Monotone Branch Rigidity in Conditional Cellular Automata

Rule 110 switches between OR and XOR according to a control input. For arbitrary neighborhood size, a monotone zero branch, a parity one branch and vanishing mixed quadratic coefficients involving the control force the corresponding OR–XOR rule. Values at zero and the unit vectors determine the family and yield closed formulas for rule numbers, truth-table weights, coordinate influences and permutation symmetries. For elementary rules with the positive algebraic-normal-form lift, vanishing centered first-order transport is equivalent to reversal symmetry. Branch coefficients and characteristic formulas describe the associated formal differential equations. At the orbit level, first-contact and degree bounds control Rule 30 supports, while a two-step identity determines the single-seed Rule 22 rows and their diagonal supports. These results connect the branch structure of local rules to explicit algebraic classifications and discrete orbit formulas.