Moebius Function
number_theory · 36 views
number_theory
What It Is
Moebius mu(n) in {-1, 0, +1} --- 0 on non-squarefree n (~39.2% by density), (-1)^omega(n) on squarefree n. Generator of multiplicative number theory via the identity sum_{d|n} mu(d) = [n=1]; stationary ternary sequence whose partial sum is Mertens.
Interpretation
Standard analysis sees: bounded / light-tailed; few distinct values; aperiodic / broadband; homoskedastic; monofractal; high-dimensional / space-filling; stationary. The atlas finds no named structure, but the source is distinctively extreme on Persistent Homology:h1_total_persistence (+3.6z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.14
asymmetry0.36
occupancy0.09
short-range corr0.24
long-range memory0.30
spectral colour0.79
periodicity0.15
complexity0.62
time-irreversibility0.31
volatility clustering0.06
multifractality0.07
dimensionality0.99
nonstationarity0.12
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Persistent Homology:h1_total_persistence | +3.6z | bank-miss 1.2σ |
SL(2,ℝ) (Thurston):parabolic_fraction | +3.3z | bank-miss 2.7σ |
Boltzmann:dominant_lag | +3.3z | bank-miss 1.8σ |
Zariski:algebraic_residual | +3.1z | bank-miss 3.6σ |
Composition
dtypeint8
range[-1, 1]
unique values3 / 16384
mean ± std-0.00397 ± 0.78
Fixed alphabet — only 3 distinct symbols across 16384 samples.
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Moebius_Function.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Moebius_Function.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Moebius_Function.png)
Inflation (Substitution) > barcode
/d_curve/Moebius_Function.png)
Inflation (Substitution) > d_curve
Information Theory > default
Julia Set > escape_histogram
Klein Bottle > default
Laplacian > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Ordinal Partition > default
/phi_spectrum/Moebius_Function.png)
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > h1_diagram
Predictability > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > phase_space
/default/Moebius_Function.png)
Ulam Spiral (Sacks) > default
/default/Moebius_Function.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Free Group F₂ Walk | 4.35 | cross-domain |
| Categorical Sensor | 4.52 | cross-domain |
| Codon Usage | 4.52 | cross-domain |
Which Geometries Light Up
Cayley › Cayley:delta_hyp_norm | rank 2/298 | 0.2628 |
Mostow Rigidity › Mostow Rigidity:spectral_rigidity | rank 1/298 | 1.0000 |
Mostow Rigidity › Mostow Rigidity:volume_rigidity | rank 1/298 | 0.9741 |
Persistent Homology › Persistent Homology:h1_total_persistence | rank 3/298 | 0.8284 |
Zariski › Zariski:algebraic_residual | rank 1/298 | 0.0716 |
Zariski › Zariski:residual_slope | rank 298/298 | -5.7252 |
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