We give answer to an argument trying to show the divergence of Assembly Theory from LZ compression. We formally proved that any implementation of the concept of `copy number' underlying Assembly Theory (AT) and its assembly index (Ai) is equivalent to Shannon Entropy and not fundamentally or methodologically different from algorithms like ZIP/PNG via LZ compression. We show that the weak empirical correlation between Ai and LZW, which the authors offered as a defence against the proof that the assembly index calculation method is an LZ scheme, is based on an incomplete and misleading experiment. When the experiment is completed and conducted properly, the asymptotic convergence to LZ compression and Shannon Entropy is evident and aligned with the proof previously offered. Therefore, this completes both the theoretical and empirical demonstrations that any variation of the copy-number concept underlying AT, which resorts to counting the number of object repetitions `to arrive at a measure for life', is equivalent to statistical compression and Shannon Entropy. We demonstrate that the authors' `we-are-better-because-we-are-worse' defence argument against compression does not withstand basic scrutiny, and that their empirical results separating organic from inorganic compounds have not only been previously reported -- sans claims to unify physics and biology -- but are also driven solely by molecular length, nota special feature of life captured by their assembly index. Finally, we show that Ai is a special case of our BDM introduced almost a decade earlier and that arguments attributing special stochastic properties to Ai are misleading, not unique, and exactly the same than those that Shannon Entropy is already not only equipped with but designed for which we have also proven to be equivalent to Ai making AT redundant even in practice when applied to their own experimental data.

翻译：我们回应了试图证明装配理论与LZ压缩存在分歧的论点。我们严格证明了装配理论（AT）及其装配指数（Ai）所基于的"拷贝数"概念的任何实现形式均等价于香农熵，且在基本原理和方法论上与通过LZ压缩实现的ZIP/PNG等算法并无本质区别。我们指出，作者为反驳"装配指数计算方法是LZ方案"的证明而提出的Ai与LZW之间微弱经验相关性，是基于不完整且具有误导性的实验。当实验被完整且规范执行时，其向LZ压缩和香农熵的渐近收敛性显而易见，这与先前提供的证明完全一致。因此，这从理论和实证两方面完整论证了：AT所基于的任何拷贝数概念变体——即通过统计对象重复次数"以获得生命度量指标"的方法——均等价于统计压缩和香农熵。我们证明作者针对压缩提出的"因劣而优"辩护论点经不起基本检验，且他们区分有机与无机化合物的实证结果不仅早有报道（未声称统一物理学和生物学），而且完全由分子长度驱动，并非其装配指数所捕捉的生命特殊特征。最后，我们证明Ai是我们近十年前提出的BDM度量的特例，那些声称Ai具有特殊随机特性的论点具有误导性且非独有，这些特性正是香农熵早已具备且为其设计目标的功能——我们已证明其与Ai完全等价，这使得AT即使在其自身实验数据的应用中也变得冗余。