Hi @Todd Rowland, I'm reaching out because I haven't found any literature referencing this specific representation of Rule 30/86, and I believe it's a significant missing piece.

My work proves that Rule 30 can be lifted to the Lucas basis where it acts as a finite-set machine. The dynamics reduce to a precise tension between OR-Convolution (which merges dyadic blocks) and Increment Carry (which fractures them). This seems to pinpoint the exact algebraic mechanism of the rule's randomness.

Given Stephen Wolfram’s search for a reducible formula or mechanism, this "Carry-vs-Convolution" duality seems like a critical missing piece. It rigorously maps the system's chaos to an iterated fracturing process strongly suggesting that predicting the n-th bit is computationally irreducible (and, intuitively, that P is different from NP given the exponential growth of the block support).

Attachments:

rule30.pdf

Hi Tigran, I am still working on my rule 30 solution, I am not sure if the prize is still running as I dont see any activity there. want to meet up and chat about your ideas?

Hi everyone, can anyone tell me if the rule 30 prize is still running? I have been working very hard on it for about a year! I have left a comment on the offical page, which is still awaiting moderation, just getting a bit nervous that everyone has forgotton about it! Thank you, David

Graham this looks neat. Can you explain a little more?

I understand that it's not in Mathematica. If it's one or two lines of math, we can translate it.

Hi, I have an algebraic equation for RULE30, it runs on octave but would be great to see it iterated on Mathematica, https://www.linkedin.com/feed/update/urn:li:activity:6609546528723877889/

I also have a turmite equation (based on my langtons ant wave equation) which would be nice to see in Mathematica, could someone please help me ?

I wanted to play with cellular automata for a while. When I saw the post about the contest, it finally got me into gear and I had some fun with rule 30:

https://brunni.de/findings30/

From the abstract:

The usual pattern starting from a single cell with state 1 / black is examined. It is contructed from a richer structure which yields a progression of polynomials for diagonals from the right. It is shown that each diagonal from the left can be expressed by the progression of polynomials for diagonals from the right and how the connection between left and right diagonals forces the diagonals from the left to eventually become periodic.

The necessary and sufficient condition for period doubling of diagonals from the right is established. It is also shown that periods cannot decrease. The result suggests that predicting period doubling is computationally expensive.

I suppose there is nothing new in there?

hi Todd,

what do you need to calculate a cell in a diagonal? The previous cell in that diagonal and two cells from the previous two diagonals. This enables one to define diagonals recursively. For diagonals from the right, the recursive definition leads to a simple picture: The previous two diagonals are combined with OR and the resulting values are combined with XOR - basically a summation over the combination up to the cell you need. It is this summation modulo 2 that causes the period doubling. I show that it will either double the basic period of the combination or leave it untouched. But combining the previous two diagonals with OR may decrease the basic period - so I also show that this cannot happen in diagonals from the right.

This is what may be new - I don't know. The rest should be well known stuff - maybe looked at from a different viewpoint. Using a richter structure by replacing A OR B with A+B+A*B did not yield any really interesting results IMO but I left it in the article because it was my starting point.