Ahmed1029 said:
That's not the correct way to think about it. I can make the measuring instrument "my eye" as close to the tape as possible. So if the effect is delivered instantaneouly to my end of the rod, I will know it happened before the light from the particle's end reaches my eye; I will only require lifht to travel a distance dx, rather than the whole distance between the particle and me. But I now know that rigid bodies doesn't exist in that deal fashon in general relativity
The problem here is that in order to measure the length of the object, you have to know the location of both ends of the object at the same instant of time. One picks some instant of time, which I'll call "now", and asks "what is the position of the left end and the right end of the ruler now?"
However, the concept of "now" turns out to be observer dependent in special relativity, something that's unexpected and causes a great deal of confusion. I'll give a reference and more details later.
If instantaneous signal transmission speed were possible via a physical mechanism, there would be a physical definition of now, and special relativity could not work properly. However, there is no such physical mechanism. It takes time to communicate information from one place to another.
Your current approach appears to be based on a desire to ignore the fact that it takes time to communicate information from one location to another. And it's going to lead to problems for your understanding of special relativity, because of the issue I mentioned previously, the relativity of simultaneity.
A reference on this issue, from one of Einstein's books, is online at
https://www.bartleby.com/173/9.html. The issue is called "The Relativity of Simultaneity", and the particular thought experiment is called "Einstein's train". There is a great deal of information written about it, but it is hard to get people to listen to it. Also of some interest is the paper "The challenge of changing deeply held student beliefs about the relativity of simultaneity". Google finds this currently at
https://core.ac.uk/download/pdf/25353534.pdf.
I'll give an informal summary of the issue, which may or may not be clearer than Einstein's treatment. Every inertial frame of reference in special relativity has a notion of simultaneity, a notion of "now". But when when switches frames, the notions change - events that are simultaneous in frame S are not simultaneous in some different frame S' that is moving with respect to S.
To go back to the "eye" analogy, if you have only one eye, you need to specify where it's at. Is it at the left end of the ruler, or is it at the right end?
If you have two eyes, then the pair of "eyes" need to have some concept of simultaneity. This is generally taken to be associated with the inertial frame that the pair of eyes is is in. But depending on the state of motion of the observer, there are different notions of simultaneity.