Structural account of period-two exclusion in finite Rule 30 orbits

Develop a structural account for excluding eventually period-two central traces in finite-support Rule 30 configurations.

Background

The paper completely resolves the constant-trace case, corresponding to eventual period one, by proving that no nonzero finite-support Rule 30 configuration has an eventually constant column. It then considers the next unresolved periodicity case, period two, and defines H(p,w) as the maximum length of an eventually p-periodic central-trace prefix for configurations of support radius w.

For period two, left permutivity guarantees arbitrarily long prescribed finite trace prefixes for suitable finite configurations, yielding the lower bound H(2,w) ≥ w. Thus, unlike the constant case, no bounded horizon law can hold for period two. The paper leaves open the structural explanation or proof mechanism for excluding eventually period-two traces in finite Rule 30 orbits.