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-persistent; blue spectrum (high-frequency power); time-irreversible (slow rise, sharp collapse); homoskedastic; monofractal; high-dimensional / space-filling; stationary. The atlas finds no named structure, but the source is distinctively extreme on H3 Icosahedral:nn_enrichment (+5.0z) — beyond what the standard bank predicts for it. It sits beside Tribonacci Word in the atlas (standard-bank rank 20) — a neighbor conventional features miss.
H3 Icosahedral:nn_enrichment | +5.0z | bank-miss 2.8σ |
H3 Icosahedral:temporal_coherence | +4.4z | bank-miss 2.5σ |
SL(2,ℝ) (Thurston):parabolic_fraction | +3.5z | bank-miss 2.6σ |
Zariski:residual_slope | -2.9z | bank-miss 1.2σ |
Composition
Fixed alphabet — only 3 distinct symbols across 16384 samples.
Render Gallery
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/phi_spectrum/Padovan_Word.png)
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Moebius Function | 4.59 | cross-domain |
| Tribonacci Word | 4.75 | |
| Symbolic Henon | 4.95 |
Which Geometries Light Up
Cayley › Cayley:delta_hyp_norm | rank 1/298 | 0.2656 |
H3 Icosahedral › H3 Icosahedral:nn_enrichment | rank 2/298 | 4.8223 |
H3 Icosahedral › H3 Icosahedral:temporal_coherence | rank 2/298 | 0.4055 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):area_length_ratio | rank 294/298 | 0.0000 |
Inflation (Substitution) › Inflation (Substitution):acf_geometric | rank 1/298 | 0.9545 |
Ordinal Partition › Ordinal Partition:statistical_complexity | rank 1/298 | 0.1040 |
Septagonal (Danzer) › Septagonal (Danzer):z_conjugate | rank 298/298 | -0.6544 |
Ulam Spiral (Square) › Ulam Spiral (Square):radon_anisotropy | rank 298/298 | 0.0002 |
Zariski › Zariski:heyting_gap | rank 4/298 | 0.8119 |
Zariski › Zariski:residual_slope | rank 297/298 | -5.6885 |