Any USGI slide will work with any USGI bolt. They were designed to be forward and backward compatible and interchangeable among manufacturers. However the best reliability results with the later round bolt and Type 5 or 6 slide. Check the headspace before shooting. It will probably be okay but just to be safe.... Your Type 4 slide will be okay.
Jackp has it exactly correct and thank you Jackp. However ... it is VITAL that you determine the exact head space before firing that new bolt. If the gun and new bolt were new maybe not. But our reality is that both are very old. In the Carbines case over 70 years old. The bolt maybe slightly younger. Trust but verify. Be safe. Have fun. Like Jackp already said, make sure that late round bolt is USGI.
I find it not uncommon for the run-of-the-mill well-fired carbine's that I have/had to close on a no-go. I think that gauge is primarily useful for head-spacing after a re-barrel job. Many seem to be worn enough to no-go a little loser than 1.296. As long as it dosen't close on a field...that's what matters.
If the bolt is "newer" you are probably okay. During the initial manufacture of carbines a lot of life testing along with interchangeability tests were performed and I don't recall seeing problems with headspace being reported. The tight tolerances of manufacture and subsequent in-process testing assured that these problems would be unlikely. Only where there is significant wear on both the bolt and receiver would you expect to see excessive headspace. You are more likely to see insufficient headspace when a barrel has been replaced. Just switching out bolts is not likely a major source of headspace problems. However if you have the ability to check headspace, do so.
Brand new bolt, cleaned cosmoline off of it. The .u. is under the left lug and the bolt has the hole on the bottom of the bolt.
That would be an Underwood post-war replacement and should replicate the original USGI spec. I second Jack's notion. I've never really found a USGI bolt to be measurably out of spec. You can make determinations without the need for the gauges. If the stripped bolt will close and rotate on a spec. round (not to exceed case length of 1.290)...that's your go gauge. Some of this stuff you can figure out by sight and feel.
When you get over into swapping commercial bolts with USGI bolts and vise-versa, I have found severe differences in head-space. As an example, I have noted a .006 difference between USGI bolts and Iver Johnson bolts in relation to the depth of the bolt face.