Liouville Function
number_theory · 36 views
number_theory
What It Is
Liouville lambda(n) = (-1)^Omega(n) --- parity of prime-factor count with multiplicity. Pure +/-1 sequence with P(+1) -> 1/2 (Landau). Sum lambda(n) = O(n^(1/2+eps)) iff the Riemann Hypothesis; the Polya conjecture (false at n=906_150_257) probed the same balance.
Interpretation
Standard analysis sees: bounded / light-tailed; few distinct values; aperiodic / broadband; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Nonstationarity:change_quantiles_mid (+4.0z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.02
asymmetry0.32
occupancy0.05
short-range corr0.25
long-range memory0.36
spectral colour0.73
periodicity0.03
complexity0.40
time-irreversibility0.49
volatility clustering0.18
multifractality0.08
dimensionality0.73
nonstationarity0.10
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Nonstationarity:change_quantiles_mid | +4.0z | bank-miss 1.5σ |
Möbius-S³:sector_transition_entropy | -2.6z | bank-miss 1.2σ |
Boltzmann:coupling_temporal_variance | +2.3z | bank-miss 1.4σ |
Fractal (Mandelbrot):escape_entropy | -2.2z | bank-miss 1.2σ |
Composition
dtypeint8
range[-1, 1]
unique values2 / 16384
mean ± std-0.00952 ± 1
Binary sequence — two distinct symbols.
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/Liouville_Function.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Liouville_Function.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Liouville_Function.png)
Inflation (Substitution) > barcode
/d_curve/Liouville_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/Liouville_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/Liouville_Function.png)
Ulam Spiral (Sacks) > default
/default/Liouville_Function.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Rule 30 | 2.34 | cross-domain |
| Symbolic Lorenz | 3.16 | cross-domain |
| Symbolic Henon | 3.85 | cross-domain |
Which Geometries Light Up
AutoRegressive › AutoRegressive:ar_residual_frac | rank 5/298 | 0.9996 |
Boltzmann › Boltzmann:coupling_strength | rank 297/298 | 0.0048 |
Catch24 › Catch24:DN_Spread_Std | rank 5/298 | 0.5000 |
Catch24 › Catch24:CO_HistogramAMI_even_2_5 | rank 295/298 | 0.0000 |
Klein Bottle › Klein Bottle:wht_spectral_kurtosis | rank 296/298 | 1.3823 |
Mostow Rigidity › Mostow Rigidity:distance_rigidity | rank 4/298 | 0.9842 |
Ordinal Partition › Ordinal Partition:markov_mixing | rank 4/298 | 0.9956 |
Spherical S² › Spherical S²:angular_spread | rank 5/298 | 1.5708 |
Spherical S² › Spherical S²:concentration | rank 295/298 | 0.0058 |
Visibility Graph › Visibility Graph:degree_exponent_gamma | rank 2/298 | 2.7927 |
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