For those who are puzzled by the idea of putting the Die on the spindle.
It is NOT to cut the spindle, but to use it as a gauge to adjust the Die to the correct size, to finish cut the thread on the stub intended for storing the chuck.
I regularly adjust a Die by setting it on a known thread of that size and type, It saves having an undersize thread, or having to recut the to reduce to it the correct size.
If you can find a 1/2 BSF bolt try the Die on it.
For the pedants, I am not talking in terms of making micrometers, but of holding things together with nuts and bolts.
If the Die is already too small to fit on the spindle, the thread cut on the storage stub with it would be undersize, so that the chuck would be a loose fit on it.
Sounds like time to check that the spindle nose IS 1/2 BSF (16 tpi ) and that the Die really IS 1/2 x 16 tpi (If either were 1/2 UNF the thread would be 20 tpi, or if 1/2 BSW (12 tpi ). The difference in thread pitches between the parts should immediately be visible with barely need to double check with thread gauges.
Although that might be a good ides here, to confirm the facts..
Alternative ideas.
Get a 1/2 BSF bolt / stud, fix this to a base, and screw the chuck onto it. Even simpler, turn a piece of bar or broom handle to just under the core diameter (0.420" ) and just hang the chuck on there.
If the storage is on a wall, set the stub at an angle so that the chuck slides towards the wall rather than being horizontal.
Let us know how you get on
Howard