SB wrote:

By chance is there someplace where it actually says that in the FAR that the
tower "owns" the airspace? I'll assume "own" means you must contact them if
you intend to operate in the extension and they have the right per a
particular section of the FAR to deny you access.

91.155 (d) no person may take off or land an aircraft, or enter the
traffic pattern of an airport, under VFR, within the lateral boundaries
of the surface areas of Class B, Class C, Class D, or Class E airspace
designated for an airport --

(1) Unless ground visibility at that airport is at least 3 statute miles; or

(2) If ground visibility is not reported at that airport, unless flight
visibility during landing or takeoff, or while operating in the traffic
pattern is at least 3 statute miles.


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The extensions are the "class E airspace designated for an airport."
That's why they exist, they are left over from the old control zones
which had such extensions and the same rules (Airport Traffic Areas:
control towered space was previously not controlled airspace, just
a communications requirement).