What is new with Koide sum rules?

[arivero]

Time ago the hard printed edition of the pdf had a last page with an illustration of how the measures had evolved across editions. Remembering that, I have setup a similar evolution for the Koide tuples

https://arivero.github.io/pdghist/predictions.html

 

[mitchell porter]

How do these papers fit into a larger research program of either of the authors?

Neither author has prior work on Koide that I can see. Hübner did some QCD and thus he knows QFT topics like renormalization schemes. Shulga worked on qubits from a physical rather than computational perspective (e.g. superconducting qubits), and maybe something there inspired his mechanism. But otherwise it seems a coincidence that two independent authors had their (very different) first Koide papers released on arxiv on the same day.

 

[arivero]

it seems a coincidence that two independent authors had their (very different) first Koide papers released on arxiv on the same day.

Actually three Mine was removed because just resending now causes an autodelete, even if the resend includes doi and journal. It is a sort of peak surely caused by the AI leverage, but still funny.

 

[Roberto Pavani]

I've been looking at the geometric origin of the phase $\delta = \pi/12$ (15°) that appears in the Koide parametrization for charged leptons.

In 4D Euclidean space, the rotation group decomposes as $SO(4) \cong SU(2)+ \times SU(2)-$ (isoclinic decomposition).
If you have a helical structure with $C_3$ symmetry (three-phase, 120° shifts), the closure condition at a nodal surface forces a total twist of $\pi/2$.
The three-phase offset accounts for $\pi/3$ of that, leaving a residual of $\pi/6$, distributed equally between the two isoclinic planes: $\pi/12$ per plane.

This gives $\delta = \pi/12$ as a geometric necessity, not a fit. The factor 3 between leptons ($\pi/12$) and quarks ($\pi/4 = 3 \times \pi/12$) would then correspond to the three color directions in the transverse space.

Has anyone explored the connection between the isoclinic decomposition of $SO(4)$ and the Koide phase? The coincidence $\pi/12 \leftrightarrow 15°$ seems too precise to be accidental, especially given that $SO(4) \cong SU(2) \times SU(2)$ is unique to 4 dimensions.

 

[mitchell porter]

π/12 appears as part of a formula for a Koide phase in the neutral leptons (see Brannen's "Koide Mass Formula for Neutrinos"), not charged leptons. The phase for the latter is 2/9 radians (no appearance of π). The neutrino masses are not known with great precision and so there is no empirical confirmation of this π/12 formula, the way there is for the 2/9 formula, it is more theory-driven. It would be interesting to try to adapt Shulga's mechanism, mentioned in immediately preceding posts, and which introduces and then cancels factors of π, to the Brannen neutrino phase. And it would be miraculous to find an "isoclinic" origin of 2/9!

 

[Roberto Pavani]

Thank you for the correction — you're right that the Brannen phase for charged leptons is 2/9 radians (no π), not π/12. I was imprecise.

Let me reformulate the observation more carefully.
The connection I'm exploring is indirect: in 4D Euclidean space with a three-phase helical structure ($C_3$ symmetry, 120° shifts), the closure of the helix at a nodal surface forces the gradient perpendicular to the axis (total angle $\pi/2$).
The three-phase offset accounts for $\pi/3$ of that rotation. The residual $\pi/6$, distributed equally between the two isoclinic planes of $SO(4) \cong SU(2)_+ \times SU(2)_-$, gives $\pi/12$ per plane.

This $\pi/12$ doesn't claim to be the Brannen phase. Rather, it enters the coherent closure condition as $7\pi/12 = 2\pi/3 - \pi/12$ (the target phase for counter-rotating closure). When this closure is imperfect (small mismatch $\delta$), the forced precession onto one of the three $C_3$ directions introduces a correction of the form $\sqrt{m_i} \propto 1 + A\cos(\theta_0 + 2\pi(i{-}1)/3)$, and the trifase identity $\sum_{\tau=0}^{2}\cos^2(\theta_0 + 2\pi\tau/3) = 3/2$ then gives $K = 3/2$ exactly independently of $A$ and $\theta_0$.

So the question I should have asked is: has anyone explored the isoclinic decomposition of $SO(4)$ as the geometric origin of the closure condition from which a Koide-type parametrization can emerge?
The factor 3 between leptons and quarks would then correspond to the number of transverse directions (color), not to the phase itself.

Regarding Shulga's mechanism, interesting. His compact dimension with antiperiodic boundary conditions and the Casimir-like shift producing 2/9 is structurally reminiscent of a closure condition on a cycle.
I'd be curious whether his "a-field" dressing could be reinterpreted as a frame connection on a helical structure.

 

[arivero]

they are a lot of elegant results that are just generic of the theory of quadrics. Basically it is possible, except for the integrality, to repreoduce all the symmetries of https://en.wikipedia.org/wiki/Markov_number Markow trees of tuples, find the critical coefficient where the atractor of the tree becomes fractal, look for accumulation points. A loot of very pretty geometry, but that really one can do with different kinds of quadrics.

 

[jedishrfu]

Your post seems to be related to your Zenodo papers. I’d like to remind you that these questions are okay as stand-alone questions but not if they bring in your unpublished Zenodo papers. PF explicitly state that we only discuss peer-reviewed papers from reputable journals. Sadly, zenodo does not count as either a published paper nor as a reputable journal.

If the discussion does extend into your Zenodo papers then the PF mentors will be forced to delete your relevant threads and give you a personal speculation warning or personal theory site ban.

Please take some time to read our site global guidelines.

 

[Roberto Pavani]

Noted. My posts in this thread discuss the isoclinic decomposition of SO(4) and its relation to closure conditions, standard differential geometry. No personal papers are referenced or needed to follow the argument.

 

[arivero]

Linking the question of Pavani with mine, the point is that here there is a fine thread for Mathematics that somehow is not fully exposed in online forums: to characterise the solutions of permutation invariant equations Q(x_1,x_2,...x_n)=0. Markov's
$$x^2 + y^2 + z^2 - K x y z = 0$$
have been very studied because it has Diophante solutions when K=3.
Koide generalisations
$$(x^2 + y^2 + z^2 )+ K \times (xy + yz + zx) = 0$$
are fine objects too, but they have no such elegant diophantine solutions and it is not easy to find studies going concrete with this, or at least digitised ones. This history goes as deep as Newton and Girard, and of course linked to the problem of having all the solutions as roots of a single cubic equation. Pure maths, anyway.