I quite extensively researched the subject of a G-reg in the US, but could not find any evidence that you had to notify any authorities about your movement within the US. However, crossing any border and coming in to the US requires prior permission and a special form.

For sure, nothing in the FARs limits permanent G-reg basing in the USA.

I also don't recall seeing anything saying that you cannot do (what they call here) aerial work in it, including training for US papers.

Hi

I do seem to remember on a flightsafety int operational procedures recurrent the instructor telling us that US Customs (not the FAA) needed prior notice before arrival into the US of any intended airports likely to be used WITHIN the US for a foreign registered aircraft. I.E not just the first port of landing.

Also, the 'feds' wanted to know all possible crew and pax names and passport details along with the aircraft's registered owner(s) and usual cargo. I will try to fish out the references from the fsi manuals if time permits.