Aperiodic Structures Never Collapse: Fibonacci Hierarchies for Lossless Compression

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:

1. Golden Compensation property. 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 → φ / √5.

2. Maximal codebook coverage. Using the Sturmian complexity law p(n) = n + 1, we show that Fibonacci/Sturmian hierarchies maximize codebook coverage efficiency among binary aperiodic tilings.

3. Lower coding entropy. Under long-range dependence, the resulting hierarchy achieves lower coding entropy than comparable periodic hierarchies.

4. Super-exponential redundancy decay. 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.