Given the importance of the claim, we want to start by exposing the following consideration: this claim comes out more than a year after the article "Practical applications of Set Shaping Theory in Huffman coding" which reports the program that carried out an experiment of data compression in which the coding limit NH0(S) of a single sequence was questioned. We waited so long because, before making a claim of this type, we wanted to be sure of the consistency of the result. All this time the program has always been public; anyone could download it, modify it and independently obtain the reported results. In this period there have been many information theory experts who have tested the program and agreed to help us, we thank these people for the time dedicated to us and their precious advice. Given a sequence S of random variables i.i.d. with symbols belonging to an alphabet A; the parameter NH0(S) (the zero-order empirical entropy multiplied by the length of the sequence) is considered the average coding limit of the symbols of the sequence S through a uniquely decipherable and instantaneous code. Our experiment that calls into question this limit is the following: a sequence S is generated in a random and uniform way, the value NH0(S) is calculated, the sequence S is transformed into a new sequence f(S), longer but with the symbols belonging to the same alphabet, finally we code f(S) using Huffman coding. By generating a statistically significant number of sequences we obtain that the average value of the length of the encoded sequence f(S) is less than the average value of NH0(S). In this way, a result is obtained which is incompatible with the meaning given to NH0(S).

翻译：鉴于该主张的重要性，我们首先要阐述以下考虑：这一主张是在《集合整形理论在霍夫曼编码中的实际应用》一文发表一年后提出的，该文报告了开展数据压缩实验的程序，其中对单一序列的编码极限 NH0(S) 提出了质疑。我们等待如此之久，是因为在提出此类主张之前，希望确保结果的一致性。在此期间，该程序始终公开；任何人都可以下载、修改并独立获得所报告的结果。这段时间里，许多信息论专家对该程序进行了测试，并同意提供帮助，我们感谢他们为我们付出的时间以及宝贵的建议。给定一个由属于字母表 A 的符号组成的独立同分布随机变量序列 S；参数 NH0(S)（零阶经验熵乘以序列长度）被认为是序列 S 的符号通过唯一可译即时码的平均编码极限。我们的实验对该极限提出了质疑，具体如下：随机均匀生成一个序列 S，计算 NH0(S) 的值，将序列 S 转换为一个更长的、但符号属于同一字母表的新序列 f(S)，最后使用霍夫曼编码对 f(S) 进行编码。通过生成统计显著数量的序列，我们得到编码后序列 f(S) 长度的平均值小于 NH0(S) 的平均值。这样，就得到了一个与 NH0(S) 的既定含义不相容的结果。