These totally semantically separate categories are a direct result of the
circumstance that PIE initial *a- is shortened from *a:- deriving from
pre-PIE *Ha(:)-, itself the result of Nostratic *?a, *ha, and *Ha (Nostratic
*H = voiceless pharyngal fricative, Arabic dotted h; contrasting with
pre-PIE *H, which is simply [h]).
Nostratic (and PIE) *la:- means 'move back and forth'; and *Ha, which
becomes pre-PIE *Ha(:) means 'water'; combined, this root, Nostratic *Hala:-
means to 'wander back and forth as light does on water in gentle motion, and
is the source of *al- listed under D.
C. also is built on this second element: *la:-, 'move back and forth'; but
the first element, pre-PIE *Ha(:) derives from Nostratic *?a, 'top'; so, to
'move back and forth on top of' = 'grind'.
B. has the same elements as C. but interpreted different: *Ha(:)la:-, 'move
back and forth _at_ the top' = 'gently sway at the top as _tall_ grass does
in a gentle breeze'.
A. is built on *la:- again but with the first element Nostratic *ha, 'be
across from': pre-PIE *Ha(:)la:-, 'move back and forth at a point across
from' = 'be (in motion) on the other side'.
It is annoying that none of these roots can be attested in the canonical
citation form: *Ha(:)l- / *Ha(:)lə- / *Ha(:)la:- but that is the luck of the
I am frankly at a loss, Brian, to begin to understand how the standard
'Laryngeal' Theory could explain these similarities more cogently.