Mike Weeks wrote:
Assuming in the first place that there were indications there was such a
sea-borne threat to Hawaii, how would it be done if there weren't even enough
PBYs in Hawaii to cover "every direction" of the compass -- for what sector
width and for what range ...????

Well, look at what they were actually doing.

http://www.history.navy.mil/docs/wwii/pearl/ph11.htm
VP-21 7 planes - in the air conducting search 120° to 170° to 450
miles from Midway.
....
VP-14 3 planes - in the air on morning security patrol armed with
depth charges.
....
VP-24 4 planes - in the air conducting inter-type tactics with
submarines.
....
To summarize the foregoing, at the moment the first bomb dropped,
aircraft of this command were in the following condition:

14 - in the air (7 on a search from Midway).
58 - on the surface ready for flight in four hours or less.
9 - undergoing repairs.
81 - Total.

The only early warning system Hawaii had (besides the overlooked radar)
weren't on patrol, instead they were doing ASW exercises.

Applying VP-21's numbers we get 51 planes required to do a 360° search
to 450 miles so just 20 planes in the air could have done a search
beyond the strike radius of any Japanese carrier based aircraft.

-HJC