Has a very light (less than 1eV) Higgs mass been excluded by theory and/or experiment?
Yes. Masses up to at least some 90 GeV have been excluded by existing experiments, see for example HIGGS BOSONS: THEORY AND SEARCHES; Bernardi, Carena, Junk.
An extremely light Higgs (less than < 1ev) would be a catastrophe in astrophysics. Top amongst many other problems, it would destabilize stars like our own Sun.
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