Titchener et al. (2005) asked whether non-simple T-prescriptions can achieve larger T-complexity than simple T-prescriptions for infinitely many finite values of the maximum codeword length. We answer this question affirmatively for every fixed alphabet of size at least two. In fact, we prove a stronger upper-envelope statement: for infinitely many lengths $N$, there exists a valid non-simple T-prescription of exact maximum codeword length $N$ whose T-complexity exceeds the best value attainable by any simple prescription with maximum codeword length at most $N$. Hence the unrestricted exact-length maximum over prescriptions is strictly larger than the corresponding simple exact-length maximum for infinitely many $N$. The proof is elementary. Once a copy pattern has been used, it cannot be selected again. Thus simple prescriptions with more and more steps must use ever more distinct words, which forces the minimal simple length thresholds to have infinitely many strict upward jumps. At each such jump, changing the last copy factor of a minimal simple prescription from $1$ to $2$ yields a valid non-simple prescription, thereby increasing T-complexity by $\log_2 3-1$ while staying below the next simple threshold.

翻译：Titchener等人（2005）曾提出疑问：对于无穷多个最大码字长度的有限值，非简单T-处方能否比简单T-处方获得更大的T-复杂度。我们对每个固定字母表（大小至少为2）给出肯定回答。事实上，我们证明了一个更强的上界结论：对于无穷多个长度$N$，存在一个精确最大码字长度为$N$的有效非简单T-处方，其T-复杂度超过任何最大码字长度至多为$N$的简单处方所能达到的最佳值。因此，对于无穷多个$N$，无限制精确长度下的处方最大值严格大于相应的简单精确长度最大值。该证明是初等的。一旦使用某个复制模式，该模式便不可再被选中。因此，随着步骤增多，简单处方必须使用更多不同的单词，这迫使最小简单长度阈值出现无穷多个严格跃升。在每个这样的跃升处，将最小简单处方的最后一个复制因子从$1$改为$2$，即可得到一个有效的非简单处方，从而在不超过下一个简单阈值的前提下，将T-复杂度增加$\log_2 3-1$。