IMO, you're definitely on the right track

Consider the satellite weapon theory. Satellites in geo-synchronous orbit finish 1 orbit in about 108 mins. So every 108 minutes there is an opportunity OR a requirement to contact it. (This IMO is the likeliest source of "the incident"). So where is the sat at the various points along it's orbit, if the "numbers" are merely corresponding to minutes of travel?

Assume the satellite is traveling East to West and is starting somewhere in the Pacific Ocean. The "numbers" correspond with major world capitals/cities:

0 minutes = Above the island
4 mins: Tokyo
8 min: Bejing
15 mins: India
16 min: Middle east Cities
23 min: European capitals
42 mins: DC/New York
108 mins: island again

This is my theory. The hatches are missile silos now used as science labs and world peace enforcement silos. Hanso built them. He then told the world's leaders that they best behave, and not threaten mankind with nuke war or he'd blast them with an EMP from his own satellite. The control point of that satellite is off the grid on an island WHICH IS HIDDEN FROM DETECTION, due to it's electromagnetic anomalies and technology. So:

The entry of the code into the hatch computer does potentially a few different things...but the following is my pet fave:

It causes the islands invisibilty to be maintained by 'zapping' the spy sattelites that are twirling around the planet looking for the island; the worlds powers will not be happy to have some rogue power hold a gun to it's collective forehead, and would be out to find the island....WHEN THE BUTTON IS NOT PUSHED, THE BLAST DOORS DROP DOWN to protect the hatchees from attack, as their location will be learned.

This has happened once already...now referred to as the "Incident"...the resulting attack poisons the island's environment is, for a while quaranteened...lest ye risk acquiring the "sickness". This same sickness renders you infertile, so that for your tribe to go on, so you must 'acquire' any available children...

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Sorry...I think my Kharma just ran over your Dogma...