Mian-Chowla

number_theory · 36 views
number_theory

What It Is

Greedy Sidon (B₂) sequence: a₁=1, a_{n+1} = smallest integer strictly greater than a_n such that all pairwise sums {a_i + a_j : i ≤ j} remain distinct. First terms 1, 2, 5, 10, 11, 13, 37, 42, 75, 96, ... (Mian–Chowla 1944). Emitted as length-N binary indicator (255 at positions a_n < N, 0 elsewhere) cyclically shifted by an rng-chosen offset --- cyclic shift preserves the Sidon property on ℤ/N. Density ~1.1% at N=16384. Canonical positive control for sumset-isolation metrics (Erdős #152 family): by construction every pairwise sum is uniquely realized.

Interpretation

Standard analysis sees: heavy-tailed; right-skewed; few distinct values; aperiodic / broadband; low-complexity (predictable, not noise-like); multifractal; nonstationary / drifting. The atlas additionally detects critical slowing down.

What standard analysis sees
tail heaviness0.99
asymmetry0.99
occupancy0.00
short-range corr0.34
long-range memory0.61
spectral colour0.66
periodicity0.13
complexity0.01
time-irreversibility0.49
volatility clustering0.30
multifractality0.91
nonstationarity0.95
What the atlas adds
critical slowing down+2.4z
lag-1 autocorrelation rises across the series — the early-warning signature of an approaching tipping point / regime shift (also fired by frequency chirps)
also fires on nonstationary arithmetic sequences whose autocorrelation drifts for non-dynamical reasons
Atlas-extreme metrics the standard bank can’t predict for this source
Ulam Spiral (Square):arm_density_variance+10.6zbank-miss 3.4σ
Mostow Rigidity:spectral_rigidity-5.0zbank-miss 1.8σ
SL(2,ℝ) (Thurston):lyapunov_exponent+3.3zbank-miss 1.1σ
Boltzmann:coupling_temporal_variance+3.2zbank-miss 2.8σ
Mostow Rigidity:margulis_ratio+3.1zbank-miss 1.5σ
Cayley:spectral_gap+2.9zbank-miss 3.3σ
Ulam Spiral (Square):polynomial_concentration-2.8zbank-miss 1.8σ
Boltzmann:spectral_gap_J-2.8zbank-miss 1.7σ

Composition

dtypeuint8
range[0, 255]
unique values2 / 16384
mean ± std1.14 ± 17

Binary sequence — two distinct symbols.

Render Gallery

Native > phase_plot
Native > phase_plotraw
Native > timeseries
Native > timeseriesraw
Native > triple_lattice
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > lag_map
Boltzmann > spin_grid
Boltzmann > spin_grid
Cayley > default
Cayley > default
Chladni > plate_bottom
Chladni > plate_bottom
Chladni > plate_top
Chladni > plate_top
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > xy_path
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
Hodge–Laplacian > default
Inflation (Substitution) > barcode
Inflation (Substitution) > barcode
Inflation (Substitution) > d_curve
Inflation (Substitution) > d_curve
Information Theory > default
Information Theory > default
Julia Set > escape_histogram
Julia Set > escape_histogram
Klein Bottle > default
Klein Bottle > default
Laplacian > default
Laplacian > default
Logarithmic Spiral > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Möbius-S³ > s2_base_map
Ordinal Partition > default
Ordinal Partition > default
Penrose (Quasicrystal) > phi_spectrum
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > barcode
Persistent Homology > h1_diagram
Persistent Homology > h1_diagram
Predictability > default
Predictability > default
Recurrence Quantification > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Analysis > default
Spectral Graph > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > flux_grid
Symplectic > phase_space
Symplectic > phase_space
Ulam Spiral (Sacks) > default
Ulam Spiral (Sacks) > default
Ulam Spiral (Square) > default
Ulam Spiral (Square) > default
Visibility Graph > default
Visibility Graph > default
Wavelet Cascade > default
Wavelet Cascade > default

Atlas Position

Nearest neighborDistance
Rainfall (ORD Hourly)4.75cross-domain
Neural Net (Pruned 90%)4.83cross-domain
Devil's Staircase5.32cross-domain

