Hofstadter Q
What It Is
Hofstadter Q-sequence (A005185): Q(1)=Q(2)=1, Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2)). Quasi-chaotic nested recurrence with no closed form; not proven well-defined for all n, though empirically holds far past the ranges sampled here. Q(n) ~ n/2 with Mallows-conjectured O(√n) fluctuations, so the emitted first differences Q[n]-Q[n-1] form an integer stream with AC1 ~ -0.5 (oscillatory: big drops followed by big rises) and a slowly-ramping variance over the window --- nonstationary by design, not a generator artifact. Trending metrics (variance_trend, vol_of_vol) respond to the ramp; that is faithful detection of Q's mathematical structure, not noise.
Interpretation
Standard analysis sees: anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power). The atlas finds no named structure, but the source is distinctively extreme on Spectral Analysis:spectral_slope (+2.4z) — beyond what the standard bank predicts for it.
Spectral Analysis:spectral_slope | +2.4z | bank-miss 1.3σ |
Composition
Render Gallery
_(centered)/signed_log_z/Hofstadter_Q.png)
_(centered)/xy_path/Hofstadter_Q.png)
/barcode/Hofstadter_Q.png)
/d_curve/Hofstadter_Q.png)
/phi_spectrum/Hofstadter_Q.png)
/default/Hofstadter_Q.png)
/default/Hofstadter_Q.png)
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Blue Noise | 3.68 | cross-origin |
| Coupled Map Lattice | 3.71 | cross-origin |
| IMS Bearing Degraded | 3.91 | cross-origin |
Which Geometries Light Up
AutoRegressive › AutoRegressive:ar_coef_1 | rank 305/307 | -1.5848 |