what are the threads on the core? you could maybe do
the OD is very close to 13/64" or 5.16mm, the thread seems to be NF36... another exotic type of thread! But I certainly will explore in the future a way to make a sub-miniature valves... maybe based on existing cores...
- why exactly did you use the live center,
jor2daje wrote: the live center was brought in because when it's cone shape is pressed into the spinning tube it almost perfectly centers it, the part is most likely loose enough to allow the few thou of movement that the live center caused.
I couldn't have explained better... in English! Thanks jor2daje!
I get that but shouldn't the chuck then have been retightened to fix the now centred part?
This was left on the cutting room's floor

Wait for the Director's cut

But I'm very glad you follow so closely!
- why didn't you add a die holder to the shopping list? Not worth $60? Do I have to make one? ... or can I be sure of centred threads using the manual holder?
Well... I lived without it during 10 years... I just bought it, it was it's first cut! And you where in a budget remember? And in a hurry?
Didn't I told you
"wait wait hold the credit card I need to check through all my drawers !!" 
This one was still in its cardboard box!
There's other accessories you'll need too. This one, you can do well holding by hand the die as I show on the video, and you can also do it the other way around for extra precise alignment: tighten the die on the head chuck, and hold your piece on the tail chuck like it was a drill bit (max diameter 3/8"). It you tighten carefully (evenly around) it's precise enough.
You could also machine your own, it's a simple project (I'm lazy I just bought it) and you could use for example the drill chuck to hold a cylindrical support if you don't want to machine a
Morse taper MT0 ... though it is also a nice way to learn how to do it, and maybe cut a bit of mild steel. You will also have a MT0 as part of your accessories that you could use and adapt to your holder. Lots of ways...
Machining some adapters is part of the fun... and may even make you save both time and money in the long run; for example here's my "universal" saw holder, that can auto center any kind of small saw's hole diameter (imperial, metric, 1/2", 5/8", 10mm, 15mm, whatever):

It just screw in as an extension of the existing Sherline saw holder. And you will have MT1 taper attachment too, as is also part of your chuck accessories... see, I thought a bit about it, budget wise...
I believe the original iteration was "man needs god..."
I would not dare changing the quotes of my favorite humorist
Pierre Desproges. It's bold enough from me to try to translate its very weird French humor.