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 AutoRegressive:ar_coef_3 (-3.9z) — beyond what the standard bank predicts for it.
AutoRegressive:ar_coef_3 | -3.9z | bank-miss 1.9σ |
AutoRegressive:ar_coef_5 | -3.3z | bank-miss 2.2σ |
Spectral Analysis:spectral_slope | +2.5z | bank-miss 1.0σ |
Composition
Render Gallery
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Blue Noise | 3.75 | cross-domain |
| Coupled Map Lattice | 3.86 | cross-domain |
| MFPT Outer Race | 3.97 | cross-domain |
Which Geometries Light Up
AutoRegressive › AutoRegressive:ar_coef_1 | rank 296/298 | -1.5848 |
Spectral Analysis › Spectral Analysis:spectral_slope | rank 2/298 | 2.0000 |