Probably sudden reduction in profile or outright disappearance from the public view of the prominent experts in the area, as they are recruited to work on the more clandestine research.
Tracking employment/location and publishing/conference attendance records of researchers will probably be good source data for this.
If there will be a AI Manhattan project, there will be effective cover up of it, so no obvious signs of it will be visible for layman.
My answer to part of this question is “it’s not at all obvious that it’s useful to pay attention too overtly to this sort of thing, especially to the degree of a tracking-website, because the government would just notice and that take some sort of different action instead.”
(I’m not sure it’s *not* useful either, just that this may be a somewhat anti-inductive domain)
Why the government? Perhaps a government?
Why singular? Governments are both afraid of and envious of each other, and actively monitor each other’s activities. If any government is doing this, then many governments are—either cooperatively or competitively.
An actively secret project on the scale of the Manhattan Project (in terms of people who know it’s going on, not just money/resources) is likely impossible. It can be kept from the public for quite awhile, but almost certainly not from other interested states for very long.
The Manhattan Project took four years and a lot of people and resources. What if instead of making a project big and fast, someone made a small and slow project? It would take a lot longer to complete, and perhaps be less effective, but it would be easier to keep secret.
Would having this information actually be beneficial. Perhaps it’d be good for us to know what is going on, but it might be negative for certain governments to know about this as it might increase the chance of an AI arms race.
It would be easy to hide it within another public project that plausibly would need those researchers and experts. Announce a mission to build a moon base, or put humans on Mars. Part of that will need advanced robotics and AI, so now you have a Black Budget project hidden within a public project, and a perfect cover story. Plant 42 has been cranking out Black Budget surprises for decades, though they don’t need a cover story, everyone knows it is a “secret” aerospace R&D and manufacturing facility.