Does the signal's spectrum contain peak triplets at the dodecagonal inflation ratio (2 + √3 ≈ 3.732) that satisfy an algebraic identity — the signature of square-triangle Stampfli tilings?
Dodecagonal quasicrystals (Ta-Te, V-Ni-Si alloys) have the highest rotational symmetry known in real quasicrystals. Earlier attempts at this geometry tested simple spectral self-similarity at the dodecagonal ratio or at √3 scaling; both degenerated into noise. The current implementation (from a ShinkaEvolve dodecagonal_v1 run) is tighter: find spectral peak triplets at (f, f·δ, f·δ²) and verify the algebraic identity X(f·δ²) + X(f) ≈ 4·X(f·δ) on complex FFT coefficients. Only the correct inflation ratio satisfies this identity at detected peaks, so the metric is selective by construction.
Mean cosine similarity between complex FFT coefficients at (f, f·δ, f·δ²) peak triplets and the algebraic combination required by the Pisot inflation law. Devil's Staircase (0.86), Fibonacci Quasicrystal (0.80), and Wave Height (0.80) score highest. Sine Wave, Thue-Morse, fBm, and constants all score 0.0 — their spectra either lack peak structure entirely or have peaks that don't cluster at dodecagonal ratios. The 11% floor pile-up reflects how narrow the hit condition is. Weakly discriminating as a standalone metric (F=1.16, rank 264/269) but atlas-retained because it flags dodecagonal-specific structure that no other geometry catches.
| Source | Domain | Value |
|---|---|---|
| Fibonacci Quasicrystal | number_theory | 0.7992 |
| ETH/BTC Ratio | financial | 0.6519 |
| BTC Close Price | financial | 0.6330 |
| ··· | ||
| fBm (Antipersistent) | noise | 0.0000 |
| fBm (Persistent) | noise | 0.0000 |
| Sine Wave | waveform | 0.0000 |
Dodecagonal tests for 12-fold rotational structure specifically, and the pisot-triplet test only fires when the spectrum contains three peaks in the correct algebraic ratio. Most of the atlas scores zero. Sources that do light up fall into two groups: genuine quasicrystalline/aperiodic structure (Fibonacci QC, Rudin-Shapiro) and signals with unexpectedly rich harmonic peak structure at dodecagonal ratios (Devil's Staircase, Wave Height, certain financial series). The latter are likely accidents of spectral geometry rather than evidence of 12-fold order, but the framework logs them as interesting cross-domain hits.