Diagonal Bases and Diagonal Periods of Elementary Cellular Automata

Abstract

Which cellular-automaton diagonal families form bases in every finite window? For canonical polynomial lifts of elementary rules, two truth-table bits determine triangularity, and units on the matrix diagonal determine invertibility. Exactly 24 rules give universal binary bases; all remain universal over every modulus. Among triangular binary coordinate maps, the Pascal transform is uniquely characterized by converting OR convolution into pointwise multiplication, while increment becomes strict prefix summation. Explicit inverses and coordinate comparisons distinguish sparsity from evaluation cost. A Rule 30 polynomial construction gives Fibonacci bounds on interpolation order and prime-modulus periods. Exact additive periods anchor a finite census modulo two and three. These results separate all-window basis classification from optimization of a representation and from period patterns observed in finite windows.