非周期结构永不坍塌：无损压缩的斐波那契层级体系

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结构的优势。研究表明，斐波那契准晶镶嵌能避免周期层级结构存在的有限深度坍塌问题：可用的n元语法查找位置在每一层级均保持非零，而周期镶嵌在周期为p时仅维持O(log p)层级后即发生坍塌。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：第一，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，使得潜在覆盖范围保持尺度不变性，其渐近值为Wφ/5；第二，基于斯特姆复杂度定律p(n)=n+1，我们证明斐波那契/斯特姆层级在二元非周期镶嵌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，冗余度随深度超指数衰减，而周期系统会在发生坍塌的深度锁定。我们通过Quasicryth压缩器验证这些结论——该无损文本压缩器基于十层斐波那契层级构建，短语长度为{2,3,5,8,13,21,34,55,89,144}。在控制变量实验中，相较于周期为5的基准系统，非周期优势从3MB时的36,243字节增长到1GB时的11,089,469字节，这源于深层级结构的激活。在enwik9数据集上，Quasicryth实现225,918,349字节（22.59%），其中斐波那契镶嵌相比无镶嵌结构节省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.