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If I were to make deep modifications to the Linux kernel, virtually make a 'new' kernel in a way, for my project, which is building my own operating system, (yes, I know, a lot of work, but hey, I'm quite a dedicated guy when it comes it working on projects of any kind.) and if I don't want the source disclosed, is this even possible with Linux's license?

Would I have to include the 'new' kernel's source code if I decide or even manage to release the operating system? What about if it's closed source, but not paid-for software?

What if you change it or add to it, but you don't want to make the code you added open source? Must I also open source that code I made that I would wish to keep closed?

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Firstly, IANAL/IANYL. That said, if you start from the existing kernel, it is likely that anything you create from it - even if you were to replace every single line of code - will be a derivative work of the existing kernel. To re-implement an existing work without falling foul of that requires an "arm's-length" technique known as clean-room reimplementation, and by definition that can't be done by one person.

Because your "virtually new" kernel is a derivative of the existing one, GPLv2 s2b requires that the whole new work be made available under GPLv2. So taking your questions one at a time:

Would I have to include the 'new' kernel's source code if I decide or even manage to release the operating system?

Yes.

What about if it's closed source, but not paid-for software?

The GPL makes no distinction between paid-for and zero-cost software; all must be free-as-in-freedom.

What if you change it or add to it, but you don't want to make the code you added open source?

The whole new work must be available under GPLv2, s3a of which requires that all the source be available.

Must I also open source that code I made that I would wish to keep closed?

If it is part of a derivative of a work initially covered by GPLv2, and you distribute it, then yes, you must.

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