You seem to be trying to solve a batch-production problem with single-operation tools. Large batches, too.
The thickness of the die is no guarantee of concentric cutting. The tool alignment is what matters, not its type. If the die starts off-centre or at angle to the axis, it will continue so irrespective of its number of internal thread turns.
One thing that affects the risk of a wonky thread is the condition of the end of the material. If if that is not faced square with a chamfer, it is very likely to push the die out of line right at the start.
If possible, turning a little spigot to core diameter for a couple of pitch lengths can help start the die accurately; but it is an extra operation I would guess you’d rather avoid! Whether you need face that off afterwards, obviously depends on the specific use of the finished screw. Left on, it can help subsequent assembly and gives some protection to the thread start; and this was sometimes used for applications like very large marine engines.
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There only two ways to ensure an accurately-aligned thread with an ordinary lathe and tools.
A) Use some form of guide. Either a tailstock die-holder / die-head or (if the work length permits) a hand die-holder with a fitted guide – a leading bush with a hole that is a sliding fit on the unthreaded work.
OR
B) Screw-cut the thread, if necessary cutting to near-depth then finishing with a die or chaser.
Unfortunately the only commercially-manufactured die-holders I have seen with a proper guide, were for conduit-dies, and the curious little (6BA?) dies supplied with an early construction-kit branded Juneero.
The conventional hand die holder is made to die-thickness, but it would not be difficult to make a deeper form to hold a flanged guide-bush as well as the die.
You could investigate – and would need make – a sliding die-holder akin to a tailstock die-holder but held in the tool-post. This though come with a cost: it assumes both a Quick-change tool-post and some way to ensure the die-holder is repeatably, truly on and concentric with the lathe axis at every tool change.
If you need make hundreds of the things, consider a capstan or tailstock-turret attachment, though I would not expect these to allow cutting very long lengths at a time.
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Bernard –
Do you not mean Grub-screws? A Set-screw is headed like a bolt but has a full-length thread.
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I recall one bad, and totally avoidable, result of a wonky thread. We were presented with a miniature overtype steam-wagon loosely resembling the Foden ‘C’, built from a kit sold by a certain company that soon gained a reputation for introducing new models without addressing QC errors in the previous ones.
On raising steam, we soon repeated the reported fault that had bewildered the unfortunate owner: there was no way of stopping it.
Investigation revealed the regulator slide-valve was moved by screw-thread, not lever. The vavle was tapped directly – no bridle or cross-nut, so no float action. The control-rod’s thread was visibly eccentric, so rotating it lifted the valve, and there it stayed as the steam used the thread clearance to hold it above the port-face.