Some basic remarks on obtaining the Koide relation, and its generalizations, via string theory. In the standard model, elementary masses and mixings come from yukawa couplings between two chiral fermions and the Higgs field. This is also the case in string theory realizations of the standard model. A sketch of how it works for intersecting branes may be seen in figure 1 (page 3) here. Each distinct species of elementary particle is found at a specific location in the extra dimensions, at a point where two branes intersect; the particle itself is an open string between the two branes. The left-handed fermion is at one location, the right-handed fermion at another location, the Higgs boson is at a third location. The yukawa coupling is a three-way interaction in which e.g. a string jumps from the left-handed site to the right-handed site, by stretching out until it joins with a Higgs string. The probability amplitude for this to happen is dominated by the worldsheet with the minimum surface area, which is the triangle in the picture. To a first approximation, the amplitude equals exp(-area). So if you know the mass matrices you want, this is a way to picture the stringy geometry that you need: the Higgs boson will be localized somewhere in the extra dimensions, the elementary chiral fermions will be scattered around it, and the distances and angles must be such that the areas of these triangles are - ln(yukawa). But you can't just say that you want the strings to be located at specific points, and then just place them there. Or at least, you can't do that in any stringy model that I ever heard of. In real string theory, you'll have an overall geometry for the extra dimensions, and then the branes will occupy hypersurfaces in that geometry, and all the geometric parameters (the moduli) are dynamical. They will settle into a state of lowest energy, and that will determine the relative locations of everything... Perhaps this could be avoided if the background geometry were hyperbolic and rigid, or if numerous branes form a dense mesh so that there's always an intersection point near where you want your particles to be located. But I am not aware of any brane model where that can be done. The masses and mixings present certain patterns or possible patterns, that might guide you in constructing such a brane geometry. But if we take Koide seriously, there's a very special and precise pattern present, specifically in the masses of the charged leptons. In Koide's field-theoretic models, he introduces extra fields, "yukawaons", which enter into the yukawa coupling, in order to produce his relation. In terms of string theory, it's possible that the Koide relation, if it can be produced at all, might be due solely to a special symmetry of the compact geometry and the location of branes within it - that might be enough to induce the mass relation. Or, there might be extra string states involved - the worldsheet may trace out an n-gon with n>3. A further interesting possibility is that virtual branes may be involved - branes that wrap some closed hypersurface in the compact geometry, with which the strings interact; a kind of vacuum polarization. It would be interesting indeed if yukawaons were associated with such "Euclidean branes". (I will also mention again that a Koide relation among pole masses seems to require still further interactions that produce special cancellations, like the family gauge bosons introduced by Sumino. All the mechanisms mentioned above are also potentially relevant here.) How about the generalization of the Koide relation which initiated this thread, the waterfall of quark triplets introduced by @arivero in arXiv:1111.7232? Unlike the original Koide relation, there is still no field-theoretic implementation of the full waterfall, because the triples include quarks with different hypercharges, and that's just more difficult to do. But all my comments still apply, and the paper contains some remarks on the geometry of the mass vectors involved, which, who knows, might be directly relevant to a stringy implementation. There's one more notable phenomenon, and that is the appearance of mass scales from QCD - 313 MeV, 939 MeV - in some of these Koide triples, when they are expressed using Carl Brannen's method. 939 MeV is the nucleon mass and it has been obtained from lattice QCD, but I am not aware of any simplified heuristic explanation of where that number comes from, that QCD theorists would agree with. In a few places in this thread, I have posted about papers which do purport to give a field-theoretic derivation of these quantities (Schumacher in #134, Gorsky et al in #136). The holographic QCD of Sakai and Sugimoto also gives a framework (from string theory rather than field theory) in which the nucleon mass can be obtained, once all the parameters of the brane geometry have been specified. If the QCD scales do appear in the extended Koide relations for a reason, and not just by chance, I think it has to be because there is some QCD-like theory underlying the standard model. There have been many proposals for what this could be, as has been documented throughout the thread on "the wrong turn of string theory". Presumably one should then look for a stringy implementation of QCD mechanisms like those just described, and then rerun the previous arguments about yukawa couplings on top of that.