It began with a discussion of Koide relations. "fzero" said he thought it was all numerology, just like the fact that 1/alpha, at the Higgs mass scale, is approximately equal to the Higgs mass in units of GeV.
Initially I concurred that the latter fact, at least, must be a coincidence. But then I noticed that 1 GeV is rather close to the nucleon mass (939 MeV). So I decided to think the impossible for a while. Could it actually mean something, that the Higgs boson weighs about the same as 125 protons? In fact it is somewhat more than that, but 125 would serve as a placeholder in my deliberations.
Later I recalled that the VEV of the Higgs field is about twice the Higgs boson mass. This was more promising. One of the mysteries of Koide-ology is the appearance of quantities from QCD, as the mass scales of the e-mu-tau and b-c-s triples. In the standard model, the masses of those particles are obtained as (yukawa coupling) x (Higgs VEV).
The standard model, even without a Higgs field, still has a Higgs mechanism, thanks to the quark-antiquark condensate. But the VEV is measured in MeVs rather than GeVs. I began to develop the notion of such a condensate, being somehow weighed down with proton-antiproton pairs - 125 of them...
I came up with a silly visualization based on the idea of a pentagon "cubed". A pentagon is made of five line segments, a square is a line segment times a line segment, a cube is a line segment cubed. It should be possible to multiply three pentagons in a certain sense, to produce a six-dimensional object made up of 125 cubes, the cubes consisting of every possible product of an edge from each pentagon.
In the 1990s, Witten discovered a notion of baryons as branes with strings hanging off them (attached by just one end to the brane); the strings are the quarks. So here you should imagine a torus in each cube of the pentagon-cubed - the torus represents a virtual proton-antiproton loop - and then up- and down-flavored quark-strings hanging off the tori.
Finally you should suppose that this construct exists at every point in space - perhaps in extra dimensions surrounding our brane-world - and that the virtual up and down quarks form the meson condensate of the Higgsless standard model.
And that was as far as I got. So imagine my surprise when the next day, I saw Marni Sheppeard blogging about how to get Koide triple mass scales from "three pentagons". The coincidence was not only uncanny, but also somewhat unwelcome, since that part of her theory looks messy and complicated to me.
About a week after that, I was reading the latest version (number 5) of her opus on scattering. Sheppeard's ideas defy summary, but let's say that in her theory, standard model fermions are braids (that are actually morphisms in a category), the dark sector is made of mirror braids, and rest mass comes from a cohomological composition of braids and mirror braids. ("Cohomology" is little more than a word to me, but I believe that taking the product of a vector and a 1-form is the algebraic prototype here.)
In various places, she remarks that maybe the mirror partners of SM fermions are dark baryons. That sounds crazy, I thought... then I realized it is not so far removed from the notions that I was just describing. There is even such a thing as baryonic cohomology.
So where do things stand?
I find it very hard to believe that the number 125 has any deep meaning here. Common sense says it just served to inspire a visual picture, which in turn only matters as a gateway to a more abstract idea, that the Higgs field could be a QCD meson condensate weighed down by virtual nucleons. That, I believe, has the potential to explain the coexistence of Koide numerology and the SM Higgs mechanism.
But it's interesting to note the multiple points of contact with Sheppeard's work. They represent one of the more exotic directions one could take the idea, alongside a more conservative field-theoretic approach.