I can't find any evidence to suggest this is a jazz term, and the earliest example I found is from 1995. The alternative da bomb dates from at least 1994.
There's no entry in a number of slang dictionaries (A Jazz Lexicon, A Dictionary of Slang and Unconventional English, Shorter Slang Dictionary, The Slang and Jargon of Drugs and Drink, A Dictionary of Cliches, American Heritage Dictionary of Idioms, British English A to Zed). This suggests it's a recent term.
Like your linked Jazz Slang, it's often claimed to be 1950s or jazz slang but there's no evidence for it. This list of 50's Slang includes:
The Bomb --- Very cool.
The Crusader's new disc, "Louisiana Hot Sauce" is "the bomb."
The list is undated but was first archived in 2002. Further, The Crusaders' released Louisiana Hot Sauce in 1996. The others all seem to be copies of this (or another).
Although not exactly the same, the closest thing in the OED is:
1e. A success (esp. in entertainment); also U.S., a failure. So phr. like a bomb and varr., with great speed; with considerable effectiveness or success. colloq.
Here's the first and a couple of other quotations:
1954 Amer. Speech 29 99 Like a bomb,..very fast.
1961 New Yorker 28 Oct. 43/2 What had once been called a failure became a ‘bomb’.
1963 The Beatles 5 Once, Paul McCartney and I played Reading as the Nurk Twins. Went down a bomb, I recall.
All the quotations are "a bomb" not "the bomb" and are quite different.
The only relevant quotation from your linked Lexical Investigations: Bomb is fairly recent:
“Your magazine is the bomb! I really like the comics and when you make fun of that Spears girl.”
—Spin, September 2002
The earliest "is the bomb" I found in Google Groups (there may be earlier, but Google recently redesigned Groups and crippled the search) is this exchange from August 1996:
On 21 Aug 1996, Cheezmelt wrote:
If I wrote a song, it'd be called " Beck is the Bomb"
And it would be the Bomb.
Or maybe it would be called "Loopity loo and your mom too"
I haven't decided
Geez don't you kids know anything?
whenever something is the bomb,
and you specify that it is THE bomb,
you must follow it with the phrase, YO.
for example, "that shit was the bomb, yo!"
as opposed to:
when something is merely bomb, and there
is no the, then you would say something like
"That's some bomb acid!"
stay in school!
There's nothing before 1996, and quite a few after and into the 2000s and this decade.
Subzin movie subtitles
The earliest I found in film subtitles is the 1995 Spike Lee film Clockers which uses it twice:
00:04:35 # Blue collar comes to bourgeois
00:04:37 # Depressed in your chest... #
00:04:39 - Chuck D is the bomb, boy. - What?
00:04:41 Get the fuck outta here. Chuck D ain't shit.
00:04:44 That nigger Chuck D is assed out, and the rest of Public Enemy.
01:12:18 - One drink? Wasn't drunk? - Nope.
01:12:21 - He was rude. - (Bartender) Yeah.
01:12:26 'Ooh! I'm just gettin' warmed up. But it's The Bomb.'
01:12:30 Was he with anybody?
01:12:33 He came in alone. Might've had a conversation.
The film was released in September 1995 and based on a 1992 book by Richard Price, although the book doesn't appear to use the phrase.
There's even more and earlier Google Groups results for the variant da bomb, such as this from soc.culture.filipino in December 1994 ("SCF at REDj is DA BOMB...."):
Actually, "DA BOMB" is part of the continuous urban parlance that Filipino teenagers tend to borrow from African American teenagers. "DA BOMB" is a comprable adjective to such words used in the past, like "swell," "groovy," "cool," "radical," "gnarly," "awesome," "def," and "hype."
Dr. Vicente Rafael, a Filipino-American and an associate professor at UCSD's communications department describes this phenomenon as "downward assimilation"; Filipino-American teenagers borrow the language, clothes, and mannerisms from pop black culture because they perceive it to be a more sexual, more agressive (as in violent) culture, characteristics which they perceive to be wanting in their own native culture.
A few days later in the same group ("*** DA BOMB ***"):
OKay ppl... I haven't been posting as on SCF due to the fact I'm forming a new IRC channel w/ MinMei aka NeuSpeed. This new channel will hopefully be Da BOMB.... N E Wayz... I see that ppl responded to the comment I made at REDj's party...
The Party Was Da BOMB (the best in the West)... and it seems that ppl around here (Elson, Rhett...) no names will be mention seems to put that title to me... Okay Elson... I'll go on an airplane and say to the pilot this "Da BOMB"!!!! =) NE Wayz.... look out for the new channel...
There's an earlier use in comp.sys.mac.apps from July 1994. The post discusses Apple computer crashes ("Has anyone else had problems with Appleshare causing a system bomb on startup?") but the subject seems to be a passing reference to the phrase:
Appleshare and "da BOMB"