NOTE 01Working mathematics
A local rule, a global question
A small experiment. A place to think.
Start with a single occupied cell. Apply the same local rule at every position, then look for structure in the rows that follow.
For binary states , the update is
The rule defines the experiment. A finite picture helps us ask better questions; an all-time statement needs an argument.
THE SINGLETON EXPERIMENTRule 30
THE NEXT STEP
Questions and next steps
Which patterns persist as the experiment grows? What argument would establish an all-time statement?
Keep the working mathematics, reproducible evidence, and reviewed summary together.
Markdown source
Start with a single occupied cell. Apply the same local rule at every position, then look for structure in the rows that follow.
For binary states $x_i^{(t)}$, the update is
$$
x_i^{(t+1)} = x_{i-1}^{(t)} \oplus \bigl(x_i^{(t)} \lor x_{i+1}^{(t)}\bigr).
$$
The rule defines the experiment. A finite picture helps us ask better questions; an all-time statement needs an argument.