Aperiodic Structures Never Collapse: Fibonacci Hierarchies for Lossless Compression

Abstract

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.

Figures (8)

• Figure 1: Concrete example of Quasicryth tiling on the phrase "the quick brown fox jumps over the lazy" (8 words). Level 0: the Fibonacci cut-and-project rule assigns L tiles (2-word bigram positions, blue) and S tiles (1-word unigram positions, orange), yielding the sequence $L\;S\;L\;L\;S$ for these 8 words. Level 1: each $(L,S)$ pair is deflated into $\hat{L}_1$ (trigram position, teal); isolated $L$ tiles become $\hat{S}_1$. Level 2: the pair $(\hat{L}_1,\hat{S}_1)$ yields $\hat{L}_2$ (5-gram position, violet). Arrows trace the lookup cascade at word 0 ("the"): the encoder first attempts the 5-gram lookup ("the quick brown fox jumps"), then trigram ("the quick brown"), then bigram ("the quick"), falling back to unigram only if all longer lookups miss. Word 5 ("over") reaches a trigram entry point but no 5-gram.
• Figure 2: Deep hierarchy position detection. An $L$ tile at level 0 may qualify as the entry point of super-L supertiles at levels 1--9, enabling lookups in the trigram through 144-gram codebooks respectively. The qualifying condition is verified in a single upward pass through the pre-computed hierarchy array.
• Figure 3: Deep substitution hierarchy of the Fibonacci tiling. Each level is obtained by one application of the deflation rule $\sigma^{-1}$. Both $L$- and $S$-supertile types are present at every depth $k$, enabling $n$-gram codebook lookups at phrase lengths $F_{k+2} \in \{3,5,8,13,21,34,55,89,144,\ldots\}$ words.
• Figure 4: Deep hierarchy hits (levels 4--9, Fibonacci-exclusive) per word versus corpus size (log-log). Labels show absolute hit counts. The ratio is roughly flat from enwik8_3M to enwik8 while fixed-size codebooks limit density; at enwik9, levels 8--9 activate and the ratio jumps to $2.7\times10^{-3}$, confirming new $O(W)$ terms.
• Figure 5: Word-weighted contribution of each deep hierarchy level, as share of total deep-hierarchy words covered (hits$_k \times n_k$, stacked to 100%). Unlike raw hit counts, this weights each hit by the phrase length it encodes. At 1 GB the 144-gram contributes 1.1% of covered words despite only 945 hits, because each hit encodes 144 words. The 34-gram holds a stable $\approx$10% share across all scales; 89-gram and 144-gram are absent until 1 GB.

Theorems & Definitions (37)

• Theorem 1: Aperiodic Hierarchy Advantage
• proof : Proof (assembly of supporting results)
• Theorem 2: Fibonacci Stability
• proof
• Remark 1
• Theorem 3: Periodic Collapse
• proof
• Corollary 4
• Definition 1: Pisot-Vijayaraghavan number pisot1938
• Theorem 5: Weyl weyl1910