How to verify the pitch #of nuts?

I suppose measuring threads with the wrong pitch would create an axial error for the location, which would effect the TP results 

Not sure I understand the question, but CMMs can measure any thread they can reach, including pitch diameter with true positions.  Too much to go into here but the coding isn't really that ridiculous.

I would like to confirm the pitch and direction :

We could find out the nut or thread bolt type from the drawing, then we could find out pitch # from manufacture data or google the type to get them. But how to determine plus or minus ?

If the probe moving down the thread direction, we use plus ? What happen if the probe moves against the thread direction ( Should we avoid this kind of situation?) , should we use minus?

Thank you

I have, rarely, found exceptions to this.  Depending on how threads were cut, it's possible for the minor/pitch diameters to not be concentric.

have you tried a barrel probe to find TP?

Lavezzi plugs are precision ground gages made specially to find a threads position

Functional lot acceptance of thread form can only be effectively assessed with a known good counterpart (GO/NOGO rings or plugs).
That might be a contentious statement, so hear me out.  You can verify components of the helix, (Pitch, Max/Min dia, angles, radii) to verify the threads should work, but until it passes functional gages, you could absolutely have a "good" part fail in the mating part's threads, in the instance mating part is also functionally close to specified limits.

Can you measure pitch with a CMM? YES.  You need to scan (ttp or analog) a line down the axis of the threads.  then you need to define a strategy to quantify the max/min peak/valley's of each thread. Extract the Z height of the peaks/valley's and you've got pitch.   Size of your probe vs pitch is super important here, as well as your machine's ability to probe normal to thread axis.

ya, no thanks haha

Sounds very complicated. Thanks for the explanation. 

Verifying thread pitch is pretty simple actually. Most typically use plug/ring gauges that are certified to determine pass/fail. You can use repro rubber for example to make a mold and measure the threads if your requirements call for specific size. Whenever I have a MMC condition on a threaded diameter in the past I use the major or minor diameter to determine the bonus. I have never had an issue in reporting this way. The only issues I have had was trying to report the major/minor diameter size accurately. even with the pitch function it's not easy to hit the peak of the thread especially when you have fine threads.

BigMac said:

Whenever I have a MMC condition on a threaded diameter in the past I use the major or minor diameter to determine the bonus. I have never had an issue in reporting this way.

The problem you would have would be determining the correct bonus because, as you said...

BigMac said:

The only issues I have had was trying to report the major/minor diameter size accurately. even with the pitch function it's not easy to hit the peak of the thread especially when you have fine threads.

If you have not guaranteed that the probe is contacting the crests of the threads, then you cannot be sure you are getting the correct minor/major diameter and therefor the correct bonus calculation.