The Fibonacci infinite word — the most studied object in combinatorics on words — can be generated by an automaton with five states (using Zeckendorf representation) or two states (most-significant-digit-first). The sequence is aperiodic, never repeating, yet a finite machine produces it.
Now shift the sequence by c positions. How many states does the automaton need?
O(log c). Shifting by a million positions requires roughly twenty states. Shifting by a billion requires roughly thirty. The cost of starting later in an aperiodic sequence grows logarithmically with the offset.
This approaches the information-theoretic minimum. An aperiodic sequence shifted by c carries at least log c bits of information in the shift amount alone — the automaton must encode which offset it represents. The Fibonacci word achieves this floor. It is as efficiently shiftable as any aperiodic sequence can be.
The proof combines automata state complexity with Diophantine approximation — the theory of how well irrational numbers can be approximated by rationals. The golden ratio, which governs the Fibonacci word's structure, is the hardest number to approximate (in the sense of continued fractions), and this extremal approximation property translates into extremal automata efficiency.
The worst-approximable number produces the best-shiftable sequence. The same property that makes the golden ratio pathological in number theory makes the Fibonacci word optimal in automata theory. Difficulty in one domain is economy in another.