Great project! You should see what Wiktor Macura did in 2004 https://education.wolfram.com/summer-school/alumni/2004/macura/ and published in Complex Systems https://content.wolfram.com/uploads/sites/13/2018/02/15-3-3.pdf
He called these n-skip Turing machines.
Thank you for your wonderful comment. I will definitely take a look into the paper!
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