The last 4,096 of 40,000 arrivals, answered exactly, both ways

The last 4,096 of 40,000 arrivals, answered exactly, both ways

Both routes return the same 6 heavy hitters above 2% with the same counts — the figure refuses to draw if they differ, because they are answers to one question. What differs is the bill. Forwards, the algorithm reads all 40,000 arrivals and must hold the window as it goes: 4,096 keys and 4,096 stamps of 13 bits, because it cannot know which arrivals will still be inside when the question is asked. Backwards, it reads 4,096 items, counts them, and stops — no expiry, no stamps, no bound to prove. The stamps are the clearest part of it: knowing when something arrived is only necessary if you met it before you knew whether it mattered.

Drawn at 700 × 304, wide on the page. Everything above is what direction-bill returns with no arguments; the caption is the generator's own, computed from the numbers in the drawing rather than written beside it.

1 essay calls direction-bill. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about — which is what optcheck and figfill exist to catch.