Yes, the mast always carries the compression load of the tension in the luff.

Do you agree that with a conventional halyard lock, the luff tension is the only compression load in the mast? A conventional halyard lock mechanically fixes the head car at the truck, the halyard can then be removed as it is doing nothing. If you do not agree with this, then we are really going to have to get back to basics.

If you agree to that, then, in the two part case, where does the rest of this mysterious force come from? There is only one line running down to the deck, it is the only thing that can put any force on the mast. It carries 1/2 the luff tension. The air cannot generate compression on the mast, it must be resolved by something attached to the boat. What is that thing?

You seem to want to concentrate on the mast head in isolation. Correctly analyzed, the result will be the same. The mast truck and sheave box, in the case of the halyard lock, has the luff tension pulling down on it from the headboard, nothing more. In the one part halyard case, the truck and sheave box has the line running to the headboard and the line running to the deck, each carries the luff tension, the result is twice the luff tension as compression in the masthead (and mast).

In the three part case, three things pull down on the mast truck and sheave box: the dead end of the halyard which is pinned at the truck, the running part of the halyard returning from the headboard block, and the line running to the deck. Each of these has 1/2 the luff tension, there are three, so that is 3/2 the luff tension as the compression load on the masthead. If you think there are other forces, where are they and what carries them - can you point them out for me?

There is no magic there, only grade school physics. The mast must carry the luff tension, but any additional load is an artifact of how you apply that load. You can choose to apply it statically (such as the halyard lock), or hydraulically (a hydraulic Cunningham) - neither of these add any compression load to the mast at all. If you choose a tackle, and cleat the end at the masthead without running it to the deck, again there is no additional compression load on the mast. If you run the tackle fall down the mast and cleat it at the bottom, then the tension in that fall is added to the compression load in the mast, but that tension is reduced by the mechanical advantage in the tackle.