Does the signal have sevenfold diffraction symmetry — the rarest rotational order in known quasicrystals?
Tests spectral self-similarity at the septagonal ratio rho = 1 + 2*cos(2*pi/7) ~ 2.247, the largest root of x^3 - 2x^2 - x + 1 = 0. The v2 evolution discovered a triple-conjugate ensemble approach: the three roots of the minimal polynomial (rho, its conjugate, and its reciprocal) provide independent spectral probes. Sevenfold symmetry is crystallographically forbidden and extremely rare even among quasicrystals.
Combined coherence across all three roots of the septagonal minimal polynomial. Tests the algebraic identity x^3 = 2x^2 + x - 1 in spectral space. Evolved via ShinkaEvolve.
Cubic-identity spectral self-coherence at r = ρ ≈ 2.247 (P1: x³−2x²−x+1=0).
Cubic-identity spectral self-coherence at r = ρ−1 ≈ 1.247 (P2: x³+x²−2x−1=0). Independent probe of the same algebraic structure.
| Source | Origin | Value |
|---|---|---|
| 2-Torus Quasiperiodic | coupled-oscillators | 0.3095 |
| MFPT Inner Loaded | machine-vibration | 0.2778 |
| Riemann-Hardy-Littlewood | number-theory | 0.2564 |
| ··· | ||
| Circle Map Quasiperiodic | iterated-maps | -0.2913 |
| Phyllotaxis | number-theory | -0.2676 |
| Fibonacci Word | symbolic-dynamics | -0.2314 |
| Source | Origin | Value |
|---|---|---|
| 4-Torus Quasiperiodic | coupled-oscillators | 0.7828 |
| 2-Torus Quasiperiodic | coupled-oscillators | 0.7764 |
| MFPT Inner Loaded | machine-vibration | 0.7129 |
| ··· | ||
| Takagi Function | special-functions | -0.6836 |
| Padovan Word | symbolic-dynamics | -0.6706 |
| Stochastic Resetting Walk | stochastic-process | -0.6529 |
| Source | Origin | Value |
|---|---|---|
| Bearing Outer | machine-vibration | 0.3781 |
| Bearing Inner | machine-vibration | 0.3223 |
| Riemann-Hardy-Littlewood | number-theory | 0.3143 |
| ··· | ||
| Circle Map Quasiperiodic | iterated-maps | -0.5788 |
| Phyllotaxis | number-theory | -0.5634 |
| Fibonacci Word | symbolic-dynamics | -0.4963 |
Septagonal is the most exotic geometry in the quasicrystal lens. The cubic_coherence combines three polynomial identities (P1 at ρ, P2 at ρ−1, P3 at 1/(ρ²−2ρ) ≈ 1.802) — all three must show coherence simultaneously for genuine 7-fold order. Septagonal completes the quasicrystal lens's coverage of crystallographically forbidden symmetries (5, 7, 8, 12-fold).
(The P3 component is computed internally to support cubic_coherence but is not emitted as a standalone atlas metric — its 2026-05-18 audit found it was the atlas-worst-discriminating metric (F=0.52) and its rank-1 outliers were generic quasiperiodic spectral-dip coincidences, not Danzer 7-fold signal. The fourth previously-emitted metric ratio_symmetry was DROPPED 2026-05-26 — F=1.09 (rank 308/317), and trial-spread σ_t/σ_x=1.49 puts the metric's cross-source variance below the per-trial noise floor. The shared _quasicrystal_spectral_1d helper still works for Penrose's φ and Ammann-Beenker's silver ratio; the failure is specific to atlas signals not having detectable spectral self-similarity at the Septagonal ratio 2.247.)