Mian-Chowla

number-theory · 37 views
number-theoryarithmetic

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, 4, 8, 13, 21, 31, 45, 66, 81, ... (Mian–Chowla 1944; OEIS A005282). Emitted as a length-N binary indicator (255 at positions a_n < N, 0 elsewhere) shifted by an rng-chosen offset (additive translation, NOT cyclic --- a cyclic shift would break the Sidon property on ℤ/N). Density ~0.5% 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 finds no named structure, but the source is distinctively extreme on Ulam Spiral (Square):arm_density_variance (+10.6z) — beyond what the standard bank predicts for it.

What standard analysis sees
tail heaviness0.99
asymmetry0.99
occupancy0.00
short-range corr0.33
long-range memory0.60
spectral colour0.67
periodicity0.13
complexity0.01
time-irreversibility0.48
volatility clustering0.30
multifractality0.91
nonstationarity0.95
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Ulam Spiral (Square):arm_density_variance+10.6zbank-miss 3.5σ
Hodge–Laplacian:source_fraction-6.0zbank-miss 1.2σ
SL(2,ℝ) (Thurston):lyapunov_exponent+3.2zbank-miss 1.6σ
Boltzmann:coupling_temporal_variance+3.2zbank-miss 3.0σ
Mostow Rigidity:margulis_ratio+3.2zbank-miss 2.5σ
Cayley:spectral_gap+2.9zbank-miss 3.1σ
Ulam Spiral (Square):polynomial_concentration-2.8zbank-miss 1.0σ
G2 Root System:pair_angle_kurtosis+2.8zbank-miss 1.5σ

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 > free_plate_nodal
Chladni > free_plate_nodal
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)4.26cross-origin
Neural Net (Pruned 90%)4.61cross-origin
Devil's Staircase5.06cross-origin

Open in Atlas →

Which Geometries Light Up

2-adic › 2-adic:valuation_transition_predictabilityrank 4/3070.9836
2-adic › 2-adic:multiscale_markov_predictabilityrank 5/3070.9885
2-adic › 2-adic:distance_entropyrank 307/3070.0613
2-adic › 2-adic:mean_distancerank 307/3070.0075
Boltzmann › Boltzmann:coupling_temporal_variancerank 1/3072.0000
Catch24 › Catch24:SB_TransitionMatrix_3ac_sumdiagcovrank 2/3070.3240
Catch24 › Catch24:CO_HistogramAMI_even_2_5rank 303/3070.0000
Catch24 › Catch24:DN_OutlierInclude_p_001_mdrmdrank 303/307-0.4766
Catch24 › Catch24:DN_Meanrank 304/3070.0047
Catch24 › Catch24:SB_MotifThree_quantile_hhrank 306/3070.0600
Cayley › Cayley:delta_hyp_normrank 306/3070.0000
Chladni › Chladni:nodal_gap_ratiorank 306/3070.0337
D4 Triality › D4 Triality:triplet_temporalrank 4/3070.8000
D4 Triality › D4 Triality:edge_ortho_polarityrank 305/3070.0112
Fisher Information › Fisher Information:log_det_fisherrank 3/307141.2201
Fisher Information › Fisher Information:trace_fisherrank 3/307229810.4834
G2 Root System › G2 Root System:pair_angle_kurtosisrank 3/307102.1322
G2 Root System › G2 Root System:short_long_ratiorank 3/3070.9953
Higher-Order Statistics › Higher-Order Statistics:perm_entropyrank 303/3070.0250
Hodge–Laplacian › Hodge–Laplacian:source_fractionrank 306/3070.0182
Inflation (Substitution) › Inflation (Substitution):discrepancyrank 307/3070.0000
Klein Bottle › Klein Bottle:rank_deficitrank 3/3070.9875
Klein Bottle › Klein Bottle:rank_deficit_maxrank 4/3070.9975
Klein Bottle › Klein Bottle:wht_spectral_kurtosisrank 307/3071.3812
Level Statistics › Level Statistics:spacing_gue_distancerank 1/3070.9953
Level Statistics › Level Statistics:spacing_poisson_distancerank 1/3070.9953
Mostow Rigidity › Mostow Rigidity:mean_turn_anglerank 3/3070.9868
Mostow Rigidity › Mostow Rigidity:laplacian_perturbation_stabilityrank 305/3070.6197
Möbius-S³ › Möbius-S³:phase_profile_deviationrank 1/3070.7858
Nonstationarity › Nonstationarity:ac1_trendrank 5/3070.1726
Nonstationarity › Nonstationarity:variance_trendrank 303/307-0.2063
Predictability › Predictability:sample_entropyrank 306/3070.0100
Projective ℙ² › Projective ℙ²:cross_ratio_stdrank 306/3070.0000
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):cocycle_growth_raterank 3/3071.9815
Spirograph › Spirograph:rational_grid_proximityrank 307/3070.4352
Symplectic › Symplectic:recurrence_raterank 3/3070.9772
Time Reversibility › Time Reversibility:ordinal_reversal_distancerank 307/3070.0000
Ulam Spiral (Square) › Ulam Spiral (Square):polynomial_concentrationrank 306/3070.0078
Visibility Graph › Visibility Graph:degree_r_squaredrank 1/3070.9926
Visibility Graph › Visibility Graph:assortativityrank 305/307-0.4059
Wasserstein › Wasserstein:concentrationrank 1/30731.8488
Wasserstein › Wasserstein:dist_from_uniformrank 5/3070.4798
Wasserstein › Wasserstein:entropyrank 307/3070.0433
Zariski › Zariski:nonsep_fractionrank 1/3070.9821
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