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Rule 30 exact binomial-Lucas lifting: boolean logic to integer coefficients, Stirling & support sets
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Bonus: Exact Support-Set Algebra for Rule 45!
For those interested in exploring how this binomial-Lucas lifting framework applies to other chaotic elementary cellular automata, I mapped out the exact set recurrence for Rule 45. Like Rule 30, Rule 45 is a highly complex
For Rule 45, starting with initial conditions S_1 = \{0, 1\} and S_2 = \{1, 2\}:
S_m = \mathrm{Inc}\Big( (S_{m-1} * S_{m-2}) \;\Delta\; S_{m-2} \;\Delta\; \{0\} \Big)
Key differences from Rule 30: Notice that the linear S_{m-1} term is completely absent, but we have a new constant parity injection via the \{0\} set.
If you want to test this in the Mathematica environment :
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