非周期性结构永不坍塌：无损压缩的斐波那契层级架构

Aperiodic Structures Never Collapse: Fibonacci Hierarchies for Lossless Compression

March 16, 2026

作者: Roberto Tacconelli

cs.AI

摘要

我们研究非周期层级结构是否能在无损压缩方面提供优于周期层级的结构性优势。研究表明，斐波那契准晶铺砌能避免周期层级存在的有限深度坍缩问题：可用的n元词组查找位置在各层级均保持非零值，而周期为p的周期铺砌在O(log p)层级后即发生坍缩。这形成了非周期层级优势：字典复用可在所有尺度实现，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词组长度的指数增长所平衡，使得潜在覆盖范围保持尺度不变性，其渐近值为Wφ/5；其次，基于斯特姆序列复杂度定律p(n)=n+1，我们证明斐波那契/斯特姆层级在二元非周期铺砌中实现了最大码本覆盖效率；第三，在长程依赖条件下，该层级结构比同类周期层级具有更低的编码熵；第四，冗余度随深度呈超指数衰减，而周期系统在发生坍缩的深度即陷入停滞。我们通过Quasicryth压缩器验证了这些结论——该无损文本压缩器基于十层斐波那契层级构建，词组长度为{2,3,5,8,13,21,34,55,89,144}。在控制变量实验中，使用相同码本时，非周期结构相较于周期-5基准线的优势从3MB时的36,243字节增长至1GB时的11,089,469字节，这源于更深层级结构的激活。在enwik9数据集上，Quasicryth实现225,918,349字节（22.59%）的压缩效果，其中斐波那契铺砌相较无铺砌方案节省20,735,733字节。

English

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 Wvarphi/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.