The Support-Set Calculus and the Algebra of Dyadic Blocks

We give an initialized recurrence for the Newton supports of diagonals of any binary cellular automaton and explicit formulas for propagating masked dyadic blocks through that recurrence. Products become OR convolution; discrete integration becomes increment, whose carry partition determines every output fragment. Each block evaluates by one binomial parity test. We bound the size of uncollected block lists, identify exact minimization with ESOP synthesis, and give rank lower bounds for block count. For Rule 30, a census through depth thirty compares support size, greedy compression and exact periods. At a quiescent edge, arbitrary binary rules have eventual dyadic periods at every fixed depth. Polynomial interpolation and rational generating functions connect these supports to companion recurrences, Diophantine identities, special-polynomial families and matrix powers. The resulting framework treats representation construction and evaluation as distinct, quantified operations.