Padovan Word
What It Is
Smallest-Pisot substitution: a→b, b→c, c→ab. Substitution matrix [[0,0,1],[1,0,1],[0,1,0]] has eigenvalue ρ ≈ 1.3247 (plastic number, real root of x³=x+1, smallest Pisot number greater than 1). Ternary alphabet, cubic algebraic ratio at the LOW end of the Pisot spectrum --- pairs with Tribonacci (cubic, ≈1.839) to disentangle 'cubic-Pisot' from 'ρ-magnitude' effects when probing algebraic-substitution-targeted metrics.
Interpretation
Standard analysis sees: few distinct values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); time-irreversible (slow rise, sharp collapse); homoskedastic; monofractal; high-dimensional / space-filling; stationary. The atlas additionally detects Multi-Scale Wasserstein:w_max_ratio, Time Reversibility:ordinal_reversal_distance. It sits beside Tribonacci Word in the atlas (standard-bank rank 20) — a neighbor conventional features miss.
H3 Icosahedral:nn_enrichment | +4.9z | bank-miss 2.8σ |
Zariski:heyting_gap | +4.5z | bank-miss 1.4σ |
H3 Icosahedral:temporal_coherence | +4.3z | bank-miss 3.0σ |
SL(2,ℝ) (Thurston):parabolic_fraction | +3.6z | bank-miss 2.9σ |
Zariski:residual_slope | -2.8z | bank-miss 1.0σ |
Composition
Fixed alphabet — only 3 distinct symbols across 16384 samples.
Render Gallery
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/d_curve/Padovan_Word.png)
/phi_spectrum/Padovan_Word.png)
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Tribonacci Word | 4.30 | |
| Moebius Function | 4.39 | cross-origin |
| Symbolic Henon | 4.74 | cross-origin |
Which Geometries Light Up
Cayley › Cayley:delta_hyp_norm | rank 1/307 | 0.2656 |
H3 Icosahedral › H3 Icosahedral:nn_enrichment | rank 2/307 | 4.8273 |
H3 Icosahedral › H3 Icosahedral:temporal_coherence | rank 3/307 | 0.4059 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):area_length_ratio | rank 303/307 | 0.0000 |
Inflation (Substitution) › Inflation (Substitution):acf_geometric | rank 1/307 | 0.9618 |
Ordinal Partition › Ordinal Partition:statistical_complexity | rank 1/307 | 0.1040 |
Septagonal (Danzer) › Septagonal (Danzer):z_conjugate | rank 306/307 | -0.6706 |
Ulam Spiral (Square) › Ulam Spiral (Square):radon_anisotropy | rank 306/307 | 0.0002 |
Zariski › Zariski:heyting_gap | rank 5/307 | 0.8119 |
Zariski › Zariski:residual_slope | rank 304/307 | -5.6882 |