Monotone Branch Rigidity in Conditional Cellular Automata

Rule 110 selects OR or XOR through a control input. In every neighborhood size, a monotone zero branch, a parity one branch and vanishing mixed quadratic coefficients involving the control force this family. Values at zero and unit vectors yield rule numbers, truth-table weights, coordinate influences and permutation symmetries. With n noncontrol inputs, its O(n) branch formula expands into 2^(n+1) − n − 2 monomials in algebraic normal form (ANF). For elementary rules with the positive ANF lift, vanishing centered first-order transport is equivalent to reversal symmetry. Branch coefficients and characteristic formulas describe the associated formal differential equations. First contact and exact integer degree bound Rule 30 supports. A two-step identity determines single-seed Rule 22 rows and diagonal supports: its first 2^(r+1) rows contain 4·3^r active cells, and each nonzero diagonal requires one Lucas test.