The Support-Set Calculus and the Algebra of Dyadic Blocks

Newton supports encode a binary sequence by the indices of its odd finite-difference coefficients. For cellular-automaton diagonals, the local rule becomes a recurrence using symmetric difference, parity-counted OR convolution and increment, with arbitrary initial data included. Masked dyadic blocks provide finite descriptions compatible with this calculus: their convolution is a block or the empty set, each block contributes one binomial parity test, and increment has an explicit decomposition determined by binary carries. Zeta duality identifies minimum block decomposition with exclusive-OR sum-of-products minimization. At a quiescent edge, fixed-depth binary diagonals have eventual dyadic periods and finite residue descriptions. Arithmetic applications develop pure-power and companion recurrences, Diophantine determinant identities, special-polynomial sums and matrix powers. Rule 30 illustrates the nonlinear recurrence; additive rules provide explicit comparisons. The construction separates the algebra needed to propagate a support from the representation-size bounds needed to evaluate it efficiently.