NavList:

David C you asked: In another thread Frank asked if the Nautilus had really reached the North Pole. Something occurred to me ...... could  the crew of the submarine have measured the hang of the sun at noon to prove that it was at the pole?

But there is a tan lat in the formula for hang time.. Does this mean that the hang time at the pole is infinite................

If someone set up camp exactly at the pole (as I am sure someone has) will the sun's altitude remain fixed forever? I am confused.

To answer your questions in reverse order:
Camping at the South Pole is fine.  Camping at the North Pole can be risky because the ice and leads are moving and opening and closing frequently.  When we flew there in 1974 (Well, the little, pre-led, red, glowing, wiry, numerators on the Elliott Inertial Platform readout were reading 90N), there was a lead in view.  I used to have a pre-digital photographs of lead and the numerators, but I’ve mislaid them.

The hang time is long but not infinite.  It’s only The Land of the Midnight Sun for about 6 months of the year when the Sun is above the celestial equator.

Moreover, the Sun’s declination is changing throughout the day.  The only time you could truly claim a 24-hour lunch break is at the at the northern Summer Solstice if you were at the North Pole or at the northern Winter Solstice if you were at the South Pole.

USS Nautilus SSN 571 didn’t attempt to surface at the Pole in 1958.  They just carried on underneath the ice.  If they had of done and the Sun was visible, they could have used the pole as their assumed position and checked that Ho Sun = dec Sun for the time and date (See Hinks below*).  To be certain they, might have risked wandering a mile in three directions just to take Ho Sun values to make sure they had covered all possibilities, but would it have been worth the risk of losing a crew member?  If you’d just happened to be passing, and if Nautilus had surfaced, you wouldn’t have read 571 on her sail, because Cdr Anderson had had it painted it out with black paint as part of the strict security precautions then in force.  This has led to confusion.  If you Google 'Nautilus at the North Pole' you will get a few photographs of a submarine bearing 578 on her sail, e.g.**.  That is USS Skate when she surfaced in 1959.  Please don't imagine I carry this sort of stuff around in my head.  I was preparing a comment to an earlier post when your question popped up.  DaveP 
* hinks1944southpolepaper.pdf
**https://maritimecyprus.com/2021/08/03/flashback-in-maritime-history-nautilus-submarine-travels-under-north-pole-3-aug-1958-video-2/