Aperiodic Structures Never Collapse: Fibonacci Hierarchies for Lossless Compression
This addresses the problem of limited scalability in compression hierarchies for data compression applications, offering a novel structural improvement over periodic methods.
The paper tackled the problem of finite-depth collapse in periodic hierarchies for lossless compression by showing that Fibonacci quasicrystal tilings avoid this collapse, enabling dictionary reuse across all scales. This resulted in a compression advantage, with experiments showing up to 1,349,371 bytes saved over a periodic baseline at 1 GB and achieving 35.99% compression on enwik9.
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 $WÏ/\sqrt{5}$. 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.