I would take apart that branch and start a new pot. The main plant probably needs to be repotted. Clean out as much as the old potting mix you can, trim the roots a bit and put it in a new fast draining mix. You can search 5-1-1 mix here and repotting. There are lots of explanations. If you are on Facebook there is a group called Holiday Cactus Lovers group with lots of helpful people. Most of them prefer to repot using a fifty fifty mix of potting soil and perlite. If you repot, tale your time and read up here. I'm happy to answer specific questions if you need it. My weekend will be busy so it might take a bit before I respond. If the clades on the fallen branch are withered, sometimes they can be saved by putting them i a ziplock baggie full of water for a day or two. Just flip the bag every day. I actually forgot about one for about two weeks and it had sprouted roots. Good luck!

Well to get the one out of the philo. If you want to..just get a very sharp knife and find where it is rooted into the soil and cut out a little 'cube' of soil around it,cut down about 1 inch, and pop it out. Then you can pot it into a pot of its own. You wont hurt the philodendren. That plant is indestructable. IF you 'lost' the other cactus cutting that you put in with your other christmas cactus...does it have the shape of the new hybrids? If it does, then it isnt the old antique type that I am in search of. IF it has the 'claw like' look to the stem, and they stand upright. It is a newer hybrid. The one I am talking about is much smaller,hangs nearly limp,and has smooth sides. OR maybe the one in the christmas cactus pot,didnt root and it rotted? Thats why you cant find it.? Like me. I try to root in soil and it just dissolves and dies. Thus the reason I am in search of this old one. I emailed you in private as well.

It is probably a Schlumbergera (Christmas or Thankgsgiving cactus). When people have an old "Holiday cactus" it is almost exclusively a Schlumbergera, and the Easter cactus does not produce much aerial roots, if any. These plants do not have leaves, only stem segments (cladodes). You mean that the plant dropped all green segments leaving only the oldest, corky ones, right? I have seen it happen before, and the big plant often ends up dead or growing poorly, so I really think you should take a few spare cuttings and root them. No new roots? How can you see that if you don't repot it yourself? No roots growing out of the holes in the bottom of the pot? It sounds as if it has some aerial roots growing out from some of the segment joints. I find these useful when I take cuttings from my plants, they root faster. Cut in half? How? A horizontal cut at the center of the base segment? Well, it doesn't really matter, old segments are very difficult to root. The best way is to cut off (at the joints) some green segments and root them. Then you will have some spare plants while you observe the development of the mother plant.

Yes, just ignore those air roots. Make your clipping, let it sit so the end dries up a few hours and then insert it into your new potting mix in a small pot with a hole. You do not need any honey or rooting hormone. Just make the cutting stand up as best you can. You can put more than one in a pot. There are different opinions about when to water. I wait a week or more if the potting mix had any moisture. Water sparingly, but let it run out the bottom. No sips. I would keep the newly planted cuttings wherever their "mother" plant has been. That's the light, air circulation, temperature they are used to. You should definitely repot in the suggested mix. Change it's soil, as you said above. Clean off as much of the old potting mix as possible, and toss it out. Don't use for the cuttings. I hope you can find the 3 components of the mix. At the very least, use the cactus & succulent mix and orchid bark, but if you can find pumice or perlite it will help. I've started some very large cuttings but sometimes just do 2-4 segments as shown in the photo. I could be wrong in the way I start cuttings, even tho' it works. I cut mine as shown on the left below, but in the case of your plant, you might want to gently twist the segment off as shown on the right. That way you'll cause less injury to the segment which stays on the plant. I'm not an expert. Others may chime in to correct me, but the above is what I do.