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Elon tweet “Priority/early access will be automatic for all owners with FSD”

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Can someone clarify for me the relationship between V10 and HW3. Would one need the HW3 chip to run V10?

There is no direct relationship, and this deployment model isn't really unique to Tesla, or the HW version of the car.

First, a product is released to a very limited subset of users, usually a very small pool to target new features.
There is a QA/validation period for this
Second, a product is released to Early Access adopters
Also a QA/validation period for this
Then a product goes to a wider subset of users, consider this like 5% across a wider range of of scenarios not necessarily covering new features with the product. This might also include the "advanced" switch for upgrades in the Tesla. Maybe.
Again, more QA/validation
Then the product starts a staggered "wide" release.

Note that if there are any problems with the QA/validation of any phase, this usually slows down the process to assess risk, or perhaps even increment the version number. If they're pushing 10.0.0 it might increment it to 10.0.1 to remediate the issue and then restart the whole process.

Elon / Tesla might say they'll target the 10.x release to FSD owners first, but the reality is it's just moving you to a bucket of the above. And as people rather accurately point out, getting it sooner often means accepting the risks of that feature set. But there are checks and balances all around.