—
Every chart sets the sequence (in colour) against a control (in grey). The control keeps each term's jump d and the residue of ℓ = a − d modulo M, and redraws ℓ at random within ±25 % before decomposing it again. As M runs through 1, 2, 6, 12, 60, 420, 840, 2520, 27720, 360360, the control learns the sequence's congruences one prime power at a time: 2, 3, 4, 5, 7, 8, 9, 11, 13. The step where the gap closes names the prime power that carries the structure.
Weight–level plate: the sequence
log₁₀ k across, log₁₀ L up, equal scales, so the diagonal k = L separates the two classes: weight above it, level below. Dashed lines: k·L = 10j.
The same records, controlled
The same records after the control: same jumps, ℓ redrawn in its class mod M. Columns and lines that survive here are carried by the residues the control keeps.
The residue ladder
Level share of the sequence minus that of the control, in percentage points, as the control keeps more residues; each step adds one prime power. Whiskers: ±2σ. Hollow: more than half the records frozen, so the step says little. Click a step to use that control everywhere.
| M | adds | level Δ | z | L=1 Δ | z | frozen |
|---|
Level share along the sequence
The sequence (line) against the control (grey band, ±2σ), in 30 windows of n. Drag across the chart to keep a range of n for every chart; double-click to clear.
Jumps
How often each jump d occurs (the 24 commonest). Click a bar to keep that jump for every chart; click it again to clear.
The congruence lock
For each prime r, how much more often r divides ℓ than it would at random: r·Pr[r | ℓ], over the records with d < r. 1 is chance; r/(r − 2) is the Hardy–Littlewood value for consecutive primes (ℓ ≡ 0 mod r is one admissible class in r − 2). Whiskers: ±2σ.
Columns and lines
Share of the records whose weight is k (columns of the plate) or whose level is L (its lines), for the smallest values: the sequence (bars) against the control (ticks).
The numbers
Under the filters above. The sequence's shares use every record in the filter; the control and the Δ use the control's records (the same records, paired).
| records | level | weight | ties | L = 1 | forced (ℓ ≤ d²) |
|---|
Method, and what the Lab cannot tell you
- The decomposition. d = a(n+1) − a(n), ℓ = a − d when a > 2d; the weight k is the least divisor of ℓ above d and the level L = ℓ/k; a term is in the level class when k > L (ties count as weight). The sequences' own data are the published files of decompwlj 3D, checked here against their SHA-256 fingerprints.
- The control. For each record, keep d and c = ℓ mod M; draw ℓ′ uniformly among the integers ≡ c (mod M) in [max(d + 1, ⌈0.75 ℓ⌉), ⌊1.25 ℓ⌋]; decompose ℓ′ with the same jump. The density, the jumps and the residues mod M are the sequence's own; everything else is random. Seeds are fixed (FNV-1a of the sequence id, M and the replicate), so every figure is reproducible.
- The ladder. M = lcm(1, …, n) for n = 1, 2, 3, 4, 5, 7, 8, 9, 11, 13: 1, 2, 6, 12, 60, 420, 840, 2520, 27720, 360360. Each M divides the next and adds exactly one prime power (2, 3, 4, 5, 7, 8, 9, 11, 13), so a change of Δ at a step belongs to that prime power. A modulus of your own (“your own M”, up to 107) is computed on demand and shown after the ladder; it is kept in the link.
- Frozen records. When ℓ is small next to M, its class has few members in the window and the control barely moves it. The ladder table gives the share of records with fewer than 16 members; where it is large, the Δ at that M is pushed toward 0 (a control that cannot move ℓ is the sequence itself), so it is understated, not explained. Steps with more than half the records frozen are drawn hollow and left off the line.
- Records. The control runs on a random subsample of the decomposable records (never a fixed stride: strides alias with the arithmetic). The plates show that subsample for both panels, so their densities match.
- σ. The whiskers use the larger of the binomial bound √(p(1 − p)/N) on the control's share and the spread of its replicates. A difference of 2σ in one of 7 steps is not by itself a discovery.
- The kernel. Exact below 2^53: trial division, deterministic Miller–Rabin, Pollard–Brent rho, in doubles with exact products. Checked against PARI/GP 2.15.4 and against the published rows.
- What it is not. A Δ that survives the last informative step says that something beyond those residues shapes the decomposition; it does not say what. For the primes the remainder is the Hardy–Littlewood local factor of every larger prime r, which no finite M keeps.