Binomial Decimation and State Bounds for Rule 30

Lucas' theorem makes prime decimation of a binomial sequence a scalar multiple of one binomial sequence. Within the bases generated by (1 + t + ⋯ + t^(q−1))^x, this property holds exactly when q=2. For every q ≥ 2, finite initial spans remain stable under prime decimation, and integrality, Mahler expansion and finite prime kernels persist. The highest nonzero basis index determines the same exact prime-power period and linear kernel bound throughout the family. Thus one-term digit behavior is a property of the chosen coordinates, whereas the period and kernel belong to the represented sequence. Polynomial invariants, conditioning, Pell equations, prime-representing polynomials and Fibonacci–Lucas period lifting supply arithmetic applications. For the single-seed Rule 30 center column, finite distinguishing continuations require at least 8,191 generating states with least-significant-first input and 12,950 with most-significant-first input. A finite-rank argument also excludes algebraic equations within stated degree and height bounds. The state bounds hold for every automaton generating the infinite sequence.