau mass from those of the electron and the muon — this is Koide's historical 1981 relation: m_τ = 1776.97 MeV, against 1776.93 measured. +22 ppm.

What version 2 brings (July 2026): from postulate to mechanism

The first version of the paper (June 2026) postulated the √2 amplitude and left the phase δ = 2/9 as an orphan hint — saying so frankly. The consolidations of the V6 monograph (the "rail magnetism", July 2026) changed the status of all three ingredients:

√2 found its carrier. A degeneracy theorem shows that the circuit energy cannot separate the muon from the tau: the only possible breaker is the circulation sense of the current on the knot (odd under time reversal — and the mirror exchanges μ and τ: the CPT structure is automatic). A mode coupled to both senses has two channels against one: the amplitude ratio is √2 — the same √2 that appears independently in the heavy-baryon sector of the monograph.

The phase is decomposed: δ = π/12 − ε. π/12 = 2π/24 is one step of the smallest trefoil circuit realizable on a lattice (24 steps — a theorem of knot mathematics), and ε = 0.0396 ≈ √(m_e/m₀) measures the quasi-nodality of the electron: it sits almost exactly on the zero of the amplitude. The electron's lightness is not an accident: it is a nodal protection.

The scale is identified. The common scale of the three masses equals (3/4)√σ_χ = 313.85 MeV — the same string tension that governs the whole hadronic spectrum of the monograph — against 313.85 extracted from the data: agreement at 5×10⁻⁵.

And the electron mass is computed. The allowed excitation mode of the tube (selected by line-current conservation), evaluated through a retarded Green energy on the ideal trefoil, gives m_e = 0.518–0.522 MeV, against 0.511 measured: +1.4 to +2.2%, zero adjustable parameters.

Quantity	TQNT prediction	Measurement	Deviation
Q	2/3 exact	0.666 664 5	3.3 ppm
m_τ	1776.97 MeV	1776.93 MeV	+22 ppm
scale m₀	313.85 MeV	313.85 MeV (extracted)	10⁻⁵
m_e (computed, v2)	0.518–0.522 MeV	0.511 MeV	+1.4 / +2.2%

⚖ Epistemic honesty, claimed as a feature

Every statement in the paper carries its status: derived, conditional (derivation under a declared identification), postulated or confirmed.
The v1 caveat is kept: the naive "spin-½ √2" argument is wrong (it is the spin-1 coefficient) — the identified carrier is of a different nature, and the failed derivation attempts remain documented.
Every link of the v2 chain is conditional on a declared identification — none is a fit to the data.
The result is falsifiable: any improvement in the measurement of m_τ or m_μ can refute Q = 2/3.

✓ In one sentence

Where the Standard Model has three free numbers, TQNT reads one object — the trefoil knot — whose three modes give the three generations, whose symmetry gives Q = 2/3, whose geometry gives the phase, and whose computation gives the electron mass.

🔖 Reference

L. A. Cariou, “The Koide relation from the Z₃ symmetry of the trefoil” (v2, July 2026), Zenodo, FR + EN — doi:10.5281/zenodo.20817000.

Read the paper (Zenodo, FR + EN) → The TQNT V6 monograph → The TQNT page →