"Below ground state helium" turns up zero hits on google. which is curious .
in light of the assertion that 'physics is almost complete'. 

Has this possibility not been addressed adequately ?  IOW .does anyone
recall if this (putative) species of fractional helium has been mentioned by
Mills/BLP or by others? . and if not, why not?

It could be a better candidate for a portion of 'dark matter,' if it existed
in 'deep redundancy,' than the usual suspects. Matter of fact, it is fairly
illogical, as Mills asserts, to suggest that 'hydrino-hydride' can make up a
proportion of dark matter, yet 'heliinos' or pycnohelium, do not exist.

Helium is the second most abundant element in the universe, accounting for
24-25% of the elemental mass of our galaxy (aside from dark matter). I
intentionally phrased that statement in this way because the normal view is
that hydrogen is more prevalent (90+%) in the number of atoms, is misleading
in terms of actual mass, and ultimately in terms of relative importance. 

If your intent is to make a point, then helium is clearly the most neglected
atomic-molecule in the universe, and if it was to be a preferential product
of 'redundancy' who is to say that it is not more prevalent than hydrogen,
given that dark matter itself is the wild-card in terms of total mass .
and/or that LERN, as a QM reaction operating over billions of years, would
favor pycnohelium "as the end of the line", over any other species. :-)

Jones

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