[Paper Review] Aperiodic Structures Never Collapse: Fibonacci Hierarchies for Lossless Compression

Roberto Tacconelli (Plain Text Extraction and Formatting)|arXiv (Cornell University)|Mar 16, 2026

Quasicrystal Structures and Properties0 citations

TL;DR

This paper proves that Fibonacci quasicrystal-based hierarchies for word-level compression avoid collapse at all depths, yielding scale-invariant reuse, higher codebook efficiency, and lower coding entropy than periodic tilings, demonstrated with a ten-level hierarchy and multi-tiling engine.

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 $Wφ/\sqrt{5}$. 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.

Motivation & Objective

• Demonstrate that the Fibonacci hierarchical tiling never collapses across all depths unlike periodic tilings.
• Show how Sturmian minimality and balance translate into compression advantages.
• Develop and validate a multi-structure Fibonacci tiling compressor with deep lookups (up to 144-gram).
• Quantify the aperiodic advantage through theoretical results and controlled experiments.

Proposed method

• Model input as word tokens and apply eleven Fibonacci-grammar level codebooks for n-grams up to 144 words.
• Construct 36 aperiodic tilings (including golden-ratio and non-golden irrationals) and select the best tiling per block via a scoring function.
• Use inverse Fibonacci deflation to build a deep substitution hierarchy enabling deep-lookup positions.
• Encode using multi-layer adaptive arithmetic coding with per-level contexts, caches, and a two-tier unigram model.
• Apply word-level LZ77 to capture repeats before arithmetic coding, plus a separate LZMA escape stream for out-of-vocabulary words.

Experimental results

Research questions

• RQ1 Can an aperiodic Fibonacci tiling preserve non-zero n-gram lookup positions at every depth?
• RQ2 What are the information-theoretic benefits (coverage, efficiency, entropy) of a Fibonacci hierarchy versus periodic tilings?
• RQ3 How does a multi-tiling engine affect compression performance on large corpora?
• RQ4 What is the scaling behavior of redundancy and overhead as hierarchy depth increases?

Key findings

• The Fibonacci tiling preserves both tile types at every depth, avoiding collapse, unlike periodic tilings that collapse after O(log p) levels.
• Potential word coverage per level converges to Wφ/√5, independent of depth (Golden Compensation).
• Sturmian codebook efficiency at level m reaches the maximum for aperiodic tilings, with exactly Fm+1 patterns and efficiency Cm/(Fm+1).
• Per-word flag entropy is bounded by 1/φ ≈ 0.618 bits/word, ensuring bounded parsing overhead across depths.
• For long-range dependent sources, Fibonacci hierarchy yields strictly lower per-word coding entropy than any collapsing periodic tiling (entropy advantage).
• Redundancy decays super-exponentially as O(exp(-φ^m/λ)) for Fibonacci levels, vs. fixed exponential decay for periodic tilings.