In BrE the past participle of get is in most cases got, while in AmE it is almost always gotten.
Does that imply that in the context of BrE "have/has got" is a genuine perfect construction, whereas in AmE it is just a standalone phrase?
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It is a known fact that the present perfect of some verbs can develop semantically and take over the sense of a present tense. There are some such cases in Latin and there is a term for this phenomenon: preterite-presents.
If you have got a letter, the consequence is you have a letter. This is the way how a grammatical perfect can adopt the sense of present tense. This phenomenon can be found in a lot of languages.
I think this is an example of drift, in the same way as past participles that are very commonly used in passive voice (worried, scared, amazed, etc.) have come to be considered adjectives in their own right. It would seem reasonable to suppose that as 'have/has got' when used for possessions adds nothing to the meaning of simply 'have/has' that at some point in the past this construction was originally used in its perfect sense and then use became extended for emphasis (perhaps when something was difficult to come by). Nowadays if we say, "I've got the tickets," this can have both senses ('I have been to get the tickets, and now I have got them'; and 'I have the tickets'). Of course, philosophically we could say that somehow both meanings are always true, after all, we are born with our hands empty, so everything we now 'have' we must 'have got' at some point since our birth!