Rule 30 Exact Binomial–Lucas Lifting: From Boolean Logic to Integer Coefficients, Stirling Transfer, and Support-Set Algebra

This paper presents a new algebraic framework for Wolfram’s cellular automaton Rule 30. By applying a geometric coordinate rotation, the classical two-sided update rule is transformed into a one-sided algebraic normal form recursion. The evolution is lifted from the Boolean field F₂ to the integers ℤ through prefix-sum integration in the binomial basis, where discrete integration becomes a simple index shift. A Stirling number transfer ensures exact integer coefficients while preserving the Boolean dynamics modulo 2 via Lucas’ theorem. This projection reveals that the nonlinear dynamics correspond to an exact OR-convolution over finite support sets. These support sets admit a geometric compression into masked dyadic blocks, yielding an evaluation algorithm with O(logn) complexity. The framework further establishes structural bounds for Rule 30 including a lower boundary and an upper Fibonacci growth ceiling for spatial expansion. Finally, the Zeta-Floor Reduction Theorem and Matsui’s Piling-Up Lemma are used to reformulate Wolfram’s open questions on central column periodicity and asymptotic density into explicit lattice summations and probability bounds. The work provides a new combinatorial and algebraic interpretation of Rule 30, linking Boolean cellular automata, binomial calculus, Lucas arithmetic, and cryptographic probability methods.