Combined Client Samples

Every walk steps to a random unvisited neighbour until it seals itself in and can go no further. The number of steps it managed is its length. Summed over the whole fleet, that distribution is steeply skewed: most walks suffocate early, a thin tail runs for thousands of steps. The chart below plots the count of walks at each length across the full range (log count-axis so the tail stays visible); the mean is the fleet's trapping length.

Length counts steps; these three measure ground covered. End-to-end distance is the straight line from where a walk began to where it died; farthest reach is the largest radius it ever touched (always ≥ the end distance, since walks often loop back before trapping); Manhattan distance is the grid-blocks-home figure. Each is a full-population histogram with its median marked.

The walk chooses uniformly among its free neighbours, so nothing in the process prefers a heading. Fairness therefore demands the pattern be invariant under a 90° turn — the four orthogonal counts equal each other, and the four diagonal counts equal each other. That four-fold symmetry is the real audit of the random stream; a dent in one direction is a bug caught in the act. It does not demand that straight and diagonal steps match: which of those a walk can take depends on the trail it has already laid, and here diagonals come out about 6.3% more often — geometry, not a biased generator.

Every cell any walk stepped on (left) and every cell a walk finally trapped itself on (right), summed over the fleet and binned into a coarse grid. The dashed crosshair is the world centre, where all walks begin. Colour is on a log scale — a single bright core fanning out into an ever-sparser frontier. The two fields share a shape because walks mostly die where they mostly wander.

The wander field from Section IV at full resolution — every one of the 382.5K cells any walk stepped on, over the 4K² world, painted by visit frequency on a log-of-max scale. This is the image that brands the project; the ladder at left recolours the identical data through each spectrum. Transparent means never visited — the checker shows through.

Basic 7-band Even Full spectrum Full spectrum (reversed) Curve Curve (reversed) Exponential Logarithmic Max detail Ratios Exponential II

Each client keeps its own top-20 lengths; the combine step merges them into a fleet-wide leaderboard, scoring by rank position across all clients so a length that lands high on many machines outranks a one-off outlier. These are the marathon walks — the far tail of Figure 1 made concrete.

Everything above derives from these eight files — the output of the combine step, one row per distinct value or cell. The builder reads them once (streaming the two X,Y heat files in a single binning pass) and bakes this board's snapshot; the page itself never touches the CSVs.

⭳ Download the snapshot (JSON) everything this board renders, in one small file