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A Rule 54 Mapping for the Terras Function
Rudi B. Stranden
Let a (๐) be the integer value of the binary string bounded by the outermost 1s of the ๐-th row generated by Elementary Cellular Automaton (ECA) Rule 54 from a single-cell initial condition. We define ๐ถ54 (๐) = wH(a (๐)) โ 1, where wH is the Hamming weight. We prove that ๐ถ54 (๐) = ๐ฏ (๐) for all ๐ โ โค+. This result establishes an equivalence between the Terras function and the global spacetime structure of Rule 54, providing a computational representation of the Collatz map within the dynamics of a Class IV cellular automaton.
Keywords: Elementary cellular automata, Collatz conjecture, rule 54, Hamming weight, number theory