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 2.0σ |
Zariski:algebraic_residual | +3.0z | bank-miss 1.1σ |
H² × ℝ (Thurston):hyperbolic_triangle_area | +2.3z | bank-miss 1.3σ |
Cantor Set:max_gap | -2.1z | bank-miss 2.9σ |
Composition
Fixed alphabet — only 5 distinct symbols across 16384 samples.
Render Gallery
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/d_curve/Partition_Function.png)
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Codon Usage | 3.99 | cross-origin |
| DNA Centromere | 4.33 | cross-origin |
| Look-and-Say | 4.48 | cross-origin |
Which Geometries Light Up
Boltzmann › Boltzmann:spectral_gap_J | rank 3/307 | 0.9962 |
Cantor Set › Cantor Set:bit_plane_autocorrelation | rank 5/307 | 0.9961 |
Cantor Set › Cantor Set:max_gap | rank 305/307 | 0.0015 |
Cantor Set › Cantor Set:mean_gap | rank 305/307 | 0.0000 |
Mostow Rigidity › Mostow Rigidity:volume_entropy | rank 2/307 | 4.5154 |
Mostow Rigidity › Mostow Rigidity:mean_turn_angle | rank 305/307 | 0.0596 |
Spectral Analysis › Spectral Analysis:spectral_peakedness | rank 306/307 | 0.0005 |
Zariski › Zariski:algebraic_residual | rank 3/307 | 0.0687 |