Square wave --- binary oscillation (±1), maximal harmonic content with 1/n odd-harmonic Fourier series
Standard analysis sees: bounded / light-tailed; few distinct values; long-range memory (persistent); strongly periodic; low-complexity (predictable, not noise-like); multifractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on G2 Root System:pair_angle_kurtosis (+10.4z) — beyond what the standard bank predicts for it. It sits beside Morse Code in the atlas (standard-bank rank 31) — a neighbor conventional features miss.
G2 Root System:pair_angle_kurtosis | +10.4z | bank-miss 1.4σ |
E8 Lattice:coset_transition | +6.5z | bank-miss 1.4σ |
Mostow Rigidity:margulis_ratio | +3.5z | bank-miss 2.0σ |
Spherical S²:angular_spread | +3.4z | bank-miss 1.9σ |
Wavelet Cascade:intermittency_slope | -3.4z | bank-miss 1.0σ |
Visibility Graph:avg_clustering_coeff | -3.0z | bank-miss 1.8σ |
Mostow Rigidity:volume_entropy | -3.0z | bank-miss 1.1σ |
Mostow Rigidity:mean_turn_angle | +2.9z | bank-miss 1.2σ |
Binary sequence — two distinct symbols.









_(centered)/signed_log_z/Square_Wave.png)
_(centered)/xy_path/Square_Wave.png)

/barcode/Square_Wave.png)
/d_curve/Square_Wave.png)








/phi_spectrum/Square_Wave.png)










/default/Square_Wave.png)
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| Nearest neighbor | Distance | |
|---|---|---|
| Pulse-Width Modulation | 3.82 | |
| Morse Code | 4.75 | |
| μ-law Sine | 4.84 |
2-adic › 2-adic:valuation_transition_predictability | rank 3/307 | 0.9882 |
2-adic › 2-adic:multiscale_markov_predictability | rank 4/307 | 0.9900 |
D4 Triality › D4 Triality:edge_ortho_polarity | rank 306/307 | 0.0074 |
E8 Lattice › E8 Lattice:coset_transition | rank 2/307 | 0.3659 |
Fisher Information › Fisher Information:velocity_spectral_gini | rank 4/307 | 0.7793 |
G2 Root System › G2 Root System:pair_angle_kurtosis | rank 2/307 | 286.3648 |
G2 Root System › G2 Root System:short_long_ratio | rank 2/307 | 0.9979 |
Higher-Order Statistics › Higher-Order Statistics:perm_entropy | rank 306/307 | 0.0120 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):curvature_structure | rank 303/307 | 1.9677 |
Information Theory › Information Theory:compression_ratio | rank 306/307 | 0.0023 |
Klein Bottle › Klein Bottle:rank_deficit | rank 4/307 | 0.9642 |
Klein Bottle › Klein Bottle:rank_deficit_max | rank 5/307 | 0.9761 |
Laplacian › Laplacian:poisson_recovery_error | rank 306/307 | 0.0020 |
Mostow Rigidity › Mostow Rigidity:margulis_ratio | rank 2/307 | 0.9801 |
Mostow Rigidity › Mostow Rigidity:mean_turn_angle | rank 2/307 | 0.9893 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_std | rank 1/307 | 0.1674 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_mean | rank 2/307 | 0.1483 |
Ordinal Partition › Ordinal Partition:forbidden_transitions | rank 1/307 | 0.8800 |
Projective ℙ² › Projective ℙ²:cross_ratio_std | rank 305/307 | 0.0000 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):sphere_concentration | rank 303/307 | 0.0063 |