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 $36{,}243$ B at 3 MB to $11{,}089{,}469$ B at 1 GB, explained by the activation of deeper hierarchy levels. On enwik9, Quasicryth achieves $225{,}918{,}349$ B $(22.59\%)$, with $20{,}735{,}733$ B saved by the Fibonacci tiling relative to no tiling.

翻译：我们研究了非周期分层结构是否能在无损压缩中提供优于周期替代方案的结构性优势。研究表明，斐波那契准晶铺砌避免了影响周期分层结构的有限深度坍塌：在每一层级上，可用的 $n$ 元组查找位置均保持非零，而周期铺砌在周期 $p$ 下经过 $O(\log p)$ 层后即发生坍塌。这带来了非周期分层优势：词典复用在整个尺度范围内始终可用，而非在有限深度后消失。我们的分析给出四个主要结论。首先，“黄金补偿”性质表明，位置数量的指数衰减恰好被短语长度的指数增长所平衡，因此潜在覆盖范围保持尺度不变，渐近值为 $W\varphi/\sqrt{5}$。其次，利用斯图尔米复杂度定律 $p(n)=n+1$，我们证明斐波那契/斯图尔米分层结构在二元非周期铺砌中实现了最优码本覆盖效率。第三，在长程依赖条件下，所得分层结构比可比的周期分层结构实现了更低的编码熵。第四，冗余度随深度超指数衰减，而周期系统则停滞在坍塌发生的深度。我们通过 Quasicryth（一种基于十级斐波那契分层结构、短语长度为 ${2,3,5,8,13,21,34,55,89,144}$ 的无损文本压缩器）验证了这些结果。在相同码本的受控 A/B 实验中，与周期-5基准相比，非周期优势从 3 MB 时的 $36{,}243$ B 增长到 1 GB 时的 $11{,}089{,}469$ B，这归因于更深层级的激活。在 enwik9 上，Quasicryth 实现了 $225{,}918{,}349$ B ($22.59\%$)，其中由斐波那契铺砌相比无铺砌节省了 $20{,}735{,}733$ B。