Open in Atlas →

Which Geometries Light Up

2-adic › 2-adic:valuation_transition_predictabilityrank 4/2980.9836
2-adic › 2-adic:multiscale_markov_predictabilityrank 5/2980.9885
2-adic › 2-adic:distance_entropyrank 298/2980.0613
2-adic › 2-adic:mean_distancerank 298/2980.0075
Boltzmann › Boltzmann:coupling_temporal_variancerank 1/2982.0000
Catch24 › Catch24:SB_TransitionMatrix_3ac_sumdiagcovrank 2/2980.3240
Catch24 › Catch24:DN_HistogramMode_5rank 5/2981.3902
Catch24 › Catch24:CO_HistogramAMI_even_2_5rank 294/2980.0000
Catch24 › Catch24:DN_Meanrank 296/2980.0047
Catch24 › Catch24:SB_MotifThree_quantile_hhrank 297/2980.0600
Cayley › Cayley:delta_hyp_normrank 297/2980.0000
Chladni › Chladni:nodal_gap_ratiorank 297/2980.0337
D4 Triality › D4 Triality:triplet_temporalrank 4/2980.8000
D4 Triality › D4 Triality:edge_ortho_polarityrank 296/2980.0112
Fisher Information › Fisher Information:log_det_fisherrank 3/298141.2201
Fisher Information › Fisher Information:trace_fisherrank 3/298229810.4834
G2 Root System › G2 Root System:short_long_ratiorank 3/2980.9953
G2 Root System › G2 Root System:kurtosis_differentialrank 294/298-2.3904
Gottwald-Melbourne › Gottwald-Melbourne:radial_spectral_structurerank 298/2980.4273
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):z_rate_spectral_entropyrank 295/2980.0473
Higher-Order Statistics › Higher-Order Statistics:skew_meanrank 5/2982.4716
Higher-Order Statistics › Higher-Order Statistics:perm_entropyrank 294/2980.0250
Hodge–Laplacian › Hodge–Laplacian:source_fractionrank 297/2980.0182
Inflation (Substitution) › Inflation (Substitution):discrepancyrank 298/2980.0000
Klein Bottle › Klein Bottle:rank_deficitrank 4/2980.9875
Klein Bottle › Klein Bottle:rank_deficit_maxrank 5/2980.9975
Klein Bottle › Klein Bottle:wht_spectral_kurtosisrank 298/2981.3812
Level Statistics › Level Statistics:spacing_gue_distancerank 1/2980.9953
Level Statistics › Level Statistics:spacing_poisson_distancerank 1/2980.9953
Mostow Rigidity › Mostow Rigidity:mean_turn_anglerank 3/2980.9868
Mostow Rigidity › Mostow Rigidity:spectral_rigidityrank 296/2980.6197
Möbius-S³ › Möbius-S³:phase_profile_deviationrank 1/2980.7858
Nonstationarity › Nonstationarity:ac1_trendrank 5/2980.1726
Nonstationarity › Nonstationarity:variance_trendrank 295/298-0.2063
Ordinal Partition › Ordinal Partition:markov_mixingrank 2/2981.0000
Persistent Homology › Persistent Homology:total_persistencerank 296/2981.0000
Persistent Homology › Persistent Homology:persistence_entropyrank 298/2980.0000
Piecewise-Linear › Piecewise-Linear:envelope_arearank 296/29877.4000
Predictability › Predictability:sample_entropyrank 297/2980.0100
Projective ℙ² › Projective ℙ²:cross_ratio_stdrank 297/2980.0000
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):lyapunov_exponentrank 3/2981.9815
Spectral Analysis › Spectral Analysis:spectral_flatnessrank 1/2980.6206
Spherical S² › Spherical S²:concentrationrank 5/2980.9906
Spirograph › Spirograph:gear_rationalityrank 298/2980.4352
Symplectic › Symplectic:recurrence_raterank 3/2980.9772
S² × ℝ (Thurston) › S² × ℝ (Thurston):sphere_concentrationrank 5/2980.9904
Time Reversibility › Time Reversibility:ordinal_reversal_distancerank 298/2980.0000
Ulam Spiral (Square) › Ulam Spiral (Square):arm_density_variancerank 1/2987.3271
Ulam Spiral (Square) › Ulam Spiral (Square):polynomial_concentrationrank 296/2980.0078
Visibility Graph › Visibility Graph:degree_r_squaredrank 1/2980.9926
Visibility Graph › Visibility Graph:assortativityrank 296/298-0.4059
Wasserstein › Wasserstein:concentrationrank 1/29831.8488
Wasserstein › Wasserstein:dist_from_uniformrank 4/2980.4798
Wasserstein › Wasserstein:entropyrank 298/2980.0433
Zariski › Zariski:nonsep_fractionrank 1/2980.9821
in bearing
alphabetical
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Structure Atlas — Open Atlas · GitHub