Witness
A witness is a permutation value P(x,y) that actually contains the term
k as a substring.
Field Guide
Quick hints for the sequence vocabulary
Depth
Depth measures where k appears inside its witness. Near 0
means close to the start; near 1 means close to the end.
Expansion Ratio
The expansion ratio is |P(x,y)| / |k|, telling you how much longer the
witness is than the term itself.
Split
A split breaks k into x and y. Different split
lengths can produce very different witness behavior.
System Summary
What the dataset looks like at a glance
Length Profile
Term lengths are heavily concentrated at six digits
Length Distribution
Compare how often three-, four-, five-, and six-digit terms appear in the current dataset.
Split Geometry
A heatmap of (|x|, |y|) shows which digit splits dominate the sequence.
Sequence Growth
The cumulative curve tracks how quickly terms accumulate as k increases.
Witness Structure ?
Every point is a term in the sequence
Expansion vs Relative Depth ?
Dots are colored by term length. Horizontal position shows how much larger
xPy is than k, while vertical position shows where
k appears within the permutation value.
Depth Distribution
How uniformly does k land inside P(x,y)?
Each bin covers a 10% slice of the witness length.
A flat histogram would indicate no positional bias.
Log-Log Growth
Cumulative term count on log-log axes reveals the power-law density exponent α (count(N) ≈ C · Nα). A slope α < 1 confirms the sequence grows sub-linearly.
Spacing
Gaps between consecutive terms
Inter-Term Gap
Each bar is the gap from a term to its predecessor. Large spikes mark stretches of integers where no term qualifies — the sequence's deserts.
Sequence Browser
Filter and inspect the canonical rows
| k | x | y | |P(x,y)| ? | Position ? | Depth ? | Ratio ? | Digit sum | Witnesses ? |
|---|