1) Crackpot idea of the day: "the bottom quark is S-dual to the rho meson".

Gorsky
et al conjecture that holographic QCD has a "flavor S-duality" in which
vector mesons are dual to baryons. This is to be realized in string
theory by a web of 5-branes.

Quark-hadron duality shows a kind of continuity between properties of quarks and properties of hadrons.

And
the mass of the rho meson has been estimated at sqrt(6) times the
constituent quark mass; while in the simplest version of Rivero's
waterfall, the bottom quark mass comes out as 2 . sqrt(6) . sqrt(6)
times the constituent quark mass. Also, the Brannen mass scale of the
Koide triple containing the bottom quark equals the mass of the proton,
the prototypical baryon.

2) Vague bonus idea: Koide relations are an echo of this S-duality.

This
just comes from Brannen and Sheppeard's discussion of the discrete
Fourier transform, in the context of circulant matrices (they obtain
Koide masses as eigenvalues of a circulant). One might start with the
realization of geometric Langlands via S-duality, and then look for
analogues over finite fields. Sheppeard has occasionally hinted at something
like this.