Partition Function
What It Is
p(n) mod 5, the number of integer partitions of n reduced modulo 5. Captures the first Ramanujan congruence p(5k+4) ≡ 0 (mod 5) (Ramanujan 1919, proved Watson 1938) as a stationary integer stream over {0,1,2,3,4}: every 5th position is deterministically zero, the other four positions look pseudo-random in {0,1,2,3,4}. Computed via Euler's pentagonal number theorem with arithmetic carried in Z/5, so the exponential growth of p(n) (Hardy-Ramanujan: p(n) ~ exp(pi sqrt(2n/3))) never appears directly --- the source carries the algebraic-periodicity content rather than the smooth envelope. Raw log p(n) emission was retired 2026-05-18: it was monotone-increasing and collapsed to the same rank sequence as Primes / Minkowski ?(x) in every rank-based geometry.
Interpretation
Standard analysis sees: few distinct values; homoskedastic. The atlas finds no named structure, but the source is distinctively extreme on Cantor Set:mean_gap (-5.0z) — beyond what the standard bank predicts for it.
Cantor Set:mean_gap | -5.0z | bank-miss 1.4σ |
Persistent Homology:n_significant_features | +3.7z | bank-miss 1.0σ |
Cantor Set:max_gap | -2.1z | bank-miss 2.1σ |
Composition
Fixed alphabet — only 5 distinct symbols across 16384 samples.
Render Gallery
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/d_curve/Partition_Function.png)
/phi_spectrum/Partition_Function.png)
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Codon Usage | 4.11 | cross-domain |
| DNA Centromere | 4.42 | cross-domain |
| DNA SARS-CoV-2 | 4.55 | cross-domain |
Which Geometries Light Up
Boltzmann › Boltzmann:spectral_gap_J | rank 3/298 | 0.9962 |
Cantor Set › Cantor Set:bit_plane_autocorrelation | rank 4/298 | 0.9961 |
Cantor Set › Cantor Set:max_gap | rank 297/298 | 0.0015 |
Cantor Set › Cantor Set:mean_gap | rank 297/298 | 0.0000 |
Mostow Rigidity › Mostow Rigidity:volume_entropy | rank 2/298 | 4.5154 |
Mostow Rigidity › Mostow Rigidity:mean_turn_angle | rank 296/298 | 0.0596 |
Persistent Homology › Persistent Homology:h1_total_persistence | rank 2/298 | 1.5740 |
Persistent Homology › Persistent Homology:total_persistence | rank 2/298 | 7.3142 |
Persistent Homology › Persistent Homology:n_significant_features | rank 3/298 | 24.6000 |
Spectral Analysis › Spectral Analysis:phase_coherence | rank 298/298 | 0.0005 |
Visibility Graph › Visibility Graph:degree_exponent_gamma | rank 5/298 | 2.6553 |
Zariski › Zariski:algebraic_residual | rank 2/298 | 0.0687 |