Fine, to remain on-topic, I won't go into the details of Buddhism's decline in India, starting from the 2nd century and including the names of Hindu rulers long before the Muslim arrival.
The point remains that the decline or disappearance of a religion does not invalidate the region's claims to its genesis.
It is my duty to inform you - and I hope you will not find it too rude a shock - that your last dozen or so posts more and more reflect my thinking on the subject of identity. I am not sure how to tackle this disconcerting situation, but trust that you will be courageous and brave and will not panic. There must be ways to overcome this, and we should work on finding those together.
It is clear that residents of geographical India share a common cultural heritage,as well as retaining exclusive aspects of their own. It is clear that artifacts relating to this heritage lie scattered over a very wide area. It is clear that individual nations within India, even within Greater India, are nettled by an all-embracing Indian claim of intellectual and cultural ownership of these artifacts. In this case, in this thread, it has been argued, not very convincingly, that the problem arises from our aligning our national name with the name of our common geography and our common shared cultural domain. As a result, the unwary are misled into believing that cultural artifacts belonging to the Indian cultural domain are inevitably to be found within the physical boundaries of the present nation bearing the common name.
The solution to this is not at all obvious. It is easy to understand the pique and irritation of those who find their treasures of the past sliding away into quite another country, which just happens to be the largest single physical mass, the country with largest number of inhabitants, and the country whose geographical extent covers many of the key civilisational nodes which formed the cultural legacy. The question is - what do we do about it? What can we do about it?
Frankly, I don't have a clue. Even after acknowledging that there is a problem, it is difficult to see any solution to that problem.
On another note altogether, for nearly three years and over 14,000 messages, a small group of us, refugees from the Internet wars on PTH' have been discussing these issues, although there is a broad mix of topics; currently, for instance, the Indians have been reduced to silence as the Pakistanis look at the legality of the Pakistan Supreme Court's direction to the Pakistani Prime Minister to violate the constitution of Pakistan. The debate is keen, the cut and thrust of argument is hypnotic, but the intensity is high, and it is a good discussion to watch from a prudent distance. That apart, this question of identity has been a major theme. It is now consensus that we are people of geographical and cultural India, and citizens of different countries, one being political India. The cultural heritage is held in common, except for those aspects that individuals have discarded in the light of their own value systems. As it is a small but mature group, it is not a forum where the theft of identity finds much resonance. That is not to say that this is not an issue; merely to say that perhaps this equanimity is a desideratum.
As for the rest, I don't know how to solve the problem.