What is the reference that states WHY military basic allowance for
housing (BAH) is not taxable.  I don't know if it will be DoD or
DoTreasury or what.  I know in the 2004 Armed Forces' Tax Guide it
says BAH is excluded from Gross Base Pay.  But I want to know why.  If
the reference could be provided for me to answer this I'd be very
greatful.

Well!  That was a lot harder than I thought it would be!
Military allowances are not considered "pay."  Prior to 1986, since
they are not "pay," they were not taxable.  37 USC Section
101(21)provides that the term ?pay? does not include ?allowances.?
http://caselaw.lp.findlaw.com/casecode/uscodes/37/chapters/1/sections/section_101.html

In 1986, Congress passed a new law that specifically permits them to
tax any new allowances that are enacted thereafter.
26 U.S.C. 134.  Link:
http://caselaw.lp.findlaw.com/scripts/ts_search.pl?title=26&sec=134

But the way it does so is tricky.  It states that, as a general rule,
"gross income shall not include any qualified military benefit."  It
then defines "military benefit" as any "military allowance" that was
non-taxable as of 1986.  Well, ALL military allowances were
non-taxable as of 1986.  The purpose of THIS statute was to make
FUTURE grants of military allowances taxable.
See http://www.defenselink.mil/militarypay/pay/tax/01_allowances.html
for a discussion.

That's the legal background.  The basis for it is that military
members are supposed to be provided with free, Government housing. 
But there isn't enough Government housing, so instead of building
more, the Government just pays them an allowance that permits them to
rent their own.
"The intent of BAH is to provide uniformed service members accurate
and equitable housing compensation based on housing costs in local
civilian housing markets, and is payable when government quarters are
not provided."
http://www.armedforces.net/Detailed/610.html

I hope this answers your question.  I had no idea searching the
Internal Revenue Code (Title 26) would be so difficult!

That's AWSOME !! thank you very much for your time.  Glad I could offer a challenge.