We study whether an aperiodic hierarchy can provide a structural advantage for lossless compression over periodic alternatives. We show that Fibonacci quasicrystal tilings avoid the finite-depth collapse that affects periodic hierarchies: usable $n$-gram lookup positions remain non-zero at every level, while periodic tilings collapse after $O(\log p)$ levels for period $p$. This yields an aperiodic hierarchy advantage: dictionary reuse remains available across all scales instead of vanishing beyond a finite depth. Our analysis gives four main consequences. First, the Golden Compensation property shows that the exponential decay in the number of positions is exactly balanced by the exponential growth in phrase length, so potential coverage remains scale-invariant with asymptotic value $W\varphi/\sqrt{5}$. Second, using the Sturmian complexity law $p(n)=n+1$, we show that Fibonacci/Sturmian hierarchies maximize codebook coverage efficiency among binary aperiodic tilings. Third, under long-range dependence, the resulting hierarchy achieves lower coding entropy than comparable periodic hierarchies. Fourth, redundancy decays super-exponentially with depth, whereas periodic systems remain locked at the depth where collapse occurs. We validate these results with Quasicryth, a lossless text compressor built on a ten-level Fibonacci hierarchy with phrase lengths ${2,3,5,8,13,21,34,55,89,144}$. In controlled A/B experiments with identical codebooks, the aperiodic advantage over a Period-5 baseline grows from $1{,}372$ B at 3 MB to $1{,}349{,}371$ B at 1 GB, explained by the activation of deeper hierarchy levels. On enwik9, Quasicryth achieves $359{,}883{,}431$ B $(35.99%)$, with $45{,}608{,}715$ B attributable to the quasicrystal tiling itself.

翻译：我们研究非周期层次结构是否能为无损压缩提供优于周期替代方案的结构性优势。我们证明斐波那契准晶铺砌避免了影响周期层次结构的有限深度坍缩：可用的 $n$-元组查找位置在每一层级均保持非零，而周期铺砌在 $O(\log p)$ 层级后即发生坍缩（$p$ 为周期）。这产生了非周期层次结构的优势：字典复用可在所有尺度上持续可用，而非在有限深度后消失。我们的分析得出四个主要结论。首先，黄金补偿特性表明，位置数量的指数衰减恰好被短语长度的指数增长所平衡，因此潜在覆盖保持尺度不变性，其渐近值为 $W\varphi/\sqrt{5}$。其次，利用 Sturmian 复杂度定律 $p(n)=n+1$，我们证明在二元非周期铺砌中，斐波那契/Sturmian 层次结构最大化了码本覆盖效率。第三，在长程依赖条件下，所得层次结构实现了比可比周期层次结构更低的编码熵。第四，冗余度随深度超指数衰减，而周期系统在发生坍缩的深度处保持锁定。我们通过 Quasicryth 验证了这些结果，这是一个基于十层斐波那契层次结构（短语长度 ${2,3,5,8,13,21,34,55,89,144}$）构建的无损文本压缩器。在使用相同码本的受控 A/B 实验中，相较于周期为 5 的基线，非周期优势从 3 MB 时的 $1{,}372$ 字节增长至 1 GB 时的 $1{,}349{,}371$ 字节，这归因于更深层次结构的激活。在 enwik9 数据集上，Quasicryth 实现了 $359{,}883{,}431$ 字节 $(35.99%)$ 的压缩结果，其中 $45{,}608{,}715$ 字节可归功于准晶铺砌本身。