Where do local ISPs' core networks connect to?
Do they connect to the Internet Backbone owned by AT&T and Sprint (among others) or do they connect to an IXP (Internet eXchange Point)?
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Yes. They connect either to the "internet backbone" or to a local IXP. Or they connect to neither. Or even connect to both.
Which one they connect to will depend (to some degree) on the specific ISP in question, though the vast majority of local ISPs connect to an IXP, because that's what an IXP is - a point where different ISPs connect to exchange traffic between their networks.
Regarding the internet backbone... you are misinformed as to what it actually is and who "owns" it, but suffice to say, it doesn't really exist in the form that you seem to think. It's really just a mish-mash of Tier 1 Networks (and Tier 2 Networks).
Tier 1 networks are commonly defined as networks "that can reach every other network on the Internet without purchasing IP transit or paying settlements," and Tier 2 networks are defined as "networks that peer with some networks, but still purchase IP transit or pay settlements to reach at least some portion of the Internet."
There's a somewhat useful image at the Wikipedia page on Tier 1 Networks, that illustrates all the various ways the ISPs and assorted Tier # networks can connect with each other. (Pasted below)
As you can see, some Tier 1 Networks also have ISP capabilities and would therefore be "connected to the internet backbone" as you put it. Some connect to IXPs, some only connect to other ISPs within their Network Tier, and some even connect to both the "internet backbone" and to an IXP. So it's a fair bit more complicated than connecting to the internet backbone, or an IXP. It can be either, both, or even none.
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