In the setting of multi-head finite-state dimensions, trailing heads lag behind a leading head, accessing past data to aid a finite-state gambler placing bets on successive bits read by the leading head. Cruz, Glashausser, Li, and Lutz (2026) proved that, for any fixed number of trailing heads, adaptive (data-dependent) movement rules can strictly outperform oblivious (data-independent) movement schedules. In this paper we strengthen that separation by proving that a single trailing head with adaptive movements can outperform, by a large and uniform margin, arbitrarily many trailing heads with oblivious movements. Formally, our main theorem states that there is a binary sequence whose adaptive two-head finite-state strong dimension is less than its oblivious multi-head finite-state dimension, and that the gap is greater than 0.3.

翻译：在多头有限状态维数的设定中，尾随头落后于主导头，访问历史数据以辅助有限状态赌徒对主导头依次读取的比特序列下注。Cruz、Glashausser、Li和Lutz（2026）证明，对于任意固定数量的尾随头，自适应（数据依赖）移动规则能够严格优于被动（数据无关）移动调度。本文通过证明一个自适应移动的单一尾随头能够以较大且一致的裕度胜过任意多个被动移动的尾随头，进一步强化了这一分离结果。形式上，我们的主要定理表明：存在一个二进制序列，其自适应双头有限状态强维数小于其被动多头有限状态维数，且两者之差大于0.3。