J.H. Schwarz, Phys.Lett.B37:315-319,1971 (also anticipated in a small comment in Phys. Rev. D 4, 1109–1111 (1971) )

EDIT: other references using "fermion-meson": http://dx.doi.org/10.1016/0550-3213(74)90529-X Nuclear Physics B Volume 74, Issue 2, 25 May 1974, Pages 321-342 L. Brink and D. B. Fairlie; http://www.slac.stanford.edu/spires/...=NUCIA,A11,749 Nuovo Cim.A11:749-773, 1972 by Edward Corrigan and David I. Olive.

http://vixra.org/abs/1102.0034

If both F(s) and G(s) are absolutely convergent for s > a and s > b then we have

[tex] \frac{1}{2T}\int_{-T}^{T}\,F(a+it)G(b-it)\,dt= \sum_{n=1}^{\infty} f(n)g(n)n^{-a-b} \text{ as }T \sim \infty. [/tex]

... in the context of dirichlet series, to decompose the Riemann zeta, finding a pair of functions where f(n)g(n)=1 for all n. Note that they not need to be multiplicate, do they?

The simplest not trivial case, had we absolute convergence, would...

Both
U(1)xSU(3)xSU(2)xSU(2)
and the full
SU(4)xSU(2)xSU(2)
live in 8 extra dimensions, as F-theory lives, and they probably need one of the extra dimensions to be infinitesimal, because U(1) B-L is not gauged, at least not at the scales we know.

The manifolds, by the way, are
S1 x CP2 x S3
and
S5 x S3
respectively.

The later is more complete and it allows to generate Witten's manifols almost automagically. But...

From a point of view, there are no Koide doublets. If we define Koide's relationship as coming from three matrix conditions:

[tex]M^{1/2}= A + B[/tex] with
1) A multiple of the identity
2) B traceless
3) [tex]Tr (A^2) = Tr (B^2)[/tex]

Then the 3 in the 3/2 factor is really the dimension of the matrix, and thus the factor is 2/2 for Koide's doublets and 1/2 for Koide "singlets". So in this sense there are no Koide doublets.

...

I though it was after 2007, because when in December 2006 the topic was raised by Alain Connes, I just answered the usual folklore: "You need at least three for CP violation".

But in July 2005 I raise the topic of composite supersymmetry in this blog post
http://conjeturas.blogia.com/2005/07...-del-muon-.php
(regretly my other blog, "El ultimo hovercraft", was lost after a wrong backup)
The my webpage has a pdf dated...