Does the signal have fivefold diffraction symmetry — the hallmark of Penrose quasicrystalline order?
Computes the power spectrum and autocorrelation, then tests whether their peak structure is invariant under scaling by the golden ratio (1.618...). True Penrose/Fibonacci quasicrystals have spectral peaks at positions related by powers of the golden ratio, creating self-similar diffraction patterns with discrete Bragg peaks — the defining property that separates quasicrystals from both periodic crystals and amorphous matter.
Multi-scale weighted coherence of detrended spectral residuals under phi scaling, with phi-specificity gap (target R² minus best null-ratio R²), cross-scale stability, and residual fit quality. Rewards signals where the phi self-similarity model is both the best fit and a good fit. Genuinely φ-specific by construction: the target ratio is hardcoded to φ and the null ratios [1.21, 1.38, 1.55, 1.85, 2.15, 2.45, 2.8, 3.2] include the Tribonacci constant (~1.84) and the plastic number (~1.32 ≈ 1.38) as distractors, so other Pisot eigenvalues are explicitly not alternative positives. Confirmed empirically: Penrose Substitution (golden², rank 1) and Fibonacci Word (golden, rank ~5) ceiling the metric, while Pell Word (silver Pisot, rank ~42), Tribonacci Word (cubic Pisot, rank ~79), and Padovan Word (cubic Pisot, rank ~92) all sit mid-pack despite being algebraic-irrational substitutions. Was previously named algebraic_tower, which overpromised general algebraic-degree detection. Evolved via ShinkaEvolve atlas v1.
| Source | Domain | Value |
|---|---|---|
| Penrose Substitution | exotic | 0.9785 |
| Circle Map Quasiperiodic | chaos | 0.8854 |
| Phyllotaxis | bio | 0.8378 |
| ··· | ||
| Gray Code Counter | exotic | -7.1242 |
| De Bruijn Sequence | number_theory | -4.4473 |
| Prime Gaps | number_theory | -1.7664 |
Penrose is the primary golden-ratio quasicrystal detector. The phi_tower metric provides atlas discrimination (score 0.924) while maintaining ratio specificity: the coherence gap penalizes signals that score equally well at non-phi ratios. Sources governed by the golden ratio (Fibonacci QC, phyllotaxis, critical circle map at golden-mean rotation, Penrose Substitution) stand out on it. A sister metric long_range_order (autocorrelation self-similarity at golden-ratio-scaled lags) is class-only — kept in compute_metrics for direct quasicrystal-detection use but dropped from the atlas because it is effectively a binary detector: F=0.365 (p=0.99), with 97% of sources at the zero floor and only true quasicrystals (Fibonacci QC, Phyllotaxis, Circle Map QP) escaping zero. It carries no domain discrimination once the question "is this a quasicrystal?" has been asked.