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 :
