| name | fibonacci-quasicryth-compression |
| title | Aperiodic Structures Never Collapse: Fibonacci Hierarchies in Lossless Compression |
| version | 0.0.3 |
| engine | skillxiv-v0.0.3-claude-opus-4.6 |
| license | MIT |
| url | https://arxiv.org/abs/2603.14999 |
| keywords | ["Lossless Compression","Quasicrystals","Hierarchical Coding","Mathematical Foundations","Algebraic Number Theory"] |
| description | Introduce Quasicryth, a text compressor using Fibonacci quasicrystal tilings for phrase-level compression. Prove that aperiodic structures never structurally collapse at depth, enabling compression at arbitrary hierarchy levels. Bridge quasicrystal mathematics with practical compression, achieving 22.59% ratio on enwik9 with unbounded scaling advantages. |
Aperiodic Structures in Lossless Compression
Problem Formulation: Hierarchy Collapse
Traditional periodic tilings used in hierarchical compression have a fundamental limitation: they collapse structurally within O(log p) levels, where p is the period. Beyond that depth, one tile type vanishes, making further compression impossible.
Example: Binary hierarchies collapse at depth log₂(W) (word vocabulary size). The codebook loses structural variety, forcing fallback to escapes.
New Paradigm: Aperiodic Fibonacci Tilings
Fibonacci quasicrystals offer a mathematically-proven alternative: structures that never collapse at any depth, enabling compression at arbitrarily deep hierarchy levels.
Founding Mathematical Results
Main Theorem (Aperiodic Hierarchy Advantage):
The Fibonacci tiling uniquely satisfies five properties simultaneously:
-
Non-Collapse at All Depths: Both tile types persist indefinitely
- Unlike periodic structures that collapse within O(log p) levels
- Guaranteed by golden-ratio eigenvalue φ (Pisot-Vijayaraghavan number)
-
Scale-Invariant Coverage: Potential word coverage remains constant across hierarchy levels
- ~0.724W words available at each level regardless of depth
- Proven via Golden Compensation Theorem
-
Maximum Codebook Efficiency: Exactly Fm+1 distinct patterns at level m
- Minimal for aperiodic sequences
- Maximizes phrase reuse probability
-
Bounded Overhead: Flag entropy capped at 1/φ ≈ 0.618 bits/word
- Constant regardless of hierarchy depth
- Periodic structures have unbounded overhead growth
-
Strict Entropy Advantage: Lower per-word coding entropy than periodic alternatives for long-range sources
- Exploits statistical correlations across deeper hierarchy levels
Mathematical Framework:
Substitution Matrix Analysis:
σ: L → LS, S → L (Fibonacci substitution)
Eigenvalue = φ (golden ratio)
Sturmian Sequences:
Minimal factor complexity p(n) = n+1
Maximum phrase reuse potential
Weyl Equidistribution:
Irrational tiling slopes guarantee uniform density
at all scales
Practical Implementation: Quasicryth v5.6
Hierarchy Configuration:
- 10-level hierarchy with phrase lengths: {2, 3, 5, 8, 13, 21, 34, 55, 89, 144} words
