Double tap main to sub panel or No breaker in main?

Main 100 amp panel. Service wire lugs double tapped to feed an A/C unit.
The lugs apparently may or may not allow a second conductor depending on their makeup/stamp. Still, I remember reading that all sub-panels feeds must originate from a breaker in the Main Panel. I can’t find the reference…
Also; To be clear…An A/C unit can be fed from two hot leads without a neutral if the a/c unit is using a balanced load, Correct?
Sorry, poor photos…
Bill O

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Most lugs are for single conductors only.

A sub does not need to originate with a breaker. Feed through lugs could be used. Overcurrent protection would need to be at the sub.

Central air is straight 240 volts. There is no neutral.

So, when I visit again Monday, I’ll try and look closely at the thru lugs at the main (designed to feed the split-bus) for a stamp stating the lug is ok with two connectors…The sub breaker size is correct. Besides the stamp, (Which are often concealed by wires) is there a visible difference in the one and two wire lugs? My copy of “Electrical inspections” by Doug Hanson, Kardon, Casey, has a photo of an allowable two wire lug. I believe the last two lines describe the wire size rules…Now the shape of the thing is slightly oval. Might that be the difference in a one conductor vs a two conductor lug? Without stamp info or shape/design evidence, I’d have to let it go…
Per usual, in a case like this, there’s no room in the panel for expansion, and having parts around to adapt the thing is unlikely…
Bill O

Yes lugs listed for two conductors have holes that are typically not round, my guess is that in this case you’re not going to find a permission that allows more than one conductor per lug on that panel. Here’s an example of a lug listed for two conductors:

This one doesn’t. single service drop (two hot legs and a neutral) go into the meter housing. Two separate sets of service entrance conductors come out of the meter and both enter this service panel. One set powers the supply bus bar through a 200 amp disconnect, the other powers a 125 amp disconnect that controls a panel located somewhere else in the home. The 200amp disconnect does not shut off power to the 125amp disconnect.

I don’t know what to say about this